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--- abstract: 'Total travel time $t$ and time delay $\Delta t$ between images of gravitational lensing (GL) in the equatorial plane of stationary axisymmetric (SAS) spacetimes for null and timelike signals with arbitrary velocity are studied. Using a perturbative method in the weak field limit, $t$ in general SAS spacetimes is expressed as a quasi-series of the impact parameter $b$ with coefficients involving the source-lens distance $r_s$ and lens-detector distances $r_d$, signal velocity $v$, and asymptotic expansion coefficients of the metric functions. The time delay $\Delta t$ to the leading order(s) were shown to be determined by the spacetime mass $M$, spin angular momentum $a$ and post-Newtonian parameter $\gamma$, and kinematic variables $r_s,~r_d,~v$ and source angular position $\beta$. When $\beta\ll \sqrt{aM}/r_{s,d}$, $\Delta t$ is dominated by the contribution linear to spin $a$. Modeling the Sgr A\* supermassive black hole as a Kerr-Newman black hole, we show that as long as $\beta\lesssim 1.5\times 10^{-5}$ \[as\], then $\Delta t$ will be able to reach the $\mathcal{O}(1)$ second level, which is well within the time resolution of current GRB, gravitational wave and neutrino observatories. Therefore measuring $\Delta t$ in GL of these signals will allow us to constrain the spin of the Sgr A\.' author: - Haotian Liu - Junji Jia title: Constraining the spacetime spin using time delay in stationary axisymmetric spacetimes --- Introducing =========== Nowadays time delay between gravitational lensing (GL) images has become a useful tool in astrophysics and cosmology. Time delay in GL of compact objects can be used to constrain their properties including mass and distance to earth, and distinguish black hole (BH) and naked singularity [@Virbhadra:2007kw; @Bozza:2003cp]. For GLs by galaxies or galaxy clusters, time delay can determine the Hubble parameter, matter density, dark matter substructure and dark universe parameters [@Oguri:2006qp; @Keeton:2008gq; @Coe:2009wt; @Linder:2011dr; @Mohammed:2014eca; @Treu:2016ljm; @Liao:2018ofi]. The observed GL events are usually from light signals. However, with the observation of supernova neutrinos [@Hirata:1987hu; @Bionta:1987qt; @IceCube:2018dnn; @IceCube:2018cha] and gravitational waves (GWs) [@Abbott:2016blz; @Abbott:2016nmj; @Abbott:2017oio; @TheLIGOScientific:2017qsa; @Monitor:2017mdv], the astronomical observation entered the multimessenger era. Consequently, the time delays of neutrino and GW signals can be viewed as important supplements to time delay of light signals. Compared with time delay of light signals alone, the difference between time delays of light and neutrinos or light and GW signals can provide stronger constraints on the cosmology parameters [@Liao:2017ioi; @Wei:2017emo; @Fan:2016swi]. In addition, the time delay of these signals can determine the properties of test particles like mass ordering of neutrinos and velocity of GW [@Fan:2016swi; @Jia:2017oar; @Jia:2019hih]. Although it is known that neutrinos as well as GWs in some gravitational theories beyond General Relativity have non-zero masses, most of the previous works on their time delay treated them as null signals [@Fan:2016swi; @Mena:2006ym; @Eiroa:2008ks; @Takahashi:2016jom]. It is obvious that time delay applicable to timelike signals will provide higher accuracy and therefore stronger constraints to spacetime and signal particle parameters when GWs and neutrinos are used as messengers. Previously, we showed that the time delay of timelike signals in spherically symmetric (SSS) spacetimes in the weak field limit is related to the asymptotic expansion coefficients of the metric functions, including spacetime mass $M$ and post-Newtonian parameter $\gamma$ etc [@Liu:2020td2]. However, the black hole (BH) no-hair conjecture implies that in general there exist another important parameter for BHs and potentially other compact objects, i.e., their spin angular momentum $a$. In this work, we will generalize our previous work in the SSS spacetimes to the time delay in the equatorial plane of arbitrary stationary axisymmetric (SAS) spacetimes. We will show that the time delay in the SAS case has a significant difference from that of the SSS spacetimes: the appearance of the spin dependant term at the very leading order. Applying the result to Kerr-Newman (KN) spacetime, we will show that the time delay between GL images due to the Sgr A\ supermassive BH (SMBH) can be used to constrain the SMBH spin. Some other works calculated the Shapiro time delay (or the total travel time) in specific spacetimes for light signals. Ref. [@Sereno:2003nd], [@Keeton:2005jd], [@Wang:2014yya] and [@He:2016xiu; @He:2016cya] calculated the Shapiro time delay in the Reissner-Nordstrom (RN), Schwarzschild, Kerr and KN spacetimes respectively for light signal. The time delay in the strong field limit was studied in Ref. [@Bozza:2003cp]. The paper is organized as follows. In Sec. \[sec:tt\], we use the perturbative method to obtain the total travel time $t$ in a quasi-series form of the impact parameter for signals with arbitrary velocities in general asymptotically flat SAS spacetimes. In Sec. \[sec:td\], time delay $\Delta t$ between two GL images is obtained to the leading order(s) using deflection angle that is accurate to the given order. In Sec. \[sec:tdkn\], we apply our results in general SAS spacetimes to the KN spacetime. We then model the Sgr A\* SMBH as a KN BH and show how its spin $a$ can be determined using time delay between different GL images. Total travel time in SAS spacetimes {#sec:tt} =================================== In this section we compute the total travel time in the equatorial plane of general SAS spacetimes using a perturbative method, which is essentially a combination and extension of those used in Refs. [@Huang:2020trl; @Liu:2020td2]. Therefore, we first recap some of the key steps of the method developed in these works and then calculate in details the total time in general SAS spacetimes. We begin with the most general SAS metric, which can be described as $$\label{eq:sasmet} \dd s^2 = -A\dd t^2 + B \dd t \dd \varphi + C \dd \varphi^2 + D \dd r^2 + F \dd \theta^2,$$ where $(t,~r,~\theta,~\varphi)$ are the coordinates and $A,~B,~C,~D,~F$ are metric functions depending only on $r$ and $\theta$. We choose the spherical coordinates $(r,~\theta)$ here rather than the cylindrical ones $(\rho,~z)$ since they allow us to reduce to SSS spacetimes by simply setting $B=0$ [@Huang:2020trl; @Sloane:1978]. We assume that the spacetime permits motion of particles in a plane with fixed $\theta$, which can always be shifted to $\theta=\pi/2$ and called the equatorial plane. We then concentrate on motions in this plane, whose metric after suppressing the $\theta$ coordinate is $$\label{eq:sasmeteqc} \dd s^2 = -A(r) \dd t^2 + B(r) \dd t \dd \varphi + C(r) \dd \varphi^2 + D(r) \dd r^2.$$ Using this metric, one can routinely obtain the geodesic equations, $$\begin{aligned} \dot{t}&= \frac{2 (L B+2 E C)}{B^2+4 A C}, \label{eq:tdeq}\\ \dot{\varphi}&=\frac{2 (2 L A-E B)}{B^2+4 A C}, \label{eq:phideq}\\ \dot{r}^2 &=\frac{\lsb E^2 - \kappa A\rsb \lsb B^2+4AC\rsb - \lsb 2LA-EB\rsb^2}{AD \left(B^2+4 A C\right)}, \label{eq:rdeq}\end{aligned}$$ where $\kappa=1,~0$ respectively for timelike and null rays, and $\dot{~}$ stands for the derivative with respect to the proper time or affine parameter. $L$ and $E$ are two constants of motion due to the independence of the metric functions on $\varphi$ and $t$ respectively. In asymptotically flat spacetimes, $L$ and $E$ can be interpreted respectively as the angular momentum and the energy of the massless particle or the unit mass of a massive particles. They can be further correlated to the impact parameter $b$ of the trajectory and asymptotic velocity $v$ of the massive particle, $$\label{eq:lerela} L=(\textbf{p}\times \textbf{r})\cdot \hat{\textbf{z}}=\frac{bv}{\sqrt{1-v^2}},~E=\frac{1}{\sqrt{1-v^2}}.$$ In the massless limit, note that although $L$ and $E$ diverges, $L/E=bv$ still holds. Throughout the paper, we allow $L$ and $b$ to carry signs: when the initial asymptotic approach of the signal is anticlockwise (or clockwise) with respect to the lens center, $L$ and $b$ are positive (or negative). Using Eqs. and , then the total travel time for a signal from source at radius $r_s$ to detector at radius $r_d$ is $$\begin{aligned} \label{eq:ttorif} t=&\lsb \int_{r_0}^{r_s}+\int_{r_0}^{r_d}\rsb \frac{\sqrt{AD}\lb 2LB+4EC\rb}{\sqrt{B^2+4 A C }} \frac{\dd r}{\sqrt{\lb E^2 - \kappa A \rb \lb B^2+4AC\rb - \lb 2LA-EB\rb^2}} ,\end{aligned}$$ where $A,~B,~C,~D$ here are functions of $r$, and $r_0$ is the closest approach of the trajectory. Furthermore, setting $\dot{r}\|_{r=r_0}=0$ in Eq. , the angular momentum $L$ can also be solved as a function of $r_0$ $$\label{eq:angmemg} L=\frac{E B(r_0) +s \sqrt{\lsb E^2 - \kappa A(r_0) \rsb \lsb B^2(r_0)+4A(r_0)C(r_0)\rsb} }{2 A(r_0)}$$ where $s=+1,~-1$ respectively for prograde and retrograde motions of the signal. Note that in the relativistic limit, $v\to c$ and $E\to\infty$, and therefore $s=\mbox{sign}(L)=\mbox{sign}(b)$. In application to GL observation, the impact parameter $b$ is often preferred over the closest approach $r_0$. Using Eqs. and , we can establish a relation between them, as &=& \[eq:oobeq\]\ && pb, . \[eq:pdef\] Here in the last step we defined the right-hand side of Eq. as a function $p$ of both $b$ and $1/r_0$. We can formally obtain $p(b,x)$’s inverse function $q(b,x)$ with respect to its second argument such that =qb , . To carry out the integration in , as in Ref. [@Huang:2020trl], we then do a key change of variable from $r$ to $u$ which are linked by the relation =qb , .\[eq:qinu\] After some simple but slightly tedious algebra, the various terms in Eq. then becomes [@Huang:2020trl; @Liu:2020td2] $$\begin{aligned} &r_0\to 1,~ r_{s,d}\to b\cdot p\lb b,\ \frac{1}{r_{s,d}}\rb\equiv \sin \theta_{s,d},\label{eq:bptotheta} \\ &\frac{\sqrt{A(r)D(r)}}{\sqrt{B^2(r)+4A(r)C(r)}}\to\frac{\sqrt{A(1/q) D(1/q) }}{\sqrt{B^2(1/q) +4A(1/q) C(1/q) }},\label{eq:factorone}\\ &\frac{2LB(r)+4EC(r)}{2LA(r)-EB(r)}\to \frac{2bvB(1/q)+4C(1/q)}{2bvA(1/q)-B(1/q)},\label{eq:factortwo}\\ &\frac{2LA(r)-EB(r)}{\sqrt{\lsb E^2-\kappa A(r)\rsb\lsb B^2(r)+4A(r)C(r)\rsb -\lsb 2LA(r)-EB(r)\rsb^2}}\to\frac{su}{\sqrt{1-u^2}},\label{eq:factorthr} \\ &\dd r\to-\frac{1}{p_2(b,q)q^2}\frac{1}{b}\dd u, \label{eq:factorfou}\end{aligned}$$ where the $\theta_{s,d}$ defined in Eq. , i.e., \_[s,d]{}=bpb,\[eq:thetasddef\] are indeed the apparent angles of the signal at the source and detector respectively [@Huang:2020trl]. $p_2(b,q)$ is the derivative of the function $p(b,q)$ with respect to it second argument $q$, which is given in Eq. . Collecting these terms together, the total travel time becomes $$\label{eq:ttwithy} t=\lsb \int_{\sin\theta_s}^{1}+\int_{\sin\theta_d}^{1} \rsb y\lb b,\frac{u}{b}\rb \frac{\dd u}{u\sqrt{1-u^2}},$$ where $$\begin{aligned} \label{eq:yform} y\lb b,\frac{u}{b} \rb =& \frac{\sqrt{A(1/q) D(1/q) }}{\sqrt{B^2(1/q) +4A(1/q) C(1/q) }} \frac{\lsb 2bvB(1/q)+4C(1/q)\rsb sb}{2bvA(1/q)-B(1/q)}\frac{1}{p_2(b,q)q^2}\lb\frac{u}{b}\rb^2 .\end{aligned}$$ This $y\lb b,~u/b\rb$ depends on $u$ only through the ratio $u/b$. Thus, we can expand it with respect to its second argument to find a series form $$\label{eq:bserub} y\lb b,\frac{u}{b}\rb=\sum_{n=n_0}^{\infty} y_n(b)\frac{u^n}{b^n},$$ where $y_n(b)$ are the expansion coefficients. The leading index of this series, $n_0$, is determined by the asymptotic behavior of the metric functions in Eq. . If they satisfy the asymptotic expansion , then one can show that $n_0=-1$. Finally, changing the integration variable in Eq. from $u$ to $\xi$ by $u=\sin \xi$ , the total time delay becomes $$\label{eq:ttynutoxi} t=\lsb \int_{\theta_s}^{\frac{\pi}{2}}+\int_{\theta_d}^{\frac{\pi}{2}} \rsb \sum_{n=n_0}^{\infty} y_n\lb b\rb \frac{\sin^{n-1} \xi}{b^n} \dd \xi.$$ The integrability of this time delay now is clear because the functions $\sin^{n-1}\xi$ can always be integrated to find $$\begin{aligned} \label{eq:lnform} l_n(\theta_s,~\theta_d) \equiv& \lsb \int_{\theta_s}^{\frac{\pi}{2}} +\int_{\theta_d}^{\frac{\pi}{2}} \rsb\sin^{n-1} \xi \dd \xi \\ =& \begin{cases} \displaystyle \sum_{i=s,d} \cot\theta_i,&n=-1,\\ \displaystyle \sum_{i=s,d} \ln\lsb\cot\lb\frac{\theta_i}{2}\rb\rsb,&n=0,\\ \displaystyle \sum_{i=s,d} \frac{(n-2)!!}{(n-1)!!} \left(\frac{\pi}{2}-\theta_i +\cos\theta_i\sum_{j=1}^{[(n-1)/2]} \frac{(2j-2)!!} {(2j-1)!!}\sin^{2j-1} \theta_i\right),&n=1,3,\cdots,\\ \displaystyle \sum_{i=s,d} \frac{(n-2)!!}{(n-1)!!} \cos\theta_i \left(1 +\sum_{j=1}^{[(n-1)/2]} \frac{(2j-1)!!}{(2j)!!} \sin^{2j}\theta_i\right),&n=2,4,\cdots. \end{cases} \nonumber\end{aligned}$$ Substituting these into Eq. , we finally find the formal total travel time in a perturbative form $$\label{eq:ttynln} t=\sum_{n=-1}^{\infty} y_n\lb b\rb \frac{l_n}{b^n}.$$ A few comments regarding the above procedure and the result are now in order. Firstly, in expanding $y(b,u/b)$ to series , the explicit form of the inverse function $q(b,u/b)$ of $p(b,x)$ is seemingly needed, while it is not always possible to do so for some even simple functions. However indeed here the true $q(b,u/b)$ is not really necessary because what we need is only its expansion and this can be achieved through the Lagrange inverse theorem using $p(b,x)$ alone. Secondly, the expansion is actually carried out in the small $u$ limit as a Laurent series. It can also be viewed as a quasi-series of large $b$, for which the range of convergence can be shown to be $(b_c,~\infty)$. Here $b_c$ is the critical impact parameter below which the particle will not escape to infinity. Mathematically, it is the largest singular $b$ of the function $y(b,u/b)$. Thirdly, as in the case of SSS spacetime in Ref. [@Liu:2020td2], here for any value of the impact parameter inside the range of convergence, the perturbative result will also be able to reach any desired accuracy if the series is truncated at high enough order. Furthermore, in the large $b$ limit, the coefficients of the series , $y_n(b)$, is completely determined by the behavior of the metric functions in the asymptotic region, as was shown in Ref. [@Huang:2020trl; @Liu:2020td2]. Next we will compute the first few coefficients $y_n(b)$ for asymptotically flat spacetimes and then the total time in the weak field limit. Metric functions of the asymptotically flat SAS spacetimes always have an asymptotic expansion of the form $$\label{eq:metasyf} A(r)\to 1+\sum_{i=1}\frac{a_i}{r^i},~rB(r)\to \sum_{i=0}\frac{b_i}{r^i},~\frac{C(r)}{r^2}\to 1+\sum_{i=1}\frac{c_i}{r^i},~D(r)\to 1+\sum_{i=1}\frac{d_i}{r^i},$$ where $a_i,~b_i,~c_i,~d_i$ are constants. Without losing generality, here $a_1$ can be identified with the minus 2 times the ADM mass $M$ of the spacetime. Substituting into $y(b,u/b)$ in Eq. , $y_n(b)$ for $n=-1,0,1,2$ are easily found to be $$\begin{aligned} \label{eq:sasynexp} y_{-1}(b)=&\frac{s}{v}, \\ y_0(b)=&\frac{a_1(1-2 v^2)+d_1 v^2}{2 v^3}, \\ y_1(b)=&s\lsb\frac{8 a_1^2-4 a_1 (c_1+d_1)-8 a_2-(c_1-d_1)^2+4 c_2+4 d_2}{8 v}\right.\nonumber\\ &\left. +\frac{2 b_0 v (-4 a_1+c_1+d_1)+4 b_1 v}{8 v^3} \frac{1}{b} + \frac{b_0^2}{4 v^3}\frac{1}{b^2}\rsb,\\ y_2(b)=&\frac{b_0b}{2} +\frac{1}{16 v^7} \lcb -a_1^3 \left(16 v^6+48 v^4+18 v^2+1\right)+a_1^2 v^2 \lsb 2 c_1 \left(8 v^4+32 v^2+11\right) \right. \right. \nonumber \\ & \left. + d_1 \left(8 v^4+8 v^2-1\right)\rsb +a_1 \lsb 4 a_2 v^2 \left(8 v^4+20 v^2+5\right)-v^4 \left(24 c_1^2+4 c_1 \left(2 d_1 v^2+d_1\right) \right. \right. \nonumber \\ &\left. \left. + 8 c_2 \left(2 v^2+7\right)-\left(2 v^2+1\right) \left(d_1^2-4 d_2\right)\right)\rsb +v^4 \lsb -4 a_2 \left(4 c_1 v^2+14 c_1+2 d_1 v^2+d_1\right) \right. \nonumber \\ & \left. \left. - 16 a_3 \left(v^2+2\right)+v^2 \left(24 c_1 c_2-2 c_1 d_1^2+8 c_1 d_2+8 c_2 d_1+40 c_3+d_1^3-4 d_1 d_2+8 d_3\right)\rsb \rcb \nonumber \\ & + \frac{1}{8 b v^6} \lcb a_1^2 b_0 \left(36 v^4+60 v^2+11\right)-2 a_1 v^2 \lsb b_0 \left(c_1 \left(20 v^2+26\right)+4 d_1 v^2+d_1\right) \right. \right. \nonumber \\ & \left. +14 b_1 \left(v^2+1\right)\rsb -28 a_2 b_0 \left(v^2+1\right) v^2+v^4 \lsb b_0 \left(12 c_1^2+4 c_1 d_1+32 c_2-d_1^2+4 d_2\right) \right. \nonumber \\ & \left. \left. +4 b_1 (8 c_1+d_1)+20 b_2\rsb \rcb +\frac{3 b_0 \lsb v^2 \lcb b_0 (12 c_1+d_1)+14 b_1\rcb -a_1 b_0 \left(16 v^2+11\right)\rsb}{8 b^2 v^5}+\frac{19 b_0^3}{8 b^3 v^4}.\end{aligned}$$ Here $y_{-1}$ and $y_0$ agree with the corresponding SSS spacetime result in Ref. [@Liu:2020td2] and the parameters containing the spin of the spacetime, including $b_{0,1,2},~c_{1,2,3}$ and $d_{2,3}$, only appear in $y_1,~y_2$ and orders above. Higher order results can also be found without difficulty but are not needed in the following calculations and therefore will not be shown. In general, one can show that these coefficients always take a form $$\begin{aligned} y_n(b)=& M^{n+1} \sum_{j=-1}^{n+1} y_{n,j} \frac{M^j}{b^j},~n\geqslant 2,\end{aligned}$$ where $y_{n,j}$ are polynomials of dimensionless quantities $a_i/M^i,~ b_i/M^{i+2},~c_i/M^i,~d_i/M^i$. In next section, we will use this total time to compute the time delay between different images of the GL in the weak field limit. In this limit, we have $r_{s,d}\gg b\gg M$ and therefore there will be two small parameters $b/r_{s,d}$ and $M/b$. Now the factors $y_n(b)$ and $1/b^n$ in series are already in power forms of $M/b$. One can straightforwardly expand the only other factors $l_n$ in Eq. into a series of small $b/r_{s,d}$ and $M/b$ too. The result for the first four $l_n~(n=-1,0,1,2)$ are $$\begin{aligned} \label{eq:saslnexp} l_{-1}=&\sum_{i=s,d}\lsb s\lb \frac{r_i}{b}-\frac{b}{2 r_i} \rb + \frac{s}{b} \frac{c_1 v^2-a_1}{2 v^2} +\frac{1}{b^2}\frac{sb_0}{2v}+ \mathcal{O}\lb \frac{b^3}{r_i^3}, ~\frac{Mb}{r_i^2} , ~\frac{M^2}{br_i} \rb \rsb, \\ l_0=&\sum_{i=s,d} \lsb \ln \lb \frac{2r_i}{b} \rb + \frac{c_1 v^2-a_1}{2 v^2} \frac{1}{b} \frac{b}{r_i} + \mathcal{O}\lb \frac{b^2}{r_i^2},~\frac{M^2}{b r_i} \rb \rsb, \\ l_1=&\sum_{i=s,d} \lsb\frac{\pi}{2}- \frac{sb}{r_i} + \mathcal{O}\lb \frac{b^3}{r_i^3},~ \frac{Mb}{r_i^2} \rb \rsb, \\ l_2=& \sum_{i=s,d}\lsb 1 + \mathcal{O}\lb \frac{b^2}{r_i^2} \rb \rsb. \end{aligned}$$ Again, expansion of high order $l_n$ are also easy but not necessary in the following computations. Substituting Eqs. and into Eq. , the total travel time expressed completely as series of $M/b$ and $b/r_{s,d}$ becomes $$\begin{aligned} \label{eq:ttgenfth} t=&\sum_{i=s,d}\lcb \frac{b}{v}\lsb \frac{r_i}{b}-\frac{b}{2r_i}+ \frac{1}{b} \frac{c_1v^2-a_1}{2 v^2}\rsb + \frac{\lb d_1-2 a_1 \rb v^2+a_1}{2 v^3} \lsb \ln \left(\frac{2 r_i}{sb}\right) +\frac{c_1 v^2-a_1}{2 v^2}\frac{1}{r_i}\rsb\right. \nonumber \\ & + \frac{8 a_1^2 - 4 a_1 (c_1+d_1)-8 a_2-(c_1-d_1)^2+4 c_2+4 d_2}{8 v} \frac{s\pi}{2} \frac{1}{b} + \frac{b_0}{2 b}\lb 1+\frac{1}{v^2}\rb \nonumber \\ &\left. + \mathcal{O}\lb \frac{M^3}{b^2},\frac{b^4}{r_i^3}\rb\rcb .\end{aligned}$$ Note that the $i$-th $(i=1,2,3,4)$ term in Eq. originates from the $i$-th term of Eq. and we have ignored the order higher than $\mathcal{O}\lb M^3/b^2, b^4/r_i^3\rb$. Clearly, this total travel time is in a full series form of $M/b$ and $b/r_i$. If higher accuracy than Eq. is needed, then one only needs to include $y_n,~l_n$ to higher orders and keeping more terms in their expansion. Moreover, going to higher order will also allow one to study the effect of higher order PPN parameters on the total time and time delay. Note if we substitute the velocity $v$ to 1 and the general SSS, RN, Kerr and KN metrics into this total time, then Eq. (107) of Ref. [@Keeton:2005jd], Eq. (12) of Ref. [@Sereno:2003nd], Eq. (22) (after expansion in the large $r/r_0$ limit) of Ref. [@Wang:2014yya], Eq. (20) of Ref. [@He:2016xiu] and (27) (after expansion in the large $X_{A,B}/b$ limit) of Ref. [@He:2016cya] will be produced respectively. Time delay in general SAS spacetimes {#sec:td} ==================================== For gravitationally lensed source in the equatorial plane of the SAS spacetime, there will always be one primary image (comparing to the multiple relativistic images in the strong field limit [@Virbhadra:2008ws]) on each side of the lens (see Fig. \[fig:lensing\]). The apparent angles $\theta_b$ and $\theta_t$ of the images and the time delay between the images are of interest to astronomy observations. Using lens equations that are accurate for the SAS spacetimes, in this section we will solve the image apparent angles and their corresponding impact parameters, from which we can further solve the time delay. The key result we will show is that the time delay $\Delta t$ to the leading order(s) will only depend on three parameters of the metric expansion , the spacetime mass $M$, the spacetime spin $a$ and the parameter $\gamma$ in the parameterized post-Newtonian (PPN) formalism of gravity. ![The GL in an SAS spacetimes. S, L, D are the source, lens and detector respectively. $b_t$ and $b_b$ are the impact parameters for the top and bottom paths. The spin angular momentum of the lens in this illustration is anticlockwise. We choose the sign $\epsilon=\mbox{sign}(\phi_0)=\mbox{sign}(\beta)=+1$ (and $-1$) when $\phi_0$ and $\beta$ are counterclockwise (and clockwise) against the observer-lens axis.[]{data-label="fig:lensing"}](lensing_fig_v2.pdf){width="60.00000%"} We first establish the GL equations which link the source’s angular position $\beta$ and its apparent angles $\theta_t$ and $\theta_b$ from the top and bottom paths respectively (see Fig. \[fig:lensing\]). Previously this was usually done using the deflection angle without the finite distance correction and in the small angle approximation, $\beta,~\phi_0\ll 1$ [@Bozza:2008ev]. However here we would like to use geometric equations and deflection angle that are as accurate as possible. To do this, we first link $\beta$ to $\phi_0$ in Fig. \[fig:lensing\]. Using the triangles $\triangle$SLA and $\triangle$SDA, we have a relation $$\label{eq:angrela} (r_d+r_s\cos \phi_0) \tan \beta = r_s \sin \phi_0 .$$ The $\phi_0$ then can be connected to both the impact parameters $b_t$ from the top side and $b_b$ from the bottom side through the change of the angular coordinate $\Delta \varphi(b_{b,t})$ from the corresponding side $$\label{eq:changeq} \pm \pi + \phi_0 = \pm \Delta \varphi (b_{b,t})$$ where upper sign “$+$” (or lower sign “$-$”) is for the subscript $b$ (or $t$). $\Delta \varphi (b_{b,t})$ to the leading non-trivial order in an SAS spacetimes were found in Ref. [@Huang:2020trl] (see also [@Jia:2020xbc]) as $$\label{eq:changmet} \Delta \varphi (b_{b,t}) \approx \pi + \frac{d_1v^2-a_1}{2 b_{b,t} v^2}-b_{b,t}\lb \frac{1}{r_s}+\frac{1}{r_d}\rb.$$ This $\Delta \varphi(b_{b,t})$ takes into account the finite distance effect of the source and detector and therefore provides a natural way to involve the source and lens distances into the GL equation. Note that ignores the terms of order $\mathcal{O}(M^2/b^2,~b^2/r_{s,d}^2)$ or higher, whose contribution to the final time delay is negligible. Solving Eqs. -, we can solve the impact parameters $b_{b,t}$, linking them to $r_{s,d}$ and $\phi_0$ $$\label{eq:impabts} b_{b,t}=\sqrt{\frac{(d_1 v^2 - a_1) r_d r_s}{\eta v^2 (r_d + r_s)}} \lsb \epsilon\pm \sqrt{\eta+1} \rsb,$$ where $\epsilon=\mbox{sign}(\phi_0)=\mbox{sign}(\beta)$ and $$\label{eq:etainb} \eta = \frac{4 (d_1 v^2 - a_1)(r_d + r_s)}{\phi_0^2 v^2 r_d r_s}.$$ Once $b_{b,t}$ is known, one can immediately obtain the apparent angle of the image though relation as \[eq:thetabt\] \_[b,t]{}=\^[-1]{}b\_[b,t]{}pb\_[b,t]{}, . For the time delay between the images from two sides, substituting $b_{b,t}$ into Eq. and subtracting each other, we obtain $$\begin{aligned} \label{eq:tdgenform} \Delta t = & \frac{\sqrt{\eta +1}}{\eta}\cdot \frac{2 \epsilon\left(d_1 v^2-a_1\right)}{ v^3} + \frac{ \left( d_1 -2 a_1\right) v^2+a_1}{v^3} \ln \left(\frac{\sqrt{\eta +1}+1 }{\sqrt{\eta +1}-1 }\right) \nonumber \\ & + \frac{\epsilon\pi \sqrt{r_s+r_d} }{4\sqrt{\eta (d_1 v^2 - a_1 ) r_s r_d} }\lsb 8 a_1^2-4 a_1 (c_1+d_1)-8 a_2-c_1^2+2 c_1 d_1+4 c_2-d_1^2+4 d_2\rsb \nonumber \\ & +\sqrt{\frac{\eta+1}{\eta}}\cdot\frac{2 b_0v \sqrt{(r_s+r_d)} }{\sqrt{(d_1 v^2 - a_1 ) r_s r_d} }\lb 1+\frac{1}{v^2}\rb + \mathcal{O}\lb \phi_0\sqrt{\frac{M^3}{r_{s,d}}},\phi_0^4r_{s,d} \rb.\end{aligned}$$ The four terms in this equation are respectively from the first to fourth term of Eq. and we only keep the results of order $\mathcal{O}\lsb (1+1/\eta)\sqrt{M^3/r_{s,d}}\rsb$ or lower. Similar to the SSS case in Ref. [@Liu:2020td2], the contributions from the first two terms in Eq. dominate the third term when $\eta\gg1$, i.e. $\|\phi_0\|\ll \sqrt{M/r_{s,d}}$. On the other hand, when $\eta\ll 1$, i.e. $\|\phi_0\|\gg \sqrt{M/r_{s,d}}$, the first term will be much larger than the second and third terms. Unlike the SSS spacetime, the fourth term of Eq. is new because of the appearance of the spin angular momentum $b_0$ in the SAS spacetime. A comparison of the fourth term with the first three terms immediately tells that when \[eq:fourlargecond\] , \|\_0\|\^[1/2]{},the former will be much larger than the latter. Since the fourth term is linear in $b_0$, this implies that when condition is satisfied, the time delay will critically depend on the spacetime spin. In Sec. \[sec:tdkn\], we will use this time delay to constrain the spin of the Sgr A\* SMBH. Combining all these analysis, we see that in the entire parameter space spanned by $(M/r_{s,d},~\phi_0)$, the time delay can be well approximated by the first, second and fourth terms of Eq. . On the other hand, since the angle $\beta$ is more readily linked to GL observables than $\phi_0$, it is also desirable to obtain a time delay expressed in terms of $\beta$. To do this, we can directly use Eq. to replace $\phi_0$ by $\beta$ and obtain the time delay to the leading order(s) $$\begin{aligned} \label{eq:tdgenformbeta} \Delta t = & \frac{\sqrt{\eta(\beta,v)+1}}{\eta(\beta,v)} \cdot \frac{2\epsilon \left(d_1 v^2-a_1\right)}{ v^3} + \frac{ \left( d_1 -2 a_1\right) v^2+a_1}{v^3} \ln \left(\frac{\sqrt{\eta(\beta,v)+1}+\epsilon}{\sqrt{\eta(\beta,v)+1}-\epsilon }\right) \nonumber \\ & +\sqrt{\frac{1+\eta(\beta,v)}{\eta(\beta,v)}}\cdot \frac{2 b_0v \sqrt{(r_s+r_d)} }{\sqrt{(d_1 v^2 - a_1 ) r_s r_d} } + \mathcal{O}\lb \beta\sqrt{\frac{M^3}{r_{s,d}}},\beta^4r_{s,d} \rb\end{aligned}$$ where $$\label{eq:etainbbeta} \eta(\beta,v)=\frac{4 (d_1 v^2 - a_1) r_s}{\beta^2 v^2 (r_s + r_d) r_d}.$$ Similarly, the previous discussion about the dominance of each term(s) in Eq. according to the relation between $M/r_{s,d}$ and $\phi_0$ can also apply to the parameters $M/r_{s,d}$ and $\beta$ in Eq. . In Fig. \[fig:parapart\] we plot the partition of parameter space spanned by $(M/r_{s,d},~\beta)$ according to the relative size of each term of Eq. . In region (A), the 3rd term $>$ the 1st term $>$ the 2nd term, while in region (B) the 3rd term $>$ the 1st term $\approx$ the 2nd term, and in region (C) the 1st term $\approx$ the 2nd term $>$ the 3rd term. Eq. is one of the key result of this paper. Only three parameters from the spacetime metric expansion , $a_1,~b_0$ and $d_1$, appear in it. As known to general SAS spacetimes, they are simply equivalent to the ADM mass $M$, spin angular momentum $a$ of the spacetime and the $\gamma$ parameter in the PPN formalism of gravity [@Bardeen:1973; @Weinberg:1972kfs] a\_1=-2M, b\_0=-4aM, d\_1=2M. Therefore we conclude that to the leading order(s), the time delay $\Delta t$ are determined by, in addition to kinematic variables $r_{s,d},~\beta,~v$, these three parameters of the spacetime. ![The partition of the parameter space spanned by $M/r_{s,d}$ and $\beta$. The regions (A) and (B) are separated by curve $\beta\approx \sqrt{M/r_{s,d}}$, and (B) and (C) by $\beta\approx \sqrt{b_0}/r_{s,d}$. The relation $b_0/M^2\lesssim \mathcal{O}(1)$ is implicitly assumed. \[fig:parapart\]](parapart.pdf){width="45.00000%"} The time delay applies to signals of all velocity. For relativistic timelike signals, in order to see more clearly the effect of velocity, we can expand $\Delta t$ around the speed of light. The result to the first non-trivial order is $$\begin{aligned} \label{eq:tdexpv} \Delta t(v\to 1)=&\Delta t(v=1){\nonumber}\\ &+\lcb \frac{2 \epsilon \lsb d_1 \lb \eta(\beta,1)+1 \rb - a_1 \lb 2 \eta(\beta,1)+1 \rb \rsb }{\eta(\beta,1) \sqrt{\eta(\beta,1)+1}}\right.{\nonumber}\\ & + \lsb (a_1+d_1) \ln \lb \frac{\sqrt{\eta(\beta,1)+1}+\epsilon}{\sqrt{\eta(\beta,1)+1}-\epsilon }\rb +\frac{2 \epsilon a_1 }{\sqrt{\eta(\beta,1)+1}}\rsb{\nonumber}\\ &\left. +\frac{4 b_0 \lsb a_1+d_1\lb \eta(\beta,1)+1 \rb\rsb \sqrt{r_d+r_s} }{(d_1-a_1)^{3/2} \sqrt{ \eta(\beta,1) \lsb \eta(\beta,1)+1\rsb r_d r_s }}\rcb (1-v)\\ &\equiv \Delta t(v=1)+\Delta t_{c-v}.\end{aligned}$$ Here we define $\Delta t_{c-v}$ as the deviation of the timelike signal’s time delay from that of the null signal. $\Delta t_{c-v}$ has a particular advantage in GW/GRB dual lensing: because it is independent of the GW and GRB emission time difference and therefore can be used to constrain the GW speed [@Fan:2016swi; @Jia:2019hih]. Time delay in KN spacetime and the spin of Sgr A\* SMBH {#sec:tdkn} ======================================================= We now apply the time delay to the KN spacetime and use it to constrain the spin of the Sgr A\* SMBH. We will first find out the time delay in KN spacetime and then substitute values of necessary parameters associated with Sgr A\* to show how the time delay is related to various parameters, especially the spacetime spin angular momentum. The metric of the KN spacetime is given by $$\begin{aligned} \dd s^2=&-\frac{\Delta-a^2\sin^2\theta}{\Sigma}\dd t^2+\frac{(a^2+r^2)^2-a^2\Delta \sin^2\theta}{\Sigma}\sin^2\theta\dd \phi^2-\frac{2a\sin^2\theta(a^2-\Delta+r^2)}{\Sigma}\dd t\dd \phi \nonumber \\ &+\frac{\Sigma}{\Delta}\dd r^2+\Sigma\dd \theta^2,\end{aligned}$$ where (r,)=r\^2+a\^2\^2, (r,)=r\^2-2Mr+a\^2+Q\^2 and $M,~Q,~a=J/M$ are respectively the total mass, total charge and the specific spin angular momentum of the spacetime. From this, one can immediately read off the metric functions in the equatorial plane and consequently find their asymptotic expansions in the form of Eq. with coefficients $$\label{eq:kncoeff1} \begin{split} &a_1=-2M,~a_2=Q^2,\\ &b_0=-4aM,~b_1=2aQ^2, \\ &c_2=a^2,~c_3=2a^2M, \\ &d_1=2M~(\text{i.e.}~\gamma=1),~d_2=4M^2-a^2-Q^2,~d_3=4M\lb 2 M^2 - a^2 - Q^2 \rb \end{split}$$ and all other coefficients equal to zero. Substituting these into Eq. , the time delay in KN spacetime becomes $$\begin{aligned} \label{eq:dtkn} \Delta t_K = & \frac{\sqrt{\eta_K+1}}{\eta_K}\cdot \frac{4\epsilon M(1+v^2)}{ v^3}+\frac{2 M \left(3 v^2-1\right) }{v^3}\ln \lb \frac{\sqrt{\eta_K+1}+\epsilon }{\sqrt{\eta_K+1}-\epsilon }\rb \nonumber \\ & - \sqrt{\frac{\eta_K+1}{\eta_K}}\cdot \frac{4 a \sqrt{2(1+v^2)M( r_s+r_d)}}{v\sqrt{ r_s r_d} } + \mathcal{O}\lb \beta\sqrt{\frac{M^3}{r_{s,d}}},\beta^4r_{s,d} \rb,\end{aligned}$$ where \_K=\_K(,v)= . \[eq:knbeta\] Note that the charge $Q$ appears only in the third term in Eq. , which is at least an order $\sqrt{M/r_{s,d}}$ smaller than the first and/or second terms and consequently is negligible in Eq. . Most importantly, it is seen that as pointed out in Sec. \[sec:td\], when $\beta\to0$, $\eta_K(\beta,v)\to\infty$ and the third term in Eq. dominates $$\begin{aligned} \label{eq:tdknps} \Delta t_K(\beta\to 0)\approx -\frac{4 a \sqrt{2(1+v^2) M (r_d+r_s)}}{v\sqrt{ r_d r_s }} + \mathcal{O}\lb \frac{aM}{r_{s,d}}\rb,\end{aligned}$$ which is linear to the spacetime spin $a$. For long time, the spin of the Sgr A\* has not been well constrained due to the relatively low accretion rate comparing to many other SMBHs [@Andreas:2018]. In this work we model the Sgr A\* SMBH as a KN BH with GL happening in its equatorial plane. The signal sources can be stars for light signal, binary mergers for GWs and supernovas for neutrinos. In particular, one can verify easily from Eq. that when other parameters ($r_d,~v,~M,~a$) are fixed and $\beta$ is small, the smaller the $r_s$, the larger the time delay. Therefore it is advantageous to consider sources that are as close to the BH as possible and which still allow the weak field limit to hold. Such sources at least include some stars in the S star category near the Sgr A\, such as the star S39 whose orbit has an edge-on inclination of $89.36\pm0.73 ^\circ$ with respect to the plane of the sky [@2017ApJ...837...30G]. Using the orbit data of S39, we can find that its $r_s\approx 1.113\times 10^{-3}~\text{[pc]}\equiv r_{S39}$ when it is on far side of the SMBH. ![The time delay for the KN spacetime as a function of $\beta$ ((a) logarithmically and (b) linearly), $a$ ((c) logarithmically and (d) linearly). The “1st, 2nd” and “3rd” stand for corresponding terms of Eq. . In plot (c), the 3rd term depends on $\beta$ very weakly and therefore the solid curves overlap for different $\beta$. In plot (d), the 3rd term dominates $\Delta t$ when $\beta$ is small and therefore the “+” lines slightly overlap the solid $\Delta t$ lines. \[fig:sgra1\]](dtobetrpc.pdf "fig:"){width="45.00000%"} ![The time delay for the KN spacetime as a function of $\beta$ ((a) logarithmically and (b) linearly), $a$ ((c) logarithmically and (d) linearly). The “1st, 2nd” and “3rd” stand for corresponding terms of Eq. . In plot (c), the 3rd term depends on $\beta$ very weakly and therefore the solid curves overlap for different $\beta$. In plot (d), the 3rd term dominates $\Delta t$ when $\beta$ is small and therefore the “+” lines slightly overlap the solid $\Delta t$ lines. \[fig:sgra1\]](dtobetallrpc.pdf "fig:"){width="45.00000%"}\ (a)(b)\ ![The time delay for the KN spacetime as a function of $\beta$ ((a) logarithmically and (b) linearly), $a$ ((c) logarithmically and (d) linearly). The “1st, 2nd” and “3rd” stand for corresponding terms of Eq. . In plot (c), the 3rd term depends on $\beta$ very weakly and therefore the solid curves overlap for different $\beta$. In plot (d), the 3rd term dominates $\Delta t$ when $\beta$ is small and therefore the “+” lines slightly overlap the solid $\Delta t$ lines. \[fig:sgra1\]](dtoarpc.pdf "fig:"){width="45.00000%"} ![The time delay for the KN spacetime as a function of $\beta$ ((a) logarithmically and (b) linearly), $a$ ((c) logarithmically and (d) linearly). The “1st, 2nd” and “3rd” stand for corresponding terms of Eq. . In plot (c), the 3rd term depends on $\beta$ very weakly and therefore the solid curves overlap for different $\beta$. In plot (d), the 3rd term dominates $\Delta t$ when $\beta$ is small and therefore the “+” lines slightly overlap the solid $\Delta t$ lines. \[fig:sgra1\]](tdrpca.pdf "fig:"){width="42.00000%"}\ (c)(d) In Fig. \[fig:sgra1\], we plot the total time delay and each term of it for the Sgr A\, using values of parameters $M=4.1\times 10^6 M_{\odot}$, $r_{d}=8.12$ \[kpc\] and $Q \leqslant 3\times 10^8 $ \[C\] [@Andreas:2018]. In Fig. \[fig:sgra1\] (a) and (b), we plot the time delay logarithmically for $\|\beta\|<10$ \[as\] and linearly for $\|\beta\|<10^{-5}$ \[as\] respectively using value of $a=0.71M$ for a source that is located at radius $r_{S39}$ for light signal $(v=1)$. This value of $a$ is the mean value of its currently favored range $[0.5M,~0.92M]$ [@Andreas:2018]. It is clear that when $\|\beta\|\lesssim 10^{-3}$ \[as\], the third term of Eq. depends on $\beta$ very weakly. More importantly, when $\|\beta\|\lesssim 1.5\times 10^{-5}$ \[as\], this term is larger than the first and second terms and therefore dominates the total time delay. While for $ 1.5\times 10^{-5}\lesssim \|\beta\|\lesssim 10^{-3}$ \[as\], the first and second terms will be of similar size which is much larger than the third term. When $\|\beta\|\gtrsim 10^{-3}$ \[as\], the first term will be larger than the second and third terms. To see more clearly the effect of the spin $a$ on the time delay, in Fig. \[fig:sgra1\] (c) and (d), the time delay are plotted logarithmically and linearly respectively for three values of $\beta$ ($10^{-8}$ \[as\], $10^{-7}$ \[as\] and $10^{-6}$ \[as\]) for $a$ from $-2M$ to $2M$ for light signal. Although it is generally believed that the spacetime in the Galaxy center is a BH rather than a naked singularity, the time delay formula found in this work can work for both BH and naked singularity spacetimes. It is seen that when $\|\beta\|<10^{-6}$ \[as\], the time delay due to the spin of the spacetime will dominate the total time delay for $\|a\|\gtrsim 0.2M$. The smaller the $\beta$, the larger the range of $\|a\|$ in which the spin term dominates. In general, for $\|\beta\|<10^{-6}$ \[as\], it is seen that the time delay depends on the value of $a/M$ linearly, which reaches about 2.0 second at $a/M=1$ for the given parameter setting, as dictated by Eq. . This value is well within the reach of current observatories for GRB, GW [@TheLIGOScientific:2017qsa; @Monitor:2017mdv] or supernova neutrino signals [@Jia:2017oar; @Suzuki:2019jby]. By measuring this time delay, therefore the spin of the corresponding Sgr A\* SMBH can be deduced. We emphasis that this linear dependance of the time delay on spin $a$ for small $\beta$ is not limited to light signals. From Eq. , we see clearly that the linear dependance is present in the time delay of signals with any fixed velocity, including that of GWs and neutrinos. Indeed, for relativistic timelike signals such as supernova neutrinos and massive GWs, as can be seen from Eq. their time delay will be very close to that of the light signal. The difference of the GW time delay and that of the GRB has been proposed to constrain the GW speed [@Fan:2016swi; @Liao:2017ioi]. However, a simple order estimation reveals that this difference, $\Delta t_{c-v}$ in Eq. , is much smaller than the main time delay $\Delta t(v=1)$, especially when $v$ is very close to 1 as for supernova neutrinos [@Tanabashi:2018oca] and GWs [@Abbott:2017oio; @TheLIGOScientific:2017qsa; @Monitor:2017mdv]. For KN spacetime, substituting the coefficients into the $\Delta t_{c-v}$ in Eq. , one obtains the time delay difference t\_[c-v]{}= - +(1-v) \[eq:tddkn\] with $\eta_K(\beta,1)$ given in Eq. with $v=1$. Simple order analysis shows that according to the values of $\beta^2r_{s,d}$, $a^2/r_{s,d}$ and $M$, there will be three cases for the dominance of term(s) of Eq. and the value of $\Delta t_{c-v}$. When (1) $\beta^2r_{s,d}< a^2/r_{s,d}$, the third term of Eq. dominates and $\Delta t_{c-v}$ is roughly a constant, $2a\sqrt{M/r_{s,d}}$. If (2) $a^2/r_{s,d}< \beta^2r_{s,d}< M$, the first and second terms contribute similarly and $\Delta t_{c-v}$ will be $2\beta^1(2Mr_{s,d})^{1/2}$. If (3) $\beta^2r_{s,d}> M$, then the first term contribute most to $\Delta t_{c-v}$, which approximately is $\beta^2 r_{s,d}$. The later two cases are similar to the SSS case in which there is no spin angular momentum [@Jia:2019hih]. ![The time delay difference for the KN spacetime as a function of $\beta$ and $r_s=r_d=D$ for a GW with speed $v=(1-3\times 10^{-15})c$. The red and blue curves are plotted using $\beta^2 r_{s,d}=a/r_{s,d}$ and $\beta^2 r_{s,d}=M$. \[fig:tddiff\]](tdvtolaave.eps){width="60.00000%"} In Fig. \[fig:tddiff\], we plot $\Delta t_{c-v}$ between time delays of light signal and a timelike signal with velocity $v=(1-3\times 10^{-15})c$ as a function $\beta$ and $r_s=r_d=D$. The $3\times 10^{-15}c$ is the maximal deviation that GW speed can get from the speed of light [@TheLIGOScientific:2017qsa; @Monitor:2017mdv]. The above three cases of dominance are then separated by the red and blue curves, which are plotted using $\beta^2 r_{s,d}=a/r_{s,d}$ and $\beta^2 r_{s,d}=M$ respectively. As discussed above, in the regions below the red curve (or above the blue curve), the third term (or first term) dominates the time delay difference of Eq. . While in the region between the two curves, the first and second terms contribute similarly. Given that the time resolution of current GRB and GW can roughly reach the 0.05 \[s\] and 0.002 \[s\] [@Monitor:2017mdv] level respectively, from Fig. \[fig:tddiff\] we observe that in order to obtain $\Delta t_{c-v}$ that is larger than these resolutions, the parameters $\beta$ and $r_{s,d}$ should be in case (3), well beyond the case (1) in which the spacetime spin plays a role. Therefore we can conclude that for the current or near future GRB and GW time resolution, the spin angular momentum will not affect the detectable time delay difference between the two kinds of signals in the weak field limit. Conclusion and discussions ========================== It is shown using a perturbative method that the total travel time for both timelike and null signals in the asymptotically flat SAS spacetimes can be expressed as a quasi-series of the impact parameter. The $n$-th order coefficient of this series is determined by coefficients up to the $n$-th order of the asymptotic expansion of the metric functions. Solving the impact parameters for both the GL images, we can obtain the time delay between them to the leading order(s) of $M/r_{s,d}$ and $\beta$ for signals with general velocity. Dominance of different terms in the parameter space spanned by $(M/r_{s,d},~\beta)$ is analyzed. One particular interesting case is when $\beta$ is very small. The time delay in this case then depends on the spacetime spin angular momentum $a$ linearly (see Eq. ). For Sgr A\* and typical values of $a$ (e.g. $a=0.71M$), $\Delta t$ can reach the $\mathcal{O}(1)$ \[s\] order for $\beta\lesssim 10^{-5}$ \[as\], well within the precision of time measurement for GW, GRB or neutrinos events. Therefore this time delay might be used to constrain the spacetime spin very precisely. Although we mainly applied the time delay to Sgr A\, in principle it can also be applied to other systems, including BHs with regular mass inside the Galaxy, or other SMBHs in nearby galaxy centers. The latter are farther from us than Sgr A\, and we see that the time delay will also be larger by a factor of $\sqrt{r_{SMBH}/r_{Sgr~A}}$. This will ease the time measurement to allow larger uncertainty of the events. We also emphasis that both the total time and the time delay are applicable to general asymptotically flat SAS spacetimes. Keeping higher orders in $M/b$ and/or $b/r_{s,d}$ than Eqs. and , higher order coefficients in the metric functions will affect the total time or time delay. Therefore by measuring the time delay to the necessary accuracy, these coefficients, which are often of great interest to many alternative gravitational theories [@Will:2014kxa], can be constrained. We thank Ke Huang for valuable discussions. This work is supported by the NNSF China 11504276. 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--- abstract: 'The indirect RKKY interaction in a layered metal with nearly nested (almost squared) Fermi surface is evaluated analytically. The final expressions are obtained in closed form as a combination of Bessel functions. We show that the expected “$2k_F$” oscillations occur as the far asymptote of our expressions, where a value of the effective Fermi momentum $k_F$ depends on the direction in $r-$space. We demonstrate the existence of the intermediate asymptote of the interaction which is of the sign-reversal antiferromagnetic type. This part of RKKY interaction is the only term surviving in the limit of exact nesting. A good accordance of our analytical formulas with numerical findings is found until the interatomic distances.' address: 'Petersburg Nuclear Physics Institute, Gatchina 188350 St.Petersburg, Russia ' author: - 'D.N. Aristov' - 'S.V. Maleyev' title: Indirect RKKY Interaction in the Nearly Nested Fermi Liquid --- , Exchange and superexchange interactions ; High-$T_c$ compounds ; Local moment in compounds and alloys. The Ruderman-Kittel-Kasuya-Yosida interaction (RKKY) [@rkky] plays an important role in the discussion of the interaction of localized moments in a metal through polarization of conduction electrons. It is well known that in three spatial dimension and for the spherical Fermi surface (FS) this interaction decays as $r^{-3}$ and has the $2k_F$ oscillations with the Fermi momentum $k_F$. In the case of anisotropic FSs the RKKY at largest distances is characterized by the direction-dependent period of oscillations $(2k_F^\ast)^{-1}$ and can be represented as a series in $1/r$. [@rkky-ani] Unfortunately, this series diverges at smaller distances $k_F^\ast r \sim 1$, where the RKKY is mostly significant. Nowadays the numerical calculations are a principal tool for the analysis of RKKY in metals with non-spherical FS. However the theoretical understanding is important here even on the qualitative level. In [@arimal] we developed an analytical approach fo the calculation of the RKKY interaction in metals with highly anisotropic FS. This study was motivated by the experimental observations, that the rare-earth (R) subsystem in the high-$T_c$ compounds RBa$_2$Cu$_3$O$_{7-\delta}$ undergoes a transition to the magnetically ordered antiferromagnetic phase below $\sim 1\,$K. [@Lynn] It can be theoretically shown that the onset of the $d-$wave superconductivity does not seriously alter the RKKY interaction at moderate distances [@ArMaYa]. The main anisotropy of RKKY below $T_c$ at those distances is thus determined by the anisotropy of the FS in the normal state. To study the possible character of RKKY in high-$T_c$ cuprates, a model of the nearly-nested (almost squared) FS in a layered metal was chosen. We analyzed this case in detail and showed a good accordance of our analytical findings with the numerical results. It turned out that the widely used notion of the “$2k_F$” oscillations could be applied to the far asymptote of the interaction. At intermediate distances the interaction has the sign-reversal antiferromagnetic character, and only this behaviour survives at the exact nesting of the FS. The essence of our method could be outlined as follows. The range function of the RKKY interaction can be represented as a sum over the Matsubara frequencies $ \omega_n = \pi T(2n+1)$ : $$\chi({\bf r}) = - T \sum\nolimits_n G(i\omega_n, {\bf r} )^2 \label{rkk-inter}$$ with the electronic Green’s function $G(i\omega_n, \bk ) = [i\omega_n - \varepsilon(\bk)]^{-1} $. Passing here to $r-$representation, we map the whole vicinity of the FS by the patches of the size $\kappa$ of order of inverse interatomic distances, where the dispersion $\varepsilon(\bk)$ can be approximated by a simpler form. Such mapping can be done for the particular case of the nearly-nested FS, when the spectrum has a tight-binding form, $\varepsilon(\bk) = -t( \cos k_x + \cos k_y) + \mu$ and $\|\mu\|\ll t$. One distinguishes here the vicinities of the Van Hove points $(0,\pm\pi)$, $(\pm\pi,0)$ and the flat parts of the spectrum near $(\pm\pi/2,\pm\pi/2)$. It is possible to obtain a closed form for the [partial]{} Green’s functions stemmed from each of the patches. This form is applicable in $r-$space roughly up to the scale $\kappa^{-1}$. The total Green’s function is estimated as a sum of the partial ones. The square of Green’s function in (\[rkk-inter\]) yields two types of contributions. First is the product of the partial Green’s functions $G_j(i\omega_n, {\bf r})^2$ from the same patch $j$. These terms define the far asymptote of $\chi({\bf r})$, established earlier, [@rkky-ani] and have the $2k_F^\ast$ oscillations. The second contribution to $\chi({\bf r})$ is determined by the interference terms $G_j(i\omega_n, {\bf r}) G_l(i\omega_n, {\bf r})$ with $j\neq l$. In addition to the smooth dependence on $\bf r$, these latter terms possess the overall prefactors $\exp i(\bk_j -\bk_l){\bf r}$, with the spanning vectors $\bk_j -\bk_l$ connecting the centers of the patches $\bk_j$ and $\bk_l$. For the case of nearly-nested FS the effective Fermi momentum $k_F^\ast$ does not exceed the value $\sqrt{\|\mu/t\|} \ll 1$ and vanishes along the diagonals $r_x=\pm r_y$. It turns out that the interference terms $G_{(0,\pm\pi)}(i\omega_n, {\bf r}) G_{(\pm\pi,0)}(i\omega_n, {\bf r})$ dominate at the intermediate distances $r < (k_F^\ast)^{-1}$ for most of the directions of $\bf r$. The spanning vector between the Van Hove points is ${\bf Q}= (\pi,\pi)$, thus the commensurate antiferromagnetic modulation $\exp i {\bf Qr}$ of the RKKY is observed. Along the diagonals the RKKY interaction is of the ferromagnetic sign, and non-oscillatory for moderate $r < \|t/\mu\|$ ; this, however, does not determine a particular type of magnetic ordering in the subsystem of localized moments. Our results suggest that definite shapes of the FS favor the commensurate antiferromagnetic ordering of localized spins. Particularly, it should take place, if the FS lies close to the symmetry points of the Brillouin zone, the case relevant to the high-$T_c$ cuprates. Thus our findings could explain the antiferromagnetism in the subsystem of rare-earth ions, systematically observed at low temperatures in these compounds. The partial financial support from the Russian State Program for Statistical Physics (grant VIII-2) is acknowledged with thanks. [9]{} M.A. Ruderman, C. Kittel, Phys.Rev. [96]{}, 99 (1954); T. Kasuya, Progr.Theoret.Phys. (Kyoto) [16]{}, 45 (1956); K. Yosida, Phys.Rev. [106]{}, 893 (1957). L.M. Roth, H.J. Zeiger, T.A. Kaplan, Phys.Rev. [149]{}, 519 (1966). D.N. Aristov and S.V. Maleyev, Phys. Rev. B [56]{}, 8841 (1997) ; see also D.N. Aristov, [ibid.]{} [55]{}, 8064 (1997). see, e.g., J.W. Lynn, Physica [B 163]{}, 69 (1990). D.N. Aristov, S.V. Maleyev, A.G. Yashenkin, Z.Phys. B [102]{}, 467 (1997).
--- abstract: 'We study the effective anisotropy induced in thin nanomagnets by the nonlocal demagnetization field (dipole-dipole interaction). Assuming a magnetization independent of the thickness coordinate, we reduce the energy to an inhomogeneneous onsite anisotropy. Vortex solutions exist and are ground states for this model. We illustrate our approach for a disk and a square geometry. In particular, we obtain good agreement between spin–lattice simulations with this effective anisotropy and micromagnetic simulations.' author: - 'Jean–Guy Caputo' - Yuri Gaididei - 'Volodymyr P. Kravchuk' - 'Franz G. Mertens' - 'Denis D. Sheka' date: '28.05.07' title: 'Effective anisotropy of thin nanomagnets: beyond the surface anisotropy approach' --- Introduction {#sec:introduction} ============ Magnetic nanoparticles and structures have recently attracted a growing interest for their physical properties and a number of possible applications. [@Hubert98; @Skomski03; @Bader06] For example the vortex (ground) state of a disk–shaped nanoparticle could provide high density storage and high speed magnetic RAM [@Cowburn02]. The theoretical models for these systems have been known for some time [@Brown63; @Aharoni96] and include the nonlocal demagnetization field. At microscopic level this field is due to the dipolar interaction $$\label{eq:H-dipolar} \!\!\!\!\! \mathcal{H}_{\text{d}} \!=\! \frac{D}{2}\!\! \sum_{\substack{{\bm{n}}, {\bm{m}}\\{\bm{n}}\neq {\bm{m}}}}\! \left[ \frac{{\bm{S}}_{{\bm{n}}}\cdot {\bm{S}}_{{\bm{m}}}}{r_{{\bm{n}} {\bm{m}}}^3} - 3 \frac{ \left({\bm{S}}_{{\bm{n}}}\cdot {\bm{r}}_{{\bm{n}} {\bm{m}}} \right) \left({\bm{S}}_{{\bm{m}}}\cdot {\bm{r}}_{{\bm{n}} {\bm{m}}} \right)}{r_{{\bm{n}} {\bm{m}}}^5} \right]\!.$$ Here ${\bm{S}}_{{\bm{n}}}\equiv\left(S^x_{{\bm{n}}}, S^y_{{\bm{n}}}, S^z_{{\bm{n}}}\right)$ is a classical spin vector with fixed length $S$ on the site ${\bm{n}}=(n_x,n_y, n_z)$ of a three–dimensional lattice. The summation runs over all magnets $({\bm{n}},{\bm{m}})$, and ${\bm{r}}_{{\bm{m}}{\bm{n}}} \equiv {\bm{r}}_{{\bm{n}}} - {\bm{r}}_{{\bm{m}}}$. The parameter $D = \mu_B^2 g^2$ is the strength of the long range dipolar interaction and $g$ is the Lande–factor. In the past analytical studies have been mainly limited to assuming a homogeneous demagnetization field distribution, uniform Stoner–Wohlfarth theory[@Stoner48] and near–uniform Brown’s linear analysis [@Skomski03]. Recent advances in nanotechnology and computing power established the complexity of magnetization distribution in nanoparticles. For example square nanoparticles exhibit buckling states, flower states, apple states, leaf states etc, [@Usov92; @Cowburn98; @Ivanov04] when their size exceeds the single–domain limit. In disk–shaped particles vortex states [@Usov94; @Hubert98], edge fractional vortex states etc [@Kireev03; @Tchernyshyov05] appear. Some of these complex states can be obtained by a small perturbation of a homogeneous state. For example @Cowburn98a showed that dipolar interactions cause flower and leaf states in square nanoparticles, which was confirmed by direct experiments[@Cowburn98]. However the linear analysis does not work for topologically nontrivial states like kinks, vortices etc. One possibility to study these structures in nanomagnets is the Ritz variational method. It was applied to analyze the vortex structure of the disk–shaped nanodot[@Usov94; @Hubert98]. A disadvantage of this method is to limit the solution to a certain class of minimizers, so that one can usually study only one type of excitation. Linear waves are left out together with their coupling to the main excitation. The various regimes were studied in Refs. . The important length scale is the magnetic exchange length $\ell=\sqrt{\textsf{A}/4\pi M_S^2}$ where $\textsf{A}$ is the exchange constant and $M_S$ is the saturation magnetization. Depending on the relation between the film diameter $2R$, its thickness $h$ and $\ell$ many scaling limits can be analyzed, see Ref. for an overview. Probably the first rigorous study was made by @Gioia97 who showed that for an infinitesimally thin–film ($h/R\to0$, $\ell/R\to\text{const}$) the magnetostatic energy tends to an effective 2D easy–plane anisotropy energy. In this case the ground state is a homogeneous in–plane magnetization state[@Gioia97]. This effective easy–plane anisotropy has a simple magnetostatic interpretation. The sources of magnetostatic field are volume and surface magnetostatic charges. For thin structures one can neglect the volume charges. Face surface charges contribute to the energy density as $2\pi M_z^2$ which is the same term one would get with an effective easy–plane anisotropy[@Ivanov05]. In the case $h/R \ll 1$ and $\ell^2\ll 2 h\,R\,\|\ln(h/2R)\|$ the magnetization develops edge defects, including fractional vortices.[@Moser04; @Tchernyshyov05; @Kurzke06] This problem has a boundary constraint and an interior penalty. It is relevant for typical Permalloy ($\text{Ni}_{80}\text{Fe}_{20}$, Py) disks where we have $h\sim20$nm, $2R\sim100$nm and $\ell\sim5.3$nm. It was shown in Refs. that in the limit $h/R\rightarrow 0$ under the scaling $$\label{eq:Kohn-limit} \frac{2hR}{\ell^2}\left\|\ln\frac{h}{2R}\right\|\rightarrow C$$ the full $3D$ micromagnetic problem reduces to a much simpler $2D$ variational problem where the magnetostatic energy tends to the effective surface anisotropy term $$\label{eq:surf} E_{\text{surf}}=\int\limits_{\partial \Omega}({\bm{S}}\cdot{\bm{\tau}})^2 \mathrm{d}S$$ where ${\bm{\tau}}$ is the local tangent vector on the domain boundary $\partial \Omega$. In this case the magnetization ${\bm{S}}$ has no out of plane component $(S_z=0)$ and does not develop walls and vortices. To study nanomagnets with curling ground states, here we develop a new analytical approach. We split the dipole-dipole spin interaction (\[eq:H-dipolar\]) into two parts. The first one is an on-site anisotropy with spatially dependent anisotropy coefficients. The second part represents an effective dispersive interaction. The anisotropy interaction consists of two terms: an easy-plane anisotropy and an in-plane anisotropy. We show that the vortex state minimizes the effective in-plane anisotropy. We also show that for ultra-thin nanomagnets $(h/R\to 0)$ the in-plane anisotropy term reduces to the surface anisotropy . For the nonhomogeneous state which is our main interest, our approach is valid if $$\label{eq:effective-correct} R\gg h \quad \text{and} \quad R\gg\ell.$$ In Sec. \[sec:eff-anis\] we introduce our discrete model together with the dipolar energy and adapt it to the plain–parallel spin–field distribution, which is our main simplification. We further simplify the model by considering only the local part of the dipolar energy, which results in an effective anisotropy. In the continuum approximation of the system we get a local energy functional with nonhomogeneous anisotropy coefficients, see Sec. \[sec:continuum\]. The dispersive interaction is discussed in Sec. \[sec:disp\]. To illustrate our method of effective anisotropy we consider in Sec. \[sec:disk\] the disk–shape nanoparticle and study its ground state spin distribution. Our simple model describes exactly the homogeneous state (see Sec. \[sec:disk-hom\]) and very precisely the vortex state (see Sec. \[sec:disk-vortex\]). In Sec. \[sec:simulations\] we confirm our analysis by numerical simulations. These are done first for the disk–shaped nanoparticle (Sec. \[sec:disk-simulations\]) and then for the prism–shaped one (Sec. \[sec:prism\]). We discuss our results in Sec. \[sec:discussion\]. Model. Effective anisotropy {#sec:eff-anis} =========================== We consider a ferromagnetic system described by the classical Heisenberg isotropic exchange Hamiltonian $$\label{eq:H-ex} \mathcal{H}_{\text{ex}} =-\frac{J}{2}\!\! \sum_{\langle {\bm{n}},{\bm{n}}'\rangle}\!{\bm{S}}_{{\bm{n}}} {\bm{S}}_{{\bm{n}}'} ,$$ where the exchange integral $J > 0$ and the summation runs over nearest–neighbor pairs $\langle {\bm{n}},{\bm{n}}'\rangle$. The total Hamiltonian is a sum of the exchange energy and the dipolar one . Our main assumption is that ${\bm{S}}_{{\bm{n}}}$ depends only on the $x$ and $y$ coordinates. Such a plane–parallel spin distribution is adequate for thin films with a constant thickness $h=N_z a_0$, ($a_0$ being the lattice constant) and nanoparticles with small aspect ratio. The exchange interaction can be written as the sum of an intra–plane $\mathcal{H}_{\text{ex}}^{\text{intra}}$ term and an inter–plane one $\mathcal{H}_{\text{ex}}^{\text{inter}}$ $$\label{eq:H-ex2} \begin{split} \mathcal{H}_{\text{ex}}^{\text{intra}} =&-\frac{(N_z+1)J}{2}\!\! \sum_{\substack{\langle {{\bm{\nu}}},{{\bm{\nu}}}'\rangle}}\! {\bm{S}}_{{{\bm{\nu}}}} {\bm{S}}_{{{\bm{\nu}}}'}, \\ \mathcal{H}_{\text{ex}}^{\text{inter}} =&-N_z J \sum_{{{\bm{\nu}}}}\!\left({\bm{S}}_{{{\bm{\nu}}}}\right)^2 = -N_z N_x N_y J S^2. \end{split}$$ Here and below the Greek index ${{\bm{\nu}}}=(n_x,n_y)$ corresponds to the XY components of the vector ${\bm{n}}=(n_x,n_y,n_z)$. One can see that the inter–plane interaction is equivalent to an on–site anisotropy. The inter–exchange term gives a constant contribution, so it can be omitted. Let us consider the dipolar energy. Using the above mentioned assumption about the plane–parallel spin distribution, the dipolar Hamiltonian can be written as (see Appendix \[sec:appendix-discrete\] for the details): $$\label{eq:H-dipolar-via-ABC} \begin{split} &\mathcal{H}_{\text{d}} = -\frac{D}{2}\!\! \sum_{\substack{{{\bm{\nu}}},{{\bm{\mu}}}}}\! \Bigl[ A_{{{\bm{\mu}}}{{\bm{\nu}}}}\left({\bm{S}}_{{{\bm{\nu}}}}\cdot {\bm{S}}_{{{\bm{\mu}}}} - 3 S_{{{\bm{\nu}}}}^zS_{{{\bm{\mu}}}}^z\right)\\ &+ B_{{{\bm{\mu}}}{{\bm{\nu}}}} \left( S_{{{\bm{\nu}}}}^x S_{{{\bm{\mu}}}}^x - S_{{{\bm{\nu}}}}^y S_{{{\bm{\mu}}}}^y \right) + C_{{{\bm{\mu}}}{{\bm{\nu}}}}\left( S_{{{\bm{\nu}}}}^x S_{{{\bm{\mu}}}}^y + S_{{{\bm{\nu}}}}^y S_{{{\bm{\mu}}}}^x \right)\Bigr].\end{split}$$ Here the sum runs only over the 2D lattice XY. All the information about the original 3D structure of our system is in the coefficients $A_{{{\bm{\mu}}}{{\bm{\nu}}}}$, $B_{{{\bm{\mu}}}{{\bm{\nu}}}}$ and $C_{{{\bm{\mu}}}{{\bm{\nu}}}}$, \[eq:A-B-C\] $$\begin{aligned} \label{eq:A-B-C-(1)} A_{{{\bm{\mu}}}{{\bm{\nu}}}}&=\frac12 \sum_{\substack{m_z,n_z\\r_{{\bm{n}}{\bm{m}}}\neq0}} \frac{r_{{\bm{m}}{\bm{n}}}^2 - 3z_{{\bm{m}}{\bm{n}}}^2}{r_{{\bm{m}}{\bm{n}}}^5}, \\ \label{eq:A-B-C-(2)} B_{{{\bm{\mu}}}{{\bm{\nu}}}}&= \frac32 \sum_{\substack{m_z,n_z\\r_{{\bm{n}}{\bm{m}}}\neq0}} \frac{x_{{\bm{m}}{\bm{n}}}^2 - y_{{\bm{m}}{\bm{n}}}^2}{r_{{\bm{m}}{\bm{n}}}^5}, \\ \label{eq:A-B-C-(3)} C_{{{\bm{\mu}}}{{\bm{\nu}}}}&= 3 \sum_{\substack{m_z,n_z\\r_{{\bm{n}}{\bm{m}}}\neq0}} \frac{x_{{\bm{m}}{\bm{n}}} y_{{\bm{m}}{\bm{n}}}}{r_{{\bm{m}}{\bm{n}}}^5}.\end{aligned}$$ To gain insight into the anisotropic properties of the system we represent the dipolar energy as a sum $$\mathcal{H}_{\text{d}}=\mathcal{H}_{\text{d}}^{\text{loc}}+ \Delta\mathcal{H}_{\text{d}},$$ where $$\label{eq:Hd-loc} \begin{split} \mathcal{H}_{\text{d}}^{\text{loc}} &= -\frac{D}{2}\!\! \sum_{{{\bm{\nu}}}}\! \Biggl\{ \bar{A}_{{{\bm{\nu}}}}\Bigl[\left({\bm{S}}_{{{\bm{\nu}}}}\right)^2 - 3\left( S_{{{\bm{\nu}}}}^z\right)^2\Bigr]\\ & + \bar{B}_{{{\bm{\nu}}}} \Bigl[\left(S_{{{\bm{\nu}}}}^x\right)^2 - \left(S_{{{\bm{\nu}}}}^y\right)^2 \Bigr] + 2\bar{C}_{{{\bm{\nu}}}} S_{{{\bm{\nu}}}}^x S_{{{\bm{\nu}}}}^y\Biggr\}. \end{split}$$ is an effective on–site anisotropic energy and $$\label{eq:Hd-disp} \begin{split} \Delta\mathcal{H}_{\text{d}} &= \frac{D}{4}\!\! \sum_{\substack{{{\bm{\nu}}},{{\bm{\mu}}}}}\! \Biggl\{ A_{{{\bm{\mu}}}{{\bm{\nu}}}}\left[\left({\bm{S}}_{{{\bm{\nu}}}}- {\bm{S}}_{{{\bm{\mu}}}}\right)^2 - 3\,\left( S_{{{\bm{\nu}}}}^z - S_{{{\bm{\mu}}}}^z\right)^2\right]\\ &+ B_{{{\bm{\mu}}}{{\bm{\nu}}}} \left[ \left(S_{{{\bm{\nu}}}}^x -S_{{{\bm{\mu}}}}^x\right)^2 - \left(S_{{{\bm{\nu}}}}^y -S_{{{\bm{\mu}}}}^y\right)^2 \right] \\ &+ 2C_{{{\bm{\mu}}}{{\bm{\nu}}}} \left(S_{{{\bm{\nu}}}}^x - S_{{{\bm{\mu}}}}^x\right) \left(S_{{{\bm{\nu}}}}^y -S_{{{\bm{\mu}}}}^y \right)\Biggr\}. \end{split}$$ is the dispersive part of the dipolar interaction. Here we introduce the coefficients of effective anisotropy $$\label{eq:A-B-C-discrete} \begin{split} \bar{A}_{{{\bm{\nu}}}} &= \sum_{{{\bm{\mu}}}}A_{{{\bm{\mu}}}{{\bm{\nu}}}},\; \bar{B}_{{{\bm{\nu}}}} = \sum_{{{\bm{\mu}}}}B_{{{\bm{\mu}}}{{\bm{\nu}}}},\; \bar{C}_{{{\bm{\nu}}}} = \sum_{{{\bm{\mu}}}}C_{{{\bm{\mu}}}{{\bm{\nu}}}}. \end{split}$$ The dipolar energy $\mathcal{H}_{\text{d}}^{\text{loc}}$ contains only local interaction; it has a form of the anisotropy energy with nonhomogeneous $\bar{A}_{{{\bm{\nu}}}}$, $\bar{B}_{{{\bm{\nu}}}}$, $\bar{C}_{{{\bm{\nu}}}}$. In next sections we discuss these quantities. For this end we need to obtain the continuum limit of our model. Continuum description {#sec:continuum} --------------------- The continuum description of the system is based on smoothing the lattice model, using the normalized magnetization $$\label{eq:S-via-M} {\bm{m}}({\bm{r}}) = \frac{g\mu_B}{a_0^3M_S} \sum_{{\bm{n}}} {\bm{S}}_{{\bm{n}}} \delta({\bm{r}} - {\bm{r}}_{{\bm{n}}}),$$ where $M_S$ is the saturation magnetization. The exchange energy, the continuum version of is $$\label{eq:E-exchange} \begin{split} \mathcal{E}_{\text{ex}} &= \tfrac12 \textsf{A} (h+a_0)\int \mathrm{d}^2x \left({\bm{\nabla}} {\bm{m}}\right)^2, \end{split}$$ where $\textsf{A}=JM_S^2 a_0^5/D$ is the exchange constant. ![(Color online) Arrangement of coordinates in the local reference frame.[]{data-label="fig:arb_shape"}](arb_shape){width="0.7\columnwidth"} Now let us consider the dipolar energy and use its approximate Hamiltonian . To present this energy in a standard phenomenological form one needs to transform the summation over the lattice to an integration over the volume. There is a singularity for ${\bm{r}}_{{\bm{m}}{\bm{n}}} \to 0$. Using a regularization similar to the one in Ref. , we find (see Appendix \[sec:appendix-continuum\] for details) that the local part of the dipolar energy is $$\label{eq:E-dd-eff} \begin{split} \mathcal{E}_d &= \pi M_S^2 h \int \mathrm{d}^2 x \Biggl\{ \mathcal{A}(x,y) \left[ 1 - 3 \cos^2\theta \right]\\ & + \sin^2\!\theta\ \text{Re}\Bigl[\mathcal{B}(x,y) e^{2\imath(\phi-\chi)}\Bigr] \Biggr\}, \end{split}$$ where we used the angular parameterization for the magnetization: $m^z=\cos\theta$ and $m^x+im^y=\sin\theta e^{\imath\phi}$. Here and below we dropped the loc superscript. One can see that the original nonlocal dipolar interaction results in an effective local anisotropy energy. The coefficients of effective anisotropy $\mathcal{A}$ and $\mathcal{B}$ are nonhomogeneous: \[eq:A-and-B\] $$\begin{aligned} \label{eq:A-fin} \mathcal{A}(x,y) &= -\frac{2}{3} - \frac{a_0}{12h}\Biggl[8 \Theta_+(h) + 3 + \frac{3a_0^3}{(a_0^2+h^2)^{3/2}} \Biggr] \nonumber\\ &+\frac{1}{2\pi}\int_0^{2\pi}\mathrm{d}\alpha \Biggl[ \frac{\sqrt{P^2+h^2}-P}{h} + \frac{a_0}{\sqrt{P^2+h^2}} \nonumber\\ & + \frac{a_0^2}{4Ph}+ \frac{a_0^2P^2}{4h(P^2+h^2)^{3/2}}\Biggr],\\ \label{eq:B-fin} \mathcal{B}(x,y) &= \frac{1}{2\pi}\int_0^{2\pi} \mathcal{F}(P,h) e^{-2\imath\alpha}\mathrm{d}\alpha, \nonumber\\ \mathcal{F}(P,h) &= \frac{P-\sqrt{P^2+h^2}}{h} - 2\left(1+\frac{a_0}{h}\right) \ln\frac{\sqrt{P^2+h^2}-h}{P} \nonumber \\ &+ \frac{a_0}{\sqrt{P^2+h^2}} + \frac{3a_0^2}{4Ph} + \frac{a_0^2}{4h}\frac{3P^2+2h^2}{(P^2+h^2)^{3/2}},\end{aligned}$$ where the Heaviside function $\Theta_+(x)$ takes the unit values for any positive $x$ and zero values for $x\leq0$. In Eqs. we used the local reference frame $$\label{eq:loc-frame} x' = x + \rho\cos(\chi+\alpha),\quad y' = y + \rho\sin(\chi+\alpha),$$ which is centered at $(x,y)$. The term $P$ is the distance from this point to the border of the system, it depends on the azimuthal angle $\alpha$ and position $(x,y)$, see Fig. \[fig:arb\_shape\]. In the limit case of the pure 2D system (monolayer with $h=0$) the total energy, normalized by the 2D area $\mathcal{S}$, takes a form $$\label{eq:W-mono} \begin{split} & W^{h=0} \equiv \frac{\mathcal{E}_{\text{ex}} + \mathcal{E}_d}{M_S^2\mathcal{S}a_0} = W_{\text{ex}}^{h=0} + W_d^{h=0},\\ & W_{\text{ex}}^{h=0} = \frac{2\pi\ell^2}{\mathcal{S}}\int \mathrm{d}^2x \left[ ({\bm{\nabla}}\theta)^2 +\sin^2\theta ({\bm{\nabla}}\phi)^2\right],\\ &W_d^{h=0} = \frac{\pi}{\mathcal{S}}\int \mathrm{d}^2x\Biggl\{ \mathcal{A}^{h=0}(x,y)\left[ 1 - 3 \cos^2\theta \right]\\ & + \sin^2\theta\, \text{Re}\Bigl[\mathcal{B}^{h=0}(x,y) e^{2\imath(\phi-\chi)}\Bigr]\Biggr\},\\ &\mathcal{A}^{h=0}(x,y) = -\frac12 + \frac{a_0}{4\pi}\int_0^{2\pi} \frac{\mathrm{d}\alpha}{P},\\ &\mathcal{B}^{h=0}(x,y) = \frac{3a_0}{4\pi}\int_0^{2\pi} \frac{e^{-2\imath\alpha}\mathrm{d}\alpha}{P}, \end{split}$$ Here the exchange length $\ell$ has the standard form [@Brown63]: $$\label{eq:exchange-length} \ell = \sqrt{\frac{\textsf{A}}{4\pi M_s^2}} = a_0\sqrt{\frac{Ja_0^3}{4\pi D}}$$ Note that the dipolar induces magnetic anisotropy was considered by @Levy01 for a pure 2D spin system from a Taylor’s series expansion of the spin field. The above case has rather an academic interest. Below in the paper we consider another limit, when $h\gg a_0$. In that case one can neglect the energy of the monolayer $W^{h=0}$. The total energy, normalized by the volume of the magnet, takes a form: \[eq:W-wol\] $$\begin{aligned} \label{eq:W-vol-1} W &\equiv \frac{\mathcal{E}_{\text{ex}} + \mathcal{E}_d^h}{M_S^2\mathcal{S}h} = W_{\text{ex}} + W_d,\\ \label{eq:W-vol-2} W_{\text{ex}} &= \frac{2\pi\ell^2}{\mathcal{S}}\int \mathrm{d}^2x \Bigl[\left({\bm{\nabla}} \theta\right)^2 +\sin^2\theta\left({\bm{\nabla}} \phi\right)^2\Bigr],\\ \label{eq:W-vol-3} W_d &= \frac{\pi}{\mathcal{S}} \int \mathrm{d}^2 x \Biggl\{ \mathcal{A}(x,y) \left[ 1 - 3 \cos^2\theta\right]\nonumber \\ & + \sin^2\theta\,\text{Re}\Bigl[\mathcal{B}(x,y) e^{2\imath(\phi-\chi)}\Bigr] \Biggr\}.\end{aligned}$$ The effective anisotropy constants can be expressed as follows: \[eq:A&B\] $$\begin{aligned} \label{eq:A-vol}\mathcal{A}(x,y) &\approx \frac{1}{2\pi}\int_0^{2\pi} \mathrm{d}\alpha \frac{\sqrt{P^2 + h^2}-P}{h} - \frac23, \\ \label{eq:B-vol} \mathcal{B}(x,y) &= \frac{1}{2\pi}\int_0^{2\pi} \mathcal{F}(P,h) e^{-2i\alpha} \mathrm{d}\alpha,\\ \label{eq:F-vol} \mathcal{F}(P,h) &\approx \frac{P - \sqrt{P^2 + h^2}}{h} - 2\ln\frac{\sqrt{P^2 + h^2}-h}{P}.\end{aligned}$$ Let us discuss the magnetization distribution of the nanoparticle on a large scale. The equilibrium magnetization configuration is mainly determined by the dipolar interaction, which takes the form of an effective anisotropy . The coefficient $\mathcal{A}$ determines the uniaxial anisotropy along the z–axis. For thin nanoparticle this coefficient is always negative (with $\mathcal{A}\to - 2/3$ when $h \to0$) favoring an easy–plane magnetization distribution in agreement with the rigorous calculations [@Gioia97]. The coefficient $\mathcal{B}$ is responsible for the in–plane anisotropy in the XY–plane. Assume that all spins lie in the plane corresponding to the thin limit case. The preferable magnetization distribution in the XY–plane is the function $\phi$, minimizing the expression $\text{Re}\Bigl[\mathcal{B} e^{2\imath(\phi-\chi)}\Bigr]$ in . This is $$\label{eq:phi-min} \phi = \chi + \frac{\pi}{2} - \frac12 \text{Arg}\mathcal{B}.$$ The angle determines the in–plane effective anisotropy direction observed on a large scale, without exchange interaction and effective uniaxial anisotropy. The analysis of the $\mathcal{B}$–term shows that the effective anisotropy favors such an in–plane spin distribution, always directed tangentially to the border near the sample edge (see Appendix \[sec:appendix-2geom\] for the details). This statement agrees with results for the pure surface anisotropy [@Kireev03]. Finer details depend on the geometry of the particle so we need to distinguish the disk shape from the square shape. Dispersive part of the dipolar interaction {#sec:disp} ------------------------------------------ In the continuum description the dispersive part of the dipolar interaction takes the form $$\label{eq:H-disp-cont} \begin{split} & \Delta\mathcal{E}_{\text{d}} = \frac{M_S^2\,a_0^6}{4}\!\! \int\!\! \mathrm{d}^2x \!\! \int\!\! \mathrm{d}^2x'\! \Biggl[ A({\bm{r}}-{\bm{r}}') \Bigl\{\left[{\bm{m}}({\bm{r}})- {\bm{m}}({\bm{r}}')\right]^2\\ &- 3\left[m^z({\bm{r}})-m^z({\bm{r}}')\right]^2 \Bigr\} + B({\bm{r}}-{\bm{r}}') \Bigl\{\left[ m^x({\bm{r}})-m^x({\bm{r}}')\right]^2\\ & - \left[ m^y({\bm{r}})-m^y({\bm{r}}')\right]^2 \Bigr\} + 2 C({\bm{r}}-{\bm{r}}') \left[ m^x({\bm{r}})-m^x({\bm{r}}')\right]\\ &\times \left[m^y({\bm{r}})-m^y({\bm{r}}')\right]\Bigr]. \end{split}$$ By applying the Fourier-transform $$\label{eq:fourier} {\bm{m}}({\bm{r}})=\frac{1}{(2\pi)^2}\,\int \mathrm{d}^2q\, \widehat{{\bm{m}}}({\bm{q}})\,e^{\imath{\bm{q}}\cdot{\bm{r}}},$$ and neglecting finite-size effects, the normalized dispersive part of the dipole-dipole interaction $\Delta W_d = {\Delta\mathcal{E}_{\text{d}}}/(M_S^2\mathcal{S}h)$ can be represented in the form $$\label{eq:disp} \Delta W_{\text{d}}=\frac{1}{2\pi\mathcal{S}}\int \mathrm{d}^2q\, \mathfrak{G}(q)\,\Biggl[-\|\widehat{m}^z_{{\bm{q}}}\|^2+ \frac{\|{\bm{q}}\cdot\widehat{{\bm{m}}}_{{\bm{q}}}\|^2}{q^2}\Biggr].$$ Here ${\bm{q}}=\left(q_x,q_y\right)$ is the two-dimensional wave vector, $\widehat{{\bm{m}}}({\bm{q}})$ is the Fourier-component of the two-dimensional magnetization ${\bm{m}}({\bm{r}})$, and the function $\mathfrak{G}(q)$ is defined by the expression $$\label{eq:func} \mathfrak{G}(q)=\frac{q h-1+e^{-q h}}{qh}.$$ Note that Eq. is obtained under assumption that the ortho-normalization relation $$\frac{1}{(2\pi)^2}\int\mathrm{d}^2x\, \,e^{\imath({\bm{q}}-{\bm{q}}')\cdot{\bm{r}}}=\delta({\bm{q}}-{\bm{q}}')$$ takes place. Being exact for the infinite domain, this relation is only approximate for the finite-size system. For $q h\to0$ the function takes the form $\mathfrak{G}(q)\approx q h/2$. Therefore we expect our approach to yield the correct results for the homogeneous and for weakly inhomogeneous states. For the general nonhomogeneous spin distribution the effective anisotropy approach gives only approximate results. In Sec. \[sec:disk\] we verify our effective anisotropy model for disk–shapes nanoparticles. Disk–shape nanoparticle {#sec:disk} ======================= Let us consider a cylindric nanoparticle of top surface radius $R$ and thickness $h$. We introduce $\varepsilon =h/(2R)$ the aspect ratio. Let us calculate first the effective anisotropy coefficients $\mathcal{A}$ and $\mathcal{B}$. For the circular system the coefficients $\mathcal{A}$ and $\mathcal{B}$ depend only on the relative distance $\xi$. We calculated analytically the coefficients $\mathcal{A}$ and $\mathcal{B}$ (see Appendix \[sec:appendix-continuum\]) and these are presented in Fig. \[fig:A-B\] and Eqs. , . First note that when $\varepsilon \gg1$ both anisotropy constants asymptotically do not depend on $\xi$: $\mathcal{A}(\xi)\to1/3$ and $\mathcal{B}(\xi)\to0$, see Fig. \[fig:A-B\]. The coefficient of effective uniaxial anisotropy $\mathcal{A}(\xi)$ slowly depends on $\xi$, namely $\mathcal{A}(0) = (\sqrt{1+4\varepsilon ^2}-1)/(2\varepsilon)-2/3$ and $\mathcal{A}(1)=1/3$. When the particle aspect ratio $\varepsilon\lesssim 1$ then $\mathcal{A}(\xi)<0$, see Fig. \[fig:A\] and we have an effective easy–plane anisotropy. When $\varepsilon\gtrsim 1$, then $\mathcal{A}(\xi)>0$ and we have an effective easy–axis anisotropy. More details are given in Sec. \[sec:disk-hom\]. In addition to the effective uniaxial anisotropy given by $\mathcal{A}(\xi)$, we have the essential $\mathcal{B}(\xi)$ term which gives an effective in–plane anisotropy. For the disk-shaped particle this anisotropy coefficient is always real, $\arg\mathcal{B}=0$. The value of $\mathcal{B}$ is almost $0$ at origin but its contribution becomes important at the boundary, see Fig. \[fig:A-B\]. We obtain the following asymptotics, valid form small $\varepsilon$ and $ 1/2 < \xi \lesssim 1$ $$\begin{aligned} \label{eq:B-as} \mathcal{B}(\xi)&\approx \frac{\arctan\Bigl( \dfrac{\varepsilon}{1-\xi}\Bigr)}{\pi\xi^2} -\frac{2\varepsilon (3\xi-2)}{3\pi} \ln\left(\frac{16}{\varepsilon^2+(1-\xi)^2}\right) \nonumber\\ &-\frac{1-\xi}{4\pi\,\varepsilon} \ln\left(\frac{(1-\xi)^2}{\varepsilon^2+(1-\xi)^2}\right)\end{aligned}$$ (see Appendix \[sec:appendix-continuum\]). Thus the $\mathcal{B}(\xi)$ term causes boundary effects and is responsible for the configurational anisotropy. In the limit $\varepsilon \to0$ (more precisely, when $a_0\ll h\ll R$) the $\mathcal{B}(\xi)$ term is concentrated near the boundary, corresponding to the surface anisotropy. The energy of the nanodisk can be derived from Eq. : $$\begin{aligned} \label{eq:W-total-res} &W = W_{\text{ex}} + W_d,\\ &W_{\text{ex}} = 2\left( \frac{\ell }{R}\right)^2 \int\limits_0^R r \mathrm{d}r \int\limits_0^{2\pi}\mathrm{d}\chi \Bigl[ ({\bm{\nabla}}\theta)^2 + \sin^2 \theta ({\bm{\nabla}}\phi)^2\Bigr],\nonumber\\ &W_d =\! \int\limits_0^{2\pi}\!\! \mathrm{d} \chi \!\! \int\limits_0^1\!\!\xi\mathrm{d}\xi \Biggl[\! \mathcal{A}(\xi)\! \left(1\! - \! 3\cos^2\theta \right) + \mathcal{B}(\xi) \sin^2\!\theta \cos2(\phi\!-\!\chi)\!\Biggr]\!. \nonumber\end{aligned}$$ In the next subsections we analyze the homogeneous state and the vortex state. Homogeneous state {#sec:disk-hom} ----------------- Let us consider a homogeneous magnetization along the $x$ direction of the disk–shaped nanodot, so that $\theta=\pi/2$, $\phi=0$. The exchange energy vanishes. The second term in the dipolar energy also vanishes because of averaging on $\chi$. The total energy $W^x$ is then $$\begin{aligned} \label{eq:Wx} &W^x = 2\pi \!\!\! \int\limits_0^1\!\!\! \mathcal{A}(\xi) \xi\mathrm{d}\xi = W_{\text{MS}}^{x}(\varepsilon) -\frac{2\pi}{3}\\ &W_{\text{MS}}^{x}(\varepsilon) = \frac{4}{3\varepsilon}\Bigl\{ -1\! + \! \sqrt{1+\varepsilon^2}\left[\varepsilon^2 \text{K}(m) + \left(1- \varepsilon^2\right)\text{E}(m) \right] \Bigr\},\nonumber\end{aligned}$$ where $m = (1+\varepsilon^2)^{-1}$, $\text{K}(m)$ and $\text{E}(m)$ are the complete elliptic integrals of the first and the second kind, respectively [@Abramowitz64]. The constant term $-2\pi/3$ is the isotropic contribution. The second term $W_{\text{MS}}^{x}$ is the well–known magnetostatic energy of the homogeneously magnetized disk, first calculated by @Joseph66. If the disk is now homogeneously magnetized along the $z$–axis, then $\theta=0$. From one sees that the corresponding total energy $W^z = -2W^x$. The transition between these two homogeneous ground states occurs when $W^x=W^z$. This happens only for $W_x=0$, i.e. for $W_{\text{MS}}^{x}(\varepsilon_c)=2\pi/3$. This gives a critical value $\varepsilon_c\approx 0.906$ which agrees with the result by @Aharoni90. ![(Color online) Comparison of the vortex profiles for the micromagnetic simulation and the effective anisotropy approximation for a Py nanodisk ($2R=212$ nm, $h=16$ nm). The red curve corresponds to the spin–lattice simulations for the effective anisotropy model with $\mathcal{H}=\mathcal{H}_{\text{ex}} + \mathcal{H}_d^{loc}$. The blue curve corresponds to the micromagnetic simulations. The black dashed curve to the gaussian ansatz $\cos\theta = \exp(-r^2/r_v^2)$.[]{data-label="fig:diagr-Sz"}](Sz){width="\columnwidth"} Vortex state {#sec:disk-vortex} ------------ Let us consider a nonhomogeneous state of the disk–shaped particle. In this state the system has a larger exchange energy compared to the homogeneous state. This should be compensated by the dipolar term. According to the dipolar interaction always favors a spin distribution of the form $$\label{eq:vortex} \phi = \chi \pm \frac{\pi}{2},$$ where we take into account that the in-plane anisotropy constant $\mathcal{B}$ takes real values only. Such a configuration is called a vortex. In highly anisotropic magnets there can exist pure planar vortices with $\theta=\pi/2$. [@Wysin94] However we consider here out-of-plane vortices, realized in “soft” materials typical of nanodisks. The out-of-plane component of the magnetization has a radial symmetric shape, and it almost does not depend on $z$ for thin disks, $\theta=\theta(r)$. Now we can calculate the vortex energy. The vortex solution is characterized by $\cos2(\phi-\chi)=-1$, providing the minimum of the in–plane component of the dipolar energy: $$\label{eq:W-dip-vortex} W_d^{\text{vortex}} = W^x \!-2\pi\!\!\!\int\limits_0^1\!\! \xi\mathrm{d}\xi \Biggl[3\mathcal{A}(\xi)\cos^2\theta + \mathcal{B}(\xi) \sin^2\theta\Biggr]\!.\!$$ The exchange energy term $$\label{eq:W-ex-vortex} \begin{split} W_{\text{ex}}^{\text{vortex}} = 4\pi \left( \frac{\ell }{R}\right)^2 \int_0^{R}r \mathrm{d}r \left[ {\theta'}^2 + \frac{\sin^2 \theta}{r^2}\right]. \end{split}$$ Finally, the vortex energy is $$\label{eq:W-vortex-1} \begin{split} W^{\text{vortex}} &=W^x + W_{\text{EP}}^{\text{vortex}} - F(\varepsilon),\\ W_{\text{EP}}^{\text{vortex}} &= 4\pi \left( \frac{\ell }{R}\right)^2 \!\!\! \int_0^{R} \!\!\! r\mathrm{d}r \Biggl[ {\theta'}^2+ \frac{\sin^2 \theta}{r^2} + \frac{\cos^2\theta}{\ell^2}\Biggr],\\ F(\varepsilon) &= 2\pi\int_0^1\xi\mathrm{d}\xi \Bigl\{ \left[3\mathcal{A}(\xi)+2\right] \cos^2\theta(R\xi)\\ & + \mathcal{B}(\xi) \sin^2\theta(R\xi)\Bigr\}. \end{split}$$ Here $W_{\text{EP}}^{\text{vortex}}$ coincides with the energy of the vortex in an easy–plane magnet [@Ivanov95b], $$\label{eq:Energy-EP} W_{\text{EP}}^{\text{vortex}} = 2\pi \left( \frac{\ell }{R}\right)^2\ln\left(\frac{\pi\Lambda R^2}{\ell ^2}\right), \quad \Lambda=5.27$$ and $F(\varepsilon)$ is the configurational anisotropy term. The vortex state is energetically preferable to the homogeneous state when the configurational anisotropy term exceeds the energy of the easy–plane vortex $F(\varepsilon) > W_{\text{EP}}^{\text{vortex}}$. This relation allows to calculate the critical radius $R_c$ by solving the equation $$\label{eq:critical} 2\pi \left( \frac{\ell }{R}\right)^2\ln\left(\frac{\pi\Lambda R^2} {\ell ^2}\right)= F(\varepsilon).$$ To calculate the integral in $F(\varepsilon)$ we use the trial function for the vortex structure $$\label{eq:ansatz} m^z\equiv\cos\theta = \exp(-r^2/r_v^2).$$ The core width depends on the disk thickness [@Kravchuk07] $$\label{eq:rv-vs-h} r_v(h)\approx \ell\sqrt2 \sqrt[3]{1+ch/\ell}, \qquad c\approx 0.39.$$ The relation providing the border between the easy-plane and the out-of-plane vortex states can be analyzed in the limit $\varepsilon \to0$. Then $F(\varepsilon)\sim (2\pi\varepsilon/3) \ln\Bigl(\pi/(2\varepsilon) \Bigr)$, hence $R^{(c)}\approx \ell\sqrt{3/\varepsilon }$. This is in qualitative agreement with previous results [@Usov94; @Hoellinger03]. Let us estimate now the contribution of the dispersive part of the dipolar energy. Taking into account that for the curling state the second term in Eq. vanishes: ${\bm{q}}\cdot\widehat{{\bm{m}}}_{{\bm{q}}}\equiv \widehat{\nabla\cdot{\bm{m}}}=0$ and that the Fourier-component of the out-of-plane component has the form $\widehat{m}^z=\pi\,r^2_v\,e^{-q^2r^2_v}$, from Eq. we get $$\label{eq:disp-contr} \Delta W_{\text{d}}\approx \begin{cases} \dfrac{\sqrt{\pi}}{8}\,\varepsilon\,\dfrac{r_v}{R} & \text{for $r_v\gg h$}, \\ \dfrac{1}{2}\,\dfrac{r_v^2}{R^2} & \text{for $r_v\ll h$}. \end{cases}$$ Comparing Eq. with Eqs. and , and taking into account , one can conclude that in the limit $\varepsilon \to0$ the dispersive part of the dipolar interaction does not change significantly the vortex stability criterion. More precisely, our effective anisotropy approximation works correctly not only for $\varepsilon \to0$ but also for $R\gg h,~R\gg \ell$. Our numerical results show that it gives the vortex state as an energy minimum for disk diameters $2R\gtrsim 30\ell$. Numerical simulations {#sec:simulations} ===================== To check our effective anisotropy approximation, we performed numerical simulations. We used the publicly available three–dimensional OOMMF micromagnetic simulator code [@OOMMF]. In all micromagnetic simulations we used the following material parameters for Py: $A=1.3\times10^{-6}$ erg/cm (using SI units $A^{\text{SI}}=1.3\times 10^{-11}$ J/m), $M_s=8.6\times10^{2}$ G ($M_s^{\text{SI}}=8.6\times 10^5$ A/m), the damping coefficient $\eta = 0.006$, and the anisotropy has been neglected. This corresponds to an exchange length $\ell = \sqrt{A/4\pi M_s^2}\approx 5.3$nm ($\ell^{\text{SI}} = \sqrt{A/\mu_0 M_s^2}$). The mesh cells were cubic (2 nm). We also test our effective anisotropy approach by the original discrete spin–lattice simulator. The spin dynamics is described by the discrete version of the Landau–Lifshitz equations with Gilbert damping $$\label{eq:LL-discrete} \frac{\mathrm{d} {\bm{S}}_{{\bm{n}}} }{\mathrm{d} t} = - \left[{\bm{S}}_{{\bm{n}}}\times \frac{\partial \mathcal{H} }{\partial {\bm{S}}_{{\bm{n}}}}\right] - \frac{\eta}{S} \left[{\bm{S}}_{{\bm{n}}} \times \frac{\mathrm{d} {\bm{S}}_{{\bm{n}}} }{\mathrm{d}t}\right],$$ which we consider on a 2D square lattice of size $(2R)^2$. We have assumed a plane–parallel spin distribution homogeneous along the z–direction. Each lattice is bounded by a circle of radius $R$ on which the spins are free corresponding to a Neuman boundary condition in the continuum limit. We integrate the discrete Landau–Lifshitz equations with the Hamiltonian $\mathcal{H}=\mathcal{H}_{\text{ex}} + \mathcal{H}_d$ given by and , using a 4th–order Runge–Kutta scheme with time step $0.01/N_z$. These spin–lattice simulations were done to validate our analytical calculations for the effective anisotropy model. Throughout this work we compared the results of the spin–lattice simulations with $\mathcal{H}=\mathcal{H}_{\text{ex}} + \mathcal{H}_d$ with the results of micromagnetic simulations. We never found any noticeable difference. We present the results for a disk–shaped and a prism–shaped nanoparticle because these two geometries are the most common ones in experiments. Disk–shape nanoparticle {#sec:disk-simulations} ----------------------- Our effective anisotropy approximation provides the exact solution for all homogeneous states for a nanodisk. Therefore we do not need to justify it for the homogeneous states. We consider here the vortex state. As we analyzed before, the model can provide the preferable vortex state for disk diameters $2R>30\ell$, which is in an agreement of the model usage criterium . We compare the magnetization distribution in the vortex for our effective anisotropy model and for the the micromagnetic simulations. Since the in–plane vortex structure is characterized by the same distribution $\phi=\chi\pm\pi/2$ for both methods, we are interested in the out–of–plane vortex profiles. We performed such a comparison for a disk of size $2R/\ell=40$ and $h/\ell=3$, which satisfy the criterium . The results are presented on Fig. \[fig:diagr-Sz\]. One can see that the vortex shape from the effective anisotropy model agrees with the one obtained from the micromagnetic simulations within $0.11$ in absolute error. Prism–shape nanoparticle {#sec:prism} ------------------------ Now we check the validity of the effective anisotropy approximation for the prism–shaped nanoparticle. We chose this shape because there are numerous experiment with a square geometry, see for a review Ref. . We performed the two types of simulations for a square shaped nanoparticle, see Figs. \[fig:square-local\] and \[fig:squar-OOMMF\]. The two equilibrium magnetization distributions, obtained for the micromagnetic model and the spin-lattice simulation agree with a very high precision. As discussed above the large scale distribution of the magnetization is described by Eq. . Calculating numerically the coefficient $\mathcal{B}$ (see Appendix \[sec:appendix-2geom\] for details), we found the distribution of the configurational anisotropy lines for the square geometry. This is shown in Fig. \[fig:anisLines\]. The comparison of Figs. \[fig:diagr-squar\] shows that the effective anisotropy lines corresponds to the magnetization direction in the main part of the system. Note that the effective anisotropy approach fails near the corners: the sharp field distribution near the prism vertices (Fig. \[fig:anisLines\]) is not energetically preferable when the exchange contribution is taken into account. We can also check the validity of the effective anisotropy approach for the complicated “vortex” structure in the square geometry, by comparing the distribution of the in-plane spin angle $\phi$ to the one given by the micromagnetic simulations. This is done in Fig \[fig:squareR\]. The figure shows that the two different approaches agree very well. The $\phi(\chi)$–dependencies coincide within $0.11$ in absolute error for $r=10\ell$ and within $0.04$ for $r=20\ell$. Discussion {#sec:discussion} ========== To summarize, assuming that magnetization is independant of the thickness variable $z$, we have reduced the magnetic energy of a thin nanodot to a local 2D inhomogeneous anisotropy. The first term $\mathcal{A}$ determines the uniaxial anisotropy along the $z$–axis. The second term $\mathcal{B}$ gives the anisotropy in the $XY$–plane. For thin nanoparticles $\varepsilon \lesssim 1$ the term $\mathcal{A}\approx\text{const}<0$, gives an effective easy–plane anisotropy. This generalizes the rigorous results obtained for infinitesimally thin films [@Gioia97]. The function $\mathcal{B}(x,y)$ is localized near the edge of the particle so that spins will be tangent to the boundary. This confirms the notion of a surface edge anisotropy [@Kireev03; @Tchernyshyov05].\ When the nanoparticle is thick $\varepsilon \gtrsim 1$, the anisotropy constant $\mathcal{A}>0$, is again almost constant and the spins will tend to follow the $z$ axis (easy–axis anisotropy). The in-plane anisotropy $\mathcal{B}$ depends on the thickness, see Fig. \[fig:A\]. The special distribution of $\mathcal{B}(x,y)$ is responsible for the volume contribution of the dipolar energy. The above effective anisotropy approach: (i) shows the nature of the effective easy-plane anisotropy and the surface anisotropy, (ii) generalizes the surface anisotropy for the finite thickness, and (iii) gives a unified approach to study dipolar effects in pure 2D systems and 3D magnets of finite thickness. It is instructive to make a link between our approach and the rigorous results which were obtained in Refs. . Our equations , show that for the vortex ground state to exist, it is crucial to have both types of anisotropy: out-of-plane anisotropy and in-plane one. It is shown by @Kohn05a that the energy of a thin magnetic film with an accuracy up to $\varepsilon^2$ can be presented as the sum $$\label{eq:Kohn} \begin{split} &E=E_{\text{exch}}+E_{\text{bdry}}+E_{\text{trans}}\\ &=\ell^2 \varepsilon \!\!\int_{\omega}\!\! \|\nabla {\bm{m}}\|^2+\frac{\varepsilon^2\|\ln\varepsilon\|}{2\pi} \!\!\int_{\partial\omega} \!\! \left({\bm{m}}\cdot{\bm{n}}\right)^2+ \varepsilon \!\! \int_{\omega} \!\! \left(m^z\right)^2. \end{split}$$ Considering the limit $\varepsilon\to0$ and $\ell^2/(\varepsilon\|\ln\varepsilon\|)=\text{const}$ we see from Eq. that formally the last term is dominating and its contribution has to be accounted as a constraint $m^z=0$, see Ref. . This constraint prevents the existence of the vortex ground state of the nanodot because the energy of the vortex in the continuum limit is infinite due to divergence at $r\to0$. However this divergence is removed by the out-of-plane component of vortex which is described by a localized function with radius of localization $r_v\sim \ell$ \[see Eq. \]. This means that the last term $E_{\text{trans}}$ in scales like the exchange term $E_{\text{exch}}$. In this limit all three terms of are of the same order and provide the existence of the vortex ground state. This reduction of the nonlocal dipolar interaction to a local form is a first step towards an analytical study of nanomagnetism. We developed a method of effective anisotropy and illustrated it on a few examples. We plan to apply this method to the dynamics of vortices in nanomagnets. Yu.G., V.P.K. and D.D.S. thank the University of Bayreuth, where part of this work was performed, for kind hospitality and acknowledge the support from Deutsches Zentrum f[ü]{}r Luft- und Raumfart e.V., Internationales B[ü]{}ro des BMBF in the frame of a bilateral scientific cooperation between Ukraine and Germany, project No. UKR 05/055. J.G.C., Yu.G. and D.D.S. acknowledge support from a Ukrainian–French Dnipro grant (No. 82/240293). Yu. G. thanks the University of Cergy-Pontoise for an invited professorship during which this work was completed. D.D.S. acknowledges the support from the Alexander von Humboldt–Foundation. V.P.K. acknowledges the support from the BAYHOST project. J.G.C. thanks the Centre de Ressources Informatiques de Haute-Normandie where part of the computations were carried out. Discrete dipolar energy calculations {#sec:appendix-discrete} ==================================== Let us consider the dipolar interaction term $\mathcal{H}_{\text{d}}$. Using the notations $$\label{eq:notations} \begin{split} \frac{x_{{\bm{n}} {\bm{m}}}}{a_0} = n_x - m_x,\; \frac{y_{{\bm{n}} {\bm{m}}}}{a_0} = n_y - m_y,\; \frac{z_{{\bm{n}} {\bm{m}}}}{a_0} = n_z - m_z,\; \rho_{{{\bm{\nu}}} {{\bm{\mu}}}} = \sqrt{x_{{\bm{n}}{\bm{m}}}^2 + y_{{\bm{n}} {\bm{m}}}^2}, \; r_{{\bm{n}}{\bm{m}}} = \sqrt{\rho_{{{\bm{\nu}}} {{\bm{\mu}}}}^2 + z_{{\bm{n}}{\bm{m}}}^2}, \end{split}$$ one can rewrite this energy as follows: $$\label{eq:H-dipolar-1} \begin{split} \mathcal{H}_{\text{d}} &= \frac{D}{2}\!\! \sum_{\substack{{\bm{n}},{\bm{m}}\\r_{{\bm{n}}{\bm{m}}}\neq0}}\! \Biggl\{ \frac{\left({\bm{S}}_{{\bm{n}}}\cdot {\bm{S}}_{{\bm{m}}}\right)}{r_{{\bm{n}}{\bm{m}}}^3} - \frac{3S_{{\bm{n}}}^zS_{{\bm{m}}}^z z_{{\bm{n}}{\bm{m}}}^2}{r_{{\bm{n}}{\bm{m}}}^5} - \frac{6}{r_{{\bm{n}}{\bm{m}}}^5} S_{{\bm{n}}}^z z_{{\bm{n}}{\bm{m}}}\left(S_{{\bm{m}}}^x x_{{\bm{n}}{\bm{m}}}+ S_{{\bm{m}}}^y y_{{\bm{n}}{\bm{m}}}\right)\\ & -\frac{3}{r_{{\bm{n}}{\bm{m}}}^5}\left( S_{{\bm{n}}}^x x_{{\bm{n}}{\bm{m}}} + S_{{\bm{n}}}^y y_{{\bm{n}}{\bm{m}}} \right) \left( S_{{\bm{m}}}^x x_{{\bm{n}}{\bm{m}}} + S_{{\bm{m}}}^y y_{{\bm{n}}{\bm{m}}} \right)\Biggr\} = D\!\! \sum_{\substack{{{\bm{\nu}}},{{\bm{\mu}}}\\ \rho_{{{\bm{\nu}}}{{\bm{\mu}}}}\neq0}}\! \Biggl\{ S_{{{\bm{\nu}}}}^zS_{{{\bm{\mu}}}}^z K_z(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}) + \left( S_{{{\bm{\nu}}}}^xS_{{{\bm{\mu}}}}^x + S_{{{\bm{\nu}}}}^yS_{{{\bm{\mu}}}}^y \right) K_1(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}})\\ &- \left( S_{{{\bm{\nu}}}}^x x_{{{\bm{\nu}}}{{\bm{\mu}}}} + S_{{{\bm{\nu}}}}^y y_{{{\bm{\nu}}}{{\bm{\mu}}}} \right) \left( S_{{{\bm{\mu}}}}^x x_{{{\bm{\nu}}}{{\bm{\mu}}}} + S_{{{\bm{\mu}}}}^y y_{{{\bm{\nu}}}{{\bm{\mu}}}} \right)K_2(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}})\Biggr\}. \end{split}$$ Here we used the obvious relations $x_{{\bm{n}}{\bm{m}}}=x_{{{\bm{\nu}}}{{\bm{\mu}}}}$, $y_{{\bm{n}}{\bm{m}}}=y_{{{\bm{\nu}}}{{\bm{\mu}}}}$ and the basic assumption that the magnetization does not depend on the z-coordinate: ${\bm{S}}_{{\bm{n}}}={\bm{S}}_{{{\bm{\nu}}}}$, ${\bm{S}}_{{\bm{m}}}={\bm{S}}_{{{\bm{\mu}}}}$. This allows us to reduce the summation to the 2D lattice. The kernels $K_1$, $K_2$ and $K_z$ contain information about the original 3D structure of our system, $$\label{eq:K-1-2-3} \begin{split} K_1(s) &= \frac12 \sum_{n_z, m_z}\frac{1}{\left(s^2 + z_{{\bm{n}}{\bm{m}}}^2\right)^{3/2}},\quad K_2(s) = \frac32 \sum_{n_z, m_z}\frac{1}{\left(s^2 + z_{{\bm{n}}{\bm{m}}}^2\right)^{5/2}},\quad K_z(s) = \frac12 \sum_{n_z, m_z}\frac{s^2 - 2z_{{\bm{n}}{\bm{m}}}^2}{\left(s^2 + z_{{\bm{n}}{\bm{m}}}^2\right)^{5/2}}. \end{split}$$ Taking into account that $$\begin{split} & S_{{{\bm{\nu}}}}^x S_{{{\bm{\mu}}}}^x x_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 + S_{{{\bm{\nu}}}}^y S_{{{\bm{\mu}}}}^y y_{{{\bm{\nu}}}{{\bm{\mu}}}}^2= \tfrac12 \rho_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 \left(S_{{{\bm{\nu}}}}^x S_{{{\bm{\mu}}}}^x + S_{{{\bm{\nu}}}}^y S_{{{\bm{\mu}}}}^y\right) + \tfrac12\left( x_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 - y_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 \right) \left(S_{{{\bm{\nu}}}}^x S_{{{\bm{\mu}}}}^x - S_{{{\bm{\nu}}}}^y S_{{{\bm{\mu}}}}^y\right), \end{split}$$ one can present the dipolar energy in more symmetrical way: $$\label{eq:H-dipolar-K2&Kz} \begin{split} \mathcal{H}_{\text{d}} = -\frac{D}{2}\!\! \sum_{\substack{{{\bm{\nu}}},{{\bm{\mu}}}\\ \rho_{{{\bm{\nu}}}{{\bm{\mu}}}}\neq0}}\! \Biggl\{ & K_z(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}})\left({\bm{S}}_{{{\bm{\nu}}}}\cdot {\bm{S}}_{{{\bm{\mu}}}} - 3 S_{{{\bm{\nu}}}}^zS_{{{\bm{\mu}}}}^z\right)+ K_2(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}})\left(x_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 - y_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 \right)\left( S_{{{\bm{\nu}}}}^x S_{{{\bm{\mu}}}}^x - S_{{{\bm{\nu}}}}^y S_{{{\bm{\mu}}}}^y \right) \\ &+ 2 K_2(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}) x_{{{\bm{\nu}}}{{\bm{\mu}}}} y_{{{\bm{\nu}}}{{\bm{\mu}}}}\left( S_{{{\bm{\nu}}}}^x S_{{{\bm{\mu}}}}^y + S_{{{\bm{\nu}}}}^y S_{{{\bm{\mu}}}}^x \right)\Biggr\}. \end{split}$$ The total Hamiltonian is the sum of two terms and . Here we show that the main effect of the nonlocal dipolar interaction is an effective nonhomogeneous anisotropy. Using equality $$\label{eq:equality} \begin{split} &\sum_{{\bm{n}},{\bm{m}}} C_{{\bm{m}}{\bm{n}}}S_{{\bm{n}}} S_{{\bm{m}}} = \sum_{{\bm{n}}} \mathcal{C}_{{\bm{n}}}S_{{\bm{n}}}^2 - \frac12 \sum_{{{\bm{n}}},{{\bm{m}}}} C_{{{\bm{n}}}{{\bm{m}}}}\left( S_{{\bm{n}}} - S_{{\bm{m}}} \right)^2, \quad \mathcal{C}_{{\bm{n}}} = \sum_{{{\bm{m}}}} C_{{{\bm{n}}}{{\bm{m}}}}, \end{split}$$ where $C_{{{\bm{n}}}{{\bm{m}}}} = C_{{{\bm{m}}}{{\bm{n}}}}$, one can split the dipolar Hamiltonian into a local contribution and a nonlocal correction $$\begin{aligned} \tag{\ref{eq:Hd-loc}} \label{eq:Hd-loc-again} \mathcal{H}_{\text{d}} &= \mathcal{H}_{\text{d}}^{\text{loc}} + \Delta\mathcal{H}_{\text{d}},\quad \mathcal{H}_{\text{d}}^{\text{loc}} = -\frac{D}{2}\!\! \sum_{{{\bm{\nu}}}}\! \Biggl\{ \bar{A}_{{{\bm{\nu}}}}\Bigl[\left({\bm{S}}_{{{\bm{\nu}}}}\right)^2 - 3\left( S_{{{\bm{\nu}}}}^z\right)^2\Bigr] + \bar{B}_{{{\bm{\nu}}}} \Bigl[\left(S_{{{\bm{\nu}}}}^x\right)^2 - \left(S_{{{\bm{\nu}}}}^y\right)^2 \Bigr] + 2\bar{C}_{{{\bm{\nu}}}} S_{{{\bm{\nu}}}}^x S_{{{\bm{\nu}}}}^y\Biggr\},\\ \label{eq:H-dipolar-via-local} \Delta \mathcal{H}_{\text{d}} &= \frac{D}{4}\!\! \sum_{\substack{{{\bm{\nu}}},{{\bm{\mu}}}\\ \rho_{{{\bm{\nu}}}{{\bm{\mu}}}}\neq0}}\! \Biggl\{ K_z(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}})\Bigl[\left({\bm{S}}_{{{\bm{\nu}}}}- {\bm{S}}_{{{\bm{\mu}}}}\right)^2 - 3 \left(S_{{{\bm{\nu}}}}^z - S_{{{\bm{\mu}}}}^z\right)^2\Bigr]+ K_2(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}})\left(x_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 - y_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 \right)\Bigl[ \left( S_{{{\bm{\nu}}}}^x - S_{{{\bm{\mu}}}}^x\right)^2 - \left(S_{{{\bm{\nu}}}}^y - S_{{{\bm{\mu}}}}^y \right)^2\Bigr] \nonumber\\ &+ 4 K_2(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}) x_{{{\bm{\nu}}}{{\bm{\mu}}}} y_{{{\bm{\nu}}}{{\bm{\mu}}}}\Bigl[\left( S_{{{\bm{\nu}}}}^x - S_{{{\bm{\mu}}}}^x\right) \left(S_{{{\bm{\nu}}}}^y - S_{{{\bm{\mu}}}}^y \right)\Bigr]\Biggr\}.\end{aligned}$$ Continuum limit of the local dipolar energy {#sec:appendix-continuum} =========================================== Here we present the continuum limit of the discrete dipolar Hamiltonian corresponding to the dipolar energy $$\label{eq:Energy-dipolar} \begin{split} \mathcal{E}_{\text{d}}^{\text{loc}} &= -\frac{a_0^6 M_S^2}{2}\!\! \sum_{{{\bm{\nu}}}}\! \Biggl\{ \bar{A}_{{{\bm{\nu}}}}\Bigl[1 - 3\left( m_{{{\bm{\nu}}}}^z\right)^2\Bigr] + \bar{B}_{{{\bm{\nu}}}} \Bigl[\left(m_{{{\bm{\nu}}}}^x\right)^2 - \left(m_{{{\bm{\nu}}}}^y\right)^2 \Bigr] + 2\bar{C}_{{{\bm{\nu}}}} m_{{{\bm{\nu}}}}^x m_{{{\bm{\nu}}}}^y\Biggr\}, \end{split}$$ where ${\bm{m}}_{{\bm{\nu}}} = \frac{g\mu_B}{a_0^3 M_s}{\bm{S}}_{{\bm{\nu}}}$. Hence the continuous magnetization vector ${\bm{m}}$ according to Eq. takes the form ${\bm{m}}({\bm{r}}) = \sum_{{\bm{\nu}}}{\bm{m}}_{{\bm{\nu}}} \delta \left({\bm{r}} - {\bm{r}}_{{\bm{\nu}}} \right)$. Here $\bar{A}_{{{\bm{\nu}}}}$, $\bar{B}_{{{\bm{\nu}}}}$ and $\bar{C}_{{{\bm{\nu}}}}$ are determined as follows \[eq:A-B-C-details\] $$\begin{aligned} \label{eq:A-details} \bar{A}_{{{\bm{\nu}}}} &= \sum_{\substack{{{\bm{\mu}}}\\r_{{\bm{n}}{\bm{m}}}\neq0}} K_z(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}) = \frac12 \sum_{\substack{{{\bm{\mu}}}\\r_{{\bm{n}}{\bm{m}}}\neq0}} \sum_{n_z,m_z} \frac{\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 - 2z_{{\bm{n}}{\bm{m}}}^2}{\left(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 + z_{{\bm{n}}{\bm{m}}}^2\right)^{5/2}},\\ \label{eq:B-details} \bar{B}_{{{\bm{\nu}}}} &= \sum_{\substack{{{\bm{\mu}}}\\r_{{\bm{n}}{\bm{m}}}\neq0}} K_2(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}) \left(x_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 - y_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 \right)= \frac32 \sum_{\substack{{{\bm{\mu}}}\\r_{{\bm{n}}{\bm{m}}}\neq0}} \sum_{n_z,m_z} \frac{x_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 - y_{{{\bm{\nu}}}{{\bm{\mu}}}}^2}{\left(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 + z_{{\bm{n}}{\bm{m}}}^2\right)^{5/2}},\\ \label{eq:C-details} \bar{C}_{{{\bm{\nu}}}} &= \sum_{\substack{{{\bm{\mu}}}\\r_{{\bm{n}}{\bm{m}}}\neq0}} K_2(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}) 2x_{{{\bm{\nu}}}{{\bm{\mu}}}} y_{{{\bm{\nu}}}{{\bm{\mu}}}} = \frac32 \sum_{\substack{{{\bm{\mu}}}\\r_{{\bm{n}}{\bm{m}}}\neq0}} \sum_{n_z,m_z} \frac{2x_{{{\bm{\nu}}}{{\bm{\mu}}}} y_{{{\bm{\nu}}}{{\bm{\mu}}}} }{\left(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 + z_{{\bm{n}}{\bm{m}}}^2\right)^{5/2}}.\end{aligned}$$ The continuum version of the effective anisotropy constants can be found using a relation $$\label{eq:int-relation} \begin{split} \sum_{n_z=0}^{N_z} \sum_{m_z=0}^{N_z} F(\|z_{{\bm{nm}}}\|) &\approx \frac{1}{a_0^2} \int_0^h \mathrm{d} z \int_0^h \mathrm{d} z' F(\|z-z'\|) + \frac{1}{a_0} \int_0^h \mathrm{d} z \bigl[F(\|z\|) + F(\|h-z\|)\bigr] + \frac12 \left[ F(0) + F(\|h\|)\right] \\ & = \frac{2}{a_0^2} \int_0^h \mathrm{d} z F(\|z\|) \bigl[h-z+a_0\bigr] + \frac12 \left[ F(0) + F(\|h\|)\right], \qquad h=N_z a_0\geq0. \end{split}$$ Let us start with the calculation of the coefficient $\bar{A}_{{{\bm{\nu}}}}$ from Eq. : $$\begin{aligned} \label{eq:A-calculation} &\mathcal{A}(x,y) \equiv - \frac{a_0^4}{2\pi h}\bar{A}_{{{\bm{\nu}}}} = \frac{1}{h}\left(\mathcal{A}_1 + \mathcal{A}_2 + \mathcal{A}_3\right),\qquad \mathcal{A}_1 = \frac{\Theta_+(h)}{2\pi} \lim_{r^\star\to0} \!\!\!\int\limits_{\|{\bm{r}} - {\bm{r}}'\|>r^\star} \!\!\! \mathrm{d}^2x' \int_0^h \mathrm{d}z \frac{(2z^2-\rho^2)(h-z+a_0)}{\left( \rho^2 + z^2 \right)^{5/2}},\\ &\mathcal{A}_2 = -\frac{a_0^4}{8\pi} \sum_{\substack{{{\bm{\mu}}}\\r_{{\bm{n}}{\bm{m}}}\neq0}} \frac{1}{\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}^3} \approx \frac{a_0^2}{8\pi} \int_0^{2\pi} \frac{\mathrm{d}\alpha}{P} - \frac{a_0}{4},\quad \mathcal{A}_3 = \frac{a_0^2}{8\pi} \int \mathrm{d}^2x' \frac{2h^2-\rho^2}{\left( \rho^2 + h^2 \right)^{5/2}} \approx \frac{a_0^2}{8\pi} \int_0^{2\pi} \frac{P^2\mathrm{d}\alpha}{(P^2+h^2)^{3/2}} - \frac{a_0^4}{4(a_0^2+h^2)^{3/2}}.\nonumber\end{aligned}$$ Here $\rho = \sqrt{(x-x')^2 + (y-y')^2}$ and we used a local reference frame and the Heaviside function $\Theta_+(x)$ takes the unit values for any positive $x$ and zero values for $x\leq0$. The Heaviside function is added here to fulfil the condition $\mathcal{A}_1\equiv 0$ in a 2D case, when for $h=0$. There is a singularity in $\mathcal{A}_1$, due to the nonintegrability of the kernel $K_z$ at $r_{{\bm{n}}{\bm{m}}}=0$. To regularize it we use a method similar to the one in Ref. . Specifically, we present $\mathcal{A}_1$ in the form $\mathcal{A}_1= \widetilde{\mathcal{A}_1} - \mathcal{A}_0$. The coefficient $\widetilde{\mathcal{A}_1}$ is a regular one: $$\widetilde{\mathcal{A}_1} = \frac{\Theta_+(h)}{2\pi} \int \mathrm{d}^2x' \int_0^h \mathrm{d}z \frac{(2z^2-\rho^2)(h-z+a_0)}{\left( \rho^2 + z^2 \right)^{5/2}}= -h-a_0\Theta_+(h) + \frac{1}{2\pi} \int_0^{2\pi} \mathrm{d}\alpha \left[ \sqrt{P^2+h^2} - P + \frac{a_0h}{\sqrt{P^2+h^2}}\right].$$ The singularity is inside the $\mathcal{A}_0$ term: $$\label{eq:A0} \begin{split} \mathcal{A}_0 &= \frac{\Theta_+(h)}{2\pi} \lim_{r^\star\to0} \!\!\! \int\limits_{\substack{\|{\bm{r}} - {\bm{r}}'\|<r^\star\\{z=0,}\;z'>0}} \!\!\!\!\!\!\! \mathrm{d}^2 x' \mathrm{d}z' \frac{(2{z'}^2 - \rho^2)(h-z'+a_0)}{\left(\rho^2 + {z'}^2\right)^{5/2}} = \frac{\Theta_+(h)}{2\pi}\Bigl[(h+a_0)I_1 - I_2 \Bigr],\\ I_1&= \lim_{r^\star\to0} \!\!\! \int\limits_{\substack{\|{\bm{r}} - {\bm{r}}'\|<r^\star\\ z=0,\;z'>0}} \!\!\!\!\!\!\! \mathrm{d}^2 x' \mathrm{d}z' \frac{2z'^2-\rho^2}{\left(\rho^2 + z'^2\right)^{5/2}} = \lim_{r^\star\to0} \!\!\!\!\! \int\limits_{\substack{\|{\bm{r}} - {\bm{r}}'\|<r^\star\\ z=0,\;z'>0}} \!\!\!\!\!\!\! \mathrm{d}^2 x' \mathrm{d}z' \frac{\partial^2}{\partial {z'}^2}\frac{1}{\|{\bm{r}}-{\bm{r}}'\|} =\frac13 \lim_{r^\star\to0} \!\!\!\!\! \int\limits_{\substack{\|{\bm{r}} - {\bm{r}}'\|<r^\star\\ z=0,\;z'>0}} \!\!\!\!\!\!\! \mathrm{d}^3x'\Delta\frac{1}{\|{\bm{r}}-{\bm{r}}'\|}\\ &=-\frac{4\pi}{3} \lim_{r^\star\to0} \!\!\!\!\! \int\limits_{\substack{\|{\bm{r}} - {\bm{r}}'\|<r^\star\\ z=0,\;z'>0}} \!\!\!\!\!\!\! \mathrm{d}^3x'\delta({\bm{r}} - {\bm{r}}') =-\frac{2\pi}{3},\\ I_2&=\lim_{r^\star\to0} \!\!\! \int\limits_{\substack{\|{\bm{r}} - {\bm{r}}'\|<r^\star\\ z=0,\;z'>0}} \!\!\!\!\!\!\! \mathrm{d}^2 x' \mathrm{d}z' \frac{z'(2z'^2-\rho^2)}{\left(\rho^2 + z'^2\right)^{5/2}} =\frac{4\pi}{3}\lim_{r^\star\to0} \!\!\! \int\limits_{\substack{\|{\bm{r}} - {\bm{r}}'\|<r^\star\\ z=0,\;z'>0}} \!\!\!\!\!\!\! \mathrm{d}^3x'z'\delta({\bm{r}} - {\bm{r}}')=0. \end{split}$$ Finally, $\mathcal{A}_0=-\left[h+a_0\Theta_+(h)\right]/3$ and the coefficient of effective anisotropy $\mathcal{A}(x,y)$ takes a form . The coefficients $\bar{B}_{{{\bm{\nu}}}}$ and $\bar{C}_{{{\bm{\nu}}}}$ can be calculated in the same way, starting from Eq. : $$\begin{aligned} \label{eq:B-and-C} &\mathcal{B}(x,y) \equiv -\frac{a_0^4 e^{2\imath\chi}}{2\pi h} \Bigl[ \bar{B}_{{{\bm{\nu}}}} - \imath \bar{C}_{{{\bm{\nu}}}} \Bigr] = -\frac{3a_0^4}{4\pi h} \sum_{\substack{{{\bm{\mu}}}\\r_{{\bm{n}}{\bm{m}}}\neq0}} \rho_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 e^{-2\imath \alpha_{{{\bm{\nu}}}{{\bm{\mu}}}}} \sum_{n_z,m_z} \frac{1}{\left(\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}^2 + z_{{\bm{n}}{\bm{m}}}^2\right)^{5/2}} = \frac{1}{h}\left( \mathcal{B}_1 + \mathcal{B}_2 + \mathcal{B}_3\right),\\ &\mathcal{B}_1 = -\frac{3}{2\pi}\!\!\int\!\! \mathrm{d}^2x' \rho^2 e^{-2\imath\alpha}\!\!\! \int\limits_0^h \!\!\mathrm{d}z \frac{h-z+a_0}{(\rho^2+z^2)^{5/2}} =\frac{1}{2\pi}\!\!\! \int\limits_0^{2\pi}\!\!\mathrm{d}\alpha e^{-2\imath\alpha}\!\! \Biggl[ P \!- \! \sqrt{P^2+h^2} + \frac{a_0 h}{\sqrt{P^2+h^2}} - 2\left(h+a_0\right) \ln\frac{\sqrt{P^2+h^2}-h}{P} \Biggr],\nonumber \\ &\mathcal{B}_2 = -\frac{3 a_0^4}{8\pi} \sum_{\substack{{{\bm{\mu}}}\\r_{{\bm{n}}{\bm{m}}}\neq0}} \frac{e^{-2\imath\alpha}}{\rho_{{{\bm{\nu}}}{{\bm{\mu}}}}^3} \approx \frac{3 a_0^2}{8\pi} \int_0^{2\pi} \frac{e^{-2\imath\alpha}\mathrm{d}\alpha}{P},\qquad \mathcal{B}_3 = -\frac{3a_0^2}{8\pi} \int \mathrm{d}^2x' \frac{\rho^2 e^{-2\imath\alpha}}{\left( \rho^2 + h^2 \right)^{5/2}} = \frac{a_0^2}{8\pi} \int_0^{2\pi} \mathrm{d}\alpha e^{-2\imath\alpha}\frac{3P^2 + 2h^2}{(P^2+h^2)^{3/2}}.\nonumber\end{aligned}$$ Finally, the coefficient of effective anisotropy $\mathcal{B}(x,y)$ takes a form . As a result the dipolar energy can be expressed as . Note that for the circular system one can obtain exact expressions for the coefficients $\mathcal{A}$ and $\mathcal{B}$. Let us first find the coefficient $\mathcal{A}$. Assuming that $h\gg a_0$ (or equivalently $a_0\to0$), one can rewrite the coefficient $\mathcal{A}$, see Eq. , as follows: $$\label{eq:A-circular} \begin{split} \mathcal{A}(\xi) &= \frac13 + \frac1{4\pi\varepsilon} \Bigl[I_A(2\varepsilon) - I_A(0)\Bigr],\quad I_A(x) = \int_0^{2\pi} \mathrm{d}\alpha \int_0^1 \frac{\xi'\mathrm{d}\xi'}{ \sqrt{\xi^2 + {\xi'}^2 + x^2 - 2\xi\xi'\cos\alpha}},\\ I_A(x) &=\frac{2}{\sqrt{x^2+(\xi +1)^2}} \Bigl\{\left[x^2+(\xi +1)^2\right] \text{E}(\mu)+\left[1-x^2-\xi^2\right] \text{K}(\mu) + F_+(x) + F_-(x)\Bigr\}-2 \pi x,\\ F_\pm(x) &= x^2 \frac{\sqrt{x^2+\xi^2}\mp1}{\sqrt{x^2+\xi ^2}\pm \xi } \Pi \left(\nu_\pm\|\mu\right), \qquad \mu = \frac{4\xi}{x^2+(1+\xi)^2}, \quad \nu_\pm = \frac{2\xi}{\xi\pm\sqrt{x^2+\xi^2}}, \end{split}$$ where $\Pi(\nu_\pm\|\mu)$ is the complete elliptic integral of the third kind. [@Abramowitz64]. To calculate the in-plane anisotropy coefficient $\mathcal{B}$, see Eq. , it is convenient to use the following relations $$\begin{aligned} \label{eq:Stokes} &\text{Re}\left[\mathcal{B} e^{-2\imath\chi}\right] =-\frac{a_0^4}{2\pi h}\bar{B}_{{{\bm{\nu}}}}=-\frac{1}{2\pi h} \int_0^h \mathrm{d}z (h-z) I_z(x),\qquad I_z(x) = 3\int \mathrm{d}^2x' \,\frac{(x-x')^2-(y-y')^2}{ \left( \rho^2 + z^2\right)^{5/2}}\\ &=\int \mathrm{d}^2x'\left(\frac{\partial^2}{\partial y\,\partial y'}-\frac{\partial^2}{\partial x\,\partial x'}\right)\frac{1}{ \sqrt{\rho^2+ z^2}} \equiv\int_\Omega \left[{\bm{\nabla}}'\times {\bm{F}}\right]\cdot \mathrm{d}{\bm{\sigma}} =\oint_{\partial\Omega} {\bm{F}}\cdot \mathrm{d}{\bm{l'}}, \quad {\bm{F}} = {\bm{e}}_z \times {\bm{\nabla}}\frac{1}{\sqrt{(x-{x'})^2 + (y-{y'})^2+ z^2}}.\nonumber\end{aligned}$$ For a circular system $\mathrm{d}{\bm{l'}}=R\mathrm{d}\chi'\left(-{\bm{e}}_x\sin\chi' + {\bm{e}}_y \cos\chi'\right)$, hence $$I_z(x) = R\!\! \int\limits_0^{2\pi}\!\!\mathrm{d}\chi'\!\! \left[\frac{\partial}{\partial y}\sin\chi'\!-\! \frac{\partial}{\partial x}\cos\chi'\right] \!\! \frac1{\sqrt{r^2+R^2-2 rR \cos(\chi-\chi')+ z^2}} =rR\cos(2\chi)\frac{\partial}{\partial r}\! \left[\frac{1}{r} \! \int\limits_0^{2\pi}\!\!\frac{\cos\alpha\,\mathrm{d}\alpha}{ \sqrt{r^2+R^2-2 rR\cos\alpha+ z^2 }}\right].$$ Taking into account that $\text{Im}\mathcal{B}=0$ for the circular system, one can calculate finally the effective in-plane anisotropy coefficient $\mathcal{B}$ as follows: $$\label{eq:B-circular} \begin{split} &\mathcal{B}(\xi) = \frac{1}{2\pi\varepsilon} \Bigl[I_B(2\varepsilon) - I_B(0)\Bigr],\quad I_B(x) = c_1\text{K}(\mu)+c_2 \text{E}(\mu)+c_3 \Pi \left(\left.\frac{4\xi}{(1+\xi)^2}\right\|\mu\right),\\ &c_1=\frac{2-2 x^2-\xi ^2-\left(x^2+\xi^2\right)^2}{3\xi^2\sqrt{x^2+(1+\xi)^2}}, \quad c_2=\frac{\left(x^2+\xi ^2-2\right)\sqrt{x^2+(1+\xi)^2}}{3 \xi ^2},\quad c_3=\frac{x^2 (1-\xi )}{\xi ^2 (1+\xi)\sqrt{x^2+(\xi +1)^2}}. \end{split}$$ The dipolar energy $W_d$ \[see Eq. \] for the disk–shaped system can be presented in the form $W_d= W_d^0 + \widetilde{W_d}$, where $$\label{eq:Wd-disk} \widetilde{W_d} = \frac{1}{R^2}\int\mathrm{d}^2x \Bigl[\widetilde{\mathcal{A}}(r) + \mathcal{B}(r)\cos 2(\phi-\chi)\Bigr] \sin^2\theta$$ and $W_d^0 = -2R^{-2}\int\mathrm{d}^2x \mathcal{A}(r)$ being the isotropic part, the effective easy-plane anisotropy parameter $\widetilde{\mathcal{A}} = 3\mathcal{A}$. Configurational anisotropy for a half-plane and a square prism {#sec:appendix-2geom} ============================================================== We start here with the problem for a half-plane. Consider the large scale behavior of the dipolar energy, given by the in–plane effective anisotropy $\mathcal{B}(x,y)$, see Eq. . Straightforward calculations lead to the effective anisotropy constant for the upper half-plane $$\label{eq:B-exact} \begin{split} \mathcal{B}(x,y) &\equiv \mathcal{B}(y_0) = \frac{1}{2\pi}\int_{-\infty}^\infty \mathrm{d}x y_{0}(y_{0}^2-x^2)\frac{\mathcal{F}(P,h)}{P^4}\\ &=\frac{y_{0}}{2\pi h}\ln\frac{y_{0}^2}{y_{0}^2 + h^2} + \frac{1}{\pi} \arctan\frac{h}{y_{0}}, \end{split}$$ where we choose the origin of the local reference frame at the boundary of the domain, at $(x,y) = (0,0)$, $y_0$ denotes the distance from the boundary, and $P=\sqrt{{x}^2 + y_0^2}$. One can see that $\mathcal{B}$ does not depend on $x$, it takes only positive real values, hence $\arg\mathcal{B}=0$ for any distances $y_{0}$ from the boundary. This means that the in–plane spin angle $\phi$ is always parallel to the half-plane edge. Using Eqs. , and we found that the main contribution to is provided by the boundary domain $x\in [-R_{0};R_{0}]$ with $R_{0}\sim\sqrt{y_{0}h}$. Since this domain collapses to a point when $y_{0}\to0$, we conclude that for any geometry the in-plane spin distribution is parallel to the boundary near the edge. If the curvature radius of the sample boundary is larger than $R_{0}$, then spins are parallel to the boundary over a distance smaller than $R_{0}^2/h$. One should remember, that this conclusion is adequate for regions, where exchange interaction has no principal influence. Let us consider now the configurational anisotropy for the square prism, which has the diagonal $2R$, see Fig. \[fig:MobFrameSquare\]. ![(Color online) Arrangement of coordinates in the local reference frame for the prism shaped particle.[]{data-label="fig:MobFrameSquare"}](squareGeom){width="\columnwidth"} It is convenient to use the local reference frame in the same way as in Sec. \[sec:continuum\]. The relative polar coordinates are defined as follows: $$\label{eq:square_params} \begin{split} R_n=&R\sqrt{1+\xi^2-2\xi\cos\left(n\pi/2-\chi\right)},\\ \varphi_n=&\arccos\frac{R_n^2+R_{n+1}^2-2R^2}{2R_nR_{n+1}},\\ P_n=&\frac{R_{n}R_{n+1}}{R\sqrt{2}}\frac{\sin\varphi_n}{\cos\left(\alpha +\chi-(2n+1)\pi/4\right)}, \end{split}$$ where $\xi=\sqrt{x^2+y^2}/R$. Now we are able to compute magnetization distribution on a large scale, which follows from the minimization condition . Straightforward calculations give $$\begin{aligned} \label{eq:phi-min-1} \phi &= \chi + \frac{\pi}{2} - \frac12 \text{Arg}\mathcal{B},\\ \label{eq:phi-min-2} \mathcal{B} &= \frac{1}{2\pi} \Biggl[ \int_{\psi_0-\varphi_0}^{\psi_0}e^{-2\imath\alpha} \mathcal{F}(P_0,h)\mathrm{d}\alpha\nonumber\\ & + \sum\limits_{j=1}^3\int_{\psi_{j-1}}^{\psi_j} e^{-2\imath\alpha}\mathcal{F}(P_j,h) \mathrm{d}\alpha \Biggr],\\ \label{eq:phi-min-3} \psi_j &= \psi_0+\sum_{i=1}^j\varphi_i,\\ \label{eq:phi-min-4} \psi_0 &= \frac{3\pi}{4} - \chi - \arcsin\left(\frac{R_0 \sin\varphi_0}{R\sqrt{2}}\right),\end{aligned}$$ where $\mathcal F(P_i,h)$ is defined by . 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--- abstract: 'In our ongoing study of $\eta$ Carinae’s light echoes, there is a relatively bright echo that has been fading slowly, reflecting the 1845-1858 plateau phase of the eruption. A separate paper discusses its detailed evolution, but here we highlight one important result: the H$\alpha$ line in this echo shows extremely broad emission wings that reach $-$10,000 km s$^{-1}$ to the blue and $+$20,000 km s$^{-1}$ to the red. The line profile shape is inconsistent with electron scattering wings, so the broad wings indicate high-velocity outflowing material. To our knowledge, these are the fastest outflow speeds ever seen in a non-terminal massive star eruption. The broad wings are absent in early phases of the eruption in the 1840s, but strengthen in the 1850s. These speeds are two orders of magnitude faster than the escape speed from a warm supergiant, and 5–10 times faster than winds from O-type or Wolf-Rayet stars. Instead, they are reminiscent of fast supernova ejecta or outflows from accreting compact objects, profoundly impacting our understanding of $\eta$ Car and related transients. This echo views $\eta$ Car from latitudes near the equator, so the high speed does not trace a collimated polar jet aligned with the Homunculus. Combined with fast material in the Outer Ejecta, it indicates a wide-angle explosive outflow. The fast material may constitute a small fraction of the total outflowing mass, most of which expands at $\sim$600 km s$^{-1}$. This is reminiscent of fast material revealed by broad absorption during the presupernova eruptions of SN 2009ip.' author: - \| Nathan Smith$^{1}$[^1], Armin Rest$^{2,3}$, Jennifer E. Andrews$^1$, Tom Matheson$^4$, Federica B. Bianco$^{5,6}$, Jose L. Prieto$^{7,8}$, David J. James$^9$, R. Chris Smith$^{10}$, Giovanni Maria Strampelli$^{2,11}$, and A. Zenteno$^{10}$\ $^{1}$Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA\ $^2$Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\ $^3$Department of Physics and Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA\ $^4$National Optical Astronomy Observatory, Tucson, AZ 85719, USA\ $^5$CCPP, New York University, 4 Washington Place, New York, NY 10003, USA\ $^6$Center for Urban Science and Progress, New York University, 1 MetroTech Center, Brooklyn, NY 11201, USA\ $^7$Núcleo de Astronomia de la Facultad de Ingenieria, Universidad Diego Portales, Av. Ejercito 441, Santiago, Chile\ $^8$Millennium Institute of Astrophysics, Santiago, Chile\ $^9$Event Horizon Telescope, Smithsonian Astrophysical Observatory MS 42, Harvard-Smithsonian Center for Astrophysics, 60 Garden\ Street, Cambridge, MA 02138, USA\ $^{10}$Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Colina El Pino S/N, La Serena, Chile\ $^{11}$Universidad de La Laguna, Tenerife, Spain title: 'Exceptionally fast ejecta seen in light echoes of Eta Carinae’s Great Eruption' --- \[firstpage\] circumstellar matter — stars: evolution — stars: winds, outflows INTRODUCTION ============ The massive evolved star $\eta$ Carinae serves as a tremendous reservoir of information about episodic mass loss in the late-stage evolution of massive stars. It is uniquely valuable because it is nearby, because it underwent a spectacular “Great Eruption” event observed in the mid-19th century, and because we can now observe the spatially resolved shrapnel of that event with modern tools like the [Hubble Space Telescope]{} ([HST]{}). Added to this list is the recent discovery of light echoes from the 19th century eruption [@rest12], which now allow us to obtain spectra of light from an event that was seen directly by Earth-based observers before the invention of the astronomical spectrograph. This is similar to studies of light echoes from historical supernovae (SNe) and SN remnants in the Milky Way and Large Magellanic Cloud [@rest05a; @rest05b; @rest08]. Spectroscopy of these light echoes provides informative comparisons between $\eta$ Car and extragalactic eruptions. Based mostly on its historical light curve [@sf11], $\eta$ Car has been a prototype for understanding luminous blue variable (LBV) giant eruptions and SN impostors [@smith+11; @vdm12]. Eruptions akin to $\eta$ Car have been discussed in the context of brief precursor episodes of extreme mass loss that create circumstellar material (CSM) of super-luminous Type IIn supernovae [@smith+07; @sm07]. In addition to extreme $\eta$ Car-like mass loss, several lines of evidence connect LBVs and SNe with dense CSM (see @smith14 for a review; also e.g., @gl09 [@groh13; @groh14; @justham14; @kv06; @mauerhan13; @so06; @smith07; @smith+08; @smith+11; @trundle08]). LBVs have the highest known mass-loss rates of any stars before death, where LBV giant eruptions can lose as much as several $M_{\odot}$ in a few years (see @smith14). The physical trigger and mechanism of these LBV-like giant eruptions are still highly uncertain. Eruptive mass loss is usually discussed in the context of super-Eddington winds [@davidson87; @og97; @owocki04; @os16; @q16; @so06; @vanmarle08]. This framework addresses how mass can be lost at such a high rate, but it does not account for where the extra energy comes from. There are also (sometimes overlapping) scenarios that have been discussed, involving binary mergers, stellar collisions, violent common envelope events, accretion events onto a companion (perhaps including compact object companions, although this has not been discussed much for $\eta$ Car), violent pulsations, extreme magnetic activity, pulsational pair instability eruptions, unsteady or explosive nuclear burning, and wave driving associated with late nuclear burning phases approaching core collapse [@jsg89; @fuller17; @gl12; @hs09; @ks09; @pz16; @piro11; @qs12; @sq14; @smith11; @sa14; @smith+11; @smith+16; @soker01; @soker04; @woosley17]. In any case, a tremendous amount of mass (several $M_{\odot}$) leaves the star in a brief window of time (a few years), and observational constraints on the outflow properties provide a key way to guide theoretical interpretation. In general, quasi-steady radiation-driven winds are expected to leave a star with a speed that is within a factor of order unity compared to the escape speed from the star’s surface. That is why red supergiant winds are slow (10s of km s$^{-1}$), blue supergiant winds are a few hundred km s$^{-1}$, O-type stars have winds around 1000 km s$^{-1}$, and more compact H-poor Wolf-Rayet star winds are 2000-3000 km s$^{-1}$ (see @smith14 for a review). For example, line-driven winds of hot O-type stars have a ratio of their terminal wind to the star’s escape speed of $v_{\infty}/v_{\rm esc} \approx 2.6$, and cooler stars below about 21,000 K have $v_{\infty}/v_{\rm esc} \approx 1.3$ [@lamers95; @vink99]. Line driven wind theory and observations indicate that $v_{\infty}/v_{\rm esc} \approx 2.6$ is expected to become lower as the star’s temperature drops [@cak; @abbott82; @pauldrach86; @pp90; @lamers95; @vink99]. For a strongly super-Eddington wind in an LBV, where $\Gamma$ substantially exceeds 1, the effective gravity is low, the stellar envelope may inflate, the wind may show a complicated pattern of outflow and infall, and material may ultimately leak out slowly. The atmosphere may be porous [@owocki04], perhaps leading to a range of outflow speeds, but we don’t expect a steady wind-driven outflow with a high mass-loss rate to be many times faster than a star’s escape speed [@owocki04; @vanmarle08; @owocki17]. Numerical simulations of super-Eddington continuum-driven winds predict terminal wind speeds below the star’s surface escape speed [@vanmarle08; @vanmarle09]. Observationally, a wide range of outflow speeds are seen in the $\eta$ Car system. The bulk outflow of the present-day wind is around 400-500 km s$^{-1}$ [@hillier01], although with some faster speeds up to around 1000 km s$^{-1}$ in the polar wind [@smith+03]. The bipolar Homunculus nebula, which contains most of the mass ejected in the 19th century eruption [@morse01; @smith17] has a range of speeds that vary with latitude from 650 km s$^{-1}$ at the poles to about 50 km s$^{-1}$ in the pinched waist at the equator [@smith06]. However, there is also faster material in the system. Hard X-ray emission from the colliding-wind binary suggests that a companion star has a very fast wind of 2000-3000 km s$^{-1}$ [@corcoran01; @pc02; @parkin11; @russell16]. In spectra of the central star system, speeds as fast as $-$2000 km s$^{-1}$ are only seen in absorption at certain phases, attributed to the companion’s wind shocking the primary star’s wind along our line of sight [@groh10]. So far, the fastest material associated with $\eta$ Car has been seen in the Outer Ejecta, where filaments have speeds based on Doppler shifts and projection angles as high as 5000 km s$^{-1}$ [@smith08]. (Most of the Outer Ejecta seen in images are slower, moving at a few hundred km s$^{-1}$; @kiminki16 [@weis12].) This fast material, combined with the high ratio of kinetic energy to total radiated energy in the eruption [@smith03], has led to speculation that the Great Eruption may have been partly caused by a hydrodynamic explosion [@smith06; @smith08; @smith13]. Spectra of $\eta$ Car’s light echoes [@rest12; @prieto14] also seemed inconsistent with traditional expectations for a simple wind pseudo-photosphere [@davidson87], although @os16 showed that proper treatment of opacity and radiative equilibrium in such a wind may lead to cool temperatures around 5000 K. Well-developed models for sub-energetic and non-terminal explosive events do not yet exist, but an explosive ejection of material and a surviving star might arise if energy is deposited in the star’s envelope that is less than the total binding energy of the core, but enough to unbind the outer layers. @dessart10 explored how stellar envelopes might respond to such energy deposition, and found some cases with partial envelope ejection and model light curves reminiscent of SN impostors. @rm17 argued that any deep energy deposition at a rate that substantially exceeds the steady luminosity of the star is likely to steepen to a shock. For the specific case of $\eta$ Car’s eruption, @smith13 argued that an explosive ejection of fast material interacting with a previous slow wind could account for the historical light curve and several properties of the Homunculus, where CSM interaction leads to efficient radiative cooling as in SNe IIn. In this paper, we present evidence based on Doppler shifts in light echo spectra from the Great Eruption, which show that there was in fact an explosive ejection of very fast material relatively late in the eruption. The observed speeds in excess of 10,000 km s$^{-1}$ suggest that a small fraction of the mass was accelerated to very high speeds by a blast wave, confirming similar conclusions based on fast nebular ejecta observed around the star at the present epoch [@smith08; @smith13]. UT Date Tel./Intr. grating $\Delta\lambda$ (Å) slit PA ------------- ---------------- --------- --------------------- ------------ --------------- -- 2014 Nov 03 Gemini/GMOS R400 5000-9200 1$\farcs$0 293$^{\circ}$ 2015 Jan 20 Baade/IMACS f4 1200 5500-7200 0$\farcs$7 293$^{\circ}$ 2015 Jan 20 Baade/IMACS f4 300 4000-9000 0$\farcs$7 293$^{\circ}$ : Optical Spectroscopy of Light Echo EC2 \[tab:spec\] Optical Spectroscopy ==================== Following the discovery of light echoes from $\eta$ Carinae [@rest12], we have continued to follow the slow spectral evolution of several echoes. So far in previous papers, we have discussed the initial spectra and spectral evolution of a close group of echoes (called “EC1”) thought to arise from pre-1845 peaks in the light curve [@rest12; @prieto14], but we have monitored a number of other echo systems as well. One echo, which we refer to henceforth as “EC2”, captured our attention because it was relatively bright in our first-epoch image in 2003, and has faded very slowly over several years since then. EC2 arises from the reflection off of a cometary shaped dust cloud. We infer from its slow rate of fading that EC2 reflects light from the main plateau of the Great Eruption in 1845-1858; this is explained in more detail in our companion paper (@smith18 S18 hereafter), where we analyze the imaging photometry of the echo. Based on the probable time delay of about 160 yr, spectra that we have been obtaining in the last few years trace light emitted by $\eta$ Car in its mid-1850s plateau. From the known geometry of light echo paraboloids [@couderc39] and with a known time delay, one can deduce the viewing angle for any echo. EC2’s position on the sky is near the EC1 echoes discussed previously by @rest12 and @prieto14. Like those echoes, EC2 views $\eta$ Car from a vantage point that is near the equatorial plane of the Homunculus (within $\sim$10$^{\circ}$), with the uncertainty dominated by the value of the adopted time delay of the echo. The fact that this viewing angle is near the equator and not the polar axis is an important detail, to which we return later. The full photometric and spectroscopic evolution of EC2 will be described in detail in a more lengthy paper referenced above (S18). Here we focus on just one aspect of these data that stands out as an important and independent result. Briefly, EC2 spectra show faint but extremely broad line wings that indicate faster ejecta than has ever been seen before in $\eta$ Car or in any other massive star eruption. The result described below was quite astonishing to us, and profoundly impacts our understanding of the nature of $\eta$ Car’s eruption. Implications are discussed in the next section. We obtained low- and moderate-resolution spectra of the EC2 echo on a number of dates from 2011 to the present, but we only mention a few of those in this paper. We obtained one spectrum on 2014 Nov 13 using the Gemini Multi-Object Spectrograph (GMOS) [@hook02] at Gemini South on Cerro Pachón. Nod-and-shuffle techniques [@gb01] were used with GMOS to improve sky subtraction. Standard CCD processing and spectrum extraction were accomplished with IRAF[^2]. The spectrum covers the range $4540-9250$ Å with a resolution of $\sim9$ Å. We used an optimized version[^3] of the LA Cosmic algorithm [@vandokkum01] to eliminate cosmic rays. We optimally extracted the spectrum using the algorithm of @horne86. Low-order polynomial fits to calibration-lamp spectra were used to establish the wavelength scale. Small adjustments derived from night-sky lines in the object frames were applied. We employed our own IDL routines to flux calibrate the data using the well-exposed continua of spectrophotometric standards [@wade88; @matheson00]. An extensive series of spectra was obtained using the Inamori-Magellan Areal Camera and Spectrograph (IMACS; @dressler11) mounted on the 6.5m Baade telescope of the Magellan Observatory. Most of these spectra are presented in our more lengthy paper, but we selected a few epochs here to demonstrate the clear detection of the broad line wings. The chosen slit width for these observations was 0$\farcs$7. With the f/4 camera, we used the 300 lines per mm (lpm) grating to sample a wide wavelength range at moderate $R \simeq 500$ resolution, and the 1200 lpm grating to sample a smaller wavelength range with higher resolution of $R \simeq 6000$. The 1200 lpm grating was centered on H$\alpha$. The 2-D spectra were reduced and extracted using routines in the IMACS package in IRAF. With the f/4 camera of IMACS, the spectrum is dispersed across 4 CCD chips, so small gaps in the spectrum appear at different wavelengths depending on the grating and tilt used. We correct all our spectra for a reddening of $E(B-V)$=1.0 mag. This includes the minimum line-of-sight reddening in the interstellar medium (ISM) toward the Carina Nebula of $E(B-V)$=0.47 mag [@walborn95; @smith02], plus an additional $\sim$0.5 mag to account for extra extinction from dust clouds within the southern part of the Carina Nebula. This choice is explained in more detail in S18, but briefly, this reddening allows the continuum shape of various echoes to match spectral diagnostics of the temperature [@rest12]. When we deredden our spectra, the continuum shape is consistent with a temperature of about 6000 K. A different choice of $E(B-V)$ would alter the inferred continuum temperature, but would not significantly alter our conclusions about the fast ejecta, which are based on line profile shape. ![Low-resolution spectra of light echo EC2 corresponding to late times in the plateau phase of the eruption, showing very broad H$\alpha$ line wings. The relative Doppler shift is plotted in units of 10$^3$ km s$^{-1}$, and the broad wings of H$\alpha$ appear to extend to at least $\pm$10,000 km s$^{-1}$. The blue curve shows a 6000 K blackbody matched to the continuum.[]{data-label="fig:fast"}](fig1.eps){width="3.4in"} ![Same as Figure \[fig:fast\], but replacing the IMACS low-resolution spectrum (300 lpm grating) with a higher-resolution spectrum (1200 lpm grating) obtained on the same observing run. The same broad component appears with roughly the same strength even though a different grating is used. The same GMOS spectrum as in Figure \[fig:fast\] is shown again (in red) for comparion.[]{data-label="fig:fast1200"}](fig2.eps){width="3.4in"} ![Similar to Figures 1 and 2, but here we plot the higher-resolution EC2 spectrum (1200 lpm grating) from Figure \[fig:fast1200\] in orange. In black, we show the spectrum of a different echo (EC1), discussed previously [@rest12; @prieto14], which reflects light from an early peak in the Great Eruption (1843 or 1838), also obtained with IMACS. This echo that traces an earlier phase of the eruption [does not]{} show the broad wings, so the fast material seems to have appeared at late phases in the eruption. Merely as an illustrative comparison, we also show the broad component in the Type IIn/II-L core-collapse supernova PTF11iqb from @smith15, with velocities multiplied by 1.5.[]{data-label="fig:fast1843"}](fig3.eps){width="3.4in"} Results and Discussion ====================== Broad H$\alpha$ Wings Trace Fast Material ----------------------------------------- Examining our low-resolution spectra of EC2, we noticed that the continuum shape in raw spectra appeared to have an unusual kink around H$\alpha$ at some epochs. After flux calibration and correcting for a modest amount of reddening of $E(B-V)$=1.0 mag, this turned out to be low-level excess emission above the smooth continuum level. Our IMACS spectra all have chip gaps near H$\alpha$, so at first we were suspicious of a relative flux calibration offset on adjacent chips. Then in 2014 we obtained a low-resolution spectrum of EC2 with Gemini GMOS, which has no detector chip gap, and we saw the same excess. Subsequent low-resolution IMACS spectra show that this excess appeared to be growing in strength with time, more so on the red wing. Figure \[fig:fast\] shows the low-resolution spectrum of EC2 near H$\alpha$, taken in 2014 with Gemini/GMOS (red-orange) and in 2015 with Magellan/IMACS (black) using the low-resolution 300 lpm grating. These epochs of the echo correspond to a date during the eruption in the mid-1850s. Broad emission wings of H$\alpha$ are seen in both spectra, and although the signal to noise in the continuum is low, the two spectra agree quite well – except that the broad emission is perhaps a bit stronger in 2015. The emission wings are interrupted by Fe [ii]{} P Cygni profiles intrinsic to $\eta$ Car, as well as oversubtracted nebular \[S [ii]{}\] emission, telluric absorption (the B band), and the IMACS chip gap. Reducing and extracting the spectra of faint light echoes can be a bit tricky, especially in cases like $\eta$ Car where we have extremely faint reflected light in the echo that is embedded in a bright H [ ii]{} region. To double check that the faint broad emission wings are not due to instrumental scattering from the bright nebular H$\alpha$ emission from the Carina Nebula H [ii]{} region included in the same slit (which was, in principle, carefully subtracted with the sky emission by sampling adjacent regions along the slit), we also examined our higher resolution spectra of EC2 taken with IMACS using the 1200 lpm grating. At first glance, the broad wings were not apparent in our higher-resolution spectra, but this is because the wings are so broad that they are dispersed thinly across almost the entire sampled wavelength range and they look like continuum. In fact, the same broad wings are seen clearly in the higher resolution 1200 lpm grating IMACS spectra when they are plotted on the same intensity scale as the low-resolution spectra. Figure \[fig:fast1200\] is the same as Figure \[fig:fast\], except that we replaced the 300 lpm IMACS spectrum with the 1200 lpm IMACS spectrum obtained on the same night. The two IMACS spectra are consistent within the limitations of signal to noise. Since we see the same broad emission wings in two different instruments on two different telescopes, and also with two different gratings in the same instrument, this broad emission must be real and intrinsic to $\eta$ Carinae’s light echo. ![The effects of telluric absorption on the broad redshifted emission wing of H$\alpha$ in the EC2 spectrum. The orange tracing (with no correction for telluric absorption) is the same IMACS 1200 lpm spectrum of EC2 as in Figure \[fig:fast1200\]. The black tracing shows this same H$\alpha$ line profile after correcting for telluric absorption by the B-band (marked by $\earth$). The blue tracing is the spectrum of a field star that was included about one arcminute away in the same slit, in the same exposures as the observations of EC2. This is not a proper telluric standard, and the strong H$\alpha$ absorption corrupts the H$\alpha$ emission of EC2 at low velocities, but it is useful to correct for the telluric absorption of the B-band on the red wing of the line. After correcting for telluric absorption (black) the red wing drops smoothly from $+$12,000 km s$^{-1}$ to beyond $+$20,000 km s$^{-1}$. Exactly how much farther the red wings extends beyond $+$20,000 km s$^{-1}$ depends on the choice of the admittedly noisy continuum level.[]{data-label="fig:tell"}](fig4.eps){width="3.1in"} These broad emission wings are not due to some other source of broad emission within the Carina Nebula, because the broad wings are absent in spectra of other (EC1) light echoes from $\eta$ Car in the same part of the sky (about 2$\arcmin$ away), which trace earlier peaks (1843 or before) in the Great Eruption [@rest12; @prieto14]. A spectrum of one of these earlier EC1 echoes (also obtained with the IMACS spectrograph on Magellan) is shown in Figure \[fig:fast1843\] (thick black line), as compared to the same 1200 lpm spectrum of EC2 (orange). [Clearly the broad component is absent or much weaker in the EC1 echo that corresponds to an earlier phase of the eruption.]{} This confirms that the broad emission wings appeared relatively late in the Great Eruption. Even in our spectra of this new EC2 echo, the broad wings are somewhat weaker in our earliest spectra in 2011/2012, but strengthen through 2014 and 2015 until the present. From the adopted time delay for the echo of $\sim$160 years, this suggests that the broad emission wings became prominent in the mid-1850s. It is interesting that the emergence of fast material in spectra occurs shortly after the ejection date of 1847.1 ($\pm$0.8 yr) deduced from proper motions of the Homunculus [@smith17]. The broad emission wings are also absent in the extracted spectrum of a field star that was included in the same IMACS slit aperture about an arcminute away from EC2 (Figure \[fig:tell\]), confirming that they are not instrumental. The broad H$\alpha$ emission wings in Figures \[fig:fast\] and \[fig:fast1200\] are unprecedented for an LBV eruption. The blue wing extends to $-$10,000 km s$^{-1}$. The red wing extends past +10,000 km s$^{-1}$, to at least +14,000. It is difficult to determine the farthest extent of the red wing due to the telluric B band absorption (marked by $\earth$ in the figures) overlapping that part of the line profle. Figure \[fig:tell\] presents a correction for the telluric absorption on the red wing of H$\alpha$. Although we did not obtain an appropriate set of telluric standard star observations to correct the full spectra, one of our slit positions using the IMACS 1200 lpm grating to observe EC2 also included a bright field star in the same slit about an arcminute away. This star appears to be a late B-type or early A-type star, so the H$\alpha$ absorption intrinsic to this star is fairly strong and corrupts the telluric correction at low velocities ($\pm$1500 km/s). However, the continuum in the vicinity of the atmospheric B-band is smooth and provides a suitable way to correct for the telluric absorption on the red wing of H$\alpha$. The extracted spectrum of this field star is shown in blue in Figure \[fig:tell\]. We divided the observed spectrum of EC2 by this field star to correct for the telluric B-band absorption. Figure \[fig:tell\] shows the IMACS 1200 lpm spectrum before (orange) and after (black) correction of the telluric absorption. After correcting for telluric absorption in this way, the resulting line profile shows that the broad wings are asymmetric, extending farther on the red side out to at least $+$20,000 km s$^{-1}$. Both the red and blue wings are extreme compared to the relatively narrow core of the H$\alpha$ line, which has a width of about $\pm$600 km s$^{-1}$. ![The observed broad wings compared to Lorentzian profiles expected for electron scattering wings. The black spectrum is the same Magellan/IMACS 1200 lpm spectrum of EC2 in Figure \[fig:fast1200\], normalized to the red continuum. If the broad wings were caused by electron scattering, they should follow a roughly symmetric Lorentzian profile. The width of any such profile is restricted by the width of the narrow line core at about 1.5 times the continuum level in this plot. The thick orange curve shows a Lorentzian profile with FWHM = 400 km s$^{-1}$ that roughly approximates the width at the base of the central line core, and it vastly underpredicts the flux of the broad wings and has the wrong shape. The thinner red curve is the same Lorentzian multiplied by a factor of 10 in flux above the continuum; this can account for the flux in the very broad wings, but then the Lorentzian profile vastly overpredicts the flux at around $\pm$1500 km s$^{-1}$. []{data-label="fig:lorentzian"}](fig5.eps){width="3.1in"} Narrow line cores can have broad electron-scattering wings, as is commonly seen in dense stellar winds and early spectra of SNe IIn, and these wings can extend to larger velocities than the true kinematic speed of ejecta. Indeed, the present-day wind spectrum of $\eta$ Car shows an H$\alpha$ profile with electron-scattering wings that extend beyond $\pm$1000 km s$^{-1}$ [@hillier01; @smith+03], even though the wind speed is only about 400-500 km s$^{-1}$. However, electron scattering cannot explain the broad emission wings seen in spectra of EC2, because they are too broad and have the wrong shape. With a characteristic temperature around 6000 K, scattering off thermal electrons will produce wings with a width of only about $\pm$500-1000 km s$^{-1}$. Even in the early phases of SNe IIn when the temperatures might be 20,000 K (as in $\eta$ Car’s present-day wind), the electron scattering wings only have widths (FWHM) of typically $\pm$1500 km s$^{-1}$. Moreover, the broad wings seen in Figures \[fig:fast\] and \[fig:fast1200\] do not have the characteristic symmetric Lorentzian profiles expected for electron scattering. Figure \[fig:lorentzian\] compares the observed H$\alpha$ profile to Lorentzian profile shapes. A Lorentzian profile is expected if the wings are produced primarily by electron scattering, as is seen in early observations and models of SNe IIn [@chugai01; @smith08; @dessart16]. It is clear that a Lorentzian profile cannot match both the broad wings and the narrow width at the base of the central narrow emission line component in $\eta$ Car’s echo. The thick orange Lorentzian, with FWHM = 400 km s$^{-1}$, roughly matches the width at the base of the narrow component (note that this is not a good approximation of the shape of the narrow component, which is somewhat irregular and asymmetric, but it does agree with the widest point at the base of the narrow component). However, when this profile is extrapolated to high velocities, it vastly underpredicts the observed flux in the broad wings. If we increase the flux of this Lorentzian (thin red profile) so that it can produce the observed flux in the broad wings at $\pm$5000 km s$^{-1}$, we find that it vastly overpredicts the flux at lower speeds around 1000-2000 km s$^{-1}$, so that the base of the narrow component is much broader than observed. This Lorentzian profile also vastly overpredicts the total H$\alpha$ line flux. We conclude that electron scattering due to high optical depths in a slower outflow cannot explain the broad wings we report in light echoes of $\eta$ Car. Instead, the broad emission must trace fast ejecta, separate from the slower material emitting the narrower line core. The observed broad component has a peak shifted to the blue, with a very long red tail. This asymmetry might be due to partial P Cygni absorption of the broad blue emission wing. Similar broad profiles with blue peaks and extended red wings are sometimes seen in fast ejecta from SNe; an example from the SN II-L/IIn PTF11iqb [@smith15] is shown in red in Figure \[fig:fast1843\]. In this comparison (meant to illustrate a possibly similar line-profile shape), we have multiplied the outflow velocity of PTF11iqb by a factor of 1.5 for comparison, because the wings in $\eta$ Car’s echo are actually broader than in this core-collapse SN. ![image](fig6.eps){width="4.7in"} Implications ------------ The broad wings of H$\alpha$ seen in light echo spectroscopy of $\eta$ Car are remarkable, and are reminiscent of SN ejecta speeds. To our knowledge, these are the fastest speeds yet seen in any non-terminal massive star eruption. The fastest speeds seen from the central star (as fast as $-$2000 km s$^{-1}$) are only seen in absorption at certain phases, and have been attributed to the companion’s wind as it shocks the primary star’s wind along our line of sight [@groh10]. The broad emission wings that we report are faint, and only trace a fraction of the total outflowing matter. It is not necessarily an insignificant fraction, though; the integrated flux of the broad component is about 1/3 of the total H$\alpha$ line flux. It appears relatively faint because it is spread across such a wide wavelength range. The brighter narrow emission line core of H$\alpha$ shows widths around 500-600 km s$^{-1}$, more in line with the bulk outflow velocity in the Homunculus nebula. Even if it is only tracing a portion of the mass, an outflow speed of $\sim$10,000 km s$^{-1}$ or more is surprising, and has important implications for understanding the basic nature of $\eta$ Car’s eruption and its energy budget. There are a few key considerations: 1\. The outflow speeds of 10,000 - 20,000 km s$^{-1}$ indicated by the broad H$\alpha$ wings are two orders of magnitude faster than the escape speed from a warm supergiant. For $\eta$ Car in its cool, bloated eruption state (a radius of a few hundred $R_{\odot}$; @smith11), we would expect a dense, super-Eddington wind to have a speed on the order of the escape velocity or less [@owocki04; @vanmarle08; @vanmarle09; @smith13], or about 200 km s$^{-1}$. This is indeed the speed observed in light echoes from the early 1840s peaks in the eruption [@rest12; @prieto14], and the speed observed in spectra of the 1890 event [@walborn+liller]. The broad lines that developed in $\eta$ Car’s echo indicate that something besides a steady wind is at work. The extremely fast speeds are also $\sim$10 times faster than the escape speed from an O-type star or Wolf-Rayet star, which might correspond to $\eta$ Car’s companion star that survives today. Instead, such high speeds are reminiscent of outflows from disks around compact objects or fast shock-accelerated ejecta in SN explosions. The relatively late emergence of the fast material may be consistent with the suggestion [@smith13] that a blast wave would accelerate as it encountered a steep drop in density upon existing the outer boundary of a dense CSM shell, or it may be indicative of the fastest ejecta getting excited near a reverse shock during later CSM interaction. 2\. The fact that this echo reflects light seen from a vantage point near the equatorial plane of the Homunculus is critical to interpreting the broad emission. Fast outflow velocities could be interpreted as evidence of a jet that arises when material is accreted onto a companion star [@ks09; @soker01; @soker04; @ts13]. In such models, it has been proposed that a main-sequence O-type star accretes matter at periaston passages and blows bipolar jets that shape the Homunculus Nebula. However, it is difficult to see how such a jet could achieve speeds well in excess of the escape speed from an O-type dwarf; the observed speeds might be more indicative of accretion-driven jets from a compact object companion. More importantly, however, even this jet scenario would not result in such fast outflow speeds in the [equatorial]{} direction, because in such models, a highly collimated polar jet is invoked to shape the bipolar Homunculus lobes. That polar axis is perpendicular to the viewing angle of the echo discussed here. Instead, the broad wings in echoes seen from low latitudes point to something more like a wide-angle explosion. 3\. The fast material indicated by the broad wings in the EC2 echo may be related to fast nebular material seen today in the Outer Ejecta of $\eta$ Car. Most of the bright material in the Outer Ejecta is composed of dense nitrogen-rich condensations moving at speeds of several hundred km s$^{-1}$ [@davidson82; @kiminki16; @mehner16; @sm04; @smith08; @walborn76; @weis01; @weis12]. Some features seen in images are moving at $\sim$10$^3$ km s$^{-1}$ [@sm04; @weis01; @weis12]. However, the fastest material can only be seen in spectra (Doppler shifted out of narrow-band imaging filters); it appears to be concentrated in polar directions, and is expanding away from the star at around 5,000 km s$^{-1}$ with a likely origin during the Great Eruption [@smith08]. It seems probable that the fastest nebular material in the Outer Ejecta seen today may be a counterpart to the fast material seen in light echo spectra. If an explosion produced ejecta with a range of speeds up to 20,000 km s$^{-1}$, then the fastest of this material would have already crashed into the reverse shock, and has therefore given up its kinetic energy to power the X-ray shell around $\eta$ Car [@seward01; @sm04]. Material expanding toward the poles at $\sim$5,000 km s$^{-1}$ or less is still in free expansion because it has not yet reached the reverse shock, consistent with its observed location inside the X-ray shell [@smith08]. Since the EC2 echo discussed here views $\eta$ Car from a latitude near the equator, it is interesting to speculate about implications for the equatorial ejecta around the Homunculus and connections to the central binary system. The brightest feature in the Outer Ejecta of $\eta$ Car is the so-called “S Condensation” (part of the “S Ridge”), located to the S/SW from the star and redshifted [@walborn76; @davidson82; @kiminki16; @sm04]. Its trajectory of ejection from the star is not far from the viewing angle of the echo discussed in this work (see Figure \[fig:collision\]). Is the S Condensation related to the fast material observed in light echo spectra? Today the S Condensation is much slower (expanding at a few hundred km s$^{-1}$), but the S Condensation we see today could be the end product of a small mass of very fast ejecta from the Great Eruption that swept up and shocked much denser and slower CSM in the equator. Proper motions of material in the S Ridge yield ejection dates several decades before the Great Eruption [@kiminki16; @morse01], consistent with older and slower CSM that may have been accelerated when hit by the fast ejecta. Perhaps even more interesting (and more speculative) is a possible connection to violent binary interaction during the eruption itself. @smith11 pointed out that with the emitting radius required to achieve the observed luminosity during the lead-up to the eruption, the bloated primary star’s effective photosphere was actually bigger than the separation of the two stars at periastron in the current binary system — this means that the secondary star plunged into the extended envelope of the primary (or some extended common envelope around the system) and came out the other side, doing so multiple times. The orientation of the present-day eccentric binary system is aligned with the equatorial plane of the Homunculus, and has the secondary star orbiting clockwise on the sky, with periaston in the direction away from Earth [@madura12]. With this geometry (shown in Figure \[fig:collision\]), the point of ingress when the secondary star collided with the bloated envelope is on the S/SW side of the star, such that the direction toward the echo is similar to the point of ingress. In other words — [the echo is situated favorably to have viewed the secondary star plunging into the bloated primary star’s envelope]{}. (Of course, the central “primary star” here may have been a close binary in the midst of a merger event surrounded by a common envelope.) Similarly, the direction of egress from the bloated envelope seems well aligned with the trajectory of the NN Jet (Figure \[fig:collision\]). Some speculation is required here, since to our knowledge there have been no 3-D hydrodynamic numerical simulations to explore such a scenario of two massive stars colliding in an eccentric binary with a bloated primary. One can imagine a small mass of high velocity ejecta accelerated by the ensuing splash, which may far exceed the orbital speed of the secondary, or a small amount of shock acclerated ejecta escaping through a chimney formed by the wake of the orbiting companion. This violent collision may have had something to do with the fast ejecta revealed in the spectra of light echoes reported here, and it may have ejected fast material preferentially in the directions of the S Condensation and NN Jet. SN 2009ip --------- There are also interesting connections to extragalactic objects. Most LBVs and SN impostors show H$\alpha$ line widths that indicate bulk outflow speeds from a few hundred up to 1000 km s$^{-1}$ [@smith+11], like the majority of the mass flux in $\eta$ Car. Similar speeds are seen in the CSM of Type IIn supernovae (see @smith14). One remarkable exception is the case of SN 2009ip, which was a luminous and eruptive LBV-like star observed in eruption in 2009 [@smith10; @foley11], but which then exploded as a SN a few years later [@mauerhan13]. Spectra of pre-SN outbursts showed some very fast material in SN 2009ip with speeds as high as 7,000-10,000 km s$^{-1}$ [@smith10; @foley11; @pastorello13]. An important point, though, is that this fast material was only seen in absorption along the line of sight, whereas the main emission line core was relatively narrow, indicating outflow speeds for the bulk of the material of 600 km s$^{-1}$. This situation, with the bulk of the mass moving more slowly and a fraction of the ejecta accelerated to very high speeds, is strikingly similar to what we see in echoes of $\eta$ Car and in its fast Outer Ejecta [@smith08]. This provides yet another empirical link between LBVs and the progenitors of some SNe IIn, and provides interesting clues to the possibly similar mechanisms of pre-SN eruptive mass loss. The existence of this fast ejecta is a strong constraint for any physical model of the eruption mechanism. Summary and Conclusions ======================= This paper presents spectra of light echoes from $\eta$ Carinae that correspond to the time period of the main plateau of the eruption during the late 1840s through the 1850s. The full spectral evolution, photometry, and other details of this echo will be discussed in a forthcoming paper (S18). Here we focus on one important aspect that is significant on its own, which is the discovery of extremely broad emission wings of H$\alpha$ that represent the fastest material ever detected in an LBV-like eruption. The main results from this work are summarized as follows. 1\. In addition to a relatively narrow (600 km s$^{-1}$) line core, H$\alpha$ displays extremely broad wings in emission, reaching to approximately $-$10,000 km s$^{-1}$ to the blue and $+$20,000 km s$^{-1}$ or more on the red wing. 2\. We demonstrate that the broad wings are not instrumental. They are not present in a nearby field star included in the same slit, and moreover, the same broad wings are seen in spectra obtained with different instruments on different telescopes as well as two different gratings on the same spectrograph. The strength of the broad wings changes with time, and the broad emission is not seen in a different echo that traces earlier epoch in the eruption seen from a similar direction. Correcting for the telluric B-band absorption, the red wing clearly extends to $+$20,000 km s$^{-1}$ or more. 3\. The shape of the broad wings is inconsistent with electron scattering wings, and we argue that the broad emission must trace Doppler shifts from bulk expansion velocities. Therefore, these are the highest outflow speeds discovered yet in an LBV or any non-terminal eruptive transient. 4\. The high velocities are too fast for any previously conceived escape velocity in the system, but similar to outflow speeds from accreting compact objects or expansion speeds of SN ejecta. The expanding material probably does not arise in a steady wind, but instead likely indicates a shock-accelerated outflow. 5\. The broad wings are seen in echo spectra that view $\eta$ Car from the equator, so these high speeds are probably not indicative of a polar jet (even one from a compact object). The high speeds in echoes seen from the equator combined with fast polar speeds in the Outer Ejecta seen today [@smith08] suggest a wide-angle explosion rather than a highly collimated jet. 6\. The viewing angle of this echo could be special, however, since it is looking from a similar direction as the “S Condensation” in the Outer Ejecta. This is also a special direction in the present day binary system, since it is situated preferably to view the wide companion plunge into a putative common envelope, for example (see text). 7\. Regardless of the physical interpretation, the dual presence of fast and slow speeds (10,000-20,000 km s$^{-1}$ and 600-1000 km s$^{-1}$, respectively) point to CSM interaction at work in the eruption. They are also similar to slow and high velocities seen in spectra of the eruptive progenitor of SN 2009ip. Therefore, the high velocities in $\eta$ Car provide yet another interesting possible link between LBVs and SNe IIn. Acknowledgements {#acknowledgements .unnumbered} ================ We thank an anonymous referee for a careful reading of the manuscript and constructive comments. We acknowledge contributions of additional collaborators who helped with imaging observations to discover and monitor light echoes, as well as for discussions and contributions to proposals for telescope time for this and related projects: the Carnegie Supernova Project, Alejandro Clocchiatti, Steve Margheim, Doug Welch, and Nolan Walborn. In particular, Nolan Walborn provided helpful comments on the manuscript just weeks before he passed away, which occurred while this paper was under review. His contributions to massive star research have been tremendous, and his unique insight will be sorely missed. NS’s research on Eta Carinae’s light echoes and related LBV-like eruptions received support from NSF grants AST-1312221 and AST-1515559. Support for JLP is provided in part by FONDECYT through the grant 1151445 and by the Ministry of Economy, Development, and Tourism’s Millennium Science Initiative through grant IC120009, awarded to The Millennium Institute of Astrophysics, MAS. 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--- author: - \| Ekin D. Cubuk [^1], Barret Zoph, Jonathon Shlens, Quoc V. Le\ Google Research, Brain Team\ `{cubuk, barretzoph, shlens, qvl}@google.com` title: 'RandAugment: Practical data augmentation with no separate search' --- Acknowledgements ================ We thank Samy Bengio, Daniel Ho, Jaehoon Lee, Hanxiao Liu, Raphael Gontijo Lopes, Ruoming Pang, Ben Poole, Mingxing Tan, and the rest of the Brain team for their help. [^1]: Authors contributed equally.
--- abstract: 'The energy release and build-up processes in the solar corona have significant implications in particular for the case of large recurrent flares, which pose challenging questions about the conditions that lead to the episodic energy release processes. It is not yet clear whether these events occur due to the continuous supply of free magnetic energy to the solar corona or because not all of the available free magnetic energy is released during a single major flaring event. In order to address this question, we report on the evolution of photospheric magnetic field and the associated net Lorentz force changes in ARs 11261 and 11283, each of which gave rise to recurrent eruptive M- and X-class flares. Our study reveals that after the abrupt downward changes during each flare, the net Lorentz force increases by $(2\textup{--}5)\times 10^{22}$ dyne in between the successive flares. This distinct rebuild-up of net Lorentz forces is the first observational evidence found in the evolution of any non-potential parameter of solar active regions (ARs), which suggests that new energy was supplied to the ARs in order to produce the recurrent large flares. The rebuild-up of magnetic free energy of the ARs is further confirmed by the observations of continuous shearing motion of moving magnetic features of opposite polarities near the polarity inversion line. The evolutionary pattern of the net Lorentz force changes reported in this study has significant implications, in particular, for the forecasting of recurrent large eruptive flares from the same AR and hence the chances of interaction between the associated CMEs.' author: - Ranadeep Sarkar - Nandita Srivastava - 'Astrid M. Veronig' title: 'Lorentz Force Evolution Reveals the Energy Buildup Processes during Recurrent Eruptive Solar Flares' --- \ Introduction ================ Solar flares and coronal mass ejections (CMEs) are the most energetic phenomena that occur in the solar atmosphere. Together they can release large amounts of radiation, accelerated high-energy particles and gigantic clouds of magnetized plasma that may have severe space-weather impacts [@Gosling; @Siscoe; @Daglis; @2018SSRv]. Therefore, understanding the source region characteristics of these solar energetic events has become a top priority in space-science research. Complex large active regions (ARs) on the Sun are the main sources of large flares and most energetic CMEs [@Zirin; @Sammis; @Falconer; @Wang_2008; @Tschernitz; @Toriumi]. Understanding the energy build-up processes in the source ARs has significant implications in particular for the case of recurrent flares, which may lead to recurrent CMEs and hence to their interaction, if the following CME has a larger speed than the preceding one. [ccccccc]{}\[!t\] AR 11261 & 2011/08/03 & 13:17 & 13:45 & 14:30 & M6.0 & N17W30\ AR 11261 & 2011/08/04 & 03:41 & 03:45 & 03:57 & M9.3 & N16W38\ AR 11283 & 2011/09/06 & 01:35 & 01:50 & 02:05 & M5.3 & N13W07\ AR 11283 & 2011/09/06 & 22:12 & 22:20 & 22:24 & X2.1 & N14W18\ AR 11283 & 2011/09/07 & 22:32 & 22:38 & 22:44 & X1.8 & N14W31\ Recurrent large flares pose challenging questions regarding the conditions that lead to the episodic energy release processes [@Nitta; @DeVore_2008; @Archontis; @Romano]. In particular, it is not yet clear whether these events occur due to the continuous supply of free magnetic energy to the solar corona or because not all of the available free magnetic energy is released during a single flaring event. Emergence of new magnetic flux [@Nitta] or photospheric shearing motions [@Romano] have been observed during recurrent flares. However, quantitatively it is difficult to study the temporal evolution of the free magnetic energy of any AR due to the absence of any practical or direct method to measure the vector magnetic field in the coronal volume [@Wiegelmann]. Therefore, the spatial and temporal evolution of source region parameters which can be solely estimated from the photospheric magnetic field becomes important to probe the energy generation processes responsible for solar flares. @HFW were the first to quantitatively estimate the back reaction forces on the solar surface resulting from the implosion of the coronal magnetic field, which is required to release the energy during flares. They predicted that the photospheric magnetic fields should become more horizontal after the flare due to the act of the vertical Lorentz forces on the solar surface. @Fisher introduced a practical method to calculate the net Lorentz force acting on the solar photosphere. Since then, it became one of the important non-potential parameters to study the flare-associated changes in the source region characteristics. Earlier studies revealed that large eruptive flares are associated with an abrupt downward change of the Lorentz force [@Petrie2010; @Petrie]. Comparing the magnitude of those changes associated with eruptive and confined flares, @Sarkar2018 reported that the change in Lorentz force is larger for eruptive flares. However, studies on the evolution of the photospheric magnetic field and the associated Lorentz force changes for the case of recurrent eruptive large flares have not been performed so far. In this Letter, we study the evolution of the photospheric magnetic field and the associated net Lorentz force change during recurrent large flares which occurred in AR 11261 and AR 11283. Tracking the evolution of the net Lorentz force over the period of all the recurrent flares under study, we address the following key questions. \(i) Are the observed changes in net Lorentz force during the flare related to the linear momentum of the associated CME? \(ii) Are there any prominent signatures related to the Lorentz force evolution which might reveal the restructuring of the magnetic field after the first flare and its associated CME? If so, these signatures might be indicative of rebuild-up of non-potentiality of the coronal magnetic field and hence the imminent more powerful flare/CME. \(iii) What causes the build-up of free magnetic energy between the successive flares? Data analysis {#first_sec} ================= All the large recurrent M- and X-class flares that occurred in ARs 11261 (SOL2011-08-03T13:17 and SOL2011-08-04T03:41) and 11283 (SOL2011-09-06T01:35, SOL2011-09-06T22:12 and SOL2011-09-07T22:32) were well observed by the Atmospheric Imaging Assembly (AIA; @Lemen) and the Helioseismic and Magnetic Imager (HMI; @Schou) onboard the Solar Dynamics Observatory (SDO; @Pesnell). To study the evolution of the photospheric magnetic field associated with the recurrent flares, we have used the HMI vector magnetogram series from the version of Space weather HMI Active Region Patches (SHARP; @Turmon) having a spatial resolution of 0.5$''$ and 12 minute temporal cadence. ![image](figure1.pdf){width="\textwidth"} ![image](figure2.pdf){width=".95\textwidth"} As the errors in the vector magnetic field increase towards the limb, we have restricted our analysis to only those flares for which the flaring location of the AR was well within $\pm$ 40$^{\circ}$ from the central meridian. Moreover, we focus on the recurrent flares that initiated in the same part of the polarity inversion line of the AR and occurred within an interval of a day or less than that. This approach allows us to study the energy release and rebuild-up processes related to the recurrent flares by tracking the magnetic properties of a same flare-productive part of an AR over a period of several days. Following the aforementioned criteria, we analyze the two recurrent M-class flares (SOL2011-08-03T13:17 and SOL2011-08-04T03:41) which occurred in AR 11261 during 2011 August 3 to 4 and three recurrent flares (SOL2011-09-06T01:35, SOL2011-09-06T22:12 and SOL2011-09-07T22:32) which occurred in AR 11283 during the period 2011 September 5 to 8 (Table \[table1\]). To calculate the net Lorentz-force changes we have used the formulation introduced by @Fisher. The change in the horizontal and radial component of the Lorentz force within a temporal window of $\delta t$ is given as $$\label{first} \delta F_{\rm r}=\frac{1}{8\pi}\int_{A_{\rm ph}} (\delta {B_{\rm r}}^2-\delta {B_{\rm h}}^2)\\\ {\mathrm d}{\rm A}$$ $$\label{second} \delta F_{\rm h}=\frac{1}{4\pi}\int_{A_{\rm ph}}\delta ({B_{\rm h}}{B_{\rm r}})\\\ {\mathrm d}{\rm A}$$ where $ B_{\rm h} $ and $ B_{\rm r} $ are the horizontal and radial components of the magnetic field, $F_{\rm h}$ and $F_{\rm r}$ are the horizontal and radial components of the Lorentz force calculated over the volume of the active region, $\rm A_{\rm ph} $ is the area of the photospheric domain containing the active region, and dA is the elementary surface area on the photosphere. Similar to @Petrie, we have reversed the signs in Equations \[first\] and \[second\] compared to the Equations 9 and 10 of @Fisher, as we are considering the forces acting on the photosphere from the above atmospheric volume instead of the equal and opposite forces acting on the above atmosphere from below. As the flare related major changes in horizontal magnetic field and Lorentz forces are expected to occur close to the polarity inversion line (PIL) [@Wang; @Petrie2010; @Petrie; @Sarkar2018], we have selected subdomains (shown by the region enclosed by the green rectangular boxes in Figure \[HMI\]) near the PIL on the flare productive part of each AR to carry out our analysis. As the recurrent flares studied in this paper occurred from the same part of the PIL, we are able to capture the evolution of the magnetic field over several days including the time of each flares within that same selected domain on the AR. In order to define the size, orientation, and location of the selected domains we examined the post-flare loops observed in the AIA 171 and 193 Å channels. Several studies have shown that the flare-reconnection process results in the simultaneous formation of a post-eruption arcade (PEA) and a flux rope above the PEA during solar eruptive events [@Leamon; @Longcope; @Qiu; @Hu]. Therefore in order to capture the magnetic imprints of the recurrent large eruptive flares on the solar photosphere, we have selected our region of interest in such a way so that the major post flare arcade structures formed during each flare can be enclosed within that domain. The choice of such subdomains enables us to assume that the magnetic field on the side-boundaries enclosing the volume over those selected regions is largely invariant with time and the field strength on the top boundary is negligible as compared to that at the lower boundary on the photosphere. Therefore, only the photospheric magnetic field change contributes to the surface integrals as shown in Equations 1 and 2 to estimate the change in net Lorentz force acting on the photosphere from the above atmospheric volume. Result and discussion {#second} ========================= Abrupt Changes in Magnetic Field and Lorentz Force -------------------------------------------------- Figure \[force\_plot\] depicts the abrupt changes in horizontal magnetic field and the radial component of net Lorentz forces calculated within the selected region of interest as shown in Figure \[HMI\]. The distinct changes in the magnetic properties of AR 11261 and AR 11283 associated with the recurrent large M- and X-class flares are discussed as follows. ### Magnetic Field Evolution in AR 11261 During the first M6.0 class flare (SOL2011-08-03T13:17) that occurred in AR 11261, the mean horizontal magnetic field increases approximately from 500 to 550 G and the associated net Lorentz force shows an abrupt downward change by approximately $2.8 \times 10^{22}$ dyne. After the M6.0 class flare the mean horizontal magnetic field started to decrease and reached about 490 G prior to the M9.3 class flare (SOL2011-08-04T03:41). During the M9.3 class flare the mean horizontal magnetic field again approximately increased to 550 G. The associated change in net Lorentz force during this flare is about $5.1 \times 10^{22}$ dyne which is almost two times larger than that associated with the previous M6.0 class flare. In order to examine whether the kinematic properties of the associated CMEs are related to the flare induced Lorentz force changes or not, we obtain the true mass and the deprojected speed of each flare associated CME from @Mishra. The two recurrent CMEs associated with the preceding M6.0 class and the following M9.3 class flares are hereinafter referred to as CME1 and CME2, respectively. Interestingly, CME2 was launched with a speed of 1700 km s${^{-1}}$, approximately 1.5 times higher than that of CME1 (v = 1100 km s${^{-1}}$). The true masses of CME1 and CME2, estimated from the multiview of STEREO-A and -B coronagraph data, were $7.4 \times 10^{12}$ kg and $10.2 \times 10^{12}$ kg, respectively. Considering an error of $\pm$ 100 km s${^{-1}}$ in determining the CME speed [@Mishra] and $\pm$ 15 % in estimating the CME mass [@Bein_2013; @Mishra2014], we derive the momentum of CME2 as $17 \times 10^{15} \pm 4 \times 10^{15}$ kg km s${^{-1}}$, approximately twice the momentum of CME1 ($8 \times 10^{15} \pm 2 \times 10^{15}$ kg km s${^{-1}} $). Therefore the magnitude of change in the net Lorentz force impulse during the two recurrent flares appears to be correlated with the associated CME momentum. This scenario is consistent with the flare related momentum balance condition where the Lorentz-force impulse is believed to be proportional to the associated CME momentum [@Fisher; @ShouWang]. As the masses of the two CMEs were comparable, the successive Lorentz force impulse within a time window of approximately 14 hr from the same PIL of the AR with a larger change in magnitude during the following flare appears to be an important characteristic of the source AR in order to launch a high speed CME preceded by a comparatively slower one. This was an ideal condition for CME-CME interaction. Eventually, the two CMEs interacted at a distance of 145 solar radii [@Mishra]. ### Magnetic Field Evolution in AR 11283 For all the three recurrent flares that occurred in AR 11283, the horizontal magnetic field and the net Lorentz force showed abrupt changes during each flare. It is noteworthy that the net Lorentz force increases substantially 2-4 hr prior to the occurrence of each flare, followed by a steep decrease of the same. The changes in net Lorentz force during the successive M5.3 (SOL2011-09-06T01:35), X2.1 (SOL2011-09-06T22:12) and X1.8-class (SOL2011-09-07T22:32) flares were approximately $4 \times 10^{22}$, $3.5 \times 10^{22}$ and $3.5 \times 10^{22}$ dyne respectively. All the three flares were eruptive and the associated deprojected CME speeds were 640, 773, and 751 km s${^{-1}}$, respectively as reported in Soojeong Jang’s Catalog (http://ccmc.gsfc.nasa.gov/requests/fileGeneration.php). For all the three flares the magnitude of change in net Lorentz force were almost comparable and the associated CME speeds also do not differ too much. As the three associated CMEs were launched within an interval of a day and with approximately similar speed, there was no chance of interactions among them in the interplanetary space within 1 AU. As the CDAW catalog (https://cdaw.gsfc.nasa.gov/CME\_list/) reports poor mass estimation for the aforementioned CMEs, we do not compare the linear momentum of those CMEs with the associated change in net Lorentz force. In strong events, flare induced artifacts in the magnetic field vectors may result in magnetic transients during the stepwise changes of the photospheric magnetic field [@Sun_2017]. However, these magnetic transients as reported by @Sun_2017 are spatially localized in nature and temporally can be resolved within a timescale of $\approx$ 10 minutes. Moreover, the transient features do not show any permanent changes in the magnetic field evolution during the flares. The evolution of the horizontal magnetic field and the net Lorentz force as shown in Figure \[force\_plot\] are estimated within a large area on the photosphere using the 12 minute cadence vector magnetogram data. Therefore, within the time window of the stepwise changes in the horizontal magnetic field, there is no discontinuity found in the field evolution during the flares under this study as potentially occurring magnetic transients would be spatially and temporally averaged out. Hence, there are no flare related artifacts involved in the derivation of the net Lorentz force in this study. ![image](figure3.pdf){width="\textwidth"} ![image](figure4.pdf){width="\textwidth"} Lorentz Force Rebuild-up in between the Successive Flares --------------------------------------------------------- After the abrupt downward change in net Lorentz force during each large flare that occurred in AR 11261 and AR 11283, the [net]{} Lorentz force started to rebuild-up in between the successive flares (see Figure \[force\_plot\]). Starting from the magnitude of $-1 \times 10^{22}$ dyne after the M6.0 class, the change in net Lorentz force reached to a magnitude of $4 \times 10^{22}$ dyne until the next M9.3 class flare occurred in AR 11261. Similarly in AR 11283, the net Lorentz force was rebuilt-up by approximately $2 \times 10^{22}$ dyne in between the M5.3 and X2.1 class flares, and again rebuilt-up by approximately $4 \times 10^{22}$ dyne before the X1.8-class flare. This rebuild-up of the Lorentz force reveals the restructuring of the magnetic field configuration in the vicinity of the PIL in order to increase the non-potentiality of the coronal magnetic field which in turn relaxes by producing the next recurrent flare. We tested the sensitivity of the obtained results on the size of the bounding boxes selected around the PIL. Increasing the bounding box (see Figure \[HMI\]) from approx 20 to 40 Mm, the evolutionary pattern of the Lorentz force remains similar. However, integrating the Lorentz force density over the whole AR area dilutes the flare associated changes in the estimated net Lorentz force profile. The rebuild-up of net Lorentz force in between the recurrent flares could be the consequence of the continuous shearing motion along the PIL. Figure \[mmf\] shows the continuous shearing motion observed for the two prominent moving magnetic features (MMFs) of opposite magnetic polarities (indicated by the red and green circles). The antiparallel motions of these MMFs along the two sides of the PIL of each AR during the recurrent flares provide evidence for rebuild-up of non-potential energy in between the successive flares. Therefore, the evolution of Lorentz force appears to be a clear indication of energy rebuild-up processes in order to produce successive flares from the same part of any AR. Importantly, for the first time we have shown the evolution of a non-potential parameter (net vertical Lorentz force change) that reveals the rebuild-up of non-potentiality of the AR in between the successive large flares. Indeed, this is a significant finding and has important implications. In particular, the evolutionary pattern of the net vertical Lorentz force change can be used for forecasting the recurrent large eruptive flares from the same AR. Furthermore, the associated successive CMEs from the same AR, will in turn enhance their chance of being launched in the same direction. In this scenario, the following faster CME may interact with the preceding slower one in the corona or interplanetary space, which can significantly enhance their geo-effectiveness [@Wang2003; @Farrugia; @Farrugia2; @LugazFarrugia]. Currently available machine-learning algorithms for flare prediction use, among many other parameters, the evolution of Lorentz force integrated over the whole AR, which does not show high skill score in the forecast verification metrics [@Bobra_2015]. However, the distinct changes in the vertical component of the Lorentz forces integrated near the PIL demonstrated in our study, could prove to be an important parameter to train and test the machine-learning algorithms in order to improve the current capability of flare-forecasting. Relative Evolution of the GOES X-ray Flux with that of the Associated Lorentz Force During the Flares ----------------------------------------------------------------------------------------------------- The temporal evolution of the GOES 1-8 Å X-ray flux and the associated change in Lorentz force shows that the Lorentz force starts to decrease at the start of rising phase of the GOES flares (Figure \[goes\_lf\]). Most interestingly, the Lorentz force decreases with a pattern similar to the decay phase of the GOES X-ray flux during all the flares. Among all the five flares (see Table \[table1\]), the decay phase of the X2.1 class flare (panel (d) of Figure \[goes\_lf\]) was significantly steeper than the other four flares. This reflects in the associated changes in Lorentz force. The Lorentz force also decreases sharply during that X2.1 class flare in comparison to the other flares. The derived rate of change in net Lorentz force associated with the X2.1 class flare is $3 \times 10^{19}$ dyne s$^{-1}$ (Figure \[goes\_lf\]), which is the highest among all the five flares studied in this work. These results suggest that the change in Lorentz force is not only related to the phase of impulsive flare energy release, but takes place over a longer interval and follows a similar evolutionary pattern like the decay phase of the GOES soft x-ray flux. This could be associated with a slower structuring of the coronal magnetic field during the decay phase of the flaring events. Conclusion {#third} ============== Studying the evolution of the photospheric magnetic field and the associated Lorentz force change during the recurrent large flares that occurred in AR 11261 and AR 11283, we find that the vertical component of Lorentz force undergoes abrupt downward changes during all the flares. This result is consistent with earlier studies [@Wang; @Petrie2010; @Petrie; @Sarkar2018]. The observed increase in horizontal magnetic field during each flare is in agreement with the conjecture given by @HFW, which suggests that the magnetic loops should undergo a sudden shrinkage or implosion due to the energy release processes during flares. This also supports the results obtained by @Romano, which show a decrease in the dip angle after each large flare that occurred in AR 11283. Interestingly, the decrease in horizontal magnetic field in between the successive flares reported in our study, could be due to the storage of newly supplied energy that increases the coronal magnetic pressure, thereby stretching the magnetic loops upward as proposed by @Hudson2000. Our study also reveals that the decrease in Lorentz force is not only related to the phase of impulsive flare energy release, but takes place over a longer interval that covers also the decay phase of the flaring events. The magnitude of change in net Lorentz forces reported in this work, appears to be correlated with the linear momentum of the associated CME. This scenario is consistent with the flare related momentum balance condition where the Lorentz force impulse is believed to be proportional to the associated CME momentum [@Fisher; @ShouWang]. It is noteworthy that the flare-associated momentum conservation is not only related to the bodily transfer of mass in the form of CMEs, but also includes the effects related to explosive chromospheric evaporation [@2012Hudson]. However, quantifying the momentum related to the chromospheric evaporation during the flares under this study is not possible, as this requires spectroscopic observations of both the hot upflowing and cool downflowing plasma. Such measurements are rarely available, due to the localized and dynamic nature of solar flares in contrast to the limited spatio-temporal coverage of spectrometers. However, comparing the values we obtain for the CME momentum, which is of the order of 10$^{15}$ kg km s$^{-1}$, with the momentum related to chromospheric evaporation flows in large flares as reported in the literature, which is of the order of $10^{13} - 10^{14}$ kg km s$^{-1}$ [@1988Zarro; @1990Canfield; @2012Hudson], we may conclude that the momentum changes related to the CME are the dominant contribution. Therefore, the correlation between the Lorentz force impulse and the CME momentum in the large recurrent eruptive flares reported in our study is valid as the effects of impulsive chromospheric evaporation are at least an order of magnitude smaller. Most importantly, after the abrupt downward changes during each flare, the net Lorentz force significantly increases to a higher value than that was observed few hours before the flaring event, and only then the subsequent (recurrent) energetic flare occurred. This rebuild-up of net Lorentz force in between the successive flares suggests that the magnetic field configuration in the vicinity of the PIL is restructured in order to increase the non-potentiality of the coronal magnetic field. Observations of the continuous shearing motions of the MMFs on the two sides of the PIL of each AR provide supporting evidence for rebuild-up of non-potential energy. @Romano have also reported the shearing motion along the PIL of AR 11283 during the recurrent large M- and X-class flares. They have attributed these photospheric horizontal motions as the possible cause of monotonic injection of magnetic helicity in the corona, which might have resulted in the episodic energy release processes, leading to the recurrent flares. However, the evolution of the horizontal magnetic field and the associated Lorentz force reported in our study, clearly indicates the energy rebuild-up processes in order to produce successive flares from the same part of the AR. 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--- address: - 'Theory Department, Lebedev Physical Institute, Moscow, Russia; Institute for Theoretical and Experimental Physics, Moscow, Russia' - 'Institute for Theoretical and Experimental Physics; Laboratoire de Mathematiques et Physique Theorique, CNRS-UMR 6083, Universite Francois Rabelais de Tours, France' - 'Department of Mathematics, Higher School of Economics, Moscow, Russia; A.N.Belozersky Institute, Moscow State University, Russia; Institute for Theoretical and Experimental Physics' --- FIAN/TD-11/10\ ITEP/TH-46/10 Algebra of differential operators associated with Young diagrams [ ]{} A.Mironov, A.Morozov, S.Natanzon [ ]{} [We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. The structure constants of the algebra are expressed through the Hurwitz numbers. ]{} [Introduction]{} ==================== Center of the group algebra $A_n$ of the symmetric group $S_n$ plays the main role in describing representations both of the symmetric group and of the matrix group $Gl(n)$. Its counterpart for the infinite symmetric group is the algebra $A_{\infty}$ of the conjugated classes of finite permutations of an infinite set [@IK]. Its natural generators are the Young diagrams of arbitrary degree. In the present paper, which is a continuation of [@MMN1], we construct an exact representation of algebra $A_{\infty}$ in the algebra of differential operators of infinitely many variables. The differential operators $\textsf{W}(\Delta)$, corresponding to the Young diagrams $\Delta$, are closely related to the Hurwitz numbers, matrix integrals and integrable systems [@MS; @BM; @MM; @M1; @M2]. We prove that the Schur functions form a complete set of the common eigenfunctions of the operators $\textsf{W}(\Delta)$, and find the corresponding eigenvalues. A key role in the construction is played by the Miwa variables, which naturally emerge in matrix models [@Miwa; @GKM]. In section 2, we define the algebra of Young diagrams, which is isomorphic to the algebra of conjugated classes of finite permutations of an infinite set, and express its structure constants through the structure constants of the algebra $A_n$. In section 3, we construct a representation of the universal enveloping algebra $U(gl(\infty))$ in the algebra of differential operators of Miwa variables. Using this representation, in section 4, we associate with any Young diagram a differential operator $\mathcal{W}(\Delta)$ of Miwa variables, which has a very simple form. This correspondence gives rise to an exact representation of the algebra $A_\infty$. The operators $\mathcal{W}(\Delta)$ preserve the subspace $\textsf{P}$ of all symmetric polynomials of the Miwa variables. We study further the differential operators $\textsf{W}(\Delta)= \mathcal{W}(\Delta)\|_{\textsf{P}}$ of the variables $p=\{p_i\}$, which form a natural basis in the space $\textsf{P}$. In section 5, we prove that the Schur functions $s_R(p)$ form a complete system of eigenfunctions for $\textsf{W}(\Delta)$ and find the corresponding eigenvalues. In section 6, we explain an algorithm of calculating the operators $\textsf{W}(\Delta)$, the simplest non-trivial operator $\textsf{W}([2])$ being nothing but the “cut-and-join” operator [@GJV], which plays an important role in the theory of Hurwitz numbers and moduli spaces. In the last section 7, we interpret the operators $\textsf{W}(\Delta)$ as counterparts of the “cut-and-join” operator for the arbitrary Young diagram. In particular, we prove that a special generating function of Hurwitz numbers satisfies a simple differential equation, which allows one to construct all the Hurwitz numbers successively. We thank M.Kazarian, A.N.Kirillov, S.Lando and S.Loktev for fruitful discussions. S.Natanzon is also grateful to IHES for perfect conditions for finishing this work. Our work is partly supported by Ministry of Education and Science of the Russian Federation under contract 14.740.11.0081, by RFBR grants 10-02-00509-a (A.Mir. & S.N.) and 10-02-00499 (A.Mor.), by joint grants 09-02-90493-Ukr, 09-01-92440-CE, 09-02-91005-ANF, 10-02-92109-Yaf-a. The work of A.Morozov was also supported in part by CNRS. [Algebra $A_{\infty}$ of Young diagrams]{} {#s1} ============================================== 1\. First we remind the standard facts that we need below. Denote through $\|\mathfrak{M}\|$ the number of elements in a finite set $\mathfrak{M}$ and through $S_n$ the symmetric group which acts by permutations on the set $\mathfrak{M}$, where $\|\mathfrak{M}\|=n$. A permutation $g\in S_n$ gives rise to a subgroup $<g>$, whose action divides $\mathfrak{M}$ into orbits $\mathfrak{M}_1,\dots,\mathfrak{M}_k$. The set of numbers $\|\mathfrak{M}_1\|,\dots,\|\mathfrak{M}_k\|$ is called cyclic type of the permutation $g$. It produces the Young diagram $\Delta(g)=[\|\mathfrak{M}_1\|,\dots, \|\mathfrak{M}_k\|]$ of degree $n$. The permutations are conjugated in $S_n$ if and only if they are of the same cyclic type. Linear combinations of the permutations from $S_n$ form the group algebra $G_n=G(S_n)$. Multiplication in this algebra is denoted as $"\circ"$. Associate with each Young diagram $\Delta$ the sum $G_n(\Delta)\in G_n$ of all permutations of the cyclic type $\Delta$. These sums form the basis of the algebra of the conjugated classes $A_n^{\circ}\subset G_n$, which coincides with the center $G_n$. Denote through $C_{\Delta_1,\Delta_2}^{\Delta}$ the structure constants of the algebra $A_n$ in this basis. In other words, $$G_n(\Delta_1)\circ G_n(\Delta_2)=\sum\limits_ {\Delta\in\mathcal{A}_n} C_{\Delta_1,\Delta_2}^{\Delta}G_n(\Delta),$$ where $\mathcal{A}_n$ is the set of all Young diagrams $\Delta$ of degree $\|\Delta\|=n$. The construction of algebra $A_n^{\circ}$ is continued in [@IK] to the algebra $A_\infty$ of the conjugated classes of finite permutations of the set of natural numbers $\mathbb{N}=\{1,2,\dots\}$. The algebra $A_\infty$ is generated by $G_{\infty}(\Delta)$, which are a formal sum of all finite permutations of the set $\mathbb{N}$ of the cyclic type $\Delta$. Multiplication in the algebra is generated by the multiplication of permutations. According to [@IK], this algebra is naturally isomorphic to the algebra of the shifted Schur functions [@OO]. 2\. We express now the structure constants of the algebra $A_\infty$ through the structure constants of the algebra $A_n^{\circ}$. First, we represent the algebra $A_n^{\circ}$ as an algebra generated by Young diagrams. In other words, we consider $A_n^{\circ}$ as a vector space with the basis $\mathcal{A}_n$ and multiplication $$\Delta_1\circ \Delta_2= \sum\limits_ {\Delta\in\mathcal{A}_n}C_{\Delta_1,\Delta_2}^{\Delta}\Delta.$$ Consider also the monomorphism of the vector space $\rho_k:A_n^{\circ}\rightarrow A_{n+k}^{\circ}$, where $\rho_k(\Delta)=\frac{(r+k)!}{r!k!}{\Delta}^k$. Here ${\Delta}^k$ is the Young diagram obtained from the Young diagram $\Delta$ by adding $k$ unit length rows and $r$ is the number of unit length rows originally present in the diagram $\Delta$. Let us define a multiplication of diagrams of arbitrary degree by the formula $$\Delta_1\Delta_2= \sum\limits_ {n=\max\{\|\Delta_1\|,\|\Delta_2\|\}} ^{\|\Delta_1\|+\|\Delta_2\|} \{\Delta_1\Delta_2\}_{n}$$ where $$\{\Delta_1\Delta_2\}_n = \left\{\begin{array}{lr}\rho_{(n-\|\Delta_1\|)}(\Delta_1)\circ \rho_{(n-\|\Delta_2\|)}(\Delta_2)& \hbox{for } n=\max\{\|\Delta_1\|,\|\Delta_2\|\}\\ \\ \rho_{(n-\|\Delta_1\|)}(\Delta_1)\circ \rho_{(n-\|\Delta_2\|)}(\Delta_2)& -\sum\limits_ {k=\max\{\|\Delta_1\|,\|\Delta_2\|\}}^{n-1} \rho_{(n-k)}(\{\Delta_1\Delta_2\}_k)\\ \\ &\hbox{for }n>\max\{\|\Delta_1\|,\|\Delta_2\|\} \end{array}\right.$$ [Put $\Delta_1=[1]$ and $\Delta_2=[2]$. Then, $\rho_1([1])=2[1,1]$, $\rho_2([1])=3[1,1,1]$, $\rho_1([2])=[2,1]$. Therefore, $\{\Delta_1\Delta_2\}_2=2[1,1]\circ[2]=2[2]$, $\{\Delta_1\Delta_2\}_3=3[1,1,1]\circ[2,1]-2[2,1]=[2,1]$]{} [Put $\Delta_1=[2]$ and $\Delta_2=[2]$. Then, $\rho_1([2])=[2,1]$, $\rho_2([2])=[2,1,1]$, $\rho_1([1,1])=3[1,1,1]$, $\rho_2([1,1])=6[1,1,1,1]$, $\rho_1([3])=[3,1]$. Therefore, $\{\Delta_1\Delta_2\}_2=[2]\circ[2]=[1,1]$, $\{\Delta_1\Delta_2\}_3=[2,1]\circ[2,1]-3[1,1,1]=3[3]$ and $\{\Delta_1\Delta_2\}_4=[2,1,1]\circ[2,1,1]-3[3,1]-6[1,1,1,1]=2[2,2]$.]{} [\[t1\] The operation $(\Delta_1,\Delta_2)\mapsto\Delta_1\Delta_2$ gives rise on $A_{\infty}^{\circ}=\bigoplus\limits_{n}A_n^{\circ}$ to the structure of a commutative associative algebra.]{} Commutativity follows from the commutativity of the algebras $A_n^{\circ}$. Associativity follows from the associativity of the algebras $A_n^{\circ}$ and the equality $$\Delta_1\Delta_2\Delta_3=\sum\limits_ {n=\max\{\|\Delta_1\|,\|\Delta_2\|, \|\Delta_3\|\}} ^{\|\Delta_1\|+\|\Delta_2\|}\{\Delta_1\Delta_2\Delta_3\}_{n}$$ where $$\{\Delta_1\Delta_2\Delta_3\}_n=\left\{\begin{array}{lr} \rho_{(n-\|\Delta_1\|)}(\Delta_1) \circ \rho_{(n-\|\Delta_2\|)}(\Delta_2)\circ \rho_{(n-\|\Delta_3\|)} (\Delta_3)&\hbox{for }n=\max\{\|\Delta_1\|,\|\Delta_2\|,\|\Delta_3\|\}\\ \\ \rho_{(n-\|\Delta_1\|)}(\Delta_1) \circ \rho_{(n-\|\Delta_2\|)}(\Delta_2)\circ \rho_{(n-\|\Delta_3\|)}(\Delta_3)-\\ -\sum\limits_ {k=\max\{\|\Delta_1\|,\|\Delta_2\|,\|\Delta_3\|\}}^{n-1} \rho_{(n-k)}(\{\Delta_1\Delta_2\Delta_3\}_k)&\hbox{for }n>\max\{\|\Delta_1\|, \|\Delta_2\|,\|\Delta_3\|\}\end{array}\right.$$ [\[t2\] The algebras $A_{\infty}$ and $A_{\infty}^{\circ}$ are naturally isomorphic.]{} Product of the formal sums $G_{\infty}(\Delta_1)$ and $G_{\infty}(\Delta_2)$ is a finite sum of the formal sums of the type $G_{\infty}(\Delta)$, where $\max\{\|\Delta_1\|,\|\Delta_2\|\}\leq \|\Delta\| \leq \|\Delta_1\|+\|\Delta_2\|$. If $\|\Delta_1\|=\|\Delta_2\|=\|\Delta\|=n$, then $G_{\infty}(\Delta)=\sum_g C^{\Delta}_{\Delta_1,\Delta_2}gG_n(\Delta)g^{-1}$, where the sum goes over all finite permutations $g$ of the set $\mathbb{N}$, which do not preserve $\{1,2,\dots,n\}$. By the same reason, the formal sum $G_{\infty}(\Delta)$ for $\|\Delta_1\|+1=\|\Delta_2\|+1=\|\Delta\|\in\mathcal{A}_n$ is equal to $\sum_g C^{\Delta}_{\rho_1({\Delta}_1),\rho_1({\Delta}_2)}gG_n(\Delta)g^{-1}$ minus $\rho_1(G_{\infty}(\hat{\Delta}))$, where $\Delta=\rho_1(\hat{\Delta})$. Similar arguments prove the statement of the theorem for all $\Delta_1$, $\Delta_2$ of coinciding degrees. If, however, $\|\Delta_1\|<\|\Delta_2\|=\|\Delta\|$, then the term $G_{\infty}(\Delta)$, for the product of $\Delta_1$ and $\Delta_2$, coincides with the term $G_{\infty}(\Delta)$, for the product of $\tilde{\Delta}_1$ and $\Delta_2$, where $\tilde{\Delta}_1=\rho_{\|\Delta_2\|-\|\Delta_1\|}(\Delta_1)$. [Differential representation of the algebra $U(gl(\infty))$]{}\[s2\] ======================================================================== Consider the set of formal differential operators $$D_{ab}= \sum\limits_{e\in\{1,\dots,N\}}X_{ae}\frac{\partial}{\partial X_{be}}$$ of the Miwa variables $\{X_{ij}\|i,j\leq N\}$ Multiplication of operators is given by the rule $$D_{ab} D_{cd}=\sum\limits_{e_1,e_2\in\mathbb{N}} X_{ae_1}X_{ce_2}\frac{\partial}{\partial X_{be_1}}\frac{\partial}{\partial X_{de_2}}+\delta_{bc}\sum\limits_{e\in\mathbb{N}} X_{ae}\frac{\partial}{\partial X_{de}}$$ Commutation relations for the operators $D_{ab}$ coincide with the commutation relations for the generators of the matrix algebra. Hence, the operators $D_{ab}$ give rise to the algebra $U(N)$ naturally isomorphic to the universal enveloping algebra $U(gl(N))$. In the limit $N\rightarrow\infty$ there emerges the algebra $U_{\infty}$ of the formal differential operators which are finite or countable sums of operators of the form $$:D_{a_1b_1}\cdots D_{a_nb_n}:=\sum\limits_{e_1,...,e_n\in\mathbb{N}} X_{a_1e_1}\cdots X_{a_ne_n}\frac{\partial}{\partial X_{b_1e_1}}\cdots \frac{\partial}{\partial X_{b_ne_n}}$$ We call the number $\|\mathcal{U}\|=n$ degree of the operator $\mathcal{U}$. Linear combinations of the operators of the same degree are called homogeneous operators. Thus, the vector space $U_{\infty}$ is decomposed into the direct sum $U_{\infty}=\sum\limits_ {n\in\mathbb{N}}U_n$ of the subspaces of homogeneous operators of degree $n$. Consider the projection $pr_n: U_{\infty}\rightarrow U_n$, preserving the operators of degree $n$ and mapping to zero all other homogeneous operators. Introduce on $U_n$ a multiplication $"\circ"$ by the formula $\mathcal{U}_1\circ \mathcal{U}_2=pr_n(\mathcal{U}_1\ \mathcal{U}_2)$. This multiplication turns $U_n$ into an associative algebra of the differential operators $U_n^{\circ}$. Consider an embedding of the vector spaces $$\varrho_k:U_n\rightarrow U_{n+k}$$ where $$\varrho_k(:D_{a_1b_1}\cdots D_{a_nb_n}:)= \frac{1}{k!}\sum\limits_ {c_1,...,c_k\in\mathbb{N}}:D_{c_1c_1} \cdots D_{c_kc_k}D_{a_1b_1}\cdots D_{a_nb_n}:$$ The operators $\mathcal{U}$ and $\varrho_k(\mathcal{U})$ acts similarly on the monomials $X$ of degree $n+k$ of the Miwa variables $\{X_{i,j}\}$. One immediately checks the following claim: \[t3\] There is an equality $$\mathcal{U}_1\mathcal{U}_2=\sum\limits_ {n=\max\{\|\mathcal{U}_1\|, \|\mathcal{U}_2\|\}} ^{\|\mathcal{U}_1\|+\|\mathcal{U}_2\| } \{\mathcal{U}_1\mathcal{U}_2\}_{n}$$ where $$\{\mathcal{U}_1 \mathcal{U}_2\}_n = \left\{\begin{array}{lr} \varrho_{(n-\|\mathcal{U}_1\|)}(\mathcal{U}_1)\circ \varrho_{(n-\|\mathcal{U}_2\|)}(\mathcal{U}_2)&\hbox{for } n=\max\{\|\mathcal{U}_1\|,\|\mathcal{U}_2\|\}\\ \\ \varrho_{(n-\|\mathcal{U}_1\|)}(\mathcal{U}_1)\circ \varrho_{(n-\|\mathcal{U}_2\|)}(\mathcal{U}_2)-\sum\limits_ {k=\max\{\|\mathcal{U}_1\|,\|\mathcal{U}_2\|\}}^{n-1} \varrho_{(n-k)}(\{\mathcal{U}_1 \mathcal{U}_2\}_k)\\&\hbox{for } n>\max\{\|\mathcal{U}_1\|,\|\mathcal{U}_2\|\}\end{array}\right.$$ \[ex2.1\] Put $\mathcal{U}_1=\sum\limits_{e\in\mathbb{N}} X_{a_1e}\frac{\partial}{\partial X_{b_1e}}$, $\mathcal{U}_2=\sum\limits_{e\in\mathbb{N}} X_{a_2e}\frac{\partial}{\partial X_{b_2e}}$.\ Then $\mathcal{U}_1 \mathcal{U}_2= \delta_{b_1,a_2} \sum\limits_{e\in\mathbb{N}}X_{a_1e}\frac{\partial} {\partial X_{b_2e}}+ \sum\limits_{e_1,e_2\in\mathbb{N}} X_{a_1e_1}X_{a_2e_2}\frac{\partial}{\partial X_{b_1e_1}} \frac{\partial}{\partial X_{b_2e_2}}$. On the other hand, $\{\mathcal{U}_1 \mathcal{U}_2\}_1=\mathcal{U}_1\circ\mathcal{U}_2= \delta_{b_1,a_2}\sum\limits_{e\in\mathbb{N}}X_{a_1e}\frac{\partial}{\partial X_{b_2e}}$ and\ $\varrho_1(\{\mathcal{U}_1 \mathcal{U}_2\}_1)= \delta_{b_1,a_2} \sum\limits_{c\in\mathbb{N}} \sum\limits_{e,f\in\mathbb{N}}X_{cf}X_{a_1e} \frac{\partial}{\partial X_{cf}}\frac{\partial}{\partial X_{b_2e}}$. Besides, $\varrho_1(\mathcal{U}_1)=\sum\limits_{c_1\in\mathbb{N}} \sum\limits_{e_1,e\in\mathbb{N}} X_{c_1f_1}X_{a_1e_1}\frac{\partial}{\partial X_{c_1f_1}} \frac{\partial}{\partial X_{b_1e_1}}\ $ and\ $\varrho_1(\mathcal{U}_2)= \sum\limits_{c_2\in\mathbb{N}} \sum\limits_{e_2,e\in\mathbb{N}} X_{c_2f_2}X_{a_2e_2}\frac{\partial}{\partial X_{c_2f_2}} \frac{\partial}{\partial X_{b_2e_1}}$. Hence,\ $\varrho_1(\mathcal{U}_1) \circ\varrho_1(\mathcal{U}_2)= \sum\limits_{e_1,e_2\in\mathbb{N}} X_{a_1e_1}X_{a_2e_2}\frac{\partial}{\partial X_{b_1e_1}} \frac{\partial}{\partial X_{b_2e_2}}+\delta_{b_1,a_2}\sum\limits_{c\in\mathbb{N}} \sum\limits_{e,f\in\mathbb{N}}X_{cf}X_{a_1e} \frac{\partial}{\partial X_{cf}}\frac{\partial}{\partial X_{b_2e}}$. Thus, $\{\mathcal{U}_1 \mathcal{U}_2\}_2=\sum\limits_{e_1,e_2\in\mathbb{N}} X_{a_1e_1}X_{a_2e_2}\frac{\partial}{\partial X_{b_1e_1}} \frac{\partial}{\partial X_{b_2e_2}}$. \[ex2.2\] Put $\mathcal{U}_1=\sum\limits_{e_1^1\in\mathbb{N}} X_{a_1^1e_1^1}\frac{\partial}{\partial X_{b^1_1e_1^1}}$, $\mathcal{U}_2= \sum\limits_{e_2^1,e_2^2\in\mathbb{N}} X_{a_2^1e_2^1}X_{a_2^2e_2^2}\frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}}$. Then $\mathcal{U}_1 \mathcal{U}_2= \delta_{b_1^1,a_2^1}\sum\limits_{e_1^1,e_2^2 \in\mathbb{N}} X_{a_1^1e_1^1}X_{a_2^2e_2^2} \frac{\partial}{\partial X_{b_2^1e_1^1}}\frac{\partial}{\partial X_{b_2^2e_2^2}}+ \delta_{b_1^1,a_2^2}\sum\limits_{e_1^1,e_2^1\in\mathbb{N}} X_{a_1^1e_1^1}X_{a_2^1e_2^1} \frac{\partial}{\partial X_{b_2^2e_1^1}}\frac{\partial}{\partial X_{b_2^1e_2^1}}+$ $\sum\limits_{e_1^1,e_2^1,e_2^2\in\mathbb{N}} X_{a_1^1e_1^1}X_{a_2^1e_2^1}X_{a_2^2e_2^2} \frac{\partial}{\partial X_{b_1^1e_1^1}} \frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}}$. On the other hand, $\{\mathcal{U}_1 \mathcal{U}_2\}_2=\varrho_1(\mathcal{U}_1) \circ\mathcal{U}_2=$\ $(\sum\limits_{c_1^1\in\mathbb{N}} \sum\limits_{f_1^1,e_1^1\in\mathbb{N}} X_{c_1^1f_1^1}X_{a_1^1e_1^1}\frac{\partial}{\partial X_{c_1^1f_1^1}} \frac{\partial}{\partial X_{b_1^1e_1^1}})\circ (\sum\limits_{e_2^1,e_2^2\in\mathbb{N}} X_{a_2^1e_2^1}X_{a_2^2e_2^2}\frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}})=$\ $\delta_{b_1^1,a_2^1}\sum\limits_{e_2^1,e_2^2\in\mathbb{N}} X_{a_1^1e_2^1}X_{a_2^2e_2^2}\frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}}+ \delta_{b_1^1,a_2^2}\sum\limits_{e_2^1,e_2^2\in\mathbb{N}} X_{a_1^1e_2^2}X_{a_2^1e_2^1}\frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}}$ and\ $\varrho_1(\{\mathcal{U}_1 \mathcal{U}_2\}_2)= \sum\limits_{c_1^1\in\mathbb{N}}\sum\limits_{f_1^1,e_1^1\in\mathbb{N}}$ $\delta_{b_1^1,a_2^1}\sum\limits_{e_2^1,e_2^2\in\mathbb{N}} X_{c_1^1f_1^1}X_{a_1^1e_2^1}X_{a_2^2e_2^2} \frac{\partial}{\partial X_{c_1^1f_1^1}} \frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}}+$ $\delta_{b_1^1,a_2^2}\sum\limits_{c_1^1\in\mathbb{N}} \sum\limits_{e_2^1,e_2^2\in\mathbb{N}} X_{c_1^1f_1^1}X_{a_1^1e_2^2}X_{a_2^1e_2^1} \frac{\partial}{\partial X_{c_1^1f_1^1}} \frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}}$. Besides,\ $\varrho_2(\mathcal{U}_1)\circ\varrho_1(\mathcal{U}_2)=$ $\sum\limits_{e_1^1,e_2^1,e_2^2\in\mathbb{N}} X_{a_1^1e_1^1}X_{a_2^1e_2^1}X_{a_2^2e_2^2} \frac{\partial}{\partial X_{b_1^1e_1^1}} \frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}}+\\$ $\sum\limits_{c_1^1\in\mathbb{N}}\sum\limits_{f_1^1,e_1^1\in\mathbb{N}} \delta_{b_1^1,a_2^1}\sum\limits_{e_2^1,e_2^2\in\mathbb{N}} X_{c_1^1f_1^1}X_{a_1^1e_2^1}X_{a_2^2e_2^2} \frac{\partial}{\partial X_{c_1^1f_1^1}} \frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}}+\\$ $\delta_{b_1^1,a_2^2}\sum\limits_{c_1^1\in\mathbb{N}} \sum\limits_{e_2^1,e_2^2\in\mathbb{N}} X_{c_1^1f_1^1}X_{a_1^1e_2^2}X_{a_2^1e_2^1} \frac{\partial}{\partial X_{c_1^1f_1^1}} \frac{\partial}{\partial X_{b_2^1e_2^1}} \frac{\partial}{\partial X_{b_2^2e_2^2}}$. Thus, $\mathcal{U}_1 \mathcal{U}_2=\{\mathcal{U}_1 \mathcal{U}_2\}_1+ \{\mathcal{U}_1 \mathcal{U}_2\}_2$. [Algebra $\mathcal{W}_{\infty}$ of the differential operators]{}\[s3\] ========================================================================== Associate with the Young diagram $\Delta=[\mu_1,\mu_2,\dots,\mu_l]$ with the ordered row lengths $\mu_1\geq\mu_2\geq\dots\geq\mu_l$ the numbers $m_k=m_k(\Delta)= \|\{i\|\mu_i=k\}\|$ and $\kappa(\Delta)=(\prod\limits_{k}m_k!k^{m_k})^{-1}$. Associate with the Young diagram $\Delta$ the operator $\mathcal{W}(\Delta)= \kappa(\Delta)\prod\limits_{k} :D_k^{m_k}:\ \in U_{\infty}$. \[ex3.1\] [$\mathcal{W}([1])= \sum\limits_{a\in\mathbb{N}} :D_{aa}:$ $\mathcal{W}([2])=\frac{1}{2}\sum\limits_{a,b\in\mathbb{N}} :D_{ab}D_{ba}:$]{} Denote through $\mathcal{W}^{\circ}_n$ the vector space generated by the operators of the form $\mathcal{W}(\Delta)$, where $\|\Delta\|=n$. \[l3.1\] [The operation $"\circ"$ provides the structure of algebra on $\mathcal{W}^{\circ}_n$. The correspondence $\Delta\mapsto\mathcal{W}(\Delta)$ gives rise to the isomorphism of algebras $\psi_n: A_n^{\circ}\rightarrow \mathcal{W}_n^{\circ}$.]{} Associate with each permutation $g\in S_n$ the operator $\mathcal{W}(g)=\kappa(\Delta(g))\sum\limits_{a_1,...,a_n\in\mathbb{N}} :D_{a_1a_{g(1)}}\cdots D_{a_na_{g(n)}}:$ Then $\mathcal{W}(\Delta(g))= \mathcal{W}(g)$. Hence, the claim of the lemma follows from the equality $\mathcal{W}(\Delta(g_1)\circ\Delta(g_2))= \mathcal{W}(g_1)\circ \mathcal{W}(g_2)$ для $g_1,g_2\in S_n$. \[l3.2\] [The embedding $\varrho_k:U_n\rightarrow U_{n+k}$ gives rise to the embedding $\varrho_k:\mathcal{W}^{\circ}_n\rightarrow \mathcal{W}^{\circ}_{n+k}$, where $\varrho_k\psi_n(\Delta)=\psi_{n+k}\rho_k(\Delta)$ at $\Delta\in A_n$.]{} The map $\varrho_k:U_n\rightarrow U_{n+k}$ gives rise to the correspondence $\varrho_k(\mathcal{W}(\Delta))=\frac{1}{k!}\mathcal{W} ({\Delta}^k)$. In accordance with our definitions, $\psi_n(\Delta)= \kappa(\Delta) \sum\limits_{a_1,...,a_n\in\mathbb{N}} :D_{a_1a_{g(1)}}\cdots D_{a_na_{g(n)}}:$ and $\varrho_k\psi_n(\Delta)= \frac{1}{k!}\kappa(\Delta) \sum\limits_{c_1,...,c_k,a_1,...,a_n\in\mathbb{N}} :D_{c_1c_1}\cdots D_{c_kc_k} D_{a_1a_{g(1)}}\cdots D_{a_na_{g(n)}}:$ On the other hand, $\rho_k(\Delta)=\frac{(m_1+k)!}{m_1!k!}{\Delta}^k$ and\ $\psi_{n+k}\rho_k(\Delta)= \frac{(m_1+k)!}{m_1!k!} \kappa(\Delta) (\frac{(m_1+k)!}{m_1!})^{-1}\sum\limits_{c_1,...,c_k,a_1,...,a_n\in\mathbb{N}} :D_{c_1c_1}\cdots D_{c_kc_k} D_{a_1a_{g(1)}}\cdots D_{a_na_{g(n)}}:= \\ \frac{1}{k!}(\prod\limits_{k}m_k!k^{m_k})^{-1} \sum\limits_{c_1,...,c_k,a_1,...,a_n\in\mathbb{N}} :D_{c_1c_1}\cdots D_{c_kc_k} D_{a_1a_{g(1)}}\cdots D_{a_na_{g(n)}}:$ \[ex3.2\] Put $\Delta=[1]$. Then $\rho_k(\Delta)=(k+1)[1,1,\dots,1]$ and\ $\psi_{k+1}(\rho_k(\Delta))= \psi_{k+1}((k+1)[1,1,\dots,1])=\frac{(k+1)}{(k+1)!} \sum\limits_{c_1,...,c_{k+1} \in\mathbb{N}} :D_{c_1c_1}\cdots D_{c_nc_n}:$ On the other hand, $\psi_{1}([1])=\sum\limits_{e\in\mathbb{N}} :D_{cc}:$ and\ $\varrho_{k}(\sum\limits_{e\in\mathbb{N}} :D_{cc}:)=\frac{1}{k!} \sum\limits_{e,e_1,...,e_k\in\mathbb{N}} \sum\limits_{e\in\mathbb{N}} :D_{cc}D_{c_1c_1}\cdots D_{c_nc_n}:$ \[ex3.3\] Put $\Delta=[2]$. $\psi_{2}([2])= \frac{1}{2}\sum\limits_{a,b\in\mathbb{N}} :D_{ab}D_{ba}:$ and\ $\varrho_{k}(\frac{1}{2}\sum\limits_{a,b\in\mathbb{N}} :D_{ab}D_{ba}:)=$ $\frac{1}{2k!}\sum\limits_{a,b,e_1,...,e_k\in\mathbb{N}} :D_{ab}D_{ba}D_{c_1c_1} \cdots D_{c_nc_n}:$ On the other hand, $\rho_k(\Delta)= [2,1,\dots,1]$ and\ $\psi_{k+2}(\rho_k(\Delta))=\psi_{k+2}([2,1,\dots,1])= \frac{1}{2k!}\sum\limits_{a,b,e_1,...,e_k\in\mathbb{N}} :D_{ab}D_{ba}D_{c_1c_1} \cdots D_{c_nc_n}:$ Denote through $\mathcal{W}_{\infty}\subset U_{\infty}$ the subalgebra generated by the differential operators $\mathcal{W}(\Delta)$. Confronting Theorems \[t1\], \[t2\], \[t3\] with Lemmas \[l3.1\], \[l3.2\], one obtains \[t4\] [The isomorphisms $\psi_n$ gives rise to the isomorphism of the algebras $\psi: A_{\infty}\rightarrow W_{\infty}$.]{} [Schur functions\[s4\]]{} ============================= The Schur function of $n$ variables corresponds to the Young diagram $R=\{R_1\geq R_2\geq \dots\geq R_m>0\}$, where $n\geq m$. It is defined by the formula $$s_{R}(x_1,\dots,x_n)=\frac{\det[x_i^{R_j+n-j}]_{1\leq i, j \leq n}}{\det[x_i^{n-j}]_{1\leq i, j\leq n}},$$ where $R_i=0$ at $m<i\leq n$. The property of stability $s_{R}(x_1,\dots,x_n,0)=s_{R}(x_1,\dots,x_n)$ allows one to define $s_{R}$ on an arbitrary finite set of variables. Define the Schur functions on finite matrices $X\in gl(n)\subset gl(\infty)$ by the formula $s_{R}(X)= s_{R}(x_1,\dots,x_n)$, where $x_1,\dots,x_n$ are the eigenvalues of the matrix $X$. The functions $s_{R}(X)$ form a basis in the vector space $\textsf{P}$ of all symmetric polynomials of the Miwa variables. Polynomials of the variables $p_i={\mathop{\sf tr}\nolimits}X^i$ form another natural basis of the space $\textsf{P}$, $$\tilde s_R(p)=\det [P_{R_i+j-i}(p)]_{1\leq i, j \leq n}\,,\ \ \ \ \ \ \ \exp\big(\sum_k p_kx^k\big)\equiv\sum_iP_i(p)x^i$$ where $p=(p_1,p_2\dots)$. Then $s_{R}(X)=\tilde s_R(p)$ [@M]. Associate with the Young diagram $\Delta$ the monomial $p(\Delta)= \kappa(\Delta)p_1^{m_1(\Delta)} p_2^{m_2(\Delta)}\dots p_n^{m_n(\Delta)}$. Put $\Delta^k=[\Delta,\underbrace{1,...,1}_k]$. Let $\dim R$ be the dimension of representation of the symmetric group $S_{\|R\|}$ corresponding to the diagram $R$, and $\chi_R(X_{\Delta})$ be the value of character of this representation on the element of the cyclic type $\Delta^{\|R\|-\|\Delta\|}$. Put $d_R=\frac{\dim R}{\|R\|!}=\frac{ \prod\limits_{i < j = 1}^{\|R\|} \left( \mu_i -\mu_j - i + j \right) } {\prod\limits_{i = 1}^{\|R\|} \left(\mu_i +\|R\| - i\right)!}$ and $m_1=m_1(\Delta)$. Denote through $\varphi_R(\Delta)$ the function which is equal to $\frac{\kappa(\Delta)} {d_R m_1!(\|R\|-\|\Delta\|-m_1)!}\chi_R(X_{\Delta})$ at $\|R\|-\|\Delta\|\geq m_1$ and 0 otherwise. Then $\varphi_R(\Delta^k)= \frac{m_1!k!}{(m_1+k)!}\varphi_R(\Delta)$ at $k=\|R\|-\|\Delta\|$. \[t5\] [The functions $s_{R}(X)$ are the eigenfunctions of the operators $\mathcal{W}(\Delta)$. They form a complete system of eigenfunctions for the restrictions $\textsf{W}(\Delta)=\mathcal{W}(\Delta)\|_{\textsf{P}}$ and $\textsf{W}(\Delta)(s_R)=\varphi_R(\Delta)s_R$.]{} Consider the regular representation of the algebra $U(gl(N))$ in the algebra of polynomial functions of matrix elements of $gl(N)$. The center $Z(U(gl(N)))$ preserves the vector subspace $\textsf{P}$. The algebra $Z(U(gl(N)))$ is additively generated by the operators $T(\Delta)$ associated with the Young diagrams $\Delta$, and $T(\Delta)(f)= \mathcal{W}(\Delta)(f)$ for $f\in \textsf{P}$. Besides, in accordance with the Weyl theorem [@Zh], the Schur functions $s_R(X)$ form a complete system of the eigenfunctions of the operators $T(\Delta)$. Taking the limit $N\rightarrow\infty$, one finds that the Schur functions form a complete system of the eigenfunctions of the operators $\textsf{W}(\Delta)$. Now we find the eigenvalues of the operators. In accordance with [@M s.1.7], $p(\Delta)=\sum\limits_{R:\|R\|= \|\Delta\|}d_R\varphi_R(\Delta)\tilde s_R$. Hence, $$p(\Delta)e^{p_1}= \sum\limits_{k=0}^{\infty}p(\Delta)\frac{p_1^k}{k!} \sum\limits_{k=0}^{\infty} \frac{(m_1+k)!}{m_1!k!} p(\Delta^k)=\sum\limits_{k=0}^{\infty}\sum\limits_{\|R\|= \|\Delta\|+k}\frac{(m_1+k)!}{m_1!k!}d_R\varphi(\Delta^k)\tilde s_R=$$ $$= \sum\limits_{k=0}^{\infty}\sum\limits_{\|R\|=\|\Delta\|+k} d_R\varphi(\Delta)\tilde s_R=\sum\limits_R d_R\varphi(\Delta)\tilde s_R$$ On the other hand, in accordance with [@M s.1.4, example 3], $e^{p_1} = \sum\limits_R d_R\tilde s_R(p)$, hence, $p(\Delta)e^{p_1}= \textsf{W}(\Delta)(e^{p_1})=\sum\limits_R d_R\textsf{W}(\tilde s_R)$. Thus, $\sum\limits_R d_R\textsf{W}(s_R)=\sum\limits_R d_R\varphi(\Delta)s_R$. We have already proved that $s_R$ form a complete system of the eigenfunctions of the operator $\mathcal{W}$. Therefore, the last equality implies $\textsf{W}(\Delta)(s_R)= \varphi_R(\Delta)s_R$. [The values of $\varphi_R(\Delta)$ are related by the formula $$\varphi_R(\Delta_1)\varphi_R(\Delta_2)= \sum\limits_{\Delta} C^{\Delta}_{\Delta_1\Delta_2}\varphi_R(\Delta)$$ where $C^{\Delta}_{\Delta_1\Delta_2}$ are the structure constants of the algebra $A_{\infty}$, which are obtained in s.\[s1\].]{} [First few $\textsf{W}$-operators\[s5\]]{} ============================================== Represent now the operators $\textsf{W}(\Delta)$ as differential operators of the variables $\{p_k\}$. Then, $$D_{ab} F(p) = X_{ac}\frac{\partial}{\partial X_{bc}} F(p) = \sum_{k=1}^\infty k(X^k)_{ab} \frac{\partial F(p)}{\partial p_k}$$ Using the relation $$D_{a'b'}(X^k)_{ab} = X_{a'c'}\frac{\partial}{\partial X_{b'c'}}(X^k)_{ab} = \sum_{j=0}^{k-1} X_{a'c'} (X^j)_{ab'}(X^{k-j-1})_{c'b} = \sum_{j=0}^{k-1} (X^j)_{ab'} (X^{k-j})_{a'b},$$ one obtains $$D_{a'b'} D_{ab} F(p) = \sum_{k,l=1}^\infty kl(X^l)_{a'b'}(X^k)_{ab}\frac{\partial^2F(p)}{\partial p_k p_l} + \sum_{k=1}^\infty\sum_{j=0}^{k-1}k (X^j)_{ab'}(X^{k-j})_{a'b} \frac{\partial F(p)}{\partial p_k}$$ Thus, $$: D_{a'b'}D_{ab}:\, F(p) = \sum_k\left(k\sum_{j=1}^{k-1} (X^j)_{ab'} (X^{k-j})_{a'b}\right) {\partial F(p)\over\partial p_k}+ \sum_{k,l} kl(X^k)_{ab}(X^l)_{a'b'}{\partial ^2 F(p)\over\partial p_k\partial p_l}.$$ This relation allows one to find all the operators $\textsf{W}$. In particular, $$\textsf{W}([1]) = {\mathop{\sf tr}\nolimits}\hat D= \sum_{k=1} kp_k\frac{\partial}{\partial p_k}$$ $$\textsf{W}([2]) ={1\over 2} \, :D^2\,:\ = \frac{1}{2}\sum_{a,b=1}^\infty \left( (a+b)p_ap_b\frac{\partial}{\partial p_{a+b}} + abp_{a+b}\frac{\partial^2}{\partial p_a\partial p_b}\right)$$ $$\textsf{W}([1,1]) = \frac{1}{2!}\, :({\mathop{\sf tr}\nolimits}D)^2\,:\ = \frac{1}{2}\left(\sum_{a=1}^\infty a(a-1)p_a\frac{\partial}{\partial p_{a}} +\sum_{a,b=1}^\infty abp_{a}p_b\frac{\partial^2}{\partial p_a\partial p_b}\right)$$ $$\textsf{W}([3]) = \frac{1}{3}\, :{\mathop{\sf tr}\nolimits}D^3\,:\ = \frac{1}{3}\sum_{a,b,c\geq 1}^\infty abcp_{a+b+c} \frac{\partial^3}{\partial p_a\partial p_b\partial p_c} + \frac{1}{2}\sum_{a+b=c+d} cd\left(1-\delta_{ac}\delta_{bd}\right) p_ap_b\frac{\partial^2}{\partial p_c\partial p_d} +$$ $$+ \frac{1}{3} \sum_{a,b,c\geq 1} (a+b+c)\left(p_ap_bp_c + p_{a+b+c}\right)\frac{\partial}{\partial p_{a+b+c}}$$ $$\textsf{W}([2,1]) = \frac{1}{2}\, :{\mathop{\sf tr}\nolimits}D^2\,{\mathop{\sf tr}\nolimits}D \,:\ = {1\over 2}\sum_{a,b\ge 1}(a+b)(a+b-2)p_ap_{b}{\partial\over\partial p_{a+b}}\,+ {1\over 2}\sum_{a,b\ge 1}ab(a+b-2)p_{a+b}{\partial ^2\over\partial p_a\partial p_b}\, +$$ $$+\frac{1}{2}\sum_{a,b,c\ge 1} (a+b)cp_ap_bp_c\frac{\partial^2}{\partial p_{a+b} \partial p_c} +{1\over 2}\sum_{a,b,c\ge 1}abcp_ap_{b+c}{\partial ^3\over\partial p_a\partial p_ b\partial p_c}$$ $$\textsf{W}([1,1,1]) = \frac{1}{3!}\, :({\mathop{\sf tr}\nolimits}D)^3\,:\ = {1\over 6}\sum_{a\ge 1} a(a-1)(a-2)p_a{\partial\over\partial p_a}\,+$$ $$+ {1\over 4}\sum_{a,b} ab(a+b-2)p_ap_b\frac{\partial^2}{\partial p_a\partial p_b}\, +{1\over 6}\sum_{a,b,c\ge 1}abcp_ap_bp_c{\partial ^3\over\partial p_a\partial p_b \partial p_c}$$ [Hurwitz numbers\[s7\]]{} ============================= Each holomorphic morphism of degree $n$ of Riemann surfaces $f:\widetilde{\Omega}\rightarrow\Omega$ associates with the point $s\in\Omega$ a local invariant: the Young diagram $\Delta(f,s)$ of degree $n$ with the row lengths being equal to degrees of the map $f$ at the points of complete pre-image $f^{-1}(s)= \{s^1,\dots,s^k\}$. More than 100 years ago Hurwitz [@H] formulated a problem of calculating [the Hurwitz numbers]{} $$H((s_1,\Delta_1),\dots,(s_k,\Delta_k)\|\Omega)=\sum_{f\in Cov_n(\Omega, \{\alpha_1, \dots,\alpha_s\})} \frac{1}{\|{\mathop{\sf Aut}\nolimits}(f)\|}$$ for an arbitrary set $\{\Delta_1,\dots,\Delta_k\}$ of Young diagrams of degree $n$. Here $\|{\mathop{\sf Aut}\nolimits}(f)\|$ is the order of automorphism group of the map $f$, and $Cov_n(\Omega, \{\alpha_1,\dots,\alpha_s\})$ is a set of classes of the biholomorphic equivalence of the holomorphic morphisms $f:\widetilde{\Omega}\rightarrow\Omega$ with the set of critical values $s_1,\dots,s_k\in\Omega$ and the local invariants $\alpha(f,s_i)=\alpha_i$. This number depends only on the genus $g(\Omega)$ of the surface $\Omega$ and the diagrams $\Delta_1,\dots,\Delta_k$. We define $<\Delta_1,\dots,\Delta_k>_{g(\Omega)}\ = H((s_1,\Delta_1),\dots,(s_k,\Delta_k)\|\Omega)$. The Hurwitz numbers of any genus are easily expressed through those at genus zero, $<\Delta_1,\dots,\Delta_k>\ =\ <\Delta_1,\dots,\Delta_k>_0$, [@AN]. A defining property of the Hurwitz numbers is the associativity relation $$<\Delta_1,\dots,\Delta_k>\ = \sum\limits_{\Upsilon\in\mathcal{A}_n} <\Delta_1,\dots,\Delta_r,\Upsilon>\|{\mathop{\sf Aut}\nolimits}(\Upsilon)\| <\Upsilon,\Delta_{r+1},\dots,\Delta_k>$$ The Hurwitz numbers of coverings with three critical values are related to the structure constants of the algebra $A_n$ by the formula $<\Delta_1,\Delta_2,\Delta_3>\ =C_{\Delta_1,\Delta_2}^{\Delta_3} \|{\mathop{\sf Aut}\nolimits}(\Delta_3)\|^{-1}$. Arbitrary Hurwitz numbers are expressed through these simplest Hurwitz numbers by the formula $$<\Delta_1,\dots,\Delta_k>\ = \sum\limits_{\Upsilon_1,\dots,\Upsilon_{k-1}\in\mathcal{A}_n} <\Delta_1,\Delta_2,\Upsilon_1>\|{\mathop{\sf Aut}\nolimits}(\Upsilon_1)\| <\Upsilon_1,\Delta_3,\Upsilon_2>\times$$ $$\times\|{\mathop{\sf Aut}\nolimits}(\Upsilon_2)\|\dots\|{\mathop{\sf Aut}\nolimits}(\Upsilon_{k-1})\|<\Upsilon_{k-1}, \Delta_{k-1},\Delta_k>,$$ (see, e.g., [@AN]). The Hurwitz numbers appear in different framewroks: strings and QCD [@GT], mirror symmetry [@D], theory of singularities [@A], matrix models [@KSW; @MM], integrable systems [@OP; @GKM2; @O], Yang-Mills theory [@CMR; @GKM2] and the theory of moduli of curves [@KL; @K; @MM] and other branches of string theory. Associate with Young diagrams $\Delta_1,\dots,\Delta_k$ and $\Delta$, where $\|\Delta_i\|\le \|\Delta\|$ for all $i$, the numbers $<(\Delta_1,n_1),\dots,(\Delta_k,n_k)\|\Delta>$ equal to the Hurwitz numbers $<\tilde{\Delta}_1,\dots,\tilde{\Delta}_1,\tilde{\Delta}_2, \dots,\tilde{\Delta}_2,\dots,\tilde{\Delta}_k,\dots,\tilde{\Delta}_k,\Delta>$, where the Young diagram $\tilde{\Delta}_i=\rho_{\|\Delta\|-\|\Delta_i\|}(\Delta_i)$ is met exactly $n_i$ times. We also put $<(\Delta_1,n_1),\dots,(\Delta_k,n_k)\|\Delta>\ =0$, if $\|\Delta_i\|>\|\Delta\|$ at least for one $i$. Associate a variable $\beta_{\Delta}$ with each Young diagram $\Delta$ and consider the generating function for the Hurwitz numbers $$\mathcal{Z}= \sum\limits_{k=1}^{\infty} \sum\limits_{\Delta,\Delta_1,\dots,\Delta_k\in\mathcal{A}_{\infty}} \sum\limits_{n_1,\dots,n_k\in\mathbb{N}} \frac{\beta_{\Delta_1}^{n_1}\dots\beta_{\Delta_n}^{n_k}} {n_1!\dots n_k!}<\Delta_1^{n_1},\dots,\Delta_k^{n_k}\|\Delta> p(\Delta).$$ \[t7\] [For any Young diagram $\Upsilon$ there is an equality $$\frac{\partial\mathcal{Z}}{\partial\beta_{\Upsilon}}= \textsf{W}(\Upsilon)\mathcal{Z}$$]{} The claim of the theorem implies a system of relations between the numbers $<\Delta_1^{n_1},\dots,\Delta_k^{n_k}\|\Delta>$. In accordance with Theorems \[t2\] and \[t4\], these relations are of the form $<\Delta_1^{n_1},\dots,\Delta_i^{n_i},\dots,\Delta_k^{n_k}\| \Delta>\ = \ <\Delta_1^{n_1},\dots,\Delta_i^{n_i-1},\dots,\Delta_k^{n_k}\| \Delta\circ\tilde{\Delta}_i>$ and follow from the associativity relation. 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--- abstract: 'We experimentally demonstrate a quantum walk on a line in phase space using one and two trapped ions. A walk with up to 23 steps is realized by subjecting an ion to state-dependent displacement operations interleaved with quantum coin tossing operations. To analyze the ion’s motional state after each step we apply a technique that directly maps the probability density distribution onto the ion’s internal state. The measured probability distributions and the position’s second moment clearly show the non-classical character of the quantum walk. To further highlight the difference between the classical (random) and the quantum walk, we demonstrate the reversibility of the latter. Finally, we extend the quantum walk by using two ions, giving the walker the additional possibility to stay instead of taking a step.' author: - 'F. Zähringer$^{1,2}$' - 'G. Kirchmair$^{1,2}$' - 'R. Gerritsma$^{1,2}$' - 'E. Solano$^{3,4}$' - 'R. Blatt$^{1,2}$' - 'C. F. Roos$^{1,2}$' title: Realization of a quantum walk with one and two trapped ions --- The Galton board [@Galton:1889] is a mechanical device in which a falling ball encounters a triangular lattice of pins stuck in a board that repeatedly scatter the ball to the left or right in a random way. Originally conceived for illustrating the emergence of normal probability distributions, it can also be considered as an apparatus for carrying out a random walk on a line [@Pearson:1905; @Rayleigh:1905], a notion that had not been introduced into the scientific literature at that time. Since then, random walks have become an ubiquitous concept in physics and computer science. The quantum walk [@Aharonov:1993; @Kempe:2003] is the quantum analogue of a random walk. In its discrete one-dimensional version, a spin-$\frac{1}{2}$ quantum particle initially described by a wave packet centered at position $x_0$ undergoes a one-dimensional motion governed by the particle’s internal state. The particle is state-dependently displaced by a step of length $d$ by the action of the unitary operator $U_d=\exp(-\frac{i}{\hbar}\sigma_j \hat{p}d)$ where $\sigma_j$ is a spin projection operator and $\hat{p}$ the momentum operator (see Fig. 1). This operation is followed by another unitary operation $U_i=\exp(-i\frac{\pi}{4}\sigma_k)$ with $Tr(\sigma_j\sigma_k)=0$ scrambling the particle’s internal state. After $N$ iterations of this elementary step, the particle’s initial wave function $\|\Psi_0\rangle$ has evolved into $$\|\Psi_N\rangle=\left(U_iU_d\right)^N\,\|\Psi_0\rangle=\left(e^{-i\frac{\pi}{4}\sigma_k}e^{-\frac{i}{\hbar}\sigma_j \hat{p} d}\right)^N\,\|\Psi_0\rangle. \label{eq:QuantumWalk}$$ After $N$ steps, for a wave packet initially localized at $x_0=0$, the wave packet is spread out over a distance $2Nd$. Moreover, due to quantum interference of different paths, the spatial probability distribution strongly differs from the classical case. While for the classical random walk a binomial probability distribution results with $\langle x^2 \rangle\propto N$, the distribution for the quantum walk is peaked towards the outer edge and has a second moment growing like $\langle \hat{x}^2 \rangle\propto N^2$. ![\[fig\_Method\] Quantum walk in phase space. In each step of the walk, a state-dependent displacement operation splits the wave function in phase space into two parts followed by a coin tossing operation that coherently scrambles the internal state of the ion. These operations are repeated $N$ times. To measure marginal distributions in phase space, a probe pulse is applied that state-dependently displaces the wave function in phase space in a direction orthogonal to the one to be measured.](figure1.eps){width="5cm"} There have been a number of proposals discussing experimental realizations of one-dimensional quantum walks in systems like atoms in optical lattices [@Dur:2002a], trapped ions [@Travaglione:2002], or cavity QED [@Sanders:2003]. Recently, experimental realizations with atoms in an optical lattice [@Karski:2009], a trapped ion [@Schmitz:2009a], and photons [@Schreiber:2009] have been reported. For the case of trapped ions, different techniques for analyzing the quantum walk have been discussed [@Travaglione:2002; @Xue:2009] and a proof-of-principle experimental realization was reported recently [@Schmitz:2009a] for a limited number of steps. In this paper, we demonstrate a discrete quantum walk with up to 23 steps using a single trapped ion and analyze it by a measurement technique that directly reconstructs the ion’s probability density along a line in phase space. In the experiment, a single $^{40}$Ca$^{+}$ ion is suspended in a linear Paul trap [@Kirchmair:2009] with radial and axial trap frequencies of $\omega_{\rm r} \approx$ ($2\pi$) 3 MHz and $\omega_{\rm ax}$ = ($2\pi$) 1.356 MHz, respectively. Doppler cooling, resolved sideband cooling of the axial mode and optical pumping prepare the ion in the ground state of motion and the internal state $\lvert S_{ 1/2}, m = 1/2\rangle\equiv \lvert -\rangle_z$ [^1]. A narrow linewidth laser at 729 nm coherently couples the states $\|-\rangle_z$ and $\lvert D_{5/2}, m = 3/2\rangle\equiv \lvert +\rangle_z$. State detection is done via fluorescence detection on the $S_{1/2}\leftrightarrow P_{1/2}$ transition [@Kirchmair:2009]. A general state-dependent displacement Hamiltonian is implemented using a bichromatic light field at 729 nm that is resonant with both the blue and red axial sideband of the $\lvert -\rangle_z\leftrightarrow\lvert +\rangle_z$ transition. In the Lamb-Dicke regime, the resulting Hamiltonian, which is the sum of a Jaynes-Cummings and an anti-Jaynes-Cummings Hamiltonian, is given by $$\begin{aligned} \label{H_displace} H_{D} =& \hbar\eta\Omega\left(\left(\sigma_{x}\cos{\phi_{+}} - \sigma_{y}\sin{\phi_{+}} \right)\right.\notag\\ &\ \otimes \left.\left((a + a^{\dagger})\cos{\phi_{-}} + i(a^{\dagger} - a)\sin{\phi_{-}}\right)\right). %&\ \otimes \left(\hat{x}\cos{\phi_{\rm -}} + \hat{p}\sin{\phi_{\rm -}}\right))\end{aligned}$$ Here, $\eta = 0.06$ is the Lamb-Dicke parameter, $\Omega$ the Rabi frequency and $2\phi_{+}=\phi_{\rm r}+\phi_{\rm b}$ and $2\phi_{ -}=\phi_{\rm b}-\phi_{\rm r}$ are the sum and the difference, of the phases of the light fields tuned to the red and blue sideband. To perform a symmetric quantum walk the ion is prepared in the state $\|+\rangle_y= (\lvert +\rangle_z + i\lvert -\rangle_z)/\sqrt{2}$ by a $\pi/2$-pulse on the carrier transition. Applying the bichromatic light field with $\phi_{-} = \pi/2$ and $\phi_{+} = 0$ realizes the Hamiltonian $H_{d} = 2 \eta \Omega \Delta_x \sigma_{x} \hat{p}$ with the momentum operator $\hat{p}=\frac{a^\dagger - a}{2}\frac{i\hbar}{\Delta_x}$ and $\Delta_x = \sqrt{\frac{\hbar}{2 m \omega_{\rm ax}}}$. Application of this Hamiltonian for a duration $\tau$ generates the propagator $U_d$ with step size $d=2\eta\Omega\tau\Delta_x$. ![\[fig\_Walk\] (a) Measurement of Fourier components $\left<\cos(kx)\right>$ and $\left<\sin(kx)\right>$ for a seven-step quantum walk. The data are obtained by varying the duration of the probe pulse for the ion prepared in the internal state $\|+\rangle_z$ (left) or $\|+\rangle_y$ (right) after completing the walk. The probability distribution is obtained by Fourier transforming a fit to the data (solid line). (b) Reconstruction of the symmetric part of the probability distribution $\langle\delta(\hat{x}-x)\rangle$ for up to 13 steps in the quantum walk. The blue dashed curve is a numerical calculation for the expected distribution within the Lamb-Dicke regime. The blue solid curve takes into account corrections to the Lamb-Dicke regime. In step 7, the dotted curve represents the full reconstruction using also the $\left<\sin(kx)\right>$ shown in (a).(c) Probability distribution of a five-step quantum walk after application of five additional steps which invert the walk and bring it back to the ground state.](figure2.eps){width="8.5cm"} Under the action of $H_{d}$, the ion’s wave packet coherently splits in phase space along the $x$-axis. The two emerging wave packets $\psi_1^{(m)}$, $\psi_2^{(m)}$ are associated with the internal states $\|\pm\rangle_x$. The length and the intensity of the pulse determine the width of the splitting. In our experiments we use a pulse of 40 $\mu$s with a Rabi frequency of $\Omega= (2\pi)\,68$ kHz to achieve a step size of $d = 2\Delta_x$. This step size makes the two resulting motional wave packets nearly orthogonal, $\|\langle\psi_1^{(m)}\|\psi_2^{(m)}\rangle\|^2\approx 0.02$, but still allows for a large number of steps in phase space. Next, we perform a $\pi/2$-pulse acting on the carrier transition as a symmetric coin flip. This pulse creates an equal superposition of $\sigma_x$ eigenstates for both wave packets. These two pulses are repeated according to the number of steps to be carried out. ![\[fig\_Sigma\](Color online) (a) Width $w_x$ of the probability distribution in units of ground state size $\Delta_x$ as a function of the number of steps for a quantum (${\blacksquare}$) walk. The solid curve represents a full numerical simulation of the quantum walk as realized in the experiment. The width of the $x$-distribution for a classical random walk ($\textcolor[rgb]{1,0,0.00}{\bullet}$) increases more slowly and is described (solid red line) by eq. (\[eqCl\_sigma\]). The data points (${\color{blue} \blacklozenge}$) show the measured width $w_p$ of the marginal distribution along the p-direction with $\Delta_p=\hbar/2\Delta_x$. (b) Average number of vibrational quanta after $N$ steps in the quantum walk measured by driving oscillations on the carrier transition. The solid line is based on a full simulation, the dashed line assumes the validity of the Lamb-Dicke approximation.](figure3.eps){width="7cm"} To measure the probability distribution along a line in phase space, we create two displaced copies of the state that are subsequently interfered. For this, we use of another state-dependent displacement operation $U_p=\exp(-ik\hat{x}\sigma_x/2)$ [@Wallentowitz:1995; @Gerritsma:2010]. A measurement of $\sigma_z$ following the application of $U_p$ is equivalent to measuring the observable \[eqFourier\] O(k)=U\_p\^\_z U\_p=(k)\_z+(k)\_y, with the usual position operator $\hat{x}=(a^\dagger + a) \Delta_x$ on the initial state. The propagator $U_p$ is obtained by setting $\phi_+$ and $\phi_-$ in $H_{D}$ to 0. Here, $k=2\eta\Omega_p t/\Delta_x$ is proportional to the interaction time $t$. If the ion’s internal state is $\|+\rangle_z$, we have $\langle O(k)\rangle=\langle\cos(k\hat{x})\rangle$ and for $\|+\rangle_y$, we have $\langle O(k)\rangle=\langle\sin(k\hat{x})\rangle$. A Fourier transformation of these measurements yields the probability density $\langle\delta(\hat{x}-x)\rangle$ in position space which for a pure state $\|\Psi\rangle$ amounts to $\|\Psi(x)\|^2$. Furthermore, we have that $\left.\frac{d^2}{dk^2}\langle O(k)\rangle\right\vert_{t=0}\propto\langle\hat{x}^2\sigma_z\rangle$ [@Lougovski:2006]. For eigenstates of $\sigma_z$, the initial curvature of the expectation value $\langle O(k)\rangle$ thus gives the width of the probability distribution $w_x$. The quantum walk entangles internal and motional degrees of freedom. Its analysis, however, requires the preparation of pure internal states like $\|+\rangle_z$ or $\|+\rangle_y$. Therefore, we recombine all internal state populations in $\|-\rangle_z$ before the measurement. To this end, the population in $\lvert +\rangle_z$ is transferred to $\lvert -\rangle_z$ after transferring the population in $\lvert -\rangle_z$ to the auxiliary state $\lvert D_{\rm 5/2}, m = 5/2\rangle$. A laser pulse at 854 nm excites the population from $\lvert D_{\rm 5/2}, m = 5/2\rangle$ to $\lvert P_{\rm 3/2}, m = 3/2 \rangle$ from where it spontaneously decays to $\lvert -\rangle_z$. The efficiency of this pumping process is $>99$ %, limited by a small branching ratio to the $D_{3/2}$-state. Only after the recombination step, we prepare the internal state required for measuring the even or odd Fourier components of (\[eqFourier\]). Due to the small Lamb-Dicke parameter, the probability of changing the motional state of the ion during the pumping steps is small and hardly affects measurements of observables in position space at all. In the experiment we set $\Omega_{p}=(2\pi)\,26$ kHz and measure $\langle \sigma_z \rangle$ for probe times between 0 and 300 $\mu$s in order to reconstruct the probability distribution $\langle\delta(\hat{x}-x)\rangle$ for different numbers of steps $N$. Since the walk is symmetric, it is in principle sufficient to measure only the even components of (\[eqFourier\]). For a seven-step walk, the measured odd and even Fourier components are displayed in Fig. \[fig\_Walk\](a). Panel (b) shows the reconstructed probability distribution $\langle\delta(\hat{x}-x)\rangle$ based on the even components for up to 13 steps. The uneven terms were checked to be close to zero for each number of steps $N$. The dashed lines in the plots are numerical simulations based on the Lamb-Dicke approximation. These lines deviate from the reconstructed distribution for $N>7$ due to higher order terms in $\eta$ that are not taken into account in eq. (\[H\_displace\]). The solid lines are based on a numerical simulation using all orders. A similar difficulty occurs in the measurement of observables based on eq. (\[eqFourier\]). For this reason, the reconstruction is not accomplished by a direct Fourier transformation of the data. Instead, we apply a constrained least-square fit based on convex optimization [@cvx:2009] capable of handling higher-order corrections (see the EPAPS document for more information on the reconstruction process). To get smoother distributions additional constraints were invoked by the reconstruction algorithm. A physical constraint is given by the maximal kinetic energy a one-dimensional wave packet can have. An estimate for the kinetic energy can be determined by measuring the momentum distribution in the same way as the position distribution. By changing $\phi_{\rm -} $ to $\pi/2$ in the probe pulse, the operator $\hat{x}$ appearing in (\[eqFourier\]) is replaced by an operator $\propto\hat{p}$. These measurements (see Fig. \[fig\_Sigma\](a)) indicate that the momentum distribution is not seriously affected during the walk, as expected for a pure displacement along the $x$-axis. A striking difference between classical and quantum walks is the reversibility of the latter. In the experiment we reversed a quantum walk after five steps. This was done by switching the phase of the following five displacement and coin flipping pulses by $\pi$. In this way the quantum walk is exactly reversed and the ion returns to the ground state. The corresponding reconstructed probability distribution shown in Fig. \[fig\_Walk\](c) closely resembles the one of the initial state and demonstrates once more the coherence of the quantum walk. To further highlight the differences between quantum and classical walks we also realized a classical walk by randomizing the phase between each step (while keeping the coin flip-displacement operator pair coherent for each individual step). The phase for each step was generated by a random noise generator. This mimics a completely mixed ensemble of measurement outcomes that behaves classically. A good way of quantifying the difference between the quantum and classical walks is by measuring the average width of the probability distributions. For a classical walk with a step size $d=s\Delta_x$ we have \[eqCl\_sigma\] w\_x=\_x, where the second term takes into account the initial width $\Delta_x$ of the probability distribution. By contrast, for a quantum walk the width goes as $w_x \sim N$ for high $N$. To measure $w_x$ for the random walk, the curvature of $\langle \sigma_z \rangle$ at short probe-time was analyzed. Quadratic fitting gives direct access to the width $w_x$. For the quantum walk, $w_x$ was obtained from the measured probability distributions. In Fig. \[fig\_Sigma\](a) the results of these procedures can be seen seen for both a quantum and a classical walk. ![\[TwoIonWalk\] Reconstructed probability distribution $\langle\delta(\hat{x}-x)\rangle$ for a two-ion quantum walk with up to 5 steps with a step size of $4\Delta_x$.](figure4.eps){width="8.5cm"} To avoid problems in the measurement of the motional state due to leaving the Lamb-Dicke limit for large numbers of steps, we implemented a method suggested in [@Xue:2009]. Outside the Lamb-Dicke regime the coupling strength $\Omega_{n,n}$ on the carrier depends on the phonon number $n$ as $\Omega_{n,n} = \Omega_0 L_n(\eta^2)$. Here, $L_n(\eta^2)$ is the $n$-th order Laguerre polynomial. The mean phonon number $\left<n\right>$ is determined by a constrained least-square fit of the carrier Rabi flops with the number state distribution as a fit parameter. In Fig. \[fig\_Sigma\](b), the resulting average vibrational quantum numbers are shown. As expected for the quantum walk, we observe a quadratic dependence $\langle n\rangle \propto N^2$ on the number of steps. Finally, we extend the quantum walk concept by adding a second ion to the system [@Andraca:2005]. In the two ion quantum walk we make use of the center-of-mass mode. To account for the second ion, all Pauli matrices $\sigma_i$ in eq. (\[eq:QuantumWalk\]) are replaced by $\sigma_i^{(1)}+\sigma_i^{(2)}$. This changes the coin from two sided to four sided, with three possible operations. The “side” belonging to the state $\|++\rangle_x$ ($\|--\rangle_x$) corresponds to a step to the right (left) while the sides belonging to the states $\|+-\rangle_x$ and $\|-+\rangle_x$ correspond to no step at all. The ions are prepared in the state $\|++\rangle_y$ with a $\pi/2$-pulse leading to a symmetric walk. For the two ion quantum walk all pulses are applied to both ions simultaneously. The probability distribution of the center of mass mode is obtained in the same way as for a single ion. The results for a walk of up to 5 steps are shown in Fig. \[TwoIonWalk\]. Again, the distribution deviates strongly from the classical version and shows a faster spreading. In summary, we have implemented a quantum walk using trapped ions. An experimental technique was implemented to determine the probability distribution along a line in phase space. This method might have further applications in quantum optics experiments or quantum simulations [@Gerritsma:2010]. We have highlighted the difference between a classical and a quantum walk and demonstrated the reversibility of the latter. The current limitation in number of steps is given by instabilities in the trap frequency leading to decoherence and by the change in the coupling strength due to high phonon numbers. Quantum walks are of importance as a primitive for quantum computation [@Childs:2009] and in finding search quantum algorithms that outperform their classical counterparts [@Hillery:2009]. As such the experimental implementation of the quantum walk serves as an important benchmark and points the way to further experiments. For instance, the implementation of a quantum walk with two ions opens up the interesting possibility to introduce entanglement [@Andraca:2005] and more advanced walks. Appendix: Reconstruction of the probability density {#appendix-reconstruction-of-the-probability-density .unnumbered} =================================================== For the reconstruction of the probability density $p(x)=\langle\delta(\hat{x}-x)\rangle$ of the motional quantum state $\rho_m$, we determine the expectation value of the observable \[eqFourier\] O(k)=U\_p\^\_z U\_p=(k)\_z+(k)\_y by applying the unitary $U_p=\exp(-ik\hat{x}\sigma_x/2)$ to the state $\rho=\|\Psi\rangle\langle\Psi\|\otimes\rho_m$ and measuring the operator $\sigma_z$. The right-hand-side of eq. (\[eqFourier\]) is obtained by using the equality $\exp(i\theta\sigma_x)=\cos\theta\, I+i\sin\theta\,\sigma_x$ and $\sigma_i\sigma_j=\epsilon_{ijk}\sigma_k$ for $i\neq j$. In this way, we determine $\langle\cos(k\hat{x})\rangle$ by choosing $\|\Psi\rangle=\|+\rangle_z$, and $\langle\sin(k\hat{x})\rangle$ by $\|\Psi\rangle=\|+\rangle_y$ [@Wallentowitz:1995]. In principle, a Fourier transformation of $f(k)=\langle\cos(k\hat{x})\rangle+i\langle\sin(k\hat{x})\rangle$ is sufficient for obtaining the density $\langle\delta(\hat{x}-x)\rangle$ in position space. However, with a finite number of experiments, the expectation values can only be determined for a discrete number of $k$-values and these measurements do not yield the exact expectation values but rather estimates of them. As a consequence, a reconstruction of the density based on Fourier transformation gives unphysical probability densities that are not non-negative everywhere. To overcome this problem, we reconstruct $p(x)$ by a constrained least-square optimization based on convex optimization [@cvx:2009]. We discretize the position space by using a suitable set of points $x_i$ and search among the probability distributions with $p(x_i)\ge 0$ for all $x_i$ the distribution that minimizes $$\begin{aligned} \label{eqLeastSquare} S&=&\sum_k\left(\sum_ip(x_i)\cos(kx_i)-C_k\right)^2\nonumber\\ &&+\sum_k\left(\sum_ip(x_i)\sin(kx_i)-S_k\right)^2\end{aligned}$$ where $C_k$ and $S_k$ are the experimentally determined estimates of $\langle\cos(k\hat{x})\rangle$ and$\langle\sin(k\hat{x})\rangle$, respectively. In our reconstruction, we use another physical constraint based on a measurement of the kinetic energy $\langle\frac{\hat{p}^2}{2m}\rangle$ in the following way. For a wave function $\psi=A(x)e^{i\phi(x)}$, a lower bound on the kinetic energy is given by $$\begin{aligned} \langle\frac{\hat{p}^2}{2m}\rangle&=&\frac{\hbar^2}{2m}\int_{-\infty}^\infty dx ((A(x)\phi^\prime(x))^2+A^\prime(x)^2)\nonumber\\ &\ge& \frac{\hbar^2}{2m}\int_{-\infty}^\infty dx A^\prime(x)^2\end{aligned}$$ where differentiation with respect to $x$ is indicated by primes. For $p(x)=\|\psi(x)\|^2$, we then have because of $A(x)=p(x)^{\frac{1}{2}}$ and $A^\prime(x)=\frac{1}{2}p^\prime(x)p(x)^{-\frac{1}{2}}$ that \[eq:KineticEnergyConstraint\] \_[-]{}\^dx . This constraint is also valid for mixed quantum states. In our optimization algorithm, eq. (\[eq:KineticEnergyConstraint\]) is a convex constraint that excludes distributions $p(x)$ having excessive energies. It requires a measurement of $\langle \hat{p}^2\rangle$ which is obtained by setting $\phi_-=\pi/2$ in the bichromatic Hamiltonian generating the unitary $U_p$ in eq.(\[eqFourier\]) and calculating $d^2/dk^2\langle O(k) \rangle$. Adding the constraint works particularly well for the states produced by the quantum walk. These states ideally do not have any phase gradients $\phi^\prime(x)$ in which case inequality (\[eq:KineticEnergyConstraint\]) turns into an equality. Moreover, the optimization algorithm can also handle to some extent problems related to the validity of the Lamb-Dicke approximation. In this approximation, the bichromatic laser-ion interaction, which is used for the operation $U_p$ in the reconstruction measurement, is described by the Hamiltonian \[H\_displace\] H\_D = \_[x]{}(a + a\^). This Hamiltonian is strictly valid only for $\eta\rightarrow 0$ because it is based on a Taylor expansion $e^{i\eta(a+a^\dagger)}=I+i\eta(a+a^\dagger)+{\cal O}(\eta^2)$ of the atom-light interactions that neglects terms in $\eta$ of order two or higher. If resonant terms up to third order are taken into account, the Hamiltonian becomes \[H\_displace\_thirdorder\] H\_D = \_[x]{}. Since this Hamiltonian no longer commutes with $\hat{x}$, it cannot be used instead of eq. (\[H\_displace\_thirdorder\]) for the reconstruction procedure described above. On the other hand, an analysis based on eq. (\[H\_displace\]) yields wrong results for quantum walks for large number of steps where the created states no longer fulfil the Lamb-Dicke criterion. 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--- abstract: 'We investigate the magnetic and glassy transitions of the square-lattice XY model in the presence of random phase shifts. We consider two different random-shift distributions: the Gaussian distribution and a slightly different distribution (cosine distribution) which allows the exact determination of the Nishimori line where magnetic and overlap correlation functions are equal. We perform Monte Carlo simulations for several values of the temperature and of the variance of the disorder distribution, in the paramagnetic phase close to the magnetic and glassy transition lines. We find that, along the transition line separating the paramagnetic and the quasi-long-range order phases, magnetic correlation functions show a universal Kosterlitz-Thouless behavior as in the pure XY model, while overlap correlations show a disorder-dependent critical behavior. This behavior is observed up to a multicritical point which, in the cosine model, lies on the Nishimori line. Finally, for large values of the disorder variance, we observe a universal zero-temperature glassy critical transition, which is in the same universality class as that occurring in the gauge-glass model.' address: - '$^1$ Scuola Normale Superiore and INFN, I-56126 Pisa, Italy' - '$^2$ Dipartimento di Fisica dell’Università di Roma “La Sapienza" and INFN, I-00185 Roma, Italy' - '$^3$ Dipartimento di Fisica dell’Università di Pisa and INFN, I-56127 Pisa, Italy' author: - 'Vincenzo Alba$^1$, Andrea Pelissetto$^2$ and Ettore Vicari$^3$' title: 'Magnetic and glassy transitions in the square-lattice XY model with random phase shifts' --- Introduction {#intro} ============ The two-dimensional XY model with random phase shifts (RPXY) describes the thermodynamic behavior of several disordered systems, such as Josephson junction arrays with geometrical disorder [@GK-86; @GK-89], magnetic systems with random Dzyaloshinskii-Moriya interactions [@RSN-83], crystal systems on disordered substrates [@CF-94], and vortex glasses in high-$T_c$ cuprate superconductors [@FTY-91]. See [@Korshunov-06; @KR-03] for recent reviews. The RPXY model is defined by the partition function $$\begin{aligned} &&Z(\{A\}) = \exp (-{\cal H}/T),\nonumber\\ &&{\cal H} = -\sum_{\langle xy \rangle } {\rm Re} \,\bar\psi_x U_{xy} \psi_y = - \sum_{\langle xy \rangle} {\rm cos}(\theta_x - \theta_y-A_{xy}), \label{RPXY} \end{aligned}$$ where $\psi_x\equiv e^{i\theta_x}$, $U_{xy}\equiv e^{i A_{xy}}$, and the sum runs over the bonds ${\langle xy \rangle }$ of a square lattice. The phases $A_{xy}$ are uncorrelated quenched random variables with zero average. In most studies they are distributed with Gaussian probability $$P_G(A_{xy}) \propto \exp\left(-{A_{xy}^2\over 2\sigma}\right). \label{GRPXYd}$$ We denote the RPXY model with distribution (\[GRPXYd\]) by GRPXY. We also consider the RPXY model with distribution (cosine model) $$P_C(A_{xy})\propto \exp\left({{\rm cos} A_{xy}\over \sigma}\right), \label{CRPXYd}$$ which we denote by CRPXY. Such a model is particularly interesting because the distribution (\[CRPXYd\]) allows some exact calculations along the so-called Nishimori (N) line $T \equiv 1/\beta=\sigma$ [@ON-93; @Nishimori-02]. In both GRPXY and CRPXY models the pure XY model is recovered in the limit $\sigma\rightarrow 0$, while the so-called gauge glass model [@ES-85] with uniformly distributed phase shifts is obtained in the limit $\sigma\rightarrow \infty$. The nature of the different phases arising when varying the temperature $T$ and the disorder parameter $\sigma$ and the critical behavior at the phase transitions have been investigated in many theoretical and experimental works [@RSN-83; @GK-86; @GK-89; @CF-94; @FTY-91; @Korshunov-06; @KR-03; @ON-93; @Nishimori-02; @ES-85; @FLTL-88; @CD-88; @FBL-90; @HS-90; @RTYF-91; @Li-92; @Gingras-92; @DWKHG-92; @RY-93; @Korshunov-93; @NK-93; @Nishimori-94; @NSKL-95; @CF-95; @JKC-95; @HWFGY-95; @KN-96; @Tang-96; @BY-96; @MG-97; @Scheidl-97; @KS-97; @KCRS-97; @MG-98; @CL-98; @Granato-98; @SYMOUK-98; @MW-99; @KA-99; @CP-99; @Kim-00; @CL-00; @AK-02; @HO-02; @KY-02; @Katzgraber-03; @HKM-03; @CTH-03; @NW-04; @KC-05; @TT-05; @UKMCL-06; @YBC-06; @CLL-08; @APV-09]. In spite of that, a conclusive picture of the phase diagram and of the critical behaviors has not been achieved yet. The expected $T$-$\sigma$ phase diagram for the GRPXY and CRPXY models, which is sketched in Fig. \[phdia\], presents two finite-temperature phases: a paramagnetic phase and a low-temperature phase characterized by quasi-long-range order (QLRO) for sufficiently small values of $\sigma$; see, e.g., [@APV-09] and references therein. The paramagnetic phase is separated from the QLRO phase by a transition line, which starts from the pure XY point (denoted by $P$ in Fig. \[phdia\]) at $(\sigma=0,T=T_{XY}\approx 0.893)$ and ends at a zero-temperature disorder-induced transition denoted by $D$ at $(\sigma_D,T=0)$. The QLRO phase extends up to a maximum value $\sigma_M$ of the disorder parameter, which is related to the point $M\equiv (\sigma_M,T_M)$, where the tangent to the transition line is parallel to the $T$ axis. No long-range glassy order can exist at finite temperature for any value of $\sigma$, including the gauge-glass limit $\sigma\to\infty$ [@NK-93; @Nishimori-94]. Several numerical studies of the gauge-glass XY model [@FTY-91; @RY-93; @Granato-98; @AK-02; @KY-02; @Katzgraber-03; @NW-04; @KC-05; @TT-05] support a zero-temperature glassy transition. A more complete discussion of the known features of the phase diagram will be reported below. In this paper we investigate several controversial issues concerning the critical behavior at the magnetic and glassy transitions in RPXY models. In particular, we will check whether the critical behavior along the paramagnetic-QLRO transition line is universal and belongs to the universality class of the pure XY model, whether there is a multicritical point along the paramagnetic-QLRO transition line, and, finally, whether the $T=0$ glassy transition extends from $\sigma=\infty$ to $\sigma=\sigma_D$, see Fig. \[phdia\], and belongs to the same universality class as that in the XY gauge-glass model. For this purpose, we perform Monte Carlo (MC) simulations of the GRPXY and CRPXY models for several values of the temperature and of the variance $\sigma$, approaching the magnetic and glassy transition lines from the paramagnetic phase. As we shall see, our results for the CRPXY model provide a robust evidence for a universal Kosterlitz-Thouless (KT) behavior of the magnetic correlations along the paramagnetic-QLRO transition line from the pure XY point $P$ to the point $M$ where the transition line runs parallel to the $T$ axis and magnetic and overlap correlations are equal. Along the line the magnetic correlation length $\xi$ behaves as $\ln \xi \sim u_t^{-1/2}$, where $u_t$ is the thermal scaling field, and the magnetic susceptibility as $\chi\sim \xi^{7/4}$ (corresponding to $\eta=1/4$). On the other hand, the behavior of the overlap correlations appears to be $\sigma$ dependent along this transition line. Moreover, the numerical results for the CRPXY model indicate that the point $M$ is multicritical. We conjecture that these conclusions hold for any RPXY model. In all cases we expect that the paramagnetic-QLRO transition line is divided into two parts by a multicritical point $M$, where magnetic and overlap correlations have the same critical behavior, though they are not equal. At variance with what happens in the CRPXY model, the point $M$ is not expected to coincide with the point in which the tangent to the transition line is parallel to the $T$ axis: this coincidence should be a unique feature of the CRPXY model. Then, from $P$ to $M$ we expect any RPXY model to behave as the CRPXY, that is a KT behavior for magnetic correlations and a $\sigma$ dependent behavior for disorder-related quantities. The universality of the behavior has been confirmed by our numerical results for the GRPXY model. Finally, we have investigated the critical behavior for large values of $\sigma$. Our numerical results provide strong evidence for a universal zero-temperature glassy transition for $\sigma>\sigma_D$. For $T\to 0$ overlap correlation functions are critical, and, in particular, the corresponding correlation length $\xi_o$ diverges as $\xi_o\sim T^{-\nu}$ when $T\to 0$ with $\nu=2.5(1)$. This paper is organized as follows. In Sec. \[phadia\] we review the known results for the phase diagram and for the critical behavior of the RPXY models. Sec. \[notations\] provides the definitions of the quantities considered in our numerical work. In Sec. \[paraqlro\] we study the critical behavior along the thermal paramagnetic-QLRO transition line which starts at the pure XY point $P$ and ends at multicritical point $M$. In Sec. \[HTresnline\] we discuss critical behavior along the N line of the CRPXY model and show that the point $M$ where the N line intersects the critical line is multicritical. In Sec. \[glassybeh\] we investigate the glassy critical behavior at $T=0$ for $\sigma >\sigma_D$. Finally, in Sec. \[conclusions\] we draw our conclusions. There are also several appendices. \[AppMC\] reports some details of the MC simulations. \[AppRGsigmazero\] is devoted to a careful analysis of the KT renormalization-group (RG) equations and of the corresponding RG flow. We derive the most general form of the $\beta$ function for the sine-Gordon model and discuss the structure of the scaling corrections in the XY model. These results are used in the discussion of the behavior at the paramagnetic-QLRO transition. In \[Appirrelevant\] we discuss some features of the critical behavior at a multicritical point. In \[AppRGdisordine\] we briefly discuss the RG equations in the presence of randomness. Finally, in \[App-magn\] we report some analytical results for the magnetic correlations in the gauge-glass model. The phase diagram {#phadia} ================= In Fig. \[phdia\] we show the expected $T$-$\sigma$ phase diagram of the RPXY models. In the absence of disorder ($\sigma=0$) the model shows a high-$T$ paramagnetic phase and a low-$T$ phase characterized by QLRO controlled by a line of Gaussian fixed points, where the spin-spin correlation function $\langle \bar{\psi}_x \psi_y \rangle$ decays as $1/r^{\eta(T)}$ for $r\equiv \|x-y\|\to \infty$, with $\eta$ depending on $T$. The two phases are separated by a Kosterlitz-Thouless (KT) transition [@KT-73] at [@HP-97] $\beta_{XY}\equiv 1/T_{XY}=1.1199(1)$. For $\tau \equiv T/T_{XY}-1\rightarrow 0^+$, the correlation length and the magnetic susceptibility diverge exponentially as ${\rm ln} \xi \sim \tau^{-1/2}$ and $\chi\sim \xi^{7/4}$, respectively. An interesting question is whether these features change in the presence of random phase shifts. The low-temperature phase of RPXY models shows QLRO for sufficiently small values of $\sigma$. The universal features of the long-distance behavior are explained by the random spin-wave theory [@RSN-83], obtained by replacing $${\rm cos}(\theta_x - \theta_y-A_{xy})\longrightarrow 1 - {1\over 2} (\theta_x - \theta_y+A_{xy})^2 \label{swr}$$ in Hamiltonian (\[RPXY\]). This scenario has been accurately verified by Monte Carlo (MC) simulations in both GRPXY and CRPXY models [@APV-09]. The QLRO phase disappears for large values of $\sigma$, see, e.g., [@Korshunov-06] and references therein; more precisely, as we shall see, for $\sigma\gtrsim 0.31$ in the case of the CRPXY model. For $\sigma\rightarrow \infty$ phases are uniformly distributed and one obtains the gauge-glass model. Even if this model has been much investigated [@FTY-91; @ES-85; @HS-90; @RTYF-91; @Li-92; @Gingras-92; @DWKHG-92; @RY-93; @NK-93; @Nishimori-94; @JKC-95; @HWFGY-95; @BY-96; @KS-97; @KCRS-97; @MG-98; @Granato-98; @KA-99; @CP-99; @Kim-00; @AK-02; @HO-02; @KY-02; @Katzgraber-03; @HKM-03; @CTH-03; @NW-04; @KC-05; @TT-05; @UKMCL-06; @CLL-08], its phase diagram and critical behavior are still controversial. No long-range glassy order can exist at finite temperature [@NK-93; @Nishimori-94]. However, this does not exclude more exotic low-temperature glassy phases [@CP-99; @HKM-03], for example a phase characterized by glassy QLRO. Many numerical works at finite and zero temperature support a zero-temperature transition [@FTY-91; @RY-93; @Granato-98; @AK-02; @KY-02; @Katzgraber-03; @NW-04; @KC-05; @TT-05]. According to this scenario, the correlation length determined from the overlap correlation function diverges as $\xi_o\sim T^{-\nu}$ when approaching the critical point $T=0$. The critical exponent $\nu$ has been estimated by finite-temperature Monte Carlo (MC) simulations, obtaining [@KY-02] $1/\nu=0.39(3)$ and [@NW-04] $1/\nu=0.36(3)$. The exponent $\nu$ is related to the $T=0$ stiffness exponent $\theta$ by $\theta=-1/\nu$. The $T=0$ numerical calculations of [@AK-02] and [@TT-05] provided the estimates $\theta=-0.36(1)$ and $\theta\approx -0.45$ respectively, which are consistent with the finite-temperature estimates of $\nu$. The $T=0$ transition scenario has been questioned in [@CP-99; @Kim-00; @HO-02; @HKM-03; @CTH-03; @UKMCL-06; @YBC-06; @CLL-08], which provide some numerical and experimental (for Josephson-junction arrays with positional disorder [@YBC-06]) evidence for the existence of a finite-temperature transition at $T\approx 0.2$, with a low-temperature glassy phase characterized by frozen vortices and glassy QLRO. Other features of the phase diagram are better discussed within the CRPXY model, characterized by the random phase-shift distribution (\[CRPXYd\]), because of the existence of exact results along the so-called Nishimori (N) line [@ON-93; @Nishimori-02] $$T \equiv 1/\beta=\sigma. \label{Nline}$$ Along the N line the energy density $E$ is known exactly: $$E\equiv {1\over V} [\langle {\cal H} \rangle] = -2{I_1(\beta)\over I_0(\beta)}, \label{exe}$$ where $I_0(\beta)$ and $I_1(\beta)$ are modified Bessel functions. Moreover, the spin-spin and overlap correlation functions are equal: $$\begin{aligned} [\langle \bar{\psi}_x \psi_y \rangle ] = [\|\langle \bar{\psi}_x \psi_y \rangle\|^2 ]. \label{cnl}\end{aligned}$$ As already noted in [@Nishimori-02], the N line should play an important role in the phase diagram, because it is expected to mark the crossover between the magnetic-dominated region and the disorder-dominated one. In the GRPXY and CRPXY models, the paramagnetic phase is separated from the magnetic QLRO phase by a transition line, which starts from the pure XY point (denoted by $P$ in Fig. \[phdia\]) at $(\sigma=0,T=T_{XY}\approx 0.893)$ and ends at a $T=0$ transition point induced by disorder (denoted by $D$) at $(\sigma_D,T=0)$, where $\sigma_D>0$.[^1] An important result has been proven for the CRPXY model [@ON-93]: the critical value $\sigma_M$ of $\sigma$ along the N line is an upper bound for the values of $\sigma$ where magnetic QLRO can exist. Therefore, at the critical point $M\equiv (\sigma_M,T_M)$ the tangent to the critical line should be parallel to the $T$ axis; moreover, the critical value $\sigma_D$ at $T=0$ must satisfy $\sigma_D\le\sigma_M$. It is worth noting how similar the phase diagrams of the CRPXY model and of the square-lattice $\pm J$ Ising model in the $T$-$p$ plane are, see Figs. \[phdia\] and \[phdiaisi2d\], respectively. The square lattice $\pm J$ (Edwards-Anderson) Ising model is defined by the Hamiltonian $${\cal H}_{\pm J} = - \sum_{\langle xy \rangle} J_{xy} \sigma_x \sigma_y, \label{pmj}$$ where $\sigma_x=\pm 1$, the sum is over pairs of nearest-neighbor sites of a square lattice, and $J_{xy}$ are uncorrelated quenched random variables, taking values $\pm J$ with probability distribution $P(J_{xy}) = p \delta(J_{xy} - J) + (1-p) \delta(J_{xy} + J)$. This model presents an analogous N line [@Nishimori-81] in the $T$-$p$ phase diagram, defined by $\tanh (1/T) - 2 p + 1=0$. The transition point along the N line is a multicritical point (MNP) [@HPPV-08; @PHP-06]. Moreover, the critical behavior for $T>T_{MNP}$ and $T<T_{MNP}$ is different. From the pure Ising point at $p=1$ to the MNP the critical behavior is analogous to that observed in 2D randomly dilute Ising (RDI) models [@HPPV-08-2]. From the MNP to the $T=0$ axis the critical behavior belongs to a new strong-disorder Ising (SDI) universality class [@PPV-08]. Finally, the $T=0$ end-point of the low-temperature paramagnetic-ferromagnetic transition line is the starting point of a $T=0$ transition line, characterized by a glassy universal critical behavior [@PPV-10]. In [@ON-93] it was also argued that, in the RPXY models (in particular, in the CRPXY one) the low-temperature paramagnetic-QLRO transition line from the critical point $M$ to the point $D$ runs parallel to the $T$ axis, so that $\sigma_D=\sigma_M$. The same arguments fail in the 2D $\pm J$ Ising model [@PPV-08; @HPPV-08; @PHP-06; @AH-04; @WHP-03], although they provide a good approximation. Thus, they are likely not exact also in the case of the RPXY models, although they may still provide a good approximation, suggesting that $0 < \sigma_M-\sigma_D\ll \sigma_M$. In the phase diagram reported in Fig. \[phdia\], which refers to the CRPXY, we may distinguish two transition lines meeting at point $M$: the thermal paramagnetic-QLRO transition line from $P$ to $M$, which can be approached by decreasing the temperature at fixed $\sigma$, and the transition line from $M$ to $D$, which can be instead observed by changing disorder at fixed $T$ for sufficiently low temperatures. As we shall see, our numerical results for the CRPXY model provide some evidence that the point $M$ is multicritical. We conjecture that the same conclusion holds for generic RPXY models, though in the generic case we do not expect the multicritical point $M$ to coincide with the point where the tangent to the critical line is parallel to the $T$ axis. The phase transition from the paramagnetic to the QLRO phase is generally expected to be of KT type ($\ln \xi$ is expected to have a power-law divergence), but its specific features, for instance the precise form of the power-law behavior and the value of the exponent $\eta$, have not been conclusively determined yet. Some numerical results supporting the KT-like behavior were presented in [@MG-97]. The disorder-driven $T=0$ transition at $\sigma_D$ has been argued [@NSKL-95; @CF-95; @MG-97; @CL-98; @CL-00] to show a KT-like behavior with $\ln \xi \sim (\sigma-\sigma_D)^{-1}$ and $\chi\sim \xi^{2-\eta}$ with $\eta=1/16$. However, other RG studies [@Scheidl-97; @Tang-96] obtained a different behavior: $\ln \xi \sim (\sigma-\sigma_D)^{-1/2}$. The value of $\eta$ associated with the magnetic two-point function has been believed to vary along the critical line [@RSN-83; @NSKL-95; @Tang-96; @Scheidl-97], from $\eta=1/4$ of the pure XY model at $\sigma=0$ to $\eta=1/16$ at the $T=0$ transition. As we shall see, our numerical results along the thermal paramagnetic-QLRO transition line, from $P$ to and including $M$, strongly support $\eta=1/4$, independently of $\sigma$. In the following sections we investigate some of the open issues of the RPXY models, by performing MC simulations of the GRPXY and CRPXY models close to their magnetic and glassy transition lines. In particular, we investigate the critical behavior at the thermal paramagnetic-QLRO transition line (from point $P$ to the multicritical point), along the N line in the CRPXY model, and at the $T=0$ glassy transition line for large disorder. Notations ========= We consider RPXY models defined on square lattices of size $L^2$ with periodic boundary conditions. We define the magnetic spin-spin correlation function $$G(x-y) \equiv [ \langle \bar{\psi}_x \,\psi_y \rangle ] \label{magcorr}$$ and the overlap correlation function $$G_o(x-y) \equiv [ \|\langle \bar{\psi}_x \,\psi_y \rangle\|^2 ]. \label{overcorr}$$ The angular and square brackets indicate the thermal average and the quenched average over disorder, respectively. The latter can also be written in terms of the overlap variables. Given two copies of the system with spins $\psi^{(1)}_x$ and $\psi^{(2)}_x$, we define $$q_x = \bar{\psi}_x^{(1)} \psi_x^{(2)},\qquad G_o(x-y) = [ \langle \bar{q}_x \,q_y \rangle ], \label{overlap}$$ where the thermal average is performed over the two systems with the same disorder configuration. We define the magnetic susceptibility $\chi\equiv \sum_x G(x)$, the overlap susceptibility $\chi_o\equiv \sum_x G_o(x)$, and the second-moment correlation lengths $$\begin{aligned} \xi^2 \equiv {\widetilde{G}(0) - \widetilde{G}(q_{\rm min}) \over \hat{q}_{\rm min}^2 \widetilde{G}(q_{\rm min}) }, \qquad \xi_{o}^2 \equiv {\widetilde{G}_o(0) - \widetilde{G}_o(q_{\rm min}) \over \hat{q}_{\rm min}^2 \widetilde{G}_o(q_{\rm min}) }, \label{smc}\end{aligned}$$ where $q_{\rm min} \equiv (2\pi/L,0)$, $\hat{q} \equiv 2 \sin q/2$. We also define the quartic couplings $$\begin{aligned} &&g_4 \equiv - {3\chi_4\over 2\chi^2\xi^2}, \qquad \chi_4 \equiv {1\over V} [ \langle \|\mu\|^4 \rangle - 2 \langle \|\mu\|^2 \rangle^2 ], \label{g4def} \\ &&g_{22} \equiv -{\chi_{22}\over \chi^2\xi^2},\qquad \chi_{22}\equiv {1\over V} \left( [ \langle \|\mu\|^2 \rangle^2 ] - [ \langle \|\mu\|^2 \rangle ]^2\right), \label{g22def}\\ &&g_c \equiv g_4 + 3 g_{22},\label{gcdef}\end{aligned}$$ where $\mu \equiv \sum_x \psi_x$ and $V=L^2$. Note that for the pure XY model $g_{22}=0$ and $g_c=g_4$. Finally, we define an overlap quartic coupling $g_{o}$ as $$\begin{aligned} &&g_{o} \equiv - {3 \bar\chi_{4o} \over 2\chi_o^2\xi_o^2}, \qquad\bar\chi_{4o} = {1\over V} [ \langle \|\mu_o\|^4 \rangle] - 2 [\langle \|\mu_o\|^2 \rangle ]^2 , \label{g4odef} \end{aligned}$$ where $\mu_o \equiv \sum_x q_x$. Critical behavior along the thermal para-QLRO transition line {#paraqlro} ============================================================= In this section we study the critical behavior of the RPXY models along the thermal paramagnetic-QLRO transition line, see Fig. \[phdia\], which starts at the point $P$ on the $\sigma=0$ axis and ends at the multicritical point, which belongs to the N line in the CRPXY model. For this purpose, we perform MC simulations of the GRPXY and of the CRPXY model for several values of $T$ and $\sigma$ in the paramagnetic phase, where the magnetic correlation length $\xi$ is large but finite. Fig. \[tc\] shows the points where the simulations are performed. The MC algorithm is described in \[AppMC\]. We average over a large number of samples, $N_s\approx 10^4$ in most cases. We consider large lattice sizes, satisfying $L/\xi\gtrsim 10$, in order to make finite-size effects negligible and obtain infinite-volume results. The residual finite-size effects are in all cases smaller than, or at most comparable with, the statistical errors. In the following we first discuss the critical behavior of the magnetic spin-spin correlation function (\[magcorr\]). We show that disorder is apparently irrelevant: for any $\sigma$ the correlation length diverges following the KT law valid for $\sigma = 0$ and the magnetic susceptibility diverges with critical exponent $\eta$ equal to 1/4. Then, we discuss the behavior of observables related to the overlap correlation function (\[overcorr\]), finding that the critical behavior of these quantities is apparently $\sigma$ dependent. Critical behavior approaching the pure XY transition point {#XYpuro} ---------------------------------------------------------- We wish now to understand the critical behavior along any line that lies in the paramagnetic phase and ends at the pure XY critical point at $\sigma = 0$ and $T = T_{XY}$. For $\sigma = 0$, as $T$ approaches the critical temperature $T_{XY}$ from above (paramagnetic phase), the magnetic correlation length $\xi$ diverges as $${\rm ln} (\xi/X) = C \tau^{-1/2} + O(\tau^{1/2}), \qquad \tau\equiv (T-T_{XY})/T_{XY}, \label{KTb2}$$ where $X$ and $C$ are nonuniversal constants. In the case of the square-lattice XY model with nearest-neighbor interactions [@HP-97] $\beta_{XY}\equiv 1/T_{XY}=1.1199(1)$, $X=0.233(3)$ and $C = 1.776(4)$.[^2] The magnetic susceptibility $\chi$ diverges as, see \[AppRGsigmazero\], $$\chi = A_\chi \xi^{7/4} \left[ 1 + {b_\chi\over {\rm ln} (\xi/X)} + O\left( 1/{\rm ln}^2\xi\right)\right]. \label{chibeh}$$ Note that while $A_\chi$ is a nonuniversal amplitude, the coefficient $b_\chi$ of the leading logarithmic corrections is universal. As shown in \[AppRGsigmazero\], it can be computed from the perturbative expansion of the RG dimension of the spin variable, obtaining $b_\chi=\pi^2/16$. We now consider the GRPXY model and study the critical behavior of $\chi$ and $\xi$ as one approaches the pure XY critical point along the line $\beta = \beta_{XY} = 1.1199$ by decreasing $\sigma$. We collected data for $0.46\gtrsim \sigma\gtrsim 0.14$ in the infinite-volume limit, corresponding to the quite large range of correlation lengths $4\lesssim \xi\lesssim 50$. Fig. \[xiTxy\] shows a plot of $\ln \xi$ versus $\sigma^{-1/2}$. The data fall on a straight line, showing that for $\sigma\to 0$ $$\ln \xi \sim \sigma^{-1/2}. \label{xivss}$$ This behavior can be understood within the RG framework. The general discussion presented in \[Appirrelevant\] shows that, as long as disorder is less relevant than the thermal perturbation, the critical behavior can be simply obtained by replacing $\tau$ with the nonlinear thermal scaling field. Note that it is not necessary that disorder is irrelevant to obtain the result (\[xivss\]). In general, the thermal nonlinear scaling field $u_t$ is an analytic function of the system parameters. Thus, in the presence of disorder it is a function of both $\tau = (T - T_{XY})/T_{XY}$ and $\sigma$ such that, close to the XY transition point, it behaves as $$u_t(\tau,\sigma) = \tau + c_\sigma \sigma + \ldots \label{utau}$$ where the dots stand for higher-order terms. If disorder is less relevant than the thermal perturbation, then $${\rm ln} (\xi/X) = C u_t^{-1/2} + O(u_t^{1/2}), \label{loxiut}$$ along any straight line in the $T,\sigma$ plane which ends at the XY pure transition point. Since this relation also holds for $\sigma = 0$ and $u_t(\tau,0) = \tau$, $C$ and $X$ are the same constants reported below (\[KTb2\]). Along the line $T = T_{XY}$ Equation (\[loxiut\]) implies $${\rm ln} (\xi/X) = {C\over (c_\sigma \sigma)^{1/2} } + O(\sigma^{1/2}), \label{alongkt}$$ in agreement with the observed behavior. In order to determine $c_\sigma$ we have performed fits to $${\rm ln} (\xi/X) = C_\sigma \sigma^{-1/2}\left( 1 + b \sigma\right), \label{fits}$$ using $X=0.233(3)$. We obtain the estimates $C_\sigma=2.010(2)$ and $b\approx -0.11$. In particular, a fit of the data satisfying $\xi\gtrsim 7$ gives $C_\sigma=2.0102(8)$ and $b=-0.108(2)$, with $\chi^2/{\rm DOF}\approx 1.1$ (DOF is the number of degrees of freedom of the fit). Using $C=1.776(4)$ and $C_\sigma=C/\sqrt{c_\sigma}$, we obtain $$c_\sigma = \left({C\over C_\sigma}\right)^2 = 0.781(4).$$ The constant $c_\sigma$ is nonuniversal and as such is model dependent. However, for $\sigma\to 0$ the fields $A_{xy}$ are typically very small and the distribution functions for the GRPXY and CRPXY models are identical to leading order in $A_{xy}$. We thus expect that the first correction to the thermal scaling field due to disorder is identical in the two models, i.e. $$\begin{aligned} u_{t,{\rm GRPXY}}(\tau,\sigma) = u_{t,{\rm CRPXY}}(\tau,\sigma) + O(\sigma^2), \label{utequality}\end{aligned}$$ which implies that $c_\sigma$ is the same in the GRPXY and CRPXY models. Critical behavior of the magnetic correlations at fixed $\sigma$ {#ktmagncorr} ---------------------------------------------------------------- Standard arguments that apply to critical lines and multicritical points imply that the critical temperature at fixed $\sigma$ must be the solution of the equation $$u_t[T_c(\sigma),\sigma]=0. \label{tcequt}$$ Therefore, Equation (\[utau\]) also implies that for small values of $\sigma$ the critical temperature for the GRPXY model (and also for the CRPXY model if (\[utequality\]) holds) is given by $$T_c(\sigma) = T_{XY}[1 - c_\sigma \sigma + O(\sigma^2)]. \label{taupred}$$ Equation (\[taupred\]) can be checked by analyzing data at fixed small values of $\sigma$. We have performed MC simulations of the GRPXY model at $\sigma = 0.0576$ for several values of $\beta$, from $\beta=0.95$ to $\beta=1.02$, corresponding to $10\lesssim \xi \lesssim 26$, and of the CRPXY model at the same value of $\sigma$ for $\beta=0.92,\,0.95,\,0.99$ corresponding to $7\lesssim \xi \lesssim 16$. In Fig. \[xi0p24\] we plot $\xi$ versus $t^{-1/2}$ with $t\equiv T/T_c-1$ and $T_c=0.8528$ given by (\[taupred\]) \[if we take the errors on $T_{XY}$ and $c_\sigma$ into account, we have $T_c=0.8528(3)$\]. Clearly, $\xi\to\infty$ as $t \to 0$, confirming (\[taupred\]). Moreover, they are clearly consistent with the KT behavior $${\ln \xi} = a t^{-1/2} + b. \label{genlogfit}$$ A fit of all available data for the GRPXY model to (\[genlogfit\]) gives $a=1.841(2)$ and $b=-1.511(5)$ (with $\chi^2/{\rm DOF}\approx 1.3$) keeping $T_c=0.8528$ fixed. A nonlinear fit, taking $T_c$ as a free parameter, gives $T_c=0.852(2)$, in good agreement with (\[taupred\]). Note that the estimate of the constant $b$ is close to the corresponding XY-model value $\ln X=-1.46(1)$. This is no unexpected since $X(\sigma)=X + O(\sigma)$. We also collected data at $\sigma=0.1521$ for both the GRPXY and CRPXY models, for $0.8\le \beta\le 1.1199$ (corresponding to $2\lesssim \xi\lesssim 37$) and $0.96\le \beta\le 1.145$ (corresponding to $5 \lesssim \xi\lesssim 46$), respectively. Again, the data fit well the KT behavior (\[genlogfit\]), see Fig. \[xi\]. Fits of the MC data for $\xi\gtrsim 10$ to (\[genlogfit\]) (for which $\chi^2/{\rm DOF}<1$) give the estimates $T_c=0.772(2)$ for the GRPXY model, and $T_c=0.762(1)$ for the CRPXY model. Note that (\[taupred\]) would give $T_c=0.7872$ for $\sigma=0.1521$, which is slightly larger than the above estimates. This is not unexpected since, when increasing $\sigma$, higher-order terms (which are different for the two models) may become important in (\[utau\]). We also mention the estimates $b=-1.82(7)$ and $b=-1.78(3)$ for the GRPXY and CRPXY model, respectively, from which one obtains estimates of the corresponding length scale $X(\sigma) = e^b$, $X = 0.162(11)$ and $X = 0.169(5)$. We also determined $\xi$ for other values of $\sigma$, but in a smaller range. The results are compatible with a KT behavior, but they do not allow us to get robust estimates of $T_c$. We only mention that in the case of the GRPXY at $\sigma=0.1936$, for which we have only data for $\xi\lesssim 20$, we find $T_c \approx 0.74$. At a KT transition the magnetic susceptibility behaves as in (\[chibeh\]), where $b_\chi=\pi^2/16$ is universal. In Fig. \[eta\] we show $\chi/\xi^{7/4}$ for the GRPXY and CRPXY and several values of $\sigma$ together with those of the pure XY model taken from [@BNNPSW-01]. We report the data versus $\ln \xi/X(\sigma=0)$. We could have also used $\ln \xi/X(\sigma)$, where $X(\sigma)$ is determined from the fit of $\xi$. This choice gives a plot essentially identical to the one reported, which is not unexpected since, by using $\ln \xi/X(\sigma=0)$ or $\ln \xi/X(\sigma)$ one simply changes the corrections of order $\sigma/\ln^2 \xi/X$, which are present anyway. The results appear to follow the same curve within the errors (except those obtained along the N line, which we shall discuss in Sec. \[HTresnline\]). They provide strong evidence that the value $\eta=1/4$ is universal along the thermal paramagnetic-QLRO transition line. Also the slope appears universal (the coefficient $b_\chi$ does not depend on $\sigma$), as expected on the basis of the discussion of \[AppRGsigmazero\]. The constant $A_\chi$ corresponds to the intercept of $\chi/\xi^{7/4}$ at $\ln \xi/X(\sigma)=0$. As it can be seen from the figure, this constant, which is not universal, varies very little with $\sigma$: differences are not visible within our errors, except for the CRPXY data at $\sigma=0.307$. However, note that for this value of $\sigma$ the critical behavior is controlled by the multicritical Nishimori point, i.e. by the special point $M$ which appears in Fig. \[phdia\]; we will return to it in Sec. \[HTresnline\]. In conclusion, the above numerical results provide a strong evidence that the magnetic two-point correlations show a KT behavior along the thermal paramagnetic-QLRO transition line in GRPXY and CRPXY models. Quartic couplings ----------------- We now discuss the behavior of the quartic couplings defined in (\[g4def\])-(\[gcdef\]). We recall that in the pure XY model $g_{22}=0$ while $g_4=g_c$ behaves as $$g_4 = g_4^* + {b_g\over (\ln \xi/X)^{2}} + O(1/\ln^4 \xi), \label{g4beh}$$ where $g_4^$ and $b_g$ are universal; see \[AppRGsigmazero\]. We mention the estimates $g_4^=13.65(6)$ obtained by form-factor computations in [@BNNPSW-01], and $g_4^=13.7(2)$ by field-theoretical methods [@PV-00]; other results for $g_4^$ can be found in [@PV-r] and references therein. In Fig. \[g4\] we show some MC results of $g_c$ for the CRPXY model at $\sigma=0.1521,\,0.0576$ and the GRPXY model at $\beta=\beta_{XY}=1.1199$ (within our errors of a few per mille the infinite-volume limit is reached for $L/\xi\gtrsim 10$, as in the pure XY model [@BNNPSW-01]), and compare them with MC results for the pure XY model taken from [@BNNPSW-01]. The results are identical within errors. For example, if we consider the CRPXY model for $\sigma=0.1521$, a fit to $g_c^* + b_g/(\ln \xi/X)^2$ gives $g_c^=13.57(10)$ and $b_g=-3.1(1.4)$, with $\chi^2/{\rm DOF}\approx 0.4$, to be compared with the value [@BNNPSW-01] $g_4^=13.65(6)$ of the pure XY model. Both $g_c^$ and $b_g$, which are universal in the pure-XY universality class, do not depend on $\sigma$. The quartic coupling $g_{22}$ defined in (\[g22def\]) is interesting because it is particularly sensitive to randomness effects, since in the pure XY model it vanishes trivially. The estimates of $g_{22}$ in the GRPXY model for $T=T_{XY}$ and several values of $\sigma$ are shown in Fig. \[g22Txy\]. They decrease with decreasing $\sigma$, and appear to vanish when $\sigma\to 0$ as $$g_{22}\sim c \xi^{-\varepsilon}, \label{g22txy}$$ with $\epsilon\approx 1.0$. A fit to (\[g22txy\]) gives $\varepsilon = 0.97(4)$, $c=3.1(3)$ with $\chi^2/{\rm DOF}\approx 1.1$, where DOF is the number of degrees of freedom of the fit. The fast decrease of $g_{22}$ along the line $T=T_{XY}$ \[note that $g_{22}\sim 1/\xi$ implies $g_{22} \sim \exp(-c\sigma^{-1/2})$\] might suggest the irrelevance of disorder, and therefore that the critical value $g_{22}^$ vanishes along the thermal paramagnetic-QLRO transition line. This conclusion is apparently contradicted by the results at fixed $\sigma>0$. The results for the CRPXY model at various values of $\sigma$, $\sigma=0.0576,\,0.1521,\,0.2992,\,0.307$, are shown in Fig. \[g22all\], where they are plotted versus $(\ln \xi/X)^{-2}$, which is the correction expected in the pure XY model for RG invariant quantities. The coupling $g_{22}$ is quite small, but definitely different from zero on the transition line. For $\sigma=0.1521$ an extrapolation using $g_{22}^* + b/(\ln \xi/X)^2$ suggests a nonzero critical limit. Using only data satisfying $\xi\gtrsim 10$, this fit gives $g_{22}^=-0.068(8)$ and $b=-0.080(15)$, with $\chi^2/{\rm DOF}\approx 0.4$. We should also mention that the data for the largest values of $\xi$, those satisfying $\xi\gtrsim 10$ say, may be consistent with a vanishing critical limit, but only assuming a slower logarithmic approach, i.e., $g_{22}\approx b/(\ln \xi/X)$. For instance, the data with $\xi\gtrsim 10$ are consistent with this behavior (the fit gives $b=-0.482(4)$ with $\chi^2/{\rm DOF}\approx 1.1$). At $\sigma=0.0576$ the $1/(\ln \xi)^2$ extrapolation of the data satisfying $7\lesssim \xi\lesssim 16$ gives $g_{22}^=-0.008(6)$ with $\chi^2/{\rm DOF}\approx 1.3$. The data of $g_{22}$ at $\sigma\approx 0.30$ are larger, but this can be explained by crossover effects, since this value of $\sigma$ is quite close to the critical point along the N line, where the critical behavior may change, see Sec. \[HTresnline\]. Overall the results for $g_{22}$ suggest a nonuniversal critical value. Critical behavior of the overlap correlations {#overlapcb} --------------------------------------------- We now discuss the critical behavior of overlap correlations, cf. (\[overlap\]), which are the appropriate quantities to understand the role of disorder. We consider the critical behavior of the overlap susceptibility which is expected to behave as $\chi_o\sim \xi_o^{2-\eta_o}$. In the case of the pure XY model we have $\eta_o=2\eta=1/2$. In [@APV-09] it was noted that the following relations $$\begin{aligned} &2 \eta - \eta_o \approx \displaystyle{\sigma\over \pi} \quad & {\rm for} \quad {\rm GRPXY}, \label{testdiffG}\\ &2 \eta - \eta_o \approx \displaystyle{\sigma + {1\over 2} \sigma^2\over \pi} \quad & {\rm for} \quad {\rm CRPXY} \label{testdiffC}\end{aligned}$$ approximately hold in the whole QLRO phase (within the small statistical errors), even very close to the KT transition, as long as $\sigma$ is not to large (in practice $\sigma$ should not be close to $\sigma_M$, where $M$ is the Nishimori point defined in Fig. \[phdia\]). This would suggest that they may remain valid up to the transition. Given the strong numerical evidence that the exponent $\eta$ associated with the magnetic correlation is $\eta=1/4$, see Sec. \[ktmagncorr\], the above relations imply that $\eta_o$ varies along the paramagnetic-QLRO transition line approximately as $$\begin{aligned} \eta_o\approx {1\over 2} - {\sigma\over \pi} \quad & {\rm for} \quad {\rm GRPXY},\label{etaoG}\\ \eta_o\approx {1\over 2} - {\sigma+\sigma^2/2\over \pi} \quad & {\rm for} \quad {\rm CRPXY} \label{etaoC},\end{aligned}$$ at least for sufficiently small values of $\sigma$. We wish now to verify if the high-temperature data are consistent with these predictions. In Fig. \[etao\] we plot $\chi_o/\xi^{2-\eta_o}$ versus $1/\ln(\xi/X)$. The scaling is reasonable. We also report $\chi_o/\xi^{2-\eta_o}$, fixing $\eta_o$ to the pure XY value $\eta_o=1/2$. Again the ratio is consistent with a limiting finite value. However, if $\chi_o$ behaves as in the pure XY model, we would expect a $\sigma$-independent slope, see \[AppRGsigmazero\], which is not supported by the data. to We now consider the ratio $\xi_o/\xi$ between the second-moment correlation lengths obtained from the overlap and spin correlation functions, cf. (\[smc\]).[^3] In order to estimate this ratio in the case of the pure XY model, we performed MC simulations (using the cluster algorithm) in the range $0.93\le \beta\le 1.033$ corresponding to $12\lesssim \xi \lesssim 110$. Taking into account the logarithmic scaling corrections, i.e. fitting the XY-model data satisfying $\xi\gtrsim 32$ to $a + b/(\ln \xi/X)^2$ with $X=0.233$, we obtain the estimate $\xi_o/\xi=0.417(4)$. In Fig. \[raxi\] we show the results for several values of $\sigma$. They are all consistent with a finite critical value, confirming that the paramagnetic-QLRO transitions are characterized by a single diverging length. The results can be extrapolated by assuming $\xi_o/\xi = a+b/(\ln \xi/X)^2$ for $\xi\to\infty$. We obtain $\xi_o/\xi=0.417(5),\,0.428(5),\,0.425(7),\,0.425(3)$ for the GRPXY model at $\sigma=0.0576,\,0.1521,\, 0.1936$ and the CRPXY model at $\sigma=0.1521$, respectively. A larger result is found for the CRPXY model at $\sigma\approx 0.299$, 0.307: $\xi_o/\xi\approx 0.49$. These results indicate that the ratio $\xi_o/\xi$ varies along the transition line, although it changes very weakly for small values of $\sigma$. Again, this is consistent with the observation that disorder-related quantities, like $\eta_o$ and $g_{22}$, depend on $\sigma$. Critical behavior along the N line in the CRPXY model {#HTresnline} ===================================================== We now consider the critical behavior along the N line $T=\sigma$ in the CRPXY model, approaching the transition point from the paramagnetic phase. We recall that along the N line the magnetic and overlap correlation functions are equal, so that $\eta_o=\eta$ and $\xi_o=\xi$ exactly. We performed several MC simulations along the N line, in the range $1.5\le \beta \le 2.4$, corresponding to $2\lesssim \xi \lesssim 28$, and considered large lattice sizes, in order to obtain infinite-volume results. Our MC estimates of the magnetic correlation length $\xi$ are consistent with an exponential increase, i.e. with a behavior of the form ${\ln \xi} \sim t^{-1/2}$ with $t=T/T_M-1$, see Fig. \[xi\]. A linear fit to $${\ln \xi} = a t^{-1/2} + b \label{fitmnp}$$ of the data satisfying $\xi\gtrsim 5$ gives the estimate $$T_M=\sigma_M=0.307(2), \label{tmsm}$$ with $\chi^2/{\rm DOF}\lesssim 1.0$. We also mention that alternative fits to $\xi = a t^{-b}$ and to ${\ln \xi} = a t^{-1} + b$ give rise to significantly larger $\chi^2$. In order to estimate the exponent $\eta$, we fit $\chi$ and $\xi$ to $\chi = c \xi^{2-\eta}$. Considering the MC results satisfying $\xi\gtrsim \xi_{\rm min} = 5$, we find $\eta=0.246(4)$ with $\chi^2/{\rm DOF}\approx 1.0$. If we increase $\xi_{\rm min}$, $\eta$ slightly decreases, but it is always compatible with $\eta=1/4$. These results suggest that $\eta=1/4$ also along the N line. Fig. \[g4g22nline\] shows the estimates of $g_c$. The critical limit of $g_c$ is consistent with the results for the pure XY model and those obtained along the thermal paramagnetic-QLRO line at smaller values of $\sigma$, see Fig. \[g4\]. Indeed, a fit of all data of $g_c$ to (\[g4beh\]) gives $g_c^=13.49(13)$ with $\chi^2/{\rm DOF}\approx 0.6$. If we consider only the data satisfying $\xi\gtrsim 4$, we obtain $g_c^=13.6(3)$. The above-reported results (KT behavior of $\xi$, $\eta=1/4$, and $g_c^\approx g_{4,XY}^$) suggest that the magnetic correlations behave as in the pure XY model. There is, however, a result which contradicts this hypothesis. As we discussed in Sec. \[XYpuro\], the rate of approach of $\chi \xi^{-7/4}$ to its limiting value, should be universal. As can be seen from Fig. \[eta\], this is not the case: the slope of the data along the N line is clearly different from that predicted for the pure XY model. Thus, even though at the Nishimori point the magnetic critical behavior is the same as that observed along the thermal paramagnetic-QLRO transition line, corrections are different, implying the presence of a new (probably marginal) RG operator, which only contributes to scaling corrections in magnetic quantities. A better evidence for the presence of a new, disorder-related operator is obtained by considering $g_{22}$ and $\xi_o/\xi$. In Fig. \[g4g22nline\] we also report estimates of $g_{22}$ along the N line and along the line $\sigma = 0.307$. If the estimate (\[tmsm\]) is correct, the two lines intersect the critical line at the same point, the Nishimori point. It is quite clear from the data that the limiting value of $g_{22}$ along the two lines is quite different. A fit of all available data on the Nishimori line to $g_{22}^+b/(\ln \xi/X)^2$ gives $g_{22}^=-7.00(5)$ with $\chi^2/{\rm DOF}\approx 0.9$. On the other hand, a fit of the data along the line at fixed $\sigma=0.307$ gives $g_{22}^\simeq -0.8$. The same phenomenon is observed for the ratio $\xi_o/\xi$. As can be seen in Fig. \[raxi\], for $\sigma=0.307$ this ratio is approximately equal to 0.49, which is clearly different from the result that holds exactly along the Nishimori line, $\xi_o/\xi=1$. The large differences of the values of these two RG invariant quantities along the two lines provide compelling evidence that the Nishimori point is a multicritical point as in the 2D $\pm J$ Ising model [@HPPV-08]. To understand this conclusion, let us review the basic results that apply to multicritical points. The singular part of the free energy should obey a scaling law $$\begin{aligned} {\cal F}_{\rm sing}(u_1,u_2) = b^{-d} {\cal F}_{\rm sing}(b^{y_1} u_1,b^{y_2} u_2),\end{aligned}$$ where $u_1$ and $u_2$ are two relevant scaling fields. They can be inferred by using the following facts: (i) the transition line at $M$ must be parallel to the $T$ axis, since it has been proved [@ON-93] that $\sigma_M$ is an upper bound for the values of $\sigma$ where QLRO can exist; (ii) the condition $T=\sigma$ at the N line is RG invariant. We therefore have $$u_1 = \sigma-\sigma_M + ... \label{u1scalfields}$$ where the dots indicate nonlinear corrections, which are quadratic in $\Delta\sigma\equiv \sigma-\sigma_M$ and $\Delta T\equiv T-T_M$, so that the line $u_1=0$ runs parallel to the $T$ axis at $M$. Moreover, we choose $$u_2=T-\sigma, \label{scalfields}$$ so that the N line corresponds to $u_2=0$. Close to the multicritical point, any RG invariant quantity, such as $g_{22}$, is expected to behave as $$R = f_R(u_1 u_2^{-y_1/y_2}). \label{rgincmcp}$$ Now, the N-line corresponds to $u_2=0$, so that a RG invariant quantity converges to $f_R(\infty)$. On the other hand, the line $\sigma=\sigma_M$ corresponds to $u_1 = 0$, so that a RG invariant quantity converges to $f_R(0)$ which is generically expected to be different from $f_R(\infty)$. Thus, if the Nishimori point is multicritical, we expect RG invariant quantities to have a different critical value along the two lines. This is exactly what we observe for $g_{22}$ and $\xi_o/\xi$. Thus, in view of the numerical results we conclude that the Nishimori point is a multicritical point. It is interesting to note that the multicritical behavior is not observed in the magnetic sector. For instance, $g_c^$ along the paramagnetic-QLRO line is equal to its XY value $g_{4,XY}^$. The same result holds along the Nishimori line. In terms of the scaling function $f_{g_c}$ defined in (\[rgincmcp\]) these results imply $$f_{g_c}(0) = f_{g_c}(\infty) = g_{4,XY}^~.$$ It is then natural to conjecture that $g_c^* = g_{4,XY}^$ along any line that intersects the Nishimori point, i.e. that $f_{g_c}(x) = g_{4,XY}^$ for any $x$. The absence of multicritical behavior in the magnetic sector is also supported by the fact that $\xi$ always shows a KT behavior and that the magnetic exponent $\eta$ at the Nishimori point is equal to the pure-XY value 1/4. The results we have presented should apply to generic RPXY model. In all cases we expect a multicritical point along the paramagnetic-QLRO transition line. It follows from universality that, at the multicritical point, the magnetic and the overlap correlation functions have the same critical behavior—hence, we have $\eta=\eta_o$—though they may not be necessarily equal as is the case for the CRPXY model. Note that this point is not expected in general to coincide with that in which the tangent to the transition line is parallel to the $T$ axis. Glassy critical behavior at $T=0$ {#glassybeh} ================================= In the limit $\sigma\rightarrow \infty$ the RPXY model corresponds to the gauge-glass model in which the phase shifts are uniformly distributed. This model has been extensively studied both at zero and at finite temperature [@FTY-91; @ES-85; @HS-90; @RTYF-91; @Li-92; @Gingras-92; @DWKHG-92; @RY-93; @NK-93; @Nishimori-94; @JKC-95; @HWFGY-95; @BY-96; @KS-97; @KCRS-97; @MG-98; @Granato-98; @KA-99; @CP-99; @Kim-00; @AK-02; @HO-02; @KY-02; @Katzgraber-03; @HKM-03; @CTH-03; @NW-04; @KC-05; @TT-05; @UKMCL-06; @YBC-06; @CLL-08]. [@NK-93; @Nishimori-94] showed that no long-range glassy order can exist at finite temperature. Although this result does not exclude the possibility of a finite-temperature transition with an exotic low-temperature glassy phase, for example a phase characterized by glassy QLRO, most numerical works [@FTY-91; @RY-93; @Granato-98; @AK-02; @KY-02; @Katzgraber-03; @NW-04; @KC-05; @TT-05] support a zero-temperature glassy critical behavior. The overlap correlation length $\xi_o$ diverges as $T^{-\nu}$ for $T\to 0$. We mention the estimates [@KY-02] $1/\nu=0.39(3)$ and [@NW-04] $1/\nu=0.36(3)$ from finite-temperature MC simulations, and [@AK-02] $1/\nu=0.36(1)$ and [@TT-05] $1/\nu\approx 0.45$ from $T=0$ numerical calculations. Moreover, if one assumes that the ground state is nondegenerate in the overlap variables, one obtains that at $T=0$ the finite-size overlap susceptibility satisfies the relation $\chi_o = L^2$, so that $\eta_o = 0$. We mention that this scenario was questioned in [@Kim-00; @CP-99; @HO-02; @HKM-03; @CTH-03; @UKMCL-06; @YBC-06; @CLL-08], which claimed the existence of a finite-temperature transition at $T\approx 0.2$. A natural scenario for the phase digram of the GRPXY and CRPXY models is that the glassy transition, which occurs for $\sigma=\infty$, is not isolated but that it is the endpoint of a phase transition line that starts at the paramagnetic-QLRO transition line. In particular, if the zero-temperature glassy transition scenario applies to the gauge-glass model, we expect a line of $T=0$ glassy transitions for any $\sigma>\sigma_D$, see Fig. \[phdia\]. A natural conjecture would be that all these transitions belong to the same universality class. To check this scenario we performed MC simulations of the CRPXY model at $\sigma=2/3,\,5/9,\,1/2$, $\infty$, which are larger than $\sigma_D\le \sigma_M\approx 0.31$. As we shall see, the results clearly support a glassy $T=0$ transition in the same universality glass as that of the gauge-glass model. MC simulations {#MCglassy} -------------- We performed MC simulations of the CRPXY model on square $L\times L$ lattices with periodic boundary conditions. Most of the results we shall present refer to runs with $\sigma=2/3$. In this case we considered $L=20$, 30, 40, 60, 80 and temperatures between $T=2/3$ (at the Nishimori line) and $T=0.1$ (for $L=80$ we considered $0.22 \le T \le 2/3$). We averaged over a relatively large number $N_s$ of samples: $N_s=6000$, 9000, 7000, 3000, and 2000 samples for $L=20$, 30, 40, 60 and 80, respectively. We used the MC algorithm discussed in \[AppMC\] combined with the parallel-tempering method [@raex; @par-temp]. Moreover, to check the universality of the transitions, we also performed parallel-tempering MC simulations for $\sigma=5/9$ and lattice sizes $L=60,\,70$ (5000 and 1000 disorder samples, respectively), $\sigma=1/2$ and $L=70$ (1000 samples), and $\sigma=\infty$ and $L=20$, 30, 40, 60 (5000, 5000, 2000, 2000 samples, respectively). The points in the $T$-$\sigma$ plane where we collected MC data are shown in Fig. \[tc\]. At the glassy transition the critical modes are those related to the overlap variables, while the magnetic ones are noncritical. This is clearly shown in Fig. \[xislt\], which shows $\xi$ and $\xi_o$ for $\sigma=2/3$. The overlap correlation length $\xi_o$ increases steadily with decreasing the temperature, while the magnetic correlation length $\xi$ freezes at sufficiently low temperatures at a value $\xi\approx 3.3$. Therefore, the critical temperature and exponents must be determined from quantities related to the overlap correlation functions. Evidence for a $T=0$ glassy transition at $\sigma=2/3$ {#tc2o3} ------------------------------------------------------ $L_{\rm min}$ $\chi^2$/DOF --------------- ----- -------------- ---------- 20 0.6 169/157 0.018(1) 20 0.5 99/141 0.009(1) 20 0.4 67/119 0.010(2) 20 0.3 30/92 0.010(3) 30 0.6 137/138 0.017(1) 30 0.5 66/123 0.008(2) 30 0.4 49/103 0.007(3) 30 0.3 21/79 0.005(4) 40 0.6 106/119 0.017(2) 40 0.5 42/105 0.007(2) 40 0.4 31/87 0.007(3) 40 0.3 17/66 0.007(5) : Estimates of $T_c$ obtained by fitting $R_\xi$ to (\[fitRxi\]) with $n=6$. DOF is the number of degrees of freedom of the fit. []{data-label="Tc-glass"} In order to determine the critical temperature, we analyze $R_{\xi_o}\equiv \xi_o/L$. The results, shown in Fig. \[xioovl\], show no evidence of a crossing point in the range of values of $T$ of the data, $T\ge 0.1$, and thus provide the bound $T_c<0.1$ for the critical temperature $T_c$. A more precise determination of $T_c$ can be obtained by a finite-size scaling (FSS) analysis. We fit the data to $$R_{\xi_o} = P_n[(T-T_c) L^{1/\nu}], \label{fitRxi}$$ keeping $T_c$ and $\nu$ as free parameters. Here $P_n(x)$ is a polynomial in $x$ of order $n$. The order $n$ is fixed by looking at the $\chi^2$ of the fit. For each $n$ we determine the goodness $\chi^2(n)$ of the fit. Then, we fix $n$ such that $\chi^2(n)$ is not significantly different from $\chi^2(n+1)$. The results we report correspond to $n=6$. To identify the role of the corrections to scaling we repeat the fit several times. Each time we fix two parameters $T_{\rm max}$ and $L_{\rm min}$ and we only include the data which correspond to lattices satisfying the conditions $T \le T_{\rm max}$ and $L\ge L_{\rm min}$. In Table \[Tc-glass\] we report the estimates of $T_c$ for several values of $T_{\rm max}$ and $L_{\rm min}$. We obtain estimates of $T_c$ which are quite small and satisfy the upper bound $$T_c \lesssim 0.01~.$$ Since our data satisfy $T\ge 0.1$, this estimate allows us to conclude that our results are fully consistent with a zero-temperature transition. From now on, we always assume $T_c = 0$. The critical exponent $\nu$ {#critexpnu} --------------------------- $L_{\rm min}$ $\chi^2$/DOF --------------- ------ -------------- ----------- 20 0.4 100/120 2.465(6) 20 0.3 44/93 2.496(10) 20 0.25 24/77 2.528(14) 20 0.2 17/55 2.547(22) 20 0.16 14/39 2.548(31) 30 0.4 55/104 2.446(6) 30 0.3 23/80 2.464(13) 30 0.25 13/65 2.489(20) 30 0.2 10/46 2.492(30) 30 0.16 9/32 2.488(42) 40 0.4 36/88 2.432(7) 40 0.3 19/67 2.451(15) 40 0.25 12/53 2.480(26) 40 0.2 8/37 2.490(38) 40 0.16 9/25 2.482(53) : Estimates of $\nu$ obtained by fitting $R_{\xi_o}$ to (\[fitRxi\]) with $T_c = 0$ and $n=6$. DOF is the number of degrees of freedom of the fit. []{data-label="nu-glass"} In order to determine the critical exponent $\nu$ related to the divergence of the correlation length $\xi_o$, we repeat the fit (\[fitRxi\]) at $\sigma=2/3$ setting $T_c = 0$. The results are reported in Table \[nu-glass\]. They slightly increase as $T_{\rm max}$ or $L_{\rm min}$ is lowered, but these changes are small compared to the statistical errors. In fit (\[fitRxi\]) we made two approximations. First, we neglected the nonanalytic scaling corrections, which decrease as $L^{-\omega}$. The results indicate that these corrections are small: at fixed $T_{\rm max}<0.25$ the estimates of $\nu$ obtained setting $L_{\min} = 30$ and $L_{\min} = 40$ differ by much less than the statistical errors. Second, we approximated the thermal nonlinear scaling field $u_T$ by $u_T\approx T$, neglecting the [analytic]{} corrections (see [@HPV-08] for an extensive discussion of this type of corrections). To understand their quantitative role, we performed fits to $$R_\xi = P_n(u_T L^{1/\nu}), \qquad u_T \equiv T + p T^2, \label{fitRxi-analytic}$$ where $p$ is a new free parameter. The results are reported in Table \[nu-glass-analytic\]. Corrections are tiny and we estimate $\|p\|\lesssim 0.2$, so that $\|u_T-T\|/T$ is at most 0.10, 0.02 for $T=0.5$, 0.1, respectively. The estimates of $\nu$ do not vary significantly and, for $L\ge 30$ and $T_{\rm max}\le 0.2$, are fully consistent with those obtained before. We quote $$\nu = 2.5(1)~, \qquad 1/\nu = 0.40(2) \label{nufinale}$$ as our final estimate. To show the quality of our FSS results in Fig. \[FSSxi\] we plot $R_{\xi}$ versus $TL^{1/\nu}$, using the estimate (\[nufinale\]). All data fall on top of each other with remarkable precision. $L_{\rm min}$ $\chi^2$/DOF --------------- ------ -------------- ---------- ------------- 20 0.5 98/141 2.54(1) $-$0.11(1) 20 0.4 63/119 2.62(2) $-$0.20(2) 20 0.3 28/92 2.71(4) $-$0.34(5) 20 0.25 20/76 2.67(6) $-$0.26(11) 20 0.2 16/54 2.67(10) $-$0.29(21) 30 0.5 89/123 2.42(2) $-$0.00(2) 30 0.4 47/103 2.54(2) $-$0.12(3) 30 0.3 20/79 2.58(5) $-$0.18(7) 30 0.25 13/64 2.54(7) $-$0.11(14) 30 0.2 10/45 2.50(12) $-$0.01(31) 40 0.5 50/105 2.42(2) $-$0.00(2) 40 0.4 31/87 2.50(3) $-$0.09(3) 40 0.3 17/66 2.58(6) $-$0.20(7) 40 0.25 12/52 2.54(16) $-$0.12(16) 40 0.2 10/36 2.50(13) $-$0.01(33) : Estimates of $\nu$ and $p$ obtained by fitting $R_{\xi_o}$ to (\[fitRxi-analytic\]) with $n = 6$. DOF is the number of degrees of freedom of the fit.[]{data-label="nu-glass-analytic"} The critical exponent $\eta_o$ {#critexpeta} ------------------------------ As discussed at length in [@HPV-08], the overlap susceptibility behaves in the critical limit as $$\chi_o = \overline{u}_h^2 L^{2-\eta_o} f(u_T L^{1/\nu})~.$$ Here $u_T$ is the temperature nonlinear scaling field, while $\overline{u}_h$ is related to the external [overlap-magnetic]{} scaling field $u_h$ associated with the overlap variables by $u_h=h\overline{u}_h(T)+ O(h^2)$. We have already checked that the thermal scaling field $u_T$ can be effectively approximated by $u_T = T$. Thus, neglecting nonanalytic scaling corrections, the data should behave as $$\ln \chi_o = (2-\eta_o) \ln L + \ln \overline{u}_h(T)^2 + \ln f(T L^{1/\nu})~.$$ We now estimate $\eta_o$ from the analysis of the data at $\sigma=2/3$. In a first set of fits we set $\overline{u}_h = 1$ and approximate $\ln f(x)$ with a polynomial in $x$ of order $n$, i.e., we perform fits to $$\ln \chi_o = (2-\eta) \ln L + P_n(T L^{1/\nu}). \label{fit1-eta-glass}$$ The analysis of the $\chi^2$ of the fits indicate that $n=6$ allows us to describe accurately the data. --------------- ------ -------------- --------- -------------- ------------- $L_{\rm min}$ $\chi^2$/DOF $\chi^2$/DOF 20 0.5 11570/142 0.13(2) 338/140 $-$0.01(1) 20 0.4 1439/120 0.10(2) 99/118 0.02(1) 20 0.3 498/93 0.06(1) 39/91 0.04(3) 20 0.25 182/77 0.06(1) 20/75 0.01(3) 20 0.2 43/55 0.05(1) 12/53 $-$0.04(8) 30 0.5 6592/124 0.17(2) 263/122 $-$0.03(2) 30 0.4 1096/104 0.11(2) 78/102 0.01(2) 30 0.3 330/80 0.07(1) 35/78 0.04(3) 30 0.25 89/65 0.05(1) 18/63 0.00(4) 30 0.2 28/46 0.05(2) 11/44 $-$0.06(10) 40 0.5 4237/106 0.18(2) 177/104 $-$0.05(2) 40 0.4 1096/88 0.11(2) 40/86 $-$0.03(2) 40 0.3 294/67 0.07(1) 22/65 0.02(4) 40 0.25 63/53 0.05(1) 9/51 $-$0.02(6) 40 0.2 17/37 0.04(1) 2/35 $-$0.11(12) --------------- ------ -------------- --------- -------------- ------------- : Estimates of $\eta$. On the left we report the results of the fits to (\[fit1-eta-glass\]) with $n=6$, on the right those to (\[fit2-eta-glass\]) with $n=6$ and $m = 2$. In both cases we fix $\nu = 2.5(1)$. The reported errors are the sum of the statistical error and of the variation of the estimate of $\eta$ as $\nu$ changes by one error bar. DOF is the number of degrees of freedom of the fit.[]{data-label="eta-glass"} We fix $\nu$ to the estimate (\[nufinale\]) to avoid an additional nonlinear parameter in the fit. The results are reported in Table \[eta-glass\]. We observe a significant change of the estimates as $T_{\rm max}$ decreases; moreover, the quality of the fit is quite poor. This can be explained by the presence of sizeable analytic corrections, which means that $\overline{u}_h$ is poorly approximated by a $\overline{u}_h=1$ in our range of temperatures. The same phenomenon occurs in the three-dimensional Ising spin glass [@HPV-08], where the analytic corrections cannot be neglected in the analysis of the overlap susceptibility. We thus perform a second set of fits in which we take into account the magnetic nonlinear scaling field. If we approximate $\ln \overline{u}_h^2$ with a polynomial of order $m$, we end up with the fitting form $$\ln \chi_o = (2-\eta) \ln L + P_n(T L^{1/\nu}) + Q_m(T)~, \label{fit2-eta-glass}$$ where we assume $Q_m(0) = 0$. In the following we take $m=2$ and again fix $\nu$ to the estimate (\[nufinale\]). The results are reported in Table \[eta-glass\]. The quality of the fit is now significantly better, indicating that the analytic corrections are important. The scaling function $\overline{u}_h$ is reported in Fig. \[uh\] and indeed it varies significantly in the range of values of $T$ we are considering. The estimates of $\eta_o$ do not show any systematic variation with $T_{\rm max}$ and are always consistent, within errors, with $\eta_o = 0$. Quantitatively, our data allow us to set the upper bound $$\begin{aligned} \|\eta_o\| \le 0.05.\end{aligned}$$ Results for the gauge-glass model {#gaugeglass} --------------------------------- In order to check universality we also performed runs at $\sigma=\infty$, although in this case we considered smaller lattices and the errors are significantly larger (partly because of the smaller number of samples, partly because of larger sample-to-sample fluctuations). The data were analyzed as we did in the $\sigma=2/3$ case. First, we determined the critical temperature $T_c$. A fit of $\xi_{o}/L$ to (\[fitRxi\]) gives rather small estimates of $T_c$. For $L_{\rm min} = 20$ we obtain $T_c = 0.030(2)$ \[0.020(3)\] for $T_{\rm max} = 0.4$ (resp. 0.3). Thus, we can conclude that $T_c\lesssim 0.02$, which is clearly consistent with $T_c = 0$, given that our data belong to the range $T\ge 0.1$. The claim that $T_c\approx 0.2$ is not consistent with our MC data. Then, we determined $\nu$ by assuming $T_c = 0$. The results of the fits to (\[fitRxi\]) show a significant dependence on $T_{\rm max}$. For $L_{\rm min} = 20$, $\nu$ varies between 2.50(1) and 2.80(4) as $T_{\max}$ varies between 0.4 and 0.16. If analytic scaling corrections are included, i.e. we fit the data to (\[fitRxi-analytic\]), we observe a significantly smaller dependence on $T_{\max}$, but, on the other hand, a rather large dependence on $L_{\rm min}$, with rapidly increasing error bars as $L_{\rm min}$ increases. This is probably due to the fact that we have a somewhat large statistical error on the results with the largest value of $L$, $L=60$. The estimates of $\nu$ vary between 2.8 and 3.7 if we take $L_{\rm min} = 20$, 30 and $0.2\le T_{\rm max} \le 0.5$ and thus give the final result $\nu = 3.3(5)$. This result is somewhat larger than the estimate (\[nufinale\]), but certainly not inconsistent. It supports — very weakly, though—universality. A better check is presented below. The quartic coupling $g_{o}$ and universality {#g4odata} --------------------------------------------- We computed the overlap quartic coupling $g_{o}$ defined in (\[g4odef\]). MC results at $\sigma=2/3$ are shown in Fig. \[g4o\]. The infinite-volume limit, within our statistical accuracy, is apparently reached when $L/\xi_o \gtrsim 7$, corresponding to $T\gtrsim 0.3$ for our largest lattices $L=60,\,80$. The infinite-volume results are quite stable with respect to $T$, so that we can reliably estimate the critical ($T=0$) value $g_{o}^$. We obtain $$g_{o}^=13.0(5).$$ to According to standard RG arguments, $g_{o}$ has a universal FSS limit as a function of $R_{\xi_o}\equiv \xi_o/L$, that is $$g_o(T,L) = f(R_{\xi_o}),$$ where the function $f(x)$ is universal and satisfies $f(0) = g^_o$. This scaling behavior is nicely supported by the data at $\sigma=2/3$ for various lattice sizes, see Fig. \[g4ovsrxi\]. Universality can be checked by also considering the results for $\sigma=5/9$, $\sigma=1/2$ and $\sigma=\infty$. Clearly, all points fall on top of each other. Note that here there are no free parameters to fiddle with and thus this comparison provides strong support to the hypothesis that all these models belong to the same universality class. Given the very good evidence we have that the model with $\sigma=2/3$ undergoes a $T=0$ glassy transitions, this result further confirms (and provides stronger evidence than that given in the previous paragraph) that the gauge-glass model does not have a finite-temperature exotic glassy transition. Behavior of the magnetic correlation functions {#g4g22magn} ---------------------------------------------- Let us now consider the magnetic quantities. The magnetic correlation length $\xi$ is zero in the gauge-glass model, see \[App-magn\], and increases as one approaches the QLRO region. In particular, at $T=0.159$, which is below the critical temperature $T_M\approx 0.31$ along the Nishimori line, we obtain $\xi=3.3(1),\,6.7(4),\,9.8(3)$ at $\sigma=2/3,\,5/9,\,1/2$, respectively. They are roughly consistent with a behavior like $\ln \xi\sim (\sigma-\sigma_c)^{-\kappa}$ assuming $\sigma_c\approx \sigma_M \approx 0.30$, i.e., with a KT-like behavior along the transition line that connects the Nishimori critical point $M$, see Fig. \[phdia\], and the $T=0$ transition point at $\sigma=\sigma_D$, which is expected to run almost parallel to the $T$ axis. Note, however, that while our data suggest a power-law divergence of $\ln\xi$ (therefore, $\xi$ has an exponential divergence), they are not sufficiently precise to allow us to estimate the power $\kappa$. The KT value $\kappa=1/2$ is consistent with the data, but $\kappa=1$ would be equally reasonable. It is also interesting to discuss the behavior of the quartic couplings $g_c$, $g_4$, and $g_{22}$ defined from the magnetic correlation functions in (\[g4def\])-(\[gcdef\]). In \[App-magn\], assuming universality, we predict that, in the critical limit, $g_{4}$ and $g_{22}$ should diverge as $\xi_o^2$, while $g_c \xi_o^{-2}$ should go to zero. Numerical estimates of $g_c$ are shown in Fig. \[g4lt\]. The results are clearly consistent with a finite $T=0$ limit. Note that the estimates obtained for $\sigma=2/3$, 5/9, and 1/2 are close to the XY value $g_{4,XY}^=13.65(6)$; actually, they are consistent within errors, even at small $T$, below $T_M\approx 0.31$. These results are suggestive of a KT behavior of the magnetic correlation functions also along the disorder paramagnetic-QLRO transition line from $M$ to $D$, see Fig. \[phdia\]. Indeed, for $\sigma = 2/3$, 5/9, 1/2 we have $\xi \approx 3$, 7, 10, so that along these lines one should be able to observe the critical behavior that arises when one approaches the paramagnetic-QLRO transition line at a point with $T<T_M$. In other words, these results imply that the critical limit of $g_c(\sigma,T)$ at fixed $T<T_M$ along the paramagnetic-QLRO transition line is consistent with the KT value. This fact provides some evidence that also along the disorder-driven transition line magnetic correlation functions behave as in the pure XY model. Of course, as $\sigma$ increases (thus, the magnetic correlation length $\xi$ decreases), $g_c$ changes significantly and, for $\sigma=\infty$, $g_c$ is infinite for any $T$ and $L$. The couplings $g_{22}$ and $g_4$ are instead expected to diverge as $\xi_o^2$. In Fig. \[g22vsxio\] we report $g_{22}$ for the different models. The data are clearly diverging as $\xi\to\infty$, but the asymptotic behavior $g_{22}\sim \xi_o^2$ is not clearly observed, likely because the values of $\xi_o$ are not sufficiently large. Indeed, we only observe that $g_{22}$ behaves as $\xi_o^\kappa$ with $\kappa$ rapidly increasing with $\xi_o$. More precisely, if we only include data satisfying $\xi_o\lesssim 10$ we obtain $\kappa\approx 1$. If instead we fit the data with $10\lesssim \xi_o\lesssim 20$ (we have infinite-volume data only up to $\xi_o\approx 20$) we obtain $\kappa\approx 1.5$. Conclusions =========== We have studied the magnetic and glassy transitions of the square-lattice XY model in the presence of random phase shifts and, in particular, the GRPXY and CRPXY model defined by the distributions (\[GRPXYd\]) and (\[CRPXYd\]). The latter is very useful because it allows some exact calculations along the Nishimori line $T =\sigma$ [@ON-93; @Nishimori-02], where, in particular, the magnetic and overlap two-point functions are equal. We present MC for the GRPXY and CRPXY models for several values of the temperature and of the parameter $\sigma$ controlling the disorder, approaching the magnetic and glassy transition lines from the paramagnetic phase. We substantially confirm the phase diagram shown in Fig. \[phdia\]. Our main results are the following. - We have carefully investigated the critical behavior along the transition line separating the paramagnetic and QLRO phases, from the pure XY point $P$ to the multicritical point, which, in the CRPXY model, lies on the N line and is such that the transition line runs parallel to the $T$ axis. The magnetic observables show a $\sigma$-independent KT behavior: the magnetic correlation length behaves as $\ln \xi \sim u_t^{-1/2}$, where $u_t$ is the thermal scaling field, $u_t\sim T-T_c(\sigma)$, and the magnetic susceptibility as $\chi\sim \xi^{7/4}$ (corresponding to $\eta=1/4$). Moreover, the quartic coupling $g_c$ defined in (\[gcdef\]) appears to be universal. We obtain $g^_c \approx 13.6$, which is nicely consistent with the corresponding value $g^_{4,XY}=13.65(6)$ of the pure XY model [@BNNPSW-01; @PV-r]. We have also verified the universality of the leading logarithmic correction to the critical behavior of $\chi$. On the other hand, the critical behavior of disorder-related quantities, such as those related to the overlap correlation function, depends on $\sigma$. - In the CRPXY model, the Nishimori point $M$, see Fig. \[phdia\], is a multicritical point which divides the paramagnetic-QLRO line into two parts: a thermally-driven transition line (from $P$ to $M$) and a disorder-driven transition line (from $M$ to $D$). This result should be general: a multicritical point should also exist in generic RPXY models, although in this case it is not expected to coincide with that where the transition line runs parallel to the $T$ axis. Such a multicritical point is characterized by the fact that, at criticality, magnetic and overlap functions have the same critical behavior, that is $\eta=\eta_o$: in the CRPXY model the two correlation functions are exactly equal (more generally, they are equal on the whole N line), but we do not expect this property to be generic. It is interesting to observe that the multicritical behavior is only observed in the disorder-related quantities. Magnetic observables behave, as far as the leading behavior is concerned, as in the pure XY model: the correlation length shows a KT behavior, $\eta = 1/4$, and $g^_c = g^_{4,XY}$ in the whole neighborhood of the multicritical point. However, corrections are different from those appearing in the pure XY model, providing additional evidence for the presence of an additional (probably marginal) RG operator, which is responsible for the multicritical behavior. - Little is known about the behavior along the transition line from the multicritical point to $D$. However, the fact that purely magnetic observables behave as in the pure XY model both along the thermally-driven transition line and at the multicritical point make us conjecture that the magnetic behavior is also unchanged. We have presented some very weak evidence in Sec. \[g4g22magn\]. - We have investigated the critical behavior for large values of $\sigma$. We find no evidence of a finite-temperature transition for all values of $\sigma$ we have investigated: the system is paramagnetic up to $T=0$, where a glassy transition occurs. Morever, in all cases we verify universality. We can thus conjecture that the critical behavior along the whole line that starts in $D$, see Fig. \[phdia\], is universal: for any $\sigma > \sigma_D$, one has the same critical behavior characterized by the exponents: $$\nu = 2.5(1), \quad 1/\nu = 0.40(2), \qquad \|\eta_o\| \le 0.05.$$ Our estimate of $\nu$ is consistent with earlier estimates obtained by MC simulations of the gauge-glass XY model, for examples $1/\nu=0.39(3)$ and $1/\nu=0.36(3)$ obtained in [@KY-02] and [@NW-04] respectively, and by numerical calculations of the stiffness exponent at $T=0$, for example $1/\nu=0.36(1)$ and $1/\nu\approx 0.45$ obtained in [@KY-02] and [@TT-05]. Our result for $\eta_o$ is consistent with a general argument which predicts $\eta_o = 0$. Details on the Monte Carlo simulation {#AppMC} ===================================== In the simulation we use both Metropolis and microcanonical local updates. The latter do not change the energy of the configuration and are defined as follows. Consider a site $i$; the corresponding field is $\psi_i$. The terms of the Hamiltonian that depend on $\psi_i$ can be written as $${\cal H}_i = {\rm Re}\, (\overline{\psi}_i z), \qquad \qquad z \equiv \sum_j U_{ij} \psi_j ,$$ where the sum is over all nearest neighbors $j$ of site $i$. Then, define $$\psi'_i = 2 {z\over \|z\|^2}{\rm Re}\, (\overline{\psi}_i z) - \psi_i$$ One can verify that $\|\psi'_i\| = 1$ and that $${\rm Re}\, (\overline{\psi}_i z) = {\rm Re}\, (\overline{\psi}_i' z).$$ Thus, the update $\psi_i\to\psi_i'$ does not change the energy and can therefore be always accepted. This update does not suffer the limitations of the Metropolis update: $\psi_i$ and $\psi_i'$ are not close to each other. In our simulation a MC step consists of 5 microcanonical sweeps over all the lattice followed by one Metropolis sweep. For each disorder sample we typically perform $O(10^5)$ MC steps. In some simulations of the CRPXY model we also use the parallel tempering method [@raex; @par-temp]. It allows us to obtain results for small values of $T$, in particular below the Nishimori line $T = \sigma$. In the parallel-tempering simulations we consider $N_T$ systems at the same value of $\sigma$ and at $N_T$ different inverse temperatures $\beta_{\rm min} \equiv \beta_1$, …, $\beta_{\rm max}$, where $\beta_{\rm max}$ corresponds to the minimum value of the temperature we are interested in. The value $\beta_{\rm min}$ is chosen so that thermalization at $\beta=\beta_{\rm min}$ is sufficiently fast, while the intermediate values $\beta_i$ are chosen so that the acceptance probability of the temperature exchange is at least $5\%$. Moreover, we require that, for some $i$, $\beta_i = \sigma$. This allows us to collect data on the Nishimori line. The exact results valid on it allow us to check the correctness of the MC code and perform a (weak) test of thermalization. Thermalization is checked by verifying that the averages of the observables are independent of the number of MC steps for each disorder realization. The overlap correlations and the corresponding $\chi_o$ and $\xi_o$ are measured by performing two independent runs for each disorder sample. Finally, note that the determination of $g_{22}$ defined in (\[g22def\]) requires the computation of the disorder average of products of thermal expectations. This should be done with care in order to avoid any bias due to the finite length of the run for each disorder realization. We use the essentially unbiased estimators discussed in [@HPPV-07-bias; @HPV-08]. The KT RG equations {#AppRGsigmazero} =================== In this Appendix we consider the RG flow for the sine Gordon (SG) model, with the purpose of understanding its universal features. As a results we shall obtain the critical behavior of the correlation length and of the magnetic susceptibility at the KT transition. This appendix generalizes the results presented in [@AGG-80; @BH-00; @Balog-01]. The SG model is parametrized by two couplings, $\alpha$ and $\delta$—we use the notations of [@AGG-80; @Balog-01]—whose $\beta$ functions are $$\begin{aligned} &&\beta_\alpha = 2\alpha\delta + {5\over 64} \alpha^3 + \ldots,\\ &&\beta_\delta = {1\over 32} \alpha^2 - {1\over 16} \alpha^2\delta + \ldots,\end{aligned}$$ where the dots indicate higher-order terms. To all orders the $\beta$ functions have the generic form $$\begin{aligned} &&\beta_\alpha = 2\alpha\delta + \sum_{n+m>2} b_{\alpha,nm} \alpha^n \delta^m, \\ && \beta_\delta = {1\over 32} \alpha^2 + \sum_{n+m>2} b_{\delta,nm} \alpha^n \delta^m.\end{aligned}$$ In the SG model the sign of $\alpha$ is irrelevant, which implies the symmetry relations $$\beta_\alpha(\alpha,\delta) = - \beta_\alpha(-\alpha,\delta), \qquad\qquad \beta_\delta(\alpha,\delta) = \beta_\delta(-\alpha,\delta). \qquad\qquad$$ As a consequence, $b_{\alpha,nm} = 0$ if $n$ is even and $b_{\delta,nm} = 0$ if $n$ is odd. Moreover, for $\alpha = 0$ the theory is free and $\delta$ does not flow. Hence $$\beta_\delta(\alpha=0,\delta) = 0,$$ which implies $b_{\delta,nm} = 0$ if $n=0$. Let us now consider a general nonlinear analytic redefinition of the couplings $$\begin{aligned} \alpha &=& a_{\alpha,10} u + \sum_{n+m\ge 2} a_{\alpha,nm} u^n v^m, \\ \delta &=& a_{\delta,01} v + \sum_{n+m\ge 2} a_{\delta,nm} u^n v^m.\end{aligned}$$ We have verified up to the $7^{\rm th}$ order that with a proper choice of the coefficients $a_{\alpha,nm}$ and $a_{\delta,nm}$ one can rewrite the $\beta$ functions in the form $$\begin{aligned} \beta_u(u,v) &=& - u v, \label{betau}\\ \beta_v(u,v) &=& - u^2 (1 + b_1 v + b_3 v^3 + b_5 v^5 + \ldots). \label{betav-1} \end{aligned}$$ The couplings $u$ and $v$ are not uniquely defined and indeed there is a family of transformations that do not change the $\beta$ functions (\[betau\]) and (\[betav-1\]). Extending the previous results to all orders, in the following we assume that we can choose $u$ and $v$ in such a way that $\beta_u(u,v)$ is given by (\[betau\]) and $\beta_v(u,v)$ has the form $$\beta_v(u,v) = - u^2 [1 + v f(v^2)], \label{betav}$$ where $f(v^2)$ is an analytic function in the region $v<v_0$, where $v_0$ is the starting point of the RG flow, and satisfies $1 + v f(v^2)>0$ in this domain (if this were not true, we would have another nontrivial fixed point). This parametrization is unique (universal) in the sense that there is no analytic redefinition of the couplings which allows one to write the $\beta$ functions in the form (\[betau\]), (\[betav\]) with a different function $f(v^2)$, i.e. with different coefficients $b_{2n+1}$. The perturbative calculations of [@AGG-80] allow us to determine $b_1$: $$b_1 = -{3\over2}. \label{b1calc}$$ The analysis of the flow in the general case is analogous to that presented in [@AGG-80; @Balog-01]. First, we define the RG invariant function $$\begin{aligned} &&Q(u,v) = u^2 - F(v), \label{qdef} \\\ &&F(v) = 2 \int_0^v {w dw\over 1 + w f(w^2)} = v^2 + v^3 + {9\over 8} v^4 + O(v^5), \nonumber\end{aligned}$$ which satisfies $${dQ\over dl} = {\partial Q\over \partial u} \beta_u(u,v) + {\partial Q\over \partial v} \beta_v(u,v) = 0,$$ where $l$ is the flow parameter. The RG flow follows the lines $Q =\, $constant. It is thus natural to parametrize the RG flow in terms of $Q$ and $v(l)$. Since $${dv\over dl} = \beta_v(u,v) = - [Q + F(v)][1 + v f(v^2)], \label{betavqf}$$ we obtain $$l = - \int_{v_0}^v {dw\over [Q + F(w)][1 + w f(w^2)]},$$ where $v(l=0) = v_0$. Let us now apply these results to the XY model. Repeating the discussion of [@Kosterlitz-74; @JKKN-78] the XY model can be mapped onto a line in the $(u,v)$ plane with $v > 0$. The KT transition is the intersection of this line with the line $Q=0$ and the high-temperature phase corresponds to $Q > 0$. Thus, $Q$ plays the role of thermal nonlinear scaling field, i.e. $$Q = q_1 \tau + q_2 \tau^2 + \ldots$$ where $\tau = (T-T_{XY})/T_{XY}$. To derive the expected critical behavior we consider the singular part of the free energy in a box of size $L$. It satisfies the scaling equation [@Wegner-76] $${\cal F}_{\rm sing} (\tau,L) = e^{-2l} f(Q,v(l),e^{-l}L)~,$$ where we have parametrized the flow in terms of $Q$ and $v(l)$ and we have neglected all irrelevant operators. If $Q > 0$, as discussed in [@AGG-80], $v(l)$ decreases continuously and $v(l)\to -\infty$ as $l\to \infty$. Since $v_0$, the starting point of the flow, is positive, we can fix $l$ be requiring $$v(l) = -1,$$ so that $$l = \int^{v_0}_{-1} {dw\over [Q + F(w)][1 + w f(w^2)]} = I(Q,v_0).$$ It follows $${\cal F}_{\rm sing}(\tau,L) = e^{-2I(Q,v_0)} f(Q,-1,e^{-I(Q,v_0)}L)~,$$ which gives the scaling behavior of the free energy (using $Q\sim \tau$). In the scaling limit the finite-size dependence can be parametrized in terms of $\xi/L$, where $\xi$ is the correlation length. This allows us to identify $$\xi(\tau) = \xi_0 e^{I(Q,v_0)}~,$$ where $\xi_0$ is a constant. The behavior of $\xi(\tau)$ for $\tau \to 0$ is obtained by expanding $I(Q,v_0)$ for $Q\to 0$. The generic behavior is $$I(Q,v_0) = {1\over \sqrt{Q}} \sum_n I_n Q^n + \sum_n I_{{\rm an},n}(v_0) Q^n .$$ The nonanalytic terms in the expansion depend only of the coefficients $b_{2n+1}$ which appear in (\[betav-1\]). The first two coefficients are $$\begin{aligned} I_0 &=& \pi, \nonumber \\ I_1 &=& {\pi b_1\over 4} = {9 \pi\over 16} .\end{aligned}$$ Correspondingly, we obtain $$\xi(\tau) = X \exp (\pi/\sqrt{Q}) [1 + I_1 \sqrt{Q} + O(Q)]. \label{xiKT-Q}$$ Expanding $Q$ in powers of $\tau$ we obtain the celebrated KT expression for the correlation length. Let us now consider the behavior of the susceptibility. Perturbation theory gives for the scaling dimension of the spin correlation function [@AGG-80] $$\begin{aligned} &&\gamma = -{1\over 4} + {1\over 4}\delta - {1\over 4} \delta^2 + h_1\alpha^2+ \ldots , \end{aligned}$$ where $h_1$ is an unknown coefficient. If we perform the redefinitions $(\alpha,\delta) \to (u,v)$ considered before, we can rewrite $\gamma$ as[^4] $$\begin{aligned} &&\gamma = -{1\over 4} - {1\over 8}v - {1\over 16} v^2 + \ldots \label{gammapert}\end{aligned}$$ without the $\alpha^2$ term. In the infinite-volume limit the susceptibility satisfies the scaling law $$\chi\xi^{-7/4} = A \exp\left[\int_{v_0}^{v(l)} {\gamma(w) + 1/4\over \beta_v} dw \right] G_\chi[Q,v(l)], \label{chisuxi7/4-RG}$$ the integral is computed at fixed $Q$ with $\beta_v$ given by (\[betavqf\]), and $G_\chi$ is an analytical function. Setting $v(l)=-1$ and expanding the integral in powers of $Q$, we obtain an expansion of the form $$\chi\xi^{-7/4} = A(1 + c_1 \sqrt{Q} + c_2 Q + \ldots).$$ The coefficient $c_1$ can be computed exactly using the perturbative results (\[betav-1\]), (\[b1calc\]), and (\[gammapert\]), obtaining $$c_1 = {\pi\over 16}.$$ Using (\[xiKT-Q\]) we can write $$\sqrt{Q} = {\pi\over\ln \xi/X} + O(\ln^{-3}\xi)$$ and obtain $$\chi\xi^{-7/4} = A_\chi \left[1 + {\pi^2\over 16 \ln(\xi/X)} + O(1/\ln^{2}\xi) \right]. \label{chi-expRG}$$ Note that the leading logarithmic scaling correction has a universal coefficient. We should note that in [@Balog-01] it was incorrectly claimed that $c_1 = 0$ and, as a consequence, that the leading scaling corrections in (\[chi-expRG\]) are proportional to $1/(\ln\xi)^2$. We numerically checked (\[chi-expRG\]) by fitting the infinite-volume numerical data of [@BNNPSW-01] (more precisely their data for $\beta\ge 0.92$, corresponding to $10\lesssim \xi\lesssim 420$) to $$\begin{aligned} \ln (\chi \xi^{-7/4}) = a + {b\over \ln(\xi/X)} ,\label{fita}\end{aligned}$$ obtaining $a=0.804(2)$ and $b=0.627(9)$ (with $\chi^2/{\rm DOF}\approx 0.7$), which is perfectly consistent with the value of $b$ obtained in perturbation theory, i.e. $b=\pi^2/16\approx 0.617$ (fixing $b=\pi^2/16$, we obtain $a=0.8058(1)$ with $\chi^2/{\rm DOF}\approx 0.7$, while a fit to $a + {b/\ln(\xi/X)} + {c/\ln^2(\xi/X)}$ gives $a=0.8046(9)$, $c=0.029(22)$ with $\chi^2/{\rm DOF}\approx 0.6$, which confirms that the next-to-leading correction is very small in (\[fita\])). The result (\[chi-expRG\]) is general. If ${\cal O}$ is a generic long-distance quantity which behaves as $\xi_o^{x}$ in the critical limit, we expect ${\cal O}/\xi_o^{x}$ to behave as $\chi/\xi^{7/4}$, i.e. to satisfy a relation analogous to (\[chisuxi7/4-RG\]). It is only needed to replace $\gamma(u,v) + 1/4$ with the appropriate subtracted scaling dimension. Thus, ${\cal O}/\xi_o^{x}$ also has an expansion of the form (\[chi-expRG\]), i.e. $${\cal O} = \xi_o^{x} \left[1 + {c_{\cal O}\over \ln \xi/X} + O(\ln^{-2}\xi) \right],$$ where $c_{\cal O}$ is universal and can be computed by using the perturbative expression of the scaling dimension of ${\cal O}$. More precisely, if the scaling dimension $\gamma_{\cal O}(u,v)$ has the perturbative expansion $$\gamma_{\cal O}(u,v) = g_{00} + g_{01} v + g_{02} v^2 + g_{20} u^2 + \ldots$$ we obtain $$c_{\cal O} = - \pi g_{02}.$$ Corrections proportional to $1/\ln \xi/X$ should instead be absent in RG invariant quantities. Indeed, if $R$ is such a quantity, if we neglect the scaling corrections, $R$ satisfies the scaling relation $$R(\tau) = G_R[Q,v(l)],$$ for any $l$. This implies that $R(\tau)$ is independent of $v(l)$, hence an analytic function of $Q$ and therefore of $\tau$. It follows $$R(\tau) = R^* + {c_R\over\ln^2 \xi/X} + O(\ln^{-4}\xi),$$ where the costant $c_R$ is expected to be universal. General behavior close to a critical point {#Appirrelevant} ========================================== Let us consider a multicritical point in a two-parameter space labelled by $T$ and $\sigma$ and let us assume that the correlation length behaves as $$\begin{aligned} \xi(T,\sigma) &\sim& [T-T_{c}(0)]^{-\nu_1} \qquad\qquad \sigma = 0, \label{xisigma0} \\ \xi(T,\sigma) &\sim& [T-T_{c}(\sigma)]^{-\nu_2} \qquad\qquad \sigma > 0, \label{xisigmaneq0} \end{aligned}$$ where $T_{c}(\sigma)$ is the $\sigma$-dependent critical point and $\nu_1 \not=\nu_2$. According to the RG, close to the multicritical point $\xi(T,\sigma)$ behaves as $$\xi(T,\sigma) = u_t(T,\sigma)^{-\nu_{m}} F[u_\sigma(T,\sigma) u_t(T,\sigma)^{-\phi}], \label{MCP}$$ where $u_\sigma(T,\sigma)$ and $u_t(T,\sigma)$ are the scaling fields and $\phi$ and $\nu_{m}$ two critical exponents. Since one of the two scaling fields must vanish along the transition line, we define $u_t(T,\sigma)$ as the scaling field which has this property. Therefore, we define $$\begin{aligned} u_t(T,\sigma) = {T - T_c(\sigma) \over T_c(0)}~.\end{aligned}$$ For $\sigma\to0$ and $T\to T_c(0)$, it behaves as $$u_t(T,\sigma) = \tau + c_\sigma \sigma + \ldots \qquad \tau \equiv {T - T_c(0) \over T_c(0)}.$$ We assume that $c_\sigma\not=0$, i.e. that the transition line is not perpendicular to the line $\sigma =0$, as it occurs in the RPXY model. Finally, we note that $u_\sigma(T,\sigma)$ does not vanish on the transition line, unless $\sigma = 0$. Now consider $T\to T_c(\sigma)$ at fixed novanishing $\sigma$. Since $u_\sigma(T,\sigma)\not=0$ we obtain (\[xisigmaneq0\]) only if $$F(x) \sim x^\lambda\qquad \qquad \lambda = {\nu_2 - \nu_{m}\over \phi} \label{scalF}$$ for $x \to \infty$. To go further let us distinguish two cases: (i) $u_\sigma(T,\sigma)$ vanishes identically for $\sigma=0$, i.e. $u_\sigma(T,0)=0$ for any $T$; (ii) $u_\sigma(T,0)$ is different from zero unless $T=T_c(\sigma=0)$. In case (i) (\[xisigma0\]) requires $$F(0) \not=0, \qquad\qquad \nu_{m} = \nu_1.$$ Assuming $u_\sigma(T=T_c(0),\sigma) = d_\sigma\sigma$ for $\sigma\to 0$ we obtain $$\xi(T=T_c(0),\sigma) = (c_\sigma \sigma)^{-\nu_1} F(d_\sigma c_\sigma^{-\phi} \sigma^{1-\phi}).$$ The observed behavior depends on the value of $\phi$. For $\phi< 1$, since $F(0) \not=0$ we obtain $$\xi(T=T_c(0),\sigma) = (c_\sigma \sigma)^{-\nu_1} (a + b \sigma^{1-\phi} + \ldots )$$ The corrections are correct provided that $F(x)$ is analytic for $x=0$. If $\phi > 1$, using (\[scalF\]) we obtain the behavior $$\xi(T=T_c(0),\sigma) \sim \sigma^{-\overline{\nu}} \qquad \overline{\nu} = \nu_1 - (1-\phi)\lambda = {\nu_2(\phi-1) + \nu_1\over \phi}~.$$ In case (ii), if $u_\sigma(T,\sigma=0)=d_T \tau + O(\tau^2)$ we obtain for $\sigma = 0$ $$\xi(T,0) = \tau^{-\nu_{m}} F(d_T \tau^{1-\phi}),$$ which shows that $$F(d_T \tau^{1-\phi}) \sim \tau^{\nu_m - \nu_1} \label{scal-F-App}$$ in the limit $\tau\to 0$. Let us now consider the behavior for $T = T_c(0)$ as a function of $\sigma$. For $\sigma\to 0$ we have $$\xi(T=T_c(0),\sigma) = c_\sigma^{-\nu_m} \sigma^{-\nu_m} F(d_\sigma c_\sigma^{-\phi}\sigma^{1-\phi}) \sim \sigma^{-\nu_1},$$ where we have used relation (\[scal-F-App\]). Thus, in case (ii) we have $\xi(T=T_c(0),\sigma)\sim \sigma^{-\nu_1}$ for any value of $\phi$. Let us now show that the case relevant for the RPXY model is case (i). Indeed, case (ii) can only occur if the two relevant operators which occur at the multicritical point are both present in the model at $\sigma = 0$. This does certainly not occur in our case in which $\sigma$ is associated with randomness. Therefore, our result that in the RPXY model $\xi(T=T_c(0),\sigma)$ behaves as $\sigma^{-\nu_1}$ implies that $\phi < 1$, i.e. that the RG dimension of the new operator that arises in the theory with $\sigma\not=0$ is less relevant than the thermal operator present at $\sigma = 0$. This is also the case of three-dimensional randomly dilute Ising systems or $\pm J$ Ising models at their ferromagnetic transitions at small disorder. Indeed, the crossover from the pure critical behavior to that of the randomly-dilute Ising universality class is described by the crossover exponent $\phi=\alpha_{\rm Is}=0.1096(5)$ [@PV-r; @CPRV-02], see also the discussion reported in [@HPPV-07]. Similar considerations apply to other quantities. For instance, consider a RG invariant quantity $R$. It behaves as $$R(T,\sigma) = r[u_\sigma(T,\sigma) u_t(T,\sigma)^{-\phi}].$$ If $\phi<1$, $R(T,\sigma)$ approaches the same value $R^$ along the lines $\sigma = 0$ and $T=T_c(0)$. Morover, in the second case we expect corrections of the form $$R(T_c(0),\sigma) = R^ + a \sigma^{1-\phi} + \ldots= R^* + a' \xi^{(\phi-1)/\nu_1} + \ldots$$ RG equations in the presence of randomness {#AppRGdisordine} ========================================== The RG equations in the small disorder regime and close to the paramagnetic-QLRO transition line have been derived in [@RSN-83; @NSKL-95; @Tang-96; @Scheidl-97; @CL-98]: $$\begin{aligned} &&{d T \over dl} = - 4 \pi^3 Y^2, \nonumber \\ && {d\sigma\over dl} = 0,\nonumber \\ && {dY\over dl} = (2 - \pi \beta + \pi \sigma \beta^2) Y, \nonumber\end{aligned}$$ where $Y$ is the vorticity and only terms up to $O(Y^2)$ are kept. Let us now redefine the couplings as follows: $$\begin{aligned} && T^{-1} = {1\over \pi} (2 + v + \sigma), \nonumber \\ && Y = {u\over 4\pi}.\end{aligned}$$ For $u,v\to 0$ the RG equations become $$\begin{aligned} && {du\over dl} = - uv, \nonumber \\ && {dv\over dl} = - u^2, \nonumber \\ && {d\sigma\over dl} = 0. \label{RGeq-disorder}\end{aligned}$$ We have thus reobtained the RG equations for the XY model. This implies that, in the region of couplings in which (\[RGeq-disorder\]) hold, the RG behavior is analogous to that close to the KT fixed point, apart from an analytic redefinition of the scaling fields. Magnetic correlations in the gauge-glass model {#App-magn} ============================================== For the gauge-glass model ($\sigma=\infty$) we can derive some identities which relate magnetic and overlap quantities. The basic observation is that for $\sigma = +\infty$ the distribution function of the $A_{xy}$ variables is gauge-invariant. Hence we have $$\begin{aligned} && [\langle {\psi}_{x_1}^* \ldots {\psi}_{x_n}^* \psi_{y_1} \ldots \psi_{y_n} \rangle ] = V^_{x_1} \ldots V^_{x_n} V_{y_1} \ldots V_{y_n} [\langle {\psi}_{x_1}^* \ldots {\psi}_{x_n}^* \psi_{y_1} \ldots \psi_{y_n} \rangle ],\end{aligned}$$ for any set of phases $V_x$. It implies that magnetic correlations vanish unless each $x_i$ is equal to some $y_j$. Analogously we have $$\begin{aligned} && [\langle {\psi}_{x_1}^* \ldots {\psi}_{x_n}^* \psi_{y_1} \ldots \psi_{y_n} \rangle \langle {\psi}_{z_1}^* \ldots {\psi}_{z_n}^* \psi_{t_1} \ldots \psi_{t_n} \rangle ] = \\ && = V^_{x_1} \ldots V^_{x_n} V_{y_1} \ldots V_{y_n} V^_{z_1} \ldots V^_{z_n} V_{t_1} \ldots V_{t_n} [\langle {\psi}_{x_1}^* \ldots {\psi}_{x_n}^* \psi_{y_1} \ldots \psi_{y_n} \rangle \langle {\psi}_{z_1}^* \ldots {\psi}_{z_n}^* \psi_{t_1} \ldots \psi_{t_n} \rangle]. \nonumber \end{aligned}$$ These relations allow us to write $$\begin{aligned} && [\langle {\psi}_{x}^* \psi_{y} \rangle ] = \delta_{xy}, \\ && [\langle {\psi}_{x_1}^* {\psi}_{x_2}^* \psi_{y_1} \psi_{y_2}\rangle ] = \delta_{x_1y_1} \delta_{x_2y_2} + \delta_{x_1y_2} \delta_{x_2y_1} - \delta_{x_1y_1} \delta_{x_1x_2} \delta_{x_1y_2}, \\ && [\langle {\psi}_{x_1}^* \psi_{y_1} \rangle \langle {\psi}_{x_2}^* \psi_{y_2}\rangle ] = \delta_{x_1y_1} \delta_{x_2y_2} + \delta_{x_1y_2} \delta_{x_2y_1} [\|\langle {\psi}_{x_1}^* \psi_{y_1} \rangle\|^2] - \delta_{x_1y_1} \delta_{x_1x_2} \delta_{x_1y_2}.\end{aligned}$$ It follows $$\begin{aligned} && [\langle \| \mu \|^2\rangle ] = V, \nonumber \\ && [\langle \| \mu \|^4\rangle ] = 2 V^2 - V, \nonumber \\ && [\langle \| \mu \|^2\rangle^2 ] = V^2 + V^2 \chi_o - V,\end{aligned}$$ which imply $$\begin{aligned} && \chi = 1, \nonumber \\ && \chi_4 = 1 - 2 \chi_o, \nonumber \\ && \chi_{22} = \chi_o - 1. \label{refchi}\end{aligned}$$ Moreover, it is easy to show that $\xi = 0$. Relations (\[refchi\]) show that $\chi_4$ and $\chi_{22}$ both diverge as $\chi_o$. In the critical limit we have $\chi_o\sim \xi_o^2$ because $\eta_o = 0$. Therefore we can write $$\chi_4 \approx - 2 a \xi_o^2, \qquad\qquad \chi_{22} \approx a \xi_o^2, \label{relchi}$$ for $\xi_o\to \infty$, where $a$ is constant. We shall now assume that these results are valid for the whole universality class: for any $\sigma > \sigma_D$, relations (\[relchi\]) always hold with a constant $a$ which in general depends on $\sigma$. We can reexpress these results in terms of the quartic couplings. If we use (\[relchi\]) we have $$\begin{aligned} g_4 &=& {3 a \xi_o^2\over \chi^2 \xi^2}, \\ g_{22} &=& -{ a \xi_o^2\over \chi^2 \xi^2}, \label{as-gc}\end{aligned}$$ Since the magnetic susceptibility $\chi$ and correlation length $\xi$ are finite and nonzero (except for $\sigma=\infty$, where anyhow the quartic couplings are not well-defined since $\xi=0$ for any $L$), we expect that $g_4$ and $g_{22}$ diverge as $\xi_o^2$ in the critical limit. As for $g_c=g_4+3g_{22}$, (\[as-gc\]) shows that the leading $\xi^2_o$ term cancels. Since in the calculation we have neglected the scaling corrections to (\[relchi\]), this does not necessarily imply that $g_c$ remains finite in the critical limit, but only that $g_c \xi_o^{-2} \to 0$ as $\xi_o\to\infty$. The exact behavior depends on the neglected scaling corrections. These predictions are confirmed by our numerical results, see Sec. \[g4g22magn\]. 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Phys.]{} [58]{} 801 [^1]: We mention that the first renormalization-group (RG) analyses based on a Coulomb-gas description [@RSN-83] predicted $\sigma_D = 0$, but it was later clarified that this was an artefact of the approximations. Indeed, experimental [@FLTL-88] and numerical works [@FLTL-88; @CD-88; @FBL-90; @MG-97] as well as refinings of the RG arguments [@ON-93; @NSKL-95; @KN-96; @Tang-96; @Scheidl-97; @CL-98], have shown the absence of a reentrant transition for sufficiently small values of $\sigma$. [^2]: Equation (\[KTb2\]) holds whatever the definition of the correlation length is, but of course $X$ depends on the specific choice for $\xi$. Reference [@HP-97] studied the exponential correlation length $\xi_{\rm gap}$, which is defined as the inverse of the mass gap, and determined the corresponding constant $X_{\rm gap} = 0.233(3)$. Since in the critical limit [@CPRV-96] $\xi^2/\xi_{\rm gap}^2= r=0.9985(5)$, the constant $X$ for the second-moment correlation length we use is given by $X = X_{\rm gap} \sqrt{r} = 0.233(3)$. [^3]: In a Gaussian theory without disorder, in which the magnetic correlation function is given by $\widetilde{G}(p)=(p^2+m^2)^{-1}$, one can easily find that $\xi_o/\xi=\sqrt{1/6}=0.408248...$ [^4]: The possibility of cancelling the term of order $\alpha^2$ is related to the existence of a family of transformations transformations, given at second order by $u' = u + A u v $, $v' = v + A u^2$ with arbitrary $A$, which leave invariant the $\beta$-functions (\[betau\]) and (\[betav-1\]). By properly choosing $A$ one can eliminate the $\alpha^2$ term in $\gamma(u',v')$.
\#1\#2 =0.000753 plus 2pt minus 1pt \#1 \#2 Introduction ============ Colossal magnetoresistance manganites provide an opportunity to study systems in which strong coupling among charge, spin, orbital, and lattice degrees of freedom determine bulk properties [@MILLIS]. Many experiments indicate the presence of local entities in these compounds. Neutron scattering experiments find small magnetic clusters in the paramagnetic state for which the correlation length does not diverge at T$_c$ [@DETERESA]. A large diffusive peak seen by inelastic neutron scattering [@LYNN; @BACA], and a broad distribution of muon spin relaxation rates [@HEFFNER] indicate that clusters persist far below the Curie temperature. The dynamics and trapping of these clusters may be responsible for the large magnetoresistance and other unusual transport and magnetic properties. Neutron pair distribution function [@BILLINGE; @LOUCA] and X-ray absorption fine structure measurements [@BOOTH; @TYSON] demonstrate the presence of local distortions of MnO$_6$ octahedra in which the Mn-O bond lengths differ significantly from the average value. While these distortions change during the nearly simultaneous ferromagnetic and metal-insulator transitions, they persist into the metallic regime. The precise role these clusters and distortions play in the behavior of the manganites remains unclear. Because characterization of local properties is needed to understand the manganites, information from a local probe such as Nuclear Magnetic Resonance (NMR) should be valuable. NMR measurements determine properties at a well-defined point in the system– the nuclear site– but do not require that the system exhibit long range order. In fact, NMR can often characterize the degree of disorder in a system. Analysis of the NMR spectrum determines the size and distribution of static local magnetic susceptibilities and lattice distortions. NMR spin-lattice relaxation rate measurements characterize extremely low energy fluctuations (on the order of 0.1$\mu$eV) of electron spins or of the lattice. This paper presents $^{139}$La NMR spectrum and nuclear spin-lattice relaxation measurements of\ La$_{2/3}$Ca$_{1/3}$MnO$_3$ (T$_c$=268K). Measurements were performed in the paramagnetic state (292K-575K) in high magnetic fields (2.00-9.40 Tesla). The sample is a powder synthesized by a standard solid state procedure. Chemical analysis found no traces of contaminants. T$_c$ was determined by magnetization measurements. The magnetization transition is sharp, the full width at half maximum of the $\frac{dM}{dT}$ curve at T$_c$ in a 200 Oe field being 8K. The width of the transition compares favorably with that measured in other ceramic samples [@HWANG] and indicates that the sample is single phase with small deviation from the main composition. The paper begins with an introduction to general properties of a $^{139}$La NMR spectrum. An analysis of the high temperature data demonstrates that the spectrum is a standard powder pattern, broadened to a significant degree by a variation in lattice distortions around different lanthanum sites. As the temperature lowers towards T$_c$, the spectrum shifts and broadens due to the onset of ferromagnetism. The shift is proportional to the electron spin polarization while the broadening reflects a distribution of susceptibilities. Close inspection of the spectrum indicates that the lattice distortions do not change in the temperature range studied. Spectral diffusion measurements show no signs of the freezing out of motion of magnetic polarons. After an introduction to some of the principles of nuclear spin relaxation, we provide evidence that magnetic fluctuations, and hence electron spin dynamics, cause the relaxation. Finally, the field dependence of the nuclear spin-lattice relaxation reveals unusual and slow electron spin dynamics in the paramagnetic state. MEASUREMENTS AND DISCUSSION =========================== The $^{139}$La nucleus has a spin I=$7/2$ and a magnetic dipole moment $\mu=\gamma_n\hbar\hat{I}$, where $\gamma_n$ is the gyromagnetic ratio of the nucleus. For $^{139}$La, $^{139}\gamma/2\pi$=6.014MHz/Tesla. The interaction between the magnetic dipole moment and a magnetic field, B, creates 2I+1 energy levels with energies $E_m=m\gamma_n\hbar B$, where m may adopt the values $I,I-1,I-2,\ldots,-I$. NMR measurements detect transitions between energy levels m, m’ such that $\|m-m'\|$=1 with frequencies, $\omega_0$, proportional to the magnetic field at the nucleus: $$\omega_0 = \gamma_n B.$$ Because the gyromagnetic ratio is known to a high degree of accuracy, the resonant frequency measures the magnetic field at the nuclear site. In a paramagnet, the field, B, at the nucleus is given by $B=(1+K)H_0$, where K, the fractional frequency shift of the spectrum, is proportional to the magnetic susceptibility at the nuclear site, and $H_0$ is the applied magnetic field. Measurement of the shift of the spectrum thus yields, within a constant of proportionality, the local magnetic susceptibility. The precise relation between the shift and the susceptibility depends on the properties of the system. The shift may result from intrinsic properties, such as the electron hyperfine field, and from bulk effects, such as demagnetizing fields. A distribution of shifts broadens the spectrum by an amount proportional to the product of the width of the distribution and the magnetization. The distribution may result from a variation in electronic structure among sites, variations in demagnetizing fields among differently shaped grains in a powdered sample, or, in a randomly oriented powder, from anisotropy of the shift. The $^{139}$La nucleus also possesses an electric quadrupole moment, Q=$20\times10^{-30}$meter$^2$, which interacts with electric field gradients. Electric field gradients result from the distribution of charges around the nucleus. The electric quadrupole interaction creates structure in the NMR spectrum. A site with cubic symmetry has an electric field gradient of zero, so the size of the effect of the electric quadrupole interaction on the NMR spectrum measures the degree of deviation from cubic symmetry of the surroundings of the nuclear site. For the measurements described here, the electric quadrupole interaction may be considered a perturbation of the magnetic interaction. Under the influence of the magnetic and the electric quadrupole interaction, the NMR spectrum exhibits several sharp peaks in an ordered single crystal. The frequency of the peak corresponding to the transition between level m and m-1 is: $$\omega_m=\gamma_n(1+K)H_0 + \frac{3}{2}\frac{e^2qQ}{4\hbar I(2I-1)}(3\cos^2\theta-1)(2m-1).$$ The first term results from the magnetic interaction. The second is a shift due to the electric quadrupole interaction taken to first order in perturbation theory. $e$ is the charge of the electron. $q$ is the strength of the electric field gradient. $\theta$ is the angle between the principal axis of the electric field gradient and the magnetic field. This formula applies if the electric field gradient is axially symmetric. According to the above equation, the transition between the $m=1/2$ and $m=-1/2$ levels is unaffected by the electric quadrupole interaction to first order. This transition is, however, shifted to second order by an amount given by: $$\Delta\omega^{(2)}_{1/2}=\frac{9}{64}\frac{4I^2}{2I-1}\left(\frac{e^2qQ}{\hbar}\right)^2\frac{(1-9\cos^2\theta)(1-\cos^2\theta)}{\gamma_n(1+K)H_0}$$ The electric quadrupole interaction affects the transition between the $m=1/2$ and $m=-1/2$ levels only slightly but shifts the other transitions by larger amounts to higher and lower frequencies. The $+1/2$ to $-1/2$ transition is known as the “central transition" while the other transitions are known as the “satellites." The magnetic field dependence of the quadrupole interaction is particularly important. Analysis of the spectrum in this paper utilizes the fact that the first order shift is independent of magnetic field whereas the second order shift varies inversely with magnetic field. The crystal axes determine the orientation of the electric field gradient. In a powder sample, the crystal axes, and hence the electric field gradient, adopt a distribution of angles with respect to the applied magnetic field. This angular distribution broadens the sharp peaks of a single crystal into a characteristic powder pattern which has been described and calculated in the literature [@GERSTEIN]. In the powder pattern, the satellites spread out considerably and overlap. The first order quadrupole interaction determines the satellites’ width, which is therefore proportional to the size of the electric field gradient and independent of magnetic field. The average of the first order interaction over all angles is zero, so the average shift of the satellites due to the first order quadrupole interaction is zero. The satellites do, however, exhibit a shift due to the magnetic interaction. In contrast, the central transition, feeling only a second order interaction, is much sharper than the satellites. The second order interaction varies as $q^2/H_0$, so the powder-average width of the central transition is inversely proportional to the magnetic field. The average of the second order interaction over all angles is nonzero, so the average position of the central transition is shifted, also by an amount inversely proportional to the magnetic field. The average frequency of the central transition for a spin-$7/2$ is: $$\omega_{1/2}=\gamma_n(1+K)H_0 - \frac{1}{392}\left(\frac{e^2qQ}{\hbar}\right)^2\frac{1}{\gamma_n(1+K)H_0}.$$ The frequency of the central transition thus contains a component proportional to and a component inversely proportional to the magnetic field. The effects of a deviation from axial symmetry (a non-zero asymmetry parameter) on the average frequency are small, being no more than two percent of that due to the axially symmetric part of the quadrupole interaction. The Spectrum at High Temperature -------------------------------- Figure \[HTSPECTRUM\] depicts the $^{139}$La spectra in La$_{2/3}$Ca$_{1/3}$MnO$_3$ at 575K measured at 8.19 and 4.00 Tesla. The frequency shift is measured with respect to the unshifted resonant frequency derived using the value of $H_0$ determined from the resonant frequency of deuterium in deuterated water. The spectrum consists of a narrow feature (300 kHz FWHM at 8.19 Tesla, 680kHz at 4.00 Tesla) and a broad feature (4 MHz FWHM at both fields). The inset of figure \[HTSPECTRUM\] compares the narrow feature measured at 4.00 Tesla and 8.19 Tesla after multiplying the frequency shift by the value of the field. After this scaling of the frequency axis, the narrow features lie on top of each other, demonstrating that the width scales inversely with field. We can therefore conclude that the narrow feature corresponds to the central transition of the spectrum of a spin-$7/2$ nucleus. Resolution of the broad feature at the lower field is limited by a small signal to noise ratio. However, the broad feature does not become wider upon lowering the field, indicating that it corresponds to the satellites. Further evidence for this assignment of the broad feature comes from a fit of the spectrum. The average shift of the central transition has a component which varies as $q^2/H_0$. Therefore, measurements of the frequency of the center of the narrow feature of the spectrum as a function of applied magnetic field, $H_0$, determine the average strength of the electric field gradient, $q$. Measurements at fields ranging from 2.00 to 8.19 Tesla determine a value of electric quadrupole coupling, $\nu_{Q}\equiv\frac{e^2qQ}{h}$=26 MHz. For the sake of comparison, for $^{139}$La in the layered cuprate system LaCuO$_{4+\delta}$, $\nu_{Q}$ = 89 MHz [@HAMMEL]. Unlike the cuprate, the crystal structure of the manganite is nearly cubic, with a slight orthorhombic distortion [@DAI]. However, the nonzero electric field gradient demonstrates the sensitivity of NMR to the deviation from cubic symmetry, which can be safely ignored in a number of other types of measurements. Given the measurement of the average strength of the electric field gradient, we can fit the spectrum. Figures \[UNBROADFIT\], \[MAGBROADFIT\], and \[EFGBROADFIT\] show several fits, each assuming an electric field gradient with axial symmetry, with insets showing expanded views of the narrow feature and of a portion of the wide feature. Figure \[UNBROADFIT\] depicts a calculated powder pattern spectrum which uses no adjustable parameters. The amplitude has been chosen so that the total area under the calculated powder pattern is the same as the area under the measured spectrum. The narrow feature in the calculated powder pattern is the central transition, and the broader features are the satellites. The widths of the features of the calculated powder pattern and of the measured spectrum match closely. We can also compare the areas under the different parts of the spectrum. In the calculated powder pattern, the ratio of the area under the satellites to the area under the central transition is 4.96. For comparison, we fit the measured spectrum to a superposition of a broad and a narrow gaussian. The ratio of the areas of the gaussians is $5.7\pm0.5$. Given the difficulties involved with precisely measuring absolute intensities over the large frequency range involved in this experiment, the area ratios show reasonable agreement. The theoretical spectrum shows sharp peaks which are not found experimentally. We can improve the fit by introducing broadening mechanisms. Figure \[MAGBROADFIT\] shows the effect of broadening by a distribution of magnetic shifts. The central transition of the calculated powder pattern matches the narrow feature of the measured spectrum well, but the smeared singularities of the calculated satellites (figure \[MAGBROADFIT\], inset on right) do not appear in the measured spectrum. Increasing the broadening to make the satellites match the broad feature makes the central transition too wide. A distribution of magnetic shifts does not by itself account for the spectrum shape. By including a distribution of electric field gradient as well as magnetic broadening, we can fit both the broad and narrow features. Figure \[EFGBROADFIT\] shows a powder pattern with a gaussian distribution of the electric quadrupole coupling of width $\Delta\nu_{Q}/\nu_{Q}$=15%. This substantial distribution indicates that distortions in the surroundings of different lanthanum sites vary significantly. The fit also involves a convolution with a gaussian with a width 0.23% of the average NMR frequency which represents a slight distribution of magnetic shifts. A different distribution or a deviation of the electric field gradient from axial symmetry would improve the fit. However, the quality of this simple fit confirms the interpretation of the spectrum as a central transition and satellites. The presence of a distribution in lattice distortions agrees with results of other local measurements of the lattice [@BILLINGE; @LOUCA; @BOOTH; @TYSON] which find a distribution of manganese-oxygen and lanthanum-oxygen distances. To affect the electric field gradient at the lanthanum site, the distortions need not be in bonds involving the lanthanum nucleus. Variations in distant manganese-oxygen bonds can have a measurable effect. Such distortions may result from defects, Jahn-Teller distortions of MnO$_6$ octahedra, or from distortions due to the difference between the radii of La$^{3+}$ and Ca$^{2+}$. We do not present a quantitative comparison of distortions seen by NMR and those measured by other methods because accurate calculations of electric field gradients require characterization of the charge distribution around the nucleus to a level of precision beyond the scope of this investigation. One may also reasonably expect that local lattice distortions would cause deviations from axial symmetry. However, including a distribution of such deviations in the analysis has a negligible effect on the values of the size and distribution of electric quadrupole coupling. This insensitivity also precludes quantitative characterization of the asymmetry parameter. Our analysis of the spectrum rules out an interpretation of the structure of the spectrum which invokes a number of sharp features corresponding to distinct sites. In such an interpretation, the narrow feature would, for example, correspond to a single site of high symmetry while the broad feature would correspond to a site of lower symmetry or to a number of narrower, overlapping components. Such an interpretation explains the structure of the spectrum in the ferromagnetic state [@ALLODI1; @ALLODI2], but does not apply in the paramagnetic state. Temperature Dependence of the Spectrum -------------------------------------- Figure \[TEMPDEP\] depicts the results at several temperatures ranging from 292K to 575K. The spectrum experiences a change in shape and an enormous frequency shift on the order of 20% at room temperature. The shift is comparable to the zero field resonant frequency of 20MHz observed in the ferromagnetic state [@ALLODI1; @ALLODI2]. In contrast, the bulk susceptibility creates a macroscopic demagnetizing field on the order of 1% which alone would lead to a small negative shift. The large shift and sizable zero field frequency arise from the electron-nucleus hyperfine coupling, $${\cal H}_{hf}=\vec{I}\cdot\tensor{A}\cdot\vec{S},$$ where $\vec{I}$ and $\vec{S}$ are the nuclear and electron spin operators and $\tensor{A}$ is the hyperfine coupling constant. Assuming an isotropic hyperfine coupling, the shift adopts the form: $$K=\frac{A\chi_e}{\gamma_n\gamma_e\hbar^2},$$ where $\chi_e$ is the static electron susceptibility, and $\gamma_n$, $\gamma_e$ are the gyromagnetic ratios of the nucleus and of the electron. The hyperfine coupling constant, A, provides a measurement of the electron wave function at the nuclear site. Given the hyperfine coupling constant, the shift gives an independent measurement of the susceptibility. A plot of the inverse of the shift of the spectrum center of mass versus temperature combined with its magnitude and sign confirms that hyperfine coupling to the electron system determines the shift. Figure \[CURIE\] shows that the shift obeys Curie-Weiss behavior with a Curie temperature of 269$\pm$19K, in agreement with the value of T$_c$=268K determined from magnetization measurements. The large error bars on the Curie temperature result from the small number of data points and also reflect the deviation of this system from simple Curie-Weiss behavior [@DETERESA], a hallmark of short range ferromagnetic correlations. Nevertheless, the shift clearly depends on the polarization of the electron system. Along with the shift, the spectrum changes shape. As figure \[TEMPDEP\] shows, the narrow feature disappears as the temperature drops. However, superimposing the room temperature spectrum and the broad part of the high temperature spectrum (figure \[EFGINDEPT\]) indicates that the overall width remains unchanged. As explained earlier, the size of the electric field gradient, a result of static lattice distortions, determines the overall width of the high temperature spectrum. The similarity between the high temperature and room temperature spectra indicates that, from the viewpoint of the lanthanum site, the size and distribution of lattice distortions change little with temperature in the paramagnetic state. As we explain below, the loss of the narrow central feature results from a distribution of susceptibility resulting either from a variation of local electronic structure among nuclear sites or from a distribution of demagnetizing fields among differently shaped powder grains. The resulting broadening of the narrow feature is proportional to the product of the magnetization and the width of the distribution. To characterize the size of the broadening, we start from the fit of the spectrum at high temperatures depicted in figure \[EFGBROADFIT\]. The fit includes a slight magnetic broadening calculated by convolution with a gaussian. We can fit the spectrum at all temperatures by varying the width of the gaussian. A magnetic distribution sufficient to broaden the narrow feature to match the width of the wide feature affects the wide feature only slightly. The width of the convolution gaussian exhibits Curie-Weiss behavior (figure \[CURIEWIDTH\]) with a Curie temperature of 280$\pm$30K, in agreement with the value found from the temperature behavior of the shift. Therefore, the broadening is proportional to the magnetization, consistent with a picture in which a distribution in susceptibility drives the temperature dependence of the width of the central transition. Another possible explanation for the change in the spectrum shape involves changes in the hopping rate of small polarons or other local field inhomogeneities. A number of experiments detect the presence of small magnetic polarons in the manganites [@DETERESA; @JAIMEAPL; @JAIMEPRL]. Associated with the small magnetic polarons are inhomogeneous magnetic fields which could affect the spectrum shape. Nuclei near the localized moment of the small polaron would experience a larger shift than those far away. If the polaron is fixed in space, its inhomogeneous field would broaden the spectrum. If the moment hops at a rate exceeding the root mean squared frequency shift induced by its inhomogeneous field, the nuclear system would exhibit a single, average shift instead of a distribution of shifts. As a result, the spectrum would narrow with increasing hopping rate. This phenomenon is known as motional narrowing [@SLICHTER]. To see if changes in the hopping rates of local moments affect the spectrum, we look for spectral diffusion by performing a type of hole burning experiment called an S-Wave measurement, which is described in more detail elsewhere [@BECERRA]. These measurements were performed at the peak of the central transition in a 4 Tesla field at room temperature, 350K, and 575K. To perform the measurement, we use pulsed NMR techniques to selectively invert one section of the spectrum \[figure \[SWAVE\]\] and watch the spectrum as it evolves over a time, T$_{ev}$, after the inversion. For the sake of explanation, assume that the signal at the higher frequency corresponds to nuclei near a local moment and the signal at the lower frequency corresponds to nuclei far away. We start the experiment by inverting the signal, and hence the nuclear magnetization, on the high frequency side of the spectrum. If the local moment hops while the spectrum recovers, the nuclei originally near the moment will then be far from the moment and hence their signal will move to the lower frequency. As the inverted magnetization also moves to the lower frequency, a dip in the spectrum would simultaneously appear at the lower frequency. The situation involving a distribution of local fields is more complicated, but a dip in the signal as time progresses generally indicates the presence of hopping. In the absence of hopping, the entire spectrum only grows with time via spin-lattice relaxation. Figure \[SWAVE\] shows that the magnetization only grows over time, suggesting that hopping does not affect the spectrum at room temperature. Similar measurements suggest that hopping does not affect the spectrum at 350K or 575K. The lack of spectral diffusion in the S-Wave measurement may, however, stem from limitations in the experiment. First, the excitation bandwidth is at most one-fourth the width of the central transition, so effects of hopping on parts of the spectrum outside the excitation bandwidth would not be detected. Also, if there is no correspondence between spectrum position and the spatial separation between a nucleus and a polaron, hopping of a polaron in space will not lead to spectral diffusion. Our measurements thus do not conclusively rule out hopping as a cause for the narrowing of the spectrum with increasing temperature. Furthermore, even if we accept that small polarons do not contribute to the change in the spectrum shape, such a conclusion would not contradict other measurements which detect the presence of small polarons. The density of charge carriers, and hence of small polarons, is rather large compared to that found in conventional small polaron systems [@MOTT]. Furthermore, DeTeresa [et. al.]{} [@DETERESA] find that an applied magnetic field comparable to those used in these experiments increases the correlation length of magnetic clusters. Due to their density and size, small polarons may not present an inhomogeneous field to the lanthanum sites. Also, the motion of small polarons may be too fast, over the entire temperature regime studied, to affect the spectrum shape. The hopping rate in the temperature range studied is, at the slowest, on the order of $10^{11}$Hz [@JAIMEPRL]. Motional narrowing of a spectrum occurs when the hopping rate reaches and exceeds the width of the spectrum which, in this case, is on the order of 4 MHz. We should therefore expect that changes in the dynamics of small polarons will have no effect on the spectrum shape. However, the 4 MHz lower bound on the rate of motion of a field inhomogeneity applies to any type of field inhomogeneity— an assertion independent of models which invoke the presence of small polarons. Relaxation Times ---------------- The NMR spectral shape yields information about static, or nearly static, structural properties of the system. Measurements of nuclear spin-lattice relaxation determine the intensity and correlation times of magnetic and lattice fluctuations. In typical magnetic systems, magnetic fluctuations dominate the relaxation. The manganites, however, exhibit strong dynamic lattice distortions [@DAI] which induce fluctuations in the electric field gradient that may dominate the relaxation. As explained below, measurements of nuclear spin-lattice relaxation performed at different points along the spectrum indicate that magnetic fluctuations dominate the relaxation. We start with a discussion of general aspects of nuclear spin-lattice relaxation of a spin-$7/2$ nucleus. At equilibrium, the population of nuclear energy levels obeys a Boltzmann distribution. After the populations are disturbed from equilibrium, they approach their equilibrium values in a characteristic time known as the spin-lattice relaxation time. The size of the NMR signal depends on the population difference between levels, so the rate at which the NMR signal approaches its equilibrium value after a perturbation depends on the spin-lattice relaxation time. One type of measurement of spin-lattice relaxation time is the so-called inversion-recovery experiment, in which the populations of two levels are inverted. The signal, M(t), associated with these levels is measured at several times t after the inversion. The rate at which M(t) approaches its equilibrium value depends on the spin lattice relaxation time. The intensity of the transition between levels m and m+1 of a spin-7/2 nucleus exhibits multiple- exponential spin lattice relaxation [@ANDREW; @MARTINDALE]: $$M^m(t)\propto\sum_n1-a^m_n\exp\left(-\frac{b^m_nt}{\tau}\right).$$ The spin-lattice relaxation time, $\tau$, depends on the intensity of fluctuations in the magnetic field and electric field gradient at the nucleus. The constants a$^m_n$, b$^m_n$, depend on the initial conditions of the populations of the different energy levels. Depending on whether fluctuations of the magnetic field or of the electric field gradient dominate the relaxation, these constants vary with m differently. Calculations described in the appendix indicate that, for a spin-$7/2$, the multiple-exponential spin lattice relaxation differs only slightly from a single exponential, $$M^m(t)\propto1-2\exp\left(-\frac{t}{T^m_1}\right).$$ Furthermore, as explained in the appendix, if magnetic fluctuations dominate relaxation, T$^m_1$ is smaller when measured on the central transition than on the satellites. If electric field gradient fluctuations dominate, the opposite is true. Measurements of spin-lattice relaxation at different points along a spectrum therefore determine the relaxation mechanism. Figure \[T1VSPECT\] depicts effective single-exponential relaxation times measured at different positions on the spectrum at 9.4 Tesla and at approximately 575K. The repetition time of this and all subsequent measurements has been set sufficiently long to ensure that the spin system starts at equilibrium before each measurement. The spectrum at 9.4 Tesla is similar to that shown in Figure \[HTSPECTRUM\], except that the width of central feature (265 kHz FWHM) is smaller at higher field. The values of relaxation times have been normalized to the value measured at the apex of the central transition, and the frequency shift refers to the frequency of the apex. The spin-lattice relaxation time is shortest at the apex of the central transition, indicating that magnetic fluctuations cause the relaxation. Quantitative analysis of the variation of the relaxation at different points along the spectrum depends on details of the spectrum shape. Our analysis of the spectrum indicates that the various transitions overlap. Measurements of spin-lattice relaxation at any particular frequency will therefore contain contributions from several transitions at once. From the parameters used in the fit to the spectrum shown in figure \[EFGBROADFIT\], we can determine the degree of overlap among transitions and so can calculate how the spin-lattice relaxation time varies among the different transitions. There is an average spin-lattice relaxation time due to contributions from overlapping transitions. The line in figure \[T1VSPECT\] represents the calculated average spin-lattice relaxation time as a function of position along the spectrum. The agreement between calculation and measurement supports our interpretation of the origin of spin-lattice relaxation. Since this calculation relies on the parameters used in the fit of the spectrum shape, the agreement also lends further support to our analysis of the spectrum shape. Nuclear spin-lattice relaxation measurements probe magnetic fluctuations and hence electron spin dynamics. The spin-lattice relaxation time is related to the electron spin-spin correlation function via: $$\frac{1}{\tau}\propto\int dt \cos\omega_0t<h(t)h(0)>$$ where $\omega_0$ is the NMR frequency and h(t) is the fluctuating field at the nucleus due to the electron spin. The constant of proportionality depends on the magnetic moment of the nucleus and on the transition excited in the experiment. One consequence of the above formula is that the nuclear spin-lattice relaxation time depends on fluctuations at the NMR frequency, which are of extraordinarily low energy, on the order of 0.1$\mu$eV in this system. For the sake of comparison, neutron scattering experiments can resolve fluctuations down to the meV range. NMR as a probe of spin dynamics thus complements inelastic neutron scattering. If we assume a simple exponential form for the electron spin-spin correlation function: $$<h(t)h(0)> = h_0^2\exp\left(-\frac{t}{\tau_c}\right),$$ in which $h_0$ is the magnitude of the fluctuating field and $\tau_c$ is the correlation time of the fluctuation, the spin-lattice relaxation time adopts the form [@BPP]: $$\frac{1}{\tau}=\frac{\gamma^2_n h^2_0}{3}\frac{h^2_0\tau_c}{1+\omega^2_0\tau_c^2},$$ where the value of the constant applies for measurements at the central transition for a spin-7/2 nucleus. Assuming that $h_0$ and $\tau_c$ are field-independent, the spin-lattice relaxation time varies linearly with the square of the applied magnetic field, $H_0^2$. Figure \[T1VH\] shows that, at room temperature, the field dependence of the spin-lattice relaxation time exhibits such behavior. Furthermore, we find $h_0=130$ Gauss and $\tau_c=10^{-8}$ second. These values are only slightly affected by the presence of overlapping transitions in the powder pattern spectrum. The size of the fluctuating field is the same order of magnitude as the dipolar field of an electron sitting on a near neighbor manganese site, but is several orders of magnitude smaller than the hyperfine field which describes the shift. The correlation time is also several orders of magnitude slower than that associated with small polaron hopping. The slow dynamics may be related to that seen in the ferromagnetic state by muon spin relaxation measurements [@HEFFNER] which imply spin glass-like behavior of magnetic clusters. Electron spin resonance measurements, however, do not see anomalous line broadening above T$_c$ which one would associate with glassy dynamics [@LOFLAND]. At higher temperatures, the field dependence of the nuclear spin-lattice relaxation time changes. At 375K, the spin-lattice relaxation time is proportional to $H_0$, not $H_0^2$. The simplest possible explanation for such behavior involves the same correlation function used above but with field dependent parameters: $\tau_c\propto H^{-1/2}_0$ and $h_0\propto H^{1/4}_0$. Due to limits on the signal to noise ratio, we cannot determine whether the spin-lattice relaxation time at highest measured temperature, 575K, varies as $H_0$ or H$_0^2$. However, regardless of the form used to fit the data, the correlation time at the higher temperatures is faster than that at room temperature, but still slow, on the order of 10$^{-9}$ seconds. In a typical ferromagnet, the magnitude of fluctuating fields can adopt a wide range of values. For example, the field from spins on the manganese may add constructively or destructively at the lanthanum site, leading to a random distribution of fields among different lanthanum sites. In the case where the size of the fluctuating field may adopt a range of values, the electron spin-spin correlation function should adopt the form of a gaussian. An exponential spin-spin correlation function results if the fluctuating field can adopt one of only two values. In the case of a gaussian correlation function, the spin-lattice relaxation time adopts the form: $$\ln(\tau)\propto a+bH^2_0$$ which does not fit any of the data, except possibly at the highest temperature, where the uncertainties in the data are large. The field dependence of nuclear spin-lattice relaxation thus reveals an unusual functional form for the electron spin-spin correlation function as well as unusually slow dynamics. The explanation for this unusual behavior remains unclear. conclusions =========== We have established the properties of $^{139}$La NMR of La$_{2/3}$Ca$_{1/3}$MnO$_{3}$ in the paramagnetic state which provide a basis for interpretation of $^{139}$La NMR measurements in the manganites. The spectrum is a standard powder pattern broadened to a significant degree by static lattice distortions and to a lesser degree by a distribution of susceptibility among lanthanum sites. The static lattice distortions may be due to defects or inherent disorder due to Jahn-Teller distortions or calcium substitution. In the temperature range studied, static lattice distortions around the lanthanum site do not change. Also, changes in the hopping rate of localized field inhomogeneities, such as small polarons, do not affect the spectrum. Measurements of nuclear spin-lattice relaxation at different points across the spectrum indicate that the relaxation mechanism is magnetic, implying that spin-lattice relaxation probes electron spin dynamics rather than lattice dynamics. The field dependence of the spin-lattice relaxation reveals several unusual features about the magnetic fluctuations seen at the lanthanum site. First, the fluctuations are extremely slow at all temperatures examined here. Second, the field dependence of the fluctuations crosses over from T$_1\propto H_0^2$ near the Curie temperature to T$_1\propto H_0$ at 375K. The physical origin of the unusual temperature and field dependence of the spin-lattice relaxation remains unclear. This work has been supported by the U.S. Department of Energy, Division of Materials Research under Grant DEFG02-91ER45439 through the University of Illinois at Urbana-Champaign, Frederick Seitz Materials Research Laboratory. C.P.S., M.J., and M.B.S. also acknowledge support from the National Science Foundation Grant No. DMR91-20000 through the Science and Technology Center for Superconductivity. K.E.S thanks Nick Curro, Boris Fine, Jurgen Haase, Bob Heffner, Takashi Imai, Craig Milling, Dirk Morr, Jorg Schmalian, Dylan Smith, Raivo Stern, and Kazuyoshi Yoshimura for valuable assistance and comments. distinguishing magnetic and electric field gradient mechanisms of nuclear relaxation {#distinguishing-magnetic-and-electric-field-gradient-mechanisms-of-nuclear-relaxation .unnumbered} ==================================================================================== In an applied magnetic field, $H_0$, a spin-$7/2$ nucleus with magnetic quantum number $m=-7/2,-5/2,\ldots,5/2,7/2$ adopts energy levels with energies: $$E_m=m\gamma_nH_0$$ In thermal equilibrium, the population, $N_m$, of nuclei with energy $E_m$ obeys the Boltzmann distribution: $$N_m\propto\exp\left(-\frac{E_m}{k_BT}\right)$$ For the measurements described here, perturbations to the energies, $E_m$, due to magnetic or quadrupolar shifts are too small to have a significant effect on the populations of the energy levels. The quadrupolar shifts are, however, large enough allow us to distinguish the position of the central transition from the satellites in our spectrum measurements. Our measurements actually determine the population difference between adjacent levels. For example, the size of the signal of the central transition (m=+1/2 to -1/2 transtion) is proportional to $N_{-1/2}-N_{1/2}$. When measuring the spin-lattice relaxation time, $\tau$, one disturbs the populations of the energy levels and measures the rate at which the populations return to equilibrium. In the so-called inversion-recovery measurements described in this paper, we use radiofrequency pulses to exchange the populations of two adjacent energy levels and then measure the populations of these levels during their return to equilibrium. Because the broadening of the spectrum due to the quadrupole interaction is much greater than the bandwidth of the pulses, the chances of affecting more than one energy level of the same nucleus are small, even when effects due to magnetic broadening or a non zero asymmetry parameter are taken into account. In the case in which relaxation occurs due to magnetic fluctuations, the populations of the energy levels recover from a disturbance according to a set of rate equations [@ANDREW; @MARTINDALE]: $$\frac{dN_m}{dt}=-W_mN_m + W_{m+1}N_{m+1} + W_{m-1}N_{m-1},$$ Where $W_m$ is the sum of matrix elements of the Hamiltonian describing the coupling between the magnetic fluctuations and the nucleus. Because magnetic fluctuations can only induce transitions with $\Delta m=\pm1$, $W_m$ only includes matrix elements describing the coupling between levels m and m$\pm$1. Defining the population difference between levels m and m+1: $$n_{m,m+1}\equiv N_{m+1}-N_m,$$ and solving the rate equations for the central transition in an inversion-recovery experiment, one finds: $$\begin{aligned} n_{-1/2,1/2}(t)&\propto& 1-\frac{1225}{858}\exp\left(-\frac{56t}{\tau}\right)-\frac{75}{182}\exp\left(-\frac{30t}{\tau}\right)\nonumber\\ & &-\frac{3}{22}\exp\left(-\frac{12t}{\tau}\right)-\frac{1}{42}\exp\left(-\frac{2t}{\tau}\right)\end{aligned}$$ with different coefficients and time constants for each of the satellites. It turns out that the signal, $n_{m,m+1}$(t), is well-approximated by single-exponential recovery for each of the transitions: $$n_{m,m+1}(t)\propto 1-2\exp\left(-\frac{t}{T^m_1}\right).$$ For the inversion-recovery experiment, these effective values relaxation time, $T^m_1$, adopt the following values: --------------- ------ \[0.5ex\] 1/2 1.00 3/2 1.10 5/2 1.39 7/2 2.58 --------------- ------ where we have set $T^{1/2}_1$ to 1.00 arbitary units. The time constant $T^m_1$ corresponding to the central transition is shorter than those of the satellites, and $T^m_1=T^{-m}_1$. To take into account the overlap of satellites, we average over the contributions of each satellite at any particular frequency, giving a relaxation time,$T^{average}_1(\nu)$ which depends of the frequency position on the spectrum: $$\begin{aligned} \lefteqn{1-2\exp\left(-\frac{t}{T^{average}(\nu)}\right)}\nonumber\\ &&=1-2\sum_mc_m(\nu)\exp\left(-frac{t}{T^m_1}\right).\end{aligned}$$ where the constants $c_m(\nu)$ give the intensity of the transition between levels m and m+1 at the frequency $\nu$, a quantity extracted from the fit to the spectrum. In the case in which relaxation occurs due to electric field gradient fluctuations, both $\Delta m=\pm 1$ and $\Delta m=\pm 2$ transtions are allowed, giving rate equations of the form: $$\begin{aligned} \frac{dN_m}{dt}&=&-W_mN_m + W_{m+1}N_{m+1} + W_{m-1}N_{m-1}\nonumber\\ & &+W_{m+2}N_{m+2}+W_{m-2}N_{m-2},\end{aligned}$$ In a typical system, the matrix elements with $\Delta m=\pm 1$ and $\Delta m=\pm 2$ are nearly the same [@ANDREW]. In this case, the relaxation again adopts a nearly single-exponential form. In contrast with the case of magnetic relaxation, the relaxation time of the central transition is longer than those of the satellites. It is possible for the opposite trend to occur in the situation in which electric field gradient fluctuations dominate, but only if the $\Delta m=\pm 2$ matrix elements are over an order of magnitude larger than the $\Delta m=\pm 1$ matrix elements. Present address: Los Alamos National Laboratory, MSK764, Los Alamos, NM 87545. A.J. 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--- abstract: 'The quantum electrodynamical (QED) process of Compton scattering in strong magnetic fields is commonly invoked in atmospheric and inner magnetospheric models of x-ray and soft gamma-ray emission in high-field pulsars and magnetars. A major influence of the field is to introduce resonances at the cyclotron frequency and its harmonics, where the incoming photon accesses thresholds for the creation of virtual electrons or positrons in intermediate states with excited Landau levels. At these resonances, the effective cross section typically exceeds the classical Thomson value by over 2 orders of magnitude. Near and above the quantum critical magnetic field of 44.13 TeraGauss, relativistic corrections must be incorporated when computing this cross section. This profound enhancement underpins the anticipation that resonant Compton scattering is a very efficient process in the environs of highly magnetized neutron stars. This paper presents formalism for the QED magnetic Compton differential cross section valid for both subcritical and supercritical fields, yet restricted to scattered photons that are below pair creation threshold.Calculations are developed for the particular case of photons initially propagating along the field, and in the limit of zero vacuum dispersion, mathematically simple specializations that are germane to interactions involving relativistic electrons frequently found in neutron star magnetospheres. This exposition of relativistic, quantum, magnetic Compton cross sections treats electron spin dependence fully, since this is a critical feature for describing the finite decay lifetimes of the intermediate states. Such lifetimes are introduced to truncate the resonant cyclotronic divergences via standard Lorentz profiles. The formalism employs both the traditional Johnson and Lippmann (JL) wave functions and the Sokolov and Ternov (ST) electron eigenfunctions of the magnetic Dirac equation. The ST states are formally correct for self-consistently treating spin-dependent effects that are so important in the resonances. It is found that the values of the polarization-dependent differential cross section depend significantly on the choice of ST or JL eigenstates when in the fundamental resonance, but not outside of it, a characteristic that is naturally expected. Relatively compact analytic forms for the cross sections are presented that will prove useful for astrophysical modelers.' author: - 'Peter L. Gonthier' - 'Matthew G. Baring' - 'Matthew T. Eiles' - Zorawar Wadiasingh - 'Caitlin A. Taylor' - 'Catherine J. Fitch' title: 'Compton scattering in strong magnetic fields: Spin-dependent influences at the cyclotron resonance' --- Accepted for publication (August 2014) in Physical Review D. Introduction {#sec:Intro} ============ The physics of Compton scattering in strong magnetic fields has been studied fairly extensively over the last four decades, motivated at first by the discovery of cyclotron lines in accreting x-ray binary pulsars (see [@Truemper78] for Her X-1, [@Wheaton79] for 4U 0115+634, [@Makishima90] for X0331+53, and [@Grove95] for A0535+26), a genre of neutron stars. More recently, constraints on stellar magnetic dipole moments obtained from pulse timing observations have led to the identification of the exotic and highly magnetized class of neutron stars now known as magnetars \[e.g., see [@VG97] for the Anomalous X-ray Pulsar (AXP) 1E 1841-045, [@Kouv98] for Soft Gamma-Ray Repeater (SGR) 1806-20, [@Wilson99] and references therein for AXP 4U 0142+61, and [@Kouv99] for SGR 1900+14\] — this topical development has promoted a resurgence in the interest of this intriguing physical process. In classical electrodynamics, Thomson scattering in an external field evinces a pronounced resonance for incoming photons at the cyclotron frequency in the electron rest frame (ERF), within the confines of Larmor radiation formalism [@CLR71; @GS73; @BM79]. This feature also appears in quantum formulations of magnetic Thomson scattering [@CLR71; @deRHDM74; @Herold79] appropriate for fields Gauss. For neutron star applications it is often necessary to employ forms for the magnetic Compton scattering cross section that are computed in the relativistic domain. This is dictated by the atmospheric or magnetospheric fields of such compact objects possessing strengths either approaching, or exceeding (in the case of magnetars) the quantum critical field Gauss, i.e., that for which the electron cyclotron and electron rest mass energies are equal. Such results from quantum electrodynamics (QED) have been offered in various papers [@deRHDM74; @Herold79; @DH86; @BAM86] at various levels of analytic and numerical development. In particular, Refs. [@DH86; @BAM86] highlight the essential contributions provided by relativistic quantum mechanics, namely, the appearance of multiple resonances at various “harmonics” of the cyclotron fundamental and strong Klein-Nishina reductions that are coincident with electron recoil when the incident photon has an energy exceeding around in the electron’s initial rest frame. For photons incident at nonzero angles to the magnetic field, the harmonic resonances are not equally spaced in frequency [@DH86], and they correspond to kinematic arrangements that permit excitation of the intermediate virtual electron to various Landau levels — the discrete eigenvalues of energy transverse to the field. Extant QED calculations of the magnetic Compton process in the literature [@DH86; @BAM86] emphasize frequency domains either away from the resonances, or in the wings of the resonances, and presuming infinitely long-lived intermediate states, and therefore possess divergent resonances at the cyclotron harmonics. This suffices for several astrophysical applications, for example, the consideration of Compton scattering contributions to opacity in forging atmospheric or photospheric structure in magnetars [@Ozel01; @HoLai03; @SPW09]. However, for other applications that sample the resonances preferentially, such as the resonant Compton upscattering models of magnetar spectra and associated electron cooling in [@BH07; @BWG11; @Belob13], a refined treatment of the cross section in the resonances is necessary. The divergences appear in resonant denominators that emerge from Fourier transforms of the spatial and temporal complex exponentials in the wave functions: these denominators capture the essence of precise energy conservation at the peak of the resonance. Since the intermediate state is not infinitely long-lived, its energy specification is not exact, and consequently the divergences are unphysical, and must be suitably truncated. The appropriate approach is to introduce a finite lifetime or decay width to the virtual electrons for cyclotronic transitions to lower excited Landau levels, most commonly to the ground state. This introduces a Breit-Wigner prescription and forms a Lorentz profile in energy to express the finiteness of the cross section through any resonance [@WS80; @BAM86; @HD91; @Graziani93; @GHS95]. Historically, when this approach has been adopted, spin-averaged widths (i.e., inverse decay times) for the virtual electrons have been inserted into the scattering formalism. While expedient, this is not precise in that a self-consistent treatment of the widths does not amount to a linear characterization of the overall spin dependence. In other words, averaging the spins in forming does not correctly account for the coupling of the spin dependence of the temporal decay of the intermediate electron with the spin dependence of the spatial portion of its wave functions, i.e., the spinors. Rectifying this oversight in prior work is a principal objective of this paper. Another technical issue with calculating QED interactions in strong magnetic fields is the choice of the eigenstate solutions to the magnetic Dirac equation. Historically, several choices of wave functions have been employed in determinations of the Compton scattering cross section and cyclotron decay rates. The two most widely used wave functions are those of Johnson and Lippmann (JL) [@JL49] and Sokolov and Ternov (ST) [@ST68]. The JL wave functions are derived in Cartesian coordinates and are eigenstates of the kinetic momentum operator . The ST wave functions, specifically their “transverse polarization” states, are derived in cylindrical coordinates and are eigenfunctions of the magnetic moment (or spin) operator (with ) in Cartesian coordinates within the confines of the Landau gauge . Given the different spin dependence of the ST and JL eigenstates, one must use caution in making the appropriate choice when treating spin-dependent processes. Herold, Ruder and Wunner [@HRW82] and Melrose and Parle [@MP83a] have noted that the ST eigenstates have desirable properties that the JL states do not possess, such as being eigenfunctions of the Hamiltonian including radiation corrections, having symmetry between positron and electron states, and diagonalization of the self-energy shift operator. Graziani [@Graziani93] and Baring, Gonthier and Harding [@BGH05] noted that spin states in the ST formalism for cyclotron transitions are preserved under Lorentz boosts along [B]{}, a convenient property. In contrast, the JL wave functions mix the spin states under such a Lorentz transformation, and therefore are not appropriate for a spin-dependent formulation of the cyclotron process. In [@Graziani93] it was observed that the ST wave functions are the physically correct choices for spin-dependent treatments and for incorporating widths in the scattering cross section. Although the spin-averaged ST and JL cyclotron decay rates are equal, their spin-dependent decay rates are not, except in the special case in which the initial component of momentum of the electron parallel to the magnetic field vanishes. The differential cross sections for both ST and JL formalisms of magnetic Compton scattering are developed in parallel in this paper, for all initial and final configurations of photon polarization. They apply for kinematic domains below pair creation threshold. These are implemented for both spin-average cyclotronic decay rates and spin-dependent widths for the intermediate state, which are employed in a Breit-Wigner prescription to render the cyclotron resonances finite. This element of the analysis mirrors closely that in [@HD91]. The ST formulation uses the general formalism presented by [@Sina96], while the JL case is an adaptation of the work of [@DH86] and [@Getal00]. The developments are specialized early on to the particular case of photons propagating along [B]{} in the ERF. This is actually quite an important case in astrophysical settings, since it corresponds to interactions where relativistic electrons are speeding along magnetic field lines above the stellar surface. In such cases, Lorentz boosting parallel to [B]{} collimates the interacting photon angles in the ERF almost along the local field line. The analysis spans a wide range of field strengths, focusing particularly on the regime around Gauss Gauss, and thereby magnetic domains pertinent to millisecond pulsars, young radio and gamma-ray pulsars, and magnetars. In particular, focus on the resonance regime is germane to Compton upscattering models [@BH07; @FT07; @ZTNR11; @Belob13] of the hard x-ray tails observed in quiescent emission above 10 keV from several magnetars [@kuip04; @mereg05; @goetz06; @hartog08]. The inverse Compton process, where ultrarelativistic electrons scatter seed x-ray photons from the surface of a neutron star, has gained popularity as the preferred mechanism for generating these powerful pulsed signals, primarily due to the efficiency of scattering in the cyclotron resonances for strong magnetic fields [@ZTNR11; @Belob13; @BH07; @BWG11; @NTZ08]. The cross section calculations presented in this paper will also be pertinent to future computations of Compton opacity in dynamic plasma outflows that are postulated [@dt92; @td96] to be responsible for hard x-ray flaring activity seen in magnetars (e.g., see [@Lin11] for SGR J0501+4516 and [@Lin12] for AXP 1E 1841-045 and SGR J1550-5418). After developing the general formalism for the scattering differential cross section in Sec. \[sec:csect\_formalism\], the exposition narrows the focus to the incoming photons beamed along the local field direction in the ERF, denoted by . This culminates in the relatively compact formulas for the polarization-dependent cross sections in Eq. (\[eq:dsigmas\]) for ST, JL, and spin-averaged analyses. The specialization simplifies the mathematical development dramatically, since the associated Laguerre functions that appear as the transverse dependence (with respect to [B]{}) of the eigenfunctions of the magnetic Dirac equation reduce to comparatively simple exponentials. Furthermore, only the single resonance at the cyclotron fundamental appears. Various elements of the numerical character of the differential and total cross sections are presented in Sec. \[sec:csect\_results\]. Outside the resonance, spin influences are purely linear in their contributions, and so both ST and JL formulations collapse to the spin-averaged case, as expected. In the resonance, appreciable differences between the ST, JL, and spin-averaged formulations arise, at the level of around 50% when Gauss, and rising to a factor of 3 for one polarization scattering mode in fields Gauss. The origin of this difference is in the fact that the coupling between the intermediate electron’s momentum parallel to the field and its spin in defining its decay width is dependent on the choice of eigenfunction solutions of the magnetic Dirac equation. It is in this resonant regime that the physically self-consistent, spin-dependent Sokolov and Ternov cross sections presented in this paper provide an important new contribution to the physics of magnetic Compton scattering. An interesting anomaly emerges from spin-dependent influences at very low initial photon frequencies , i.e., well below the cyclotron fundamental, and is highlighted in Sec. \[sec:low\_freq\]. In the domain , the finite lifetime of the intermediate state is inferior to the inverse frequency, and so the cross section saturates at a small constant value, times the Thomson value. This dominates the usual low-frequency behavior for the case [@CLR71; @Herold79]. In Sec. \[sec:res\_peak\], to facilitate broader utility of the new ST results offered here for use in astrophysical applications, compact analytic approximate expressions for the differential cross section are derived in Eq. (\[eq:dsig\_resonance\]), together with Eqs. (\[eq:calNs\_final\]), (\[eq:calNperp\_final\]) and (\[eq:calNpar\_final\]). These integrate to yield the approximate total cross sections in Eqs. (\[eq:sig\_resonance\_fin\]) and (\[eq:sig\_resonance\_perp\_fin\]), fairly simple results whose integrals can be efficiently computed when employing a convenient series expansion in terms of Legendre polynomials. These approximate resonant cross section results include both polarization-dependent and polarization-averaged forms, and are accurate to better than the 0.1% level. The net product is a suite of Compton scattering physics developments that can be easily deployed in neutron star radiation models. The paper concludes with a discussion of the issue of photon dispersion in the magnetized vacuum. It is indicated that for astrophysically interesting field strengths, i.e., those below around Gauss, vacuum dispersion is generally small: the refractive indices of the birefringent modes deviate from unity by a few percent, at most, and generally much less. Accordingly, the influence of such dispersion on kinematic quantities pertaining to the photons is neglected in the scattering developments offered here. Development of the Cross Section {#sec:csect_formalism} ================================ The Compton cross section can be developed along the lines of the work of Daugherty & Harding [@DH86] and more recent work of Sina [@Sina96]. In this Section, we begin with more general elements of the formalism, and then specialize to our specific developments that focus on incident photons propagating along the magnetic field. General formalism {#sec:gen_formalism} ----------------- The magnetic Compton scattering cross section can be expressed as integrations of the square of the $S$-matrix element over the pertinent phase space factors for produced electrons and photons. The protocols for its development are standard in quantum electrodynamics, and, for unmagnetized systems, can be found in works by Jauch and Rohrlich [@JR80] (see Secs. 8-6 and 11-1, therein, for nonmagnetic Compton formalism). When , the formulation is modified somewhat to take into account the quantization of momenta perpendicular to [B]{} (assumed to be in the direction throughout), and, to a large extent, parallel to the two-photon magnetic pair annihilation exposition in [@DB80] \[see Eq. (21) therein\] because of crossing-symmetry relations. The total cross section for magnetic Compton scattering can be written by adapting Eq. (11-3) of [@JR80], $$\sigma \; =\; \int \dover{L^3\, \vert S_{fi}\vert^2}{v_{\rm rel}\, T} \dover{L^3d^3k_f}{(2\pi \lambar )^3}\, \dover{L^3d^3p_f}{(2\pi \lambar )^3} \;\to\; \lambar^2 \int \dover{L^3}{1-\beta_i\cos\theta_i} \,\dover{\vert S_{fi}\vert^2}{\lambar^2\, cT}\, \dover{L^3d^3k_f}{(2\pi\lambar )^3}\, \dover{L\, dp_f}{2\pi\lambar} \, \dover{B\, L\, da_f}{2\pi\lambar} \label{eq:cross_sect_form}$$ using standard notation, where is the Compton wavelength of the electron over . In terms of the formation of the $S$-matrix element, the time denotes the duration of the temporal integral, and the spatial integrations are over a cube of side length . Hereafter, the magnetic field strength will be expressed in units of the quantum critical value (Schwinger limit) Gauss, the field at which the electron cyclotron energy equals its rest mass energy. Also, throughout this paper, all photon and electron energies and momenta will be rendered dimensionless via scalings by and , respectively. The phase space correspondence for the scattered electrons, due to the quantization of their transverse energy levels, is routinely established: see Appendix E of [@Sina96] or Sec. 4(c) of [@MP83b]. The incoming electron speed is , parallel to the magnetic field, and the initial photon makes an angle with respect to the magnetic field direction, so that is the relative speed of the colliding photons and electrons. Eventually, the main focus of this paper will specialize to the particular case where the initial electron is in the ground state (lowest Landau level), and also possesses zero parallel momentum, $p_z=0$, so that then . Following common practice, this will be referred to as the ERF, where it is understood that this applies to the initial electron throughout. One can always consider scattering in such a frame by performing a Lorentz boost parallel to [B]{} to eliminate any component of momentum of the initial electron parallel to the field. ![The two traditional Feynman diagrams for Compton scattering, with the left one labeled (1) and the right one labeled (2) corresponding to the contributions to the first and second lines in Eq. (\[eq:Sfi\_form\]) for the $S$-matrix element. Solid lines represent spin-dependent electron wave functions in a magnetic field, so the four-momenta symbolically signify energy, the momentum component parallel to [B]{}, and the transverse excitation quantum number (see text). The wavy lines represent polarized photon states.[]{data-label="fig:Feynman"}](Fig1.eps) Various definitions germane to Eq. (\[eq:cross\_sect\_form\]) and kinematic identities are now outlined. The two Feynman diagrams for the scattering are depicted in Fig. \[fig:Feynman\]. In general, the incoming electron and photon four-momenta are and , respectively, and represents the spatial location (dimensionless, i.e., in units of ) of the guiding center of the incoming electron. The corresponding quantities for the outgoing electron are , , and . The energies of the incoming and outgoing electrons in the quantizing field are generally given by $$E_j \; =\; \sqrt{1 + 2 jB + p_j^2} \quad ,\quad E_{\ell} \; =\; \sqrt{1 + 2 \ell B + p_{\ell}^2} \label{eq:elec_energies}$$ for Landau level quantum numbers and , and dimensionless momenta and parallel to the field, respectively. The quantization of leptonic momenta transverse to the field implies that the correspondences and for their four-momenta are implicit in the symbolic depiction of Fig. \[fig:Feynman\]. For most of the paper, considerations are restricted to the ERF where the momentum component of the incoming electron parallel to [B]{} is set to zero. Differential cross sections for nonzero initial electron momenta along the field can be quickly recovered via Lorentz transformation of the forms presented in this paper; the total cross section is an invariant under such boosts. In this ERF specialization, one has and . The energy of the intermediate state assumes a similar form and is denoted by . The kinematic relations between the four-momenta of the incoming and outgoing species can be expressed via \[e.g., see Eq. (15) of [@Getal00]\] $$\omega_f \; =\; \frac{2(\omega_i - \ell B)\, r} {1+\sqrt{1-2 (\omega_i - \ell B)\, r^2 \sin^2\theta_f}} \quad ,\quad r \; =\; \frac{1}{1+\omega_i\left(1-\cos\theta_f\right)} \label{eq:kinematics_photons}$$ for the photon, which initially assumes an angle relative to [B]{}, and is scattered to an angle relative to the field direction. A simple rearrangement of this kinematic relation yields the following convenient form: $$(\omega_f)^2\sin^2\theta_f - 2\omega_i\omega_f (1-\cos\theta_f) + 2(\omega_i - \ell B - \omega_f) \; =\; 0 \quad . \label{eq:res_kinematics_alt}$$ The specialization to cases is made throughout this paper, following [@Getal00]; its astrophysical relevance is discussed soon below. Relaxation of the approximation will be developed in future work. The final electron’s parallel momentum and energy are given by $$p_{\ell} \; =\; \omega_i - \omega_f\cos\theta_f \quad ,\quad E_{\ell} \; =\; 1 + \omega_i - \omega_f \;\equiv\; \sqrt{1 + 2 \ell B + \left( \omega_i - \omega_f\cos\theta_f \right)^2}\quad . \label{eq:kinematics_electrons}$$ The equivalence of the forms for can be derived from Eq. (\[eq:kinematics\_photons\]). In the appendixes, the quantities and are labeled by and for the special case of that will be generally adopted here. Note that more general kinematic identities for Eqs. (\[eq:kinematics\_photons\]) and (\[eq:kinematics\_electrons\]), applicable for arbitrary incoming electron and photon momenta, can be found in [@DH86]. To facilitate the formation of the different cross section in terms of the angles of the outgoing photon, the identification is forged in Eq. (\[eq:cross\_sect\_form\]), where . The $S$-matrix element receives two contributions, $$S_{fi} \; =\; S^{(1)}_{fi} + S^{(2)}_{fi} \quad , \label{eq:Smatrix_sum}$$ one for each of the two Feynman diagrams (e.g., see Sec. 8-2 of [@JR80]) $$\begin{aligned} S^{(1)}_{fi} & = & - 4\pi i\, \fsc \int d^4 x' \int d^4 x \, \psi _f^{\dag} (x')\, \gamma _\mu A_f^\mu (x') \, G_F (x' ,\, x)\, \gamma _\nu A_i^\nu (x)\, \psi _i (x) \nonumber\\[-10.5pt] \label{eq:Sfi_form}\\[-10.5pt] S^{(2)}_{fi}&= & - 4\pi i\, \fsc \int d^4 x' \int d^4 x \, \psi _f^{\dag} (x')\, \gamma _\mu A_i^\mu (x') \, G_F (x' ,\, x)\, \gamma _\nu A_f^\nu (x)\, \psi _i (x) \quad , \nonumber \end{aligned}$$ labeled (1) and (2) in Fig. \[fig:Feynman\], respectively. Here, and are the initial and final electron wave functions (see Appendix \[sec:wfunc\_pol\] for more details), and they are solutions of the magnetic Dirac equation: $$\gamma^{\mu} \Bigl( \hbar c\, \partial_{\mu} - ie \, A_{\mu} \Bigr) \psi + m_ec^2\, \psi \; =\; 0\quad , \label{eq:Dirac_eqn}$$ for Dirac gamma matrices and . The electron propagator or Green’s function, , satisfies the inhomogeneous counterpart Dirac equation, where the right-hand side of Eq. (\[eq:Dirac\_eqn\]) is replaced by ; it is detailed at greater length just below. In Eq. (\[eq:Sfi\_form\]), and are the initial and final photon vector potential functions, and they assume the generic form $$A^{\mu}(x) \; =\; \dover{1}{\sqrt{2\omega\, (L/\lambar)^3}}\, \epsilon^{\mu}\, \exp \{ i k^{\nu}x_{\nu} \} \;\propto\; e^{i (\mathbf{k} {\cdot} \mathbf{x} - \omega t)} \label{eq:Amu_def}$$ for photon polarization vector and four-momentum (wave vector) . These are identical to their field-free forms: see Sec. 7-7 of [@BD64] or Sec. 4-4 of [@Sakurai67] for $S$-matrix construction of unmagnetized Compton scattering in QED. Observe that due to the crossing symmetry, the photons are interchanged between contributions from the first and second Feynman diagrams. The seminal papers of [@DH86] and [@BAM86] both originally derived QED formulations for Compton scattering in strong magnetic fields using JL [@JL49] particle basis states for QED solutions to the Dirac wave equation. Later [@Sina96] employed ST [@ST68] “transverse polarization” basis states as solutions of the magnetic Dirac equation, incorporating spin-dependent widths at the cyclotron resonance. All of these works computed differential cross sections for scattering in the ERF and encompassed arbitrary angles of photon incidence relative to the magnetic field direction. In this study, we follow closely the development of Sina [@Sina96], specializing to the particular case of photon incidence angles along [B]{}, i.e., . This special case is an astrophysically important one for neutron star magnetospheres in that it applies to scatterings of x rays by ultrarelativistic electrons, when the laboratory angle of incidence of the incoming photons is Lorentz contracted to in the ERF. In particular, it is germane to Compton upscattering models of energetic x-ray production in magnetars [@BH07; @BWG11]. The specialization was explored by [@Getal00] in JL scattering formalism, where it was highlighted that the single final Landau ground state of $\ell=0$ accounts for the entire cross section up to the cyclotron resonance at , above which $\ell \geq 1$ transverse quantum numbers (i.e. excitations) begin to contribute. These connections provide ample motivation for restricting this work, our incipient study of spin-dependent resonant scattering, to ground-state–ground-state transitions. In this presentation, the spin-dependent resonant width is included in a similar fashion to that in [@BAM86; @HD91; @GHS95]: it represents the decay lifetime of the intermediate state, and therefore appears as an imaginary contribution to the energy of this virtual state. This modification therefore appears in the complex exponentials for the time dependence, and, after integration, yields complex corrections to the resonant denominators. Eventually, after squaring of the $S$-matrix elements, its inclusion generates a truncation of all cyclotronic resonances via Lorentz profiles of width that depends on the spin of the intermediate electron or positron. This is an important inclusion in Compton scattering formalism that is required for precise computations of resonant upscattering spectra and associated electron cooling rates in models of x-ray and gamma-ray emission from neutron star magnetospheres. Following the work of [@GHS95], we can describe the bound state electron propagator \[see Eq. (15) of [@DB80], which extends Eq. (6.48) of [@BD64] to accommodate the quantization associated with the external magnetic field\], including the appropriate widths in the expression $$G_F(x',\, x) \; =\; \left(\dover{L}{2\pi\lambar }\right)^2 B \int da_n \int dp_n \sum_{n=0}^\infty \Bigl\{ -i\theta(t'-t) \, \Delta_+(x',\, x) + i\theta(t-t') \, \Delta_-(x',\, x) \Bigr\}\quad , \label{eq:Greens_fn}$$ for $$\begin{aligned} \Delta_+(x',\, x) & = &\sum\limits_{s=\pm}u_n^s({\bf x'}) u_n^{\dag s}({\bf x}) \;\exp \Bigl[ -i \left( E_n-i\Gamma^s/2\right)(t'-t) \Bigr] \nonumber\\[-5.5pt] \label{eq:Greens_fn_Deltas}\\[-5.5pt] \Delta_-(x',\, x) & = & \sum\limits_{s=\pm}v_n^s({\bf x'}) v_n^{\dag s}({\bf x}) \;\exp \Bigl[ +i \left(E_n-i\Gamma^s/2\right)(t'-t) \Bigr] \quad .\nonumber\end{aligned}$$ This form for the Green’s function can be applied to any choice of electronic wave functions that satisfies the magnetic Dirac equation \[see Eq. \[6.39\] of [@BD64]\]; in the absence of decay of the intermediate state, , and this reduces to the result in [@DB80]. The $\theta(t'-t)$ in Eq. (\[eq:Greens\_fn\]) is the unit step function implemented in the standard expansions of the Green’s functions, which is zero for negative arguments and unity for positive ones. The and contributions correspond to the positive and negative frequency portions of the Fourier transform [@BD64]. The and constitute the spatial parts of the electron and positron wave functions, respectively. The quantities and denote the $x$ coordinate of the orbit center and longitudinal momentum component, respectively, of the intermediate state. It is of crucial importance to understand that there is a coupling between the wave functions and and the excited state decay width that is spin dependent: one should not sum these spin dependences separately when computing the electron propagator. Outside cyclotronic resonances, the impact of this coupling virtually disappears as one can then set in Eq. (\[eq:Greens\_fn\_Deltas\]). More particularly, for the scattering problem, the electron propagator captures motion parallel to [B]{} via the kinematics of Compton scattering. It is the presence of this parallel momentum of the intermediate state coupled intimately with spin \[deducible from Eqs. (\[eq:Em\_def\]) and (\[eq:xi\_pm\_def\]) and supporting text below\] that breaks the degeneracy between JL and ST formulations and renders the cross section in the resonance dependent on the choice of basis states, but only when decay widths are incorporated. Before proceeding, some remarks about gauge choices are in order. As in Refs. [@Herold79; @DB80], we use the standard Landau gauge to represent the field ${\bf B}=B\hat{z}$, where $A^\mu(x)=(0,{\bf A})=(0,0,xB,0)$, \[contrasting Johnson and Lippmann [@JL49] who adopted \]. This freedom exploits the fact that the total cross section is independent of the choice of gauge for specifying the electron wave functions. Changing gauge introduces a complex exponential factor with the gauge modification as its argument. In other words, the contact transformation $$\psi (\mathbf{x},\, t) \;\to\; \psi' (\mathbf{x},\, t) \; =\; \psi (\mathbf{x},\, t) \, \exp \biggl\{ \dover{ie}{\hbar c}\, \Lambda ({\mathbf x}) \biggr\} \label{eq:gauge_transf_psi}$$ yields as a solution of the transformed Dirac equation if is a solution of Eq. (\[eq:Dirac\_eqn\]) for the original gauge. This phase change property is well known. We restrict considerations to spatial gauge transformations here, assuming time independence of the external field. Under this contact transformation, it is easily seen that the Green’s function defined by Eqs. (\[eq:Greens\_fn\]) and (\[eq:Greens\_fn\_Deltas\]) transforms according to $$G_F(x',\, x) \;\to\; G_F(x',\, x) \, \exp \Bigl\{ ie \bigl[ \Lambda ({\mathbf x}') - \Lambda ({\mathbf x}) \bigr] \Bigr\} \quad , \label{eq:Greens_fn_gauge_trans}$$ since the integrations over and do not impact the spatial factors involving the s. Observe that here we have reverted to our natural unit convention . In contrast, the wave-function products in the $S$-matrix expressions in Eq. (\[eq:Sfi\_form\]) transform via an exponential factor that is precisely the complex conjugate of the one in Eq. (\[eq:Greens\_fn\_gauge\_trans\]). The quantized fields for the external photon lines only couple to the ambient magnetic field through vacuum dispersion, and, therefore, gauge-invariant absorptive processes (discussed in Sec. \[sec:disperse\] below), and so are not influenced by such a gauge transformation. Accordingly, the $S$-matrices and the scattering differential cross section are independent of the choice of gauge. Inserting the Green’s function into Eq. (\[eq:Sfi\_form\]) for the first diagram, one obtains $$\begin{split} S^{(1)}_{fi} \; =\; & - \dover{\left(2\pi\right)^2 \fsc}{\sqrt{\omega_i\omega_f}}\left(\frac{\lambar}{L}\right)^3 \int d^3 x' \int d^3 x\, e^{ - i{\bf{k}_f} \cdot {\bf{x'}}} e^{i{\bf{k}_i} \cdot {\bf{x}}} \left(\frac{L}{2\pi\lambar}\right)^2 \\ & \times B \int da_n \int dp_n \Bigl\{ \varpi_+(x',\, x) - \varpi_-(x',\, x) \Bigr\}\;\; , \label{eq:Sfi1_eval} \end{split}$$ where $$\begin{aligned} \varpi_+(x',\, x) & = & \sum_{n=0}^\infty \sum\limits_{s=\pm} \, \Bigl[ u_{\ell}^{\dag (t)} ({\bf{x'}}) M_f\, u_n^{(s)} ({\bf{x'}}) \Bigr] \; \Bigl[ u_n^{\dag (s)} ({\bf{x}}) M_i\, u_j^{(r)} ({\bf{x}}) \Bigr] \nonumber\\ && \qquad \times \int_{-\infty}^\infty dt\, e^{i\left(E_n-\Gamma^s/2-\omega_i-1\right)t} \int_{t}^\infty dt' \, e^{i\left(E_{\ell}+\omega_f-E_n+\Gamma^s/2\right)t'} \quad ,\nonumber\\[-5.5pt] \label{eq:varpi_def}\\[-5.5pt] \varpi_-(x',\, x) & = & \sum_{n=0}^\infty \sum\limits_{s=\pm} \, \Bigl[ u_{\ell}^{\dag t} ({\bf{x'}}) M_f\, v_n^{(s)} ({\bf{x'}}) \Bigr] \; \Bigl[ v_n^{\dag (s)} ({\bf{x}}) M_i\, u_j^{(r)} ({\bf{x}}) \Bigr] \nonumber\\ && \qquad \times \int_{-\infty}^\infty dt'\, e^{i\left(E_{\ell}+\omega_f+E_n-\Gamma^s/2\right)t'} \int_{t'}^\infty dt\, e^{i\left(-E_n+\Gamma^s/2 -\omega_i-1\right)t} \quad .\nonumber\end{aligned}$$ The contribution from the second Feynman diagram can be similarly transcribed. The matrices express the polarization states in terms of gamma matrices, via the relations for the incoming photon and for the scattered photon, adopting the definitions in [@Mel13] for the two orthogonal polarization vectors discussed in Appendix \[sec:wfunc\_pol\]. In this paper, we adopt the standard convention for the labeling of the photon linear polarizations: refers to the state with the photon’s [electric]{} field vector parallel to the plane containing the magnetic field and the photon’s momentum vector, while denotes the photon’s electric field vector being normal to this plane. This polarization convention is appropriate for domains where one can neglect the dispersion of light propagation in either plasma, or the birefringent vacuum that is polarized by a large-scale electromagnetic field. Such a convention is commonplace in treatments of QED processes in strong magnetic fields, but it is not absolutely accurate in that the refractive index is not precisely unity. It then becomes an approximation, , to the true eigenmodes of propagation that are eigenvalues of the polarization tensor, which satisfy . Precise treatment of photon eigenmodes \[see Eq. (46) of [@Adler71] or Eqs. (11) and (12) of [@Shabad75]\] in the magnetized vacuum generally would entail the addition of substantial or prohibitive mathematical complexity to scattering cross sections, and also rates for other processes, such as pair creation and cyclotron transitions. Fortunately, such dispersive modifications are generally a small influence for astrophysically interesting field strengths, even for magnetars. The character of vacuum dispersion and the small magnitude of its impact for the scattering problem is discussed at length in Sec. \[sec:disperse\]. There it becomes evident that the nondispersive approximation is appropriate for fields when the scattered photon perpendicular energy is below pair creation threshold, . The development of the $S$-matrix element in Eq. (\[eq:Sfi\_form\]) mirrors that leading to Eqs. (3) and (4) in [@DH86]. The temporal integrations are simply evaluated, leading to the appearance of the energy conservation function in Eq. (\[eq:S-matrix2\]) and the resonant denominators in Eq. (\[eq:T1\_orig\]) below. These steps culminate in the expression $$S_{fi} \; =\; -\dover{ i\fsc}{\sqrt {\omega _i \omega _f } } \,\frac{\lambar}{L}\, \, \delta \left( {1 + \omega _i - E_{\ell} - \omega _f } \right)\sum\limits_{n = 0}^\infty {\sum\limits_{s = \pm } B \int da_n \int dp_n \left[ {T_n^{(1)} + T_n^{(2)} } \right]} \label{eq:S-matrix2}$$ after simple integration over the temporal dimensions. Here the sum over the index $n$ captures the Landau level quantum numbers of the intermediate state, and the sum over the index accounts for the different spins of this state. The spatial integrals are encapsulated in the terms $$T_n^{(1)} \; =\; \frac{{\cal S}^{(1)}_u}{1 + \omega _i - E_n + i \Gamma^s /2} + \frac{{\cal S}^{(1)}_v}{1 + \omega _i + E_n - i \Gamma^s /2} \label{eq:T1_orig}$$ where $$\begin{aligned} {\cal S}^{(1)}_u & = & \left[ {\int {d^3 xe^{ - i{\bf{k}_f} \cdot {\bf{x}}} u_{\ell}^{\dag (t)} ({\bf{x}}) M_f\, u_n^{(s)} ({\bf{x}})} } \right] \left[ {\int {d^3 xe^{i{\bf{k}_i} \cdot {\bf{x}}} u_n^{\dag (s)} ({\bf{x}}) M_i\, u_j^{(r)} ({\bf{x}})} } \right] \nonumber\\[-5.5pt] \label{eq:S1_T1_orig}\\[-5.5pt] {\cal S}^{(1)}_v & = & \left[ {\int {d^3 xe^{ - i{\bf{k}_f} \cdot {\bf{x}}} u_{\ell}^{\dag (t)} ({\bf{x}}) M_f\, v_n^{(s)} ({\bf{x}})} } \right] \left[ {\int {d^3 xe^{i{\bf{k}_i} \cdot {\bf{x}}} v_n^{\dag (s)} ({\bf{x}}) M_i\, u_j^{(r)} ({\bf{x}})} } \right] \nonumber\end{aligned}$$ and $$T_n^{(2)} \; =\; \frac{{\cal S}^{(2)}_u}{1 - \omega _f - E_n + i \Gamma^s /2} + \frac{{\cal S}^{(2)}_v}{1 - \omega _f + E_n - i \Gamma^s /2} \label{eq:T2_orig}$$ where $$\begin{aligned} {\cal S}^{(2)}_u & = & \left[ {\int {d^3 xe^{i{\bf{k}_i} \cdot {\bf{x}}} u_{\ell}^{\dag (t)} ({\bf{x}}) M_i\, u_n^{(s)} ({\bf{x}})} } \right] \left[ {\int {d^3 xe^{ - i{\bf{k}_f} \cdot {\bf{x}}} u_n^{\dag (s)} ({\bf{x}}) M_f\, u_j^{(r)} ({\bf{x}})} } \right] \nonumber\\[-5.5pt] \label{eq:S2_T2_orig}\\[-5.5pt] {\cal S}^{(2)}_v & = & \left[ {\int {d^3 xe^{i{\bf{k}_i} \cdot {\bf{x}}} u_{\ell}^{\dag (t)} ({\bf{x}}) M_i\, v_n^{(s)} ({\bf{x}})} } \right] \left[ {\int {d^3 xe^{ - i{\bf{k}_f} \cdot {\bf{x}}} v_n^{\dag (s)} ({\bf{x}}) M_f\, u_j^{(r)} ({\bf{x}})} } \right] \quad .\nonumber\end{aligned}$$ For the numerators , the index identifies the corresponding Feynman diagram in Fig. \[fig:Feynman\], and the subscripts mark the contributions from electron () and positron () propagators. Here, the and in Eqs. (\[eq:S1\_T1\_orig\]) and (\[eq:S2\_T2\_orig\]) represent the electron and positron spinor wave functions in the Landau state with energy quantum number . The indices $r$ and $t$ therein refer to the spin of the initial and final electron states, while the $s$ index refers to the spin of the intermediate lepton. The integrals appearing in the products within the numerators , termed vertex functions by [@MP83a; @MP83b], are further developed in Appendix \[sec:Matrix\], leading to the explicit appearance of functions familiar in $S$-matrix calculations of QED processes in external magnetic fields [@ST68; @HRW82; @MP83a; @BGH05], including, specifically, expositions on Compton scattering [@Herold79; @DH86; @BAM86; @GHS95; @Getal00]. These functions include exponentials and associated Laguerre functions in the photon variables that control the ensuing mathematical character of the cross section. Observe that the crossing symmetry relations $$\omega_i \;\leftrightarrow\; - \omega_f \quad ,\quad \bf{k}_i \;\leftrightarrow\; - \bf{k}_f \quad ,\quad \pmb{\epsilon}_i \;\leftrightarrow\; \pmb{\epsilon}_f \label{eq:crossing_symmetry}$$ identify the substitutions required to form the terms from , and vice versa. It is immediately apparent from Eqs. (\[eq:T1\_orig\]) and (\[eq:T2\_orig\]) that the incorporation of widths that are dependent on the spin of the intermediate state imposes spin dependence on both the numerator and the denominator, which must first be developed separately and then summed. This is formally the correct protocol, and as we shall demonstrate, it leads to dependence of the resonant cross section on the spin of the excited virtual electron. If on the other hand, one were to implement the spin-averaged widths, then the terms are added within the denominator, thereby leading to significant simplification of the terms. This is the historically conventional approach that is employed for magnetic Compton scattering calculations away from the cyclotron resonances (i.e., when can be presumed), but is imprecise in such resonances. This is the crux of the offering here, providing the mandate for our refinements of the magnetic Compton cross section in the cyclotron resonances. Using standard squaring techniques, the norm of the $S$-matrix element can be expressed in the form $$\begin{aligned} \bigl\vert {S_{fi} } \bigr\vert^2 & = & \frac{ \left(2\pi\right)^5\fsc^2 }{{\omega _i \omega _f }} \, \left(\dover{\lambar}{L}\right)^7 \, \dover{cT}{L} \, \delta \Bigl[ {k_{yi } - k_{yf } - B\left( {a - b} \right)} \Bigr] \, \delta \left( {k_{zi } - p_{\ell} - k_{zf } } \right) \nonumber\\[-5.5pt] \label{eq:Sfi_squared}\\[-5.5pt] & \times & \;\; \delta \left( {1 + \omega _i - E_{\ell} - \omega _f } \right) \, \exp \left( { - \frac{{k_{ \perp i }^2 + k_{ \perp f }^2 }}{{2B}}} \right) \, \dover{1}{E_f} \left\vert {\sum\limits_{n = 0}^\infty {\left\lbrack {F_{n,s}^{(1)} e^{i\Phi } + F_{n,s}^{(2)} e^{-i\Phi } } \right\rbrack } } \right\vert^2 \nonumber\end{aligned}$$ with the fine-structure constant, and the time and length are those of the spacetime box for the perturbation calculation. Here standard terms and the phase factor emerges from the integrals of the products of the matrix elements in the numerators of the ${\cal S}^{(m)}$ terms, which are similar to the ones in [@DH86; @BAM86]. The specific form for the phase factor is provided in Eq. (\[eq:PhaseFac\]) in Appendix \[sec:Matrix\]. Again, the labels for the terms correspond to the associated Feynman diagrams. The delta functions in express four-momentum conservation and emerge naturally from the Fourier transform manipulations of the incoming and outgoing plane-wave portions of the wave functions for the photons and electrons. The parameters and constitute the -coordinate orbit center of the incoming and outgoing electrons, respectively, and disappear from the $S$-matrix after integration over and . Observe that the exponential portion of the cross section that depends on the photon momenta perpendicular to the field, and , is explicitly isolated in this construction. The initial electron has a parallel momentum , a quantity that does not appear explicitly in the second function in Eq. (\[eq:Sfi\_squared\]) that describes momentum conservation parallel to the magnetic field. In contrast, (the parallel momentum of the scattered electron) appears in this function. This form for the square of the $S$-matrix is just that in Eq. (6) of [@DH86], but specialized to the ERF case where the initial electron is in the ground state and possesses a zero component of momentum along [B]{}. Due to the azimuthal symmetry of the scattering, without loss of generality one can orient the coordinate system so that the initial photon momentum is along the $x$ axis by selecting $\phi_i=0$ such that $k_{yi}=0$ and . On the other hand, $\phi_f$ is nonzero in general, and $k_f$ has both $x$ and $y$ components. Inserting Eq. (\[eq:Sfi\_squared\]) into Eq. (\[eq:cross\_sect\_form\]), the differential cross section in the rest frame of the electron can be readily obtained: $$\dover{d\sigma }{d\Omega _f } \; =\; \dover{3\, \sigt }{8\, \pi } \dover{\omega _f}{\omega_i\, {\cal K}} \, e^{ - \left( \omega _i^2 \sin^2 \theta _i + \omega _f^2 \sin ^2 \theta _f \right)/2B} \left\| \sum_{n = 0}^\infty \sum_{s = \pm} \left[ {F_{n,s}^{\left( 1 \right)} e^{i\Phi } + F_{n,s}^{\left( 2 \right)} e^{-i\Phi } } \right] \right\|^2 \label{eq:diff_sig}$$ for general photon incidence angles , where $${\cal K} \; =\; E_f - p_f \cos\theta_f = \; {1 + \omega _i - \omega _f - \left( {\omega _i \cos \theta_i - \omega _f \cos \theta _f } \right)\cos \theta _f } \quad . \label{eq:kappa_def}$$ Here is the Thomson cross section, with being the Compton wavelength of the electron. Whenever the initial and final electrons are in the ground state and the initial photon is parallel to the field such that $k_{\perp i}=0$, then Eq. (\[eq:PhaseFac\]) implies that . This expression for the cross section is of a form similar to that in Eq. (11) of [@DH86], with the denominator term later corrected [@HD91]. Upon integration the resulting matrix elements represented by the integrals in Eqs. (\[eq:S1\_T1\_orig\]) and (\[eq:S2\_T2\_orig\]) are contained in the terms within the summations, which depend on the Landau level and spin quantum numbers of the intermediate states. At this juncture, a choice of electron basis states is required in order to evaluate the $F$ terms. The papers by [@DH86; @HD91] used JL spin states to evaluate the matrix elements following the work of [@DB80]. On the other hand, Sina [@Sina96] performed the requisite spatial integrals preserving the electron wave-function coefficients in general form, thereby allowing for the expedient development in [either]{} JL or ST basis states. However, Sina chose to focus on the resulting cross section within the context of the ST spin states, the preferred protocol. We take advantage of the manipulations in [@Sina96] in order to develop the differential cross section for both basis states in parallel. For the remainder of the paper, the focus is on the development of Compton scattering in strong fields in the specific case where the laboratory incident photon angles are parallel () to the external $B$ field in the electron rest frame, as was previously performed by Gonthier [et al.]{} [@Getal00]. In that study, the role of the resonance was only considered in limited fashion, resulting in analytic descriptions of the cross section below and above the resonance. The bare resonance is divergent because an infinite lifetime for the intermediate state is thereby presumed. However, introducing a finite lifetime associated with the propagators truncates the resonance according to the prescription in Eqs. (\[eq:T1\_orig\]) and (\[eq:T2\_orig\]), with the width of the resonance being necessarily dependent on the spin of the intermediate state. Spin-dependent widths were incorporated into the differential cross section in the work of [@HD91], developing $F$ terms dependent on the spin of the intermediate state using ST eigenfunctions for determining the widths, but employing a JL formulation for the wave functions of the incoming and outgoing electrons. In our previous study [@BGH05], we showed the inherent difficulties with the JL basis states in that the spin states are not preserved under a Lorentz transformation along [B]{}. In contrast, we demonstrated that the ST electron wave functions, being eigenfunctions of the magnetic moment operator, behave correctly by preserving the spin states under Lorentz transformation, and form the appropriate set of states to describe the spin dependence of the resonance width. Accordingly, a greater emphasis is placed on the ST formulation below, presenting results that have not appeared before in the literature on magnetic Compton scattering. Compton scattering for photons incident along the magnetic field {#sec:csect_specialize} ---------------------------------------------------------------- Motivated by the important astrophysical application of inverse Compton scattering in neutron stars, the focus narrows now to cases. In addition, our ensuing analysis will concentrate on the development of the main contribution to the resonant scattering, which is the final state, i.e., ground-state–ground-state transitions. Excited final electron states only become accessible when the incident photon energy exceeds the threshold [@DH86; @Getal00]. The majority of the cross section is dominated by the $\ell=0$ contribution, even somewhat above , as can been seen in Fig. 4 of [@Getal00]. Moreover, [@BWG11] highlights how cooling of relativistic electrons in neutron stars is dominated by interactions near the fundamental resonance. Accordingly, domains where contributions dominate are of the greatest interest to astrophysical applications. We note also that the nonresonant JL formalism for cases has already been presented in [@Getal00], and that away from resonances, the ST formulation will generate identical results. For , only the first excited intermediate state contributes, collapsing the sum over in Eq. (\[eq:diff\_sig\]) to one term. The spin dependence of the resonance width (rate) is strongly dependent on the strength of the magnetic field (see Fig. 1 in our previous study [@BGH05]). With the restriction, the differential cross section possesses a simple dependence on , namely just complex phase factors from the factors in Eq. (\[eq:diff\_sig\]). Since then the terms do not depend on , the integration over the final is almost trivial, with cross terms proportional to that integrate to zero over the interval \[see Eq. (4) in [@BAM86] for the analogous inference for JL scattering formalism\]. Performing this first then leads to a sum of squares of complex moduli of the matrix elements for the corresponding Feynman diagrams $1$ and $2$ appearing in the resulting form for the differential cross section: $$\dover{d\sigma }{d\cos \theta _f } \; =\; \dover{ 3\sigt }{4} \dover{\omega _f^2 \, e^{ - \omega _f^2 \sin ^2 \theta _f /2B} }{ \omega _i \left( {2\omega _i - \omega _f - \zeta } \right)} \; \sum_{s =\pm} { \left[ {\left\| {F_{n=1,s}^{\left( 1 \right)} } \right\|^2 + \left\| {F_{n=1,s}^{\left( 2 \right)} } \right\|^2 } \right]} \label{eq:Diff2}$$ where $$\zeta \; =\; \omega _i \omega _f \left( {1 - \cos \theta _f } \right)\quad . \label{eq:zeta_def}$$ Observe that in developing Eq. (\[eq:Diff2\]), the identity derived from the scattering kinematics \[See Eq. (\[eq:res\_kinematics\_alt\]) using \] has been employed in the factor out in front, with given by Eq. (\[eq:kappa\_def\]). As indicated in Appendix \[sec:Matrix\], the specialization restricts the sum over , selecting only the level of the intermediate state as contributing to the cross section. This is the leading order contribution, with a Kronecker delta evaluation of the Laguerre functions, , forcing the restriction. This simplifies the cross section dramatically, as in [@Herold79; @Getal00]. The matrix element terms comprise standard “energy-conservation” denominators, and numerators with terms, as listed in Eqs. (\[eq:S1\_T1\_orig\]) and (\[eq:S2\_T2\_orig\]), are given by $$F_{n=1,s}^{(m)} \; =\; \frac{S_u^{(m)}}{\omega _m - {\cal E}_m + i\,\Gamma^s/2} + \frac{S_v^{(m)}}{\omega _m + {\cal E}_m - i\,\Gamma^s/2} \quad ,\quad m\; =\; 1,2 \quad . \label{eq:Fs_def}$$ Note that these $S^{(m)}_k$ terms defining the $F$ in Eq. (\[eq:Fs\_def\]) differ from the calligraphic ${\cal S}^{(m)}_k$ used to define the $T$ terms in Eqs. (\[eq:T1\_orig\]) and (\[eq:T2\_orig\]) in that the calligraphic $\cal S$ terms contain factors and $\delta$ functions arising out of the spatial integrals, while these $S$ terms here are only the products of the wave-function coefficients and the associated $\Lambda$ functions discussed in Appendix \[sec:Matrix\]. The two $S_u^{(m)}$ and $S_v^{(m)}$ terms here correspond to contributions from the electron and positron spinors of the intermediate state, with positive and negative energies, respectively. Observe that the numerators depend on , the spin quantum number of the virtual pairs. We have introduced some kinematic variables that depend on the Feynman diagram number , namely, total “incoming” energies $$\omega _{m=1} \; =\; 1 + \omega _i \quad ,\quad \omega _{m=2} \; =\; 1 - \omega _f \quad , \label{eq:erg_tot_inc}$$ and energies of the intermediate electron/positron state $${\cal E}_{m=1} \; =\; \sqrt {\omega _i ^2 + \epsilon _ \perp ^2 } \quad ,\quad {\cal E}_{m=2} \; =\; \sqrt {\omega _f ^2 \cos ^2 \theta _f + \epsilon _ \perp ^2 } \label{eq:Em_def}$$ for $$\epsilon_\perp \; =\; \sqrt {1 + 2B} \label{eq:e_perp_def}$$ as the threshold energy of the first Landau level. Using the identity , $$p_{m=1} \; =\; \omega _i \quad ,\quad p_{m=2} \; =\; - \omega _f \cos \theta _f \label{eq:pm_def}$$ define the components of momenta parallel to [B]{} that correspond to the . The spin-dependent widths of the cyclotron resonance truncate the divergences at that would appear in Eq. (\[eq:Fs\_def\]) without their inclusion. In practice, because of the kinematics of scattering, only the diagram elicits such a divergence, as is evident from the inequality that is simply deduced from Eqs. (\[eq:erg\_tot\_inc\]) and (\[eq:Em\_def\]). Here is the spin-averaged width in the frame of reference where the electron possesses no component of momentum along [B]{}. It is independent of the eigenfunction solutions of the Dirac equation, and its analytic form is given in Eqs. (\[eq:Gammave\_red\]) and (\[eq:I1B\_BGH05\]) below. The spin-correction factor does depend on the basis states being employed to describe the virtual particle; using the forms for found in [@BGH05] \[Eq. (1) therein for the ST case, and Eq. (53) for the JL states\], one has $$\xi_{\pm}^{ST} \; =\; 1 \mp \frac{1}{\epsilon_{\perp}} \quad ,\quad \xi_{\pm}^{JL} \; =\; 1 \mp \frac{{\cal E}_m + \epsilon_{\perp}^2}{\epsilon_{\perp}^2 \left( {\cal E}_m + 1 \right)} \label{eq:xi_pm_def}$$ for the ST and JL basis states. Note that the Lorentz boost that would transform the intermediate electron from the zero parallel momentum () frame to the frame is employed to cast Eq. (53) of [@BGH05] into the JL version for the spin-correction factor in Eq. (\[eq:xi\_pm\_def\]). In the Lorentz profiles that will emerge when the squares of the s are taken, it will become apparent that the cross section at the peak of the truncated cyclotron resonance will scale as , so that the relative strengths of the resonant interaction for the two eigenfunction choices will scale with , and thereby be substantially spin-dependent when is not too much greater than unity. The modulus squared of the matrix element $F$ terms can be evaluated in a similar manner as in [@HD91], keeping only terms that are of the highest order in , a small quantity. Specifically, combining the terms in Eq. (\[eq:Fs\_def\]) leads to the forms for that can routinely be determined from Eq. (\[eq:Fs\_def\]). Eliminating the term proportional to in the numerators and setting in the denominators permits the squares of the terms to be cast in a compact form, with numerators that employ the functions $$\begin{aligned} \label{eq:Nterms} N_+^{(m)} & = & \omega _m \left( {S_u^ + + S_v^ + } \right) + {\cal E}_m \left( {S_u^ + - S_v^ + } \right) \nonumber\\[-5.5pt] \label{eq:Nterm}\\[-5.5pt] N_-^{(m)} & = & \omega _m \left( {S_u^ - + S_v^ - } \right) + {\cal E}_m \left( {S_u^ - - S_v^ - } \right) \quad ,\nonumber \end{aligned}$$ where is the Feynman diagram number. The error incurred with this approximation is of the order of in the cyclotron resonance, which is always small (e.g., see Fig. 1 of [@BGH05] for values of , or Fig. 3 of [@HL06]), being less than around for all field strengths, where is the fine-structure constant. With these $N$ terms defined, the spin-dependent factors can be isolated and the cross section in Eq. (\[eq:Diff2\]) can be expressed as $$\dover{{d\sigma }}{{d\cos \theta _f }} = \dover{3\sigt }{4} \dover{\omega _f^2 \, e^{ - \omega _f^2 \sin ^2 \theta _f /2B} }{ \omega _i \left( {2\omega _i - \omega _f - \zeta } \right)} \; \sum_{m = 1}^2 \Biggl[ \dover{N_+ ^{(m)} \left( N_+ ^{(m)} + \xi_+ N_-^{(m)} \right)}{\epsilon _m^2 + \xi_+^2 {\cal E}_m^2 \Gamma ^2 } + \dover{N_- ^{(m)} \left( N_- ^{(m)} + \xi_- N_+^{(m)} \right)}{\epsilon _m^2 + \xi_-^2 {\cal E}_m^2 \Gamma ^2 } \Biggr]\; , \label{eq:dsig_Nform}$$ where is the spin-averaged width, and and are spin-dependent factors given in Eq. (\[eq:xi\_pm\_def\]) that depend on the choice of basis states. It is instructive to isolate the contribution of the spin dependence of the intermediate state by using these $N$ terms to define new $T$ terms that represent the complex modulus of the matrix element $F$ terms for the spin-averaged contribution: $$T_{{\rm{ave}}}^{(m)} \; =\; \left( {N_ + ^{(m)} + N_ - ^{(m)} } \right)^2 \label{eq:Tave}$$ and the spin-dependent contribution: $$T_{{\rm{spin}}}^{\left( m \right)} \; =\; \left( N_+^{(m)}+N_-^{(m)} \right) \left(N_+^{(m)}- N_-^{(m)}\right) + \left(\xi_+ - \xi_-\right)\, N_+^{(m)}N_-^{(m)}\quad , \label{eq:Tspin}$$ The algebraic development for generic matrix element terms is outlined in Appendix \[sec:wfunc\_pol\], and then applied to the ST basis states in Appendix \[sec:STforms\] and to the JL states in Appendix \[sec:JLforms\], so as to generate the specific expressions for the . The spin-dependent differential cross section can then be divided into two contributions $$\frac{{d\sigma ^{\perp, \parallel }_{\pm} }}{{d\cos \theta _f }} \; =\; \frac{{3\sigma _{\rm{T}} }}{8}\frac{{\omega _f^2 e^{ - \omega _f^2 \sin ^2 \theta _f /2B} }}{ {\omega _i \left( {2\omega _i - \omega _f - \zeta } \right)}} \sum\limits_{m = 1}^2 {\frac{{T_{{\rm{ave}}}^{(m),\perp, \parallel } \pm T_{{\rm{spin}}}^{(m), \perp , \parallel } }}{ { \epsilon _m^2 + \xi_\pm^2 {\cal E}_m^2 \Gamma ^2 }}} \label{eq:Dspin}$$ where $+$ and $-$ correspond to the spin-up (parallel to $B$) and spin-down (antiparallel to $B$) of the virtual particle contributions; these must be summed for each photon polarization. The cross section is necessarily dependent on the photon polarization of the [final state]{}. With the initial photon having an angle of incidence of zero degrees, the cross section is independent of the linear polarization of the initial state (circular polarizations then form the preferred photon basis states). Accordingly, the designations in Eq. (\[eq:Dspin\]) apply to the polarizations of the scattered (final) photon. Also observe that here we have introduced some kinematic variables that depend on the Feynman diagram $m$, namely, $$\epsilon_m \;\equiv\; \omega _m^2 - {\cal E}_m^2 \quad \Rightarrow \quad \epsilon_1 \; =\; 2\left( {\omega _i - B} \right) \quad ,\quad \epsilon_2 \; =\; - 2\left( {\omega _i + B - \zeta } \right)\quad , \label{eq:epsilon_m_def}$$ which enable the expression of the denominators in a more compact fashion. Again, is defined in Eq. (\[eq:zeta\_def\]). The spins of the intermediate state in Eq. (\[eq:Dspin\]) can be summed over, and the result depends on how the width of the decaying cyclotron transition is accounted for. There are three cases that are highlighted here: (i) ST basis states with spin-dependent cyclotron widths, (ii) JL eigenfunctions, also with spin-dependent widths in the resonance, and (iii) the formulation where the spin-averaged width is employed in the decay of the intermediate state. This third case is the one most commonly adopted in past expositions on resonant, magnetic Compton scattering in the literature. Since spin is thereby omitted from the resonant denominators, the overall differential cross section is then independent of the choice of basis states, i.e. it does not matter whether is computed using JL (historically popular) or ST (formally more correct for approaches treating spin). These three cases are encapsulated in the forms $$\begin{aligned} \left(\dover{d\sigma }{ d\cos \theta _f } \right)^{\perp , \parallel }_{\rm ST} & = & {\cal F} \sum\limits_{m = 1}^2 \left[ \dover{T_{\rm ave}^{(m),\perp , \parallel } + T_{\rm spin }^{{\rm ST},(m),\perp, \parallel} }{\epsilon _m^2 + \xi _{+,{\rm ST}} ^2 {\cal E}_m^2 \Gamma ^2 } + \dover{ T_{\rm ave }^{(m),\perp , \parallel} - T_{\rm spin}^{{\rm ST},(m),\perp ,\parallel } } { \epsilon _m^2 + \xi _{-,{\rm ST}} ^2 {\cal E}_m^2 \Gamma ^2 } \right]\nonumber\\[0.0pt] \left(\dover{d\sigma }{ d\cos \theta _f } \right)^{\perp , \parallel }_{\rm JL} & = & {\cal F} \sum\limits_{m = 1}^2 \left[ \dover{T_{\rm ave}^{(m),\perp , \parallel } + T_{\rm spin }^{{\rm JL},(m),\perp , \parallel } }{\epsilon _m^2 + \xi _{+,{\rm JL}} ^2 {\cal E}_m^2 \Gamma ^2 } + \dover{ T_{\rm ave }^{(m),\perp , \parallel } - T_{\rm spin}^{{\rm JL},(m),\perp , \parallel } } { \epsilon _m^2 + \xi _{-,{\rm JL}} ^2 {\cal E}_m^2 \Gamma ^2 } \right] \label{eq:dsigmas}\\[0.0pt] \left(\dover{d\sigma }{ d\cos \theta _f } \right)^{\perp , \parallel }_{\rm ave} & = & 2{\cal F} \sum\limits_{m = 1}^2 \dover{T_{\rm ave}^{(m),\perp ,\parallel } }{ \epsilon _m^2 + {\cal E}_m^2 \Gamma ^2 } \quad ,\nonumber\end{aligned}$$ where $${\cal F} \; =\; \dover{ 3\sigt }{8} \dover{ \omega _f^2 e^{ - \omega _f^2 \sin ^2 \theta _f /2B} } { \omega _i \left( {2\omega _i - \omega _f - \zeta } \right) } \quad . \label{eq:Factor}$$ Observe that if , both the ST and JL spin-dependent forms in Eq. (\[eq:dsigmas\]) reduce to the third, spin-averaged form. For both JL and ST basis states, the resulting values appearing in these forms are identical for , being given by the expressions $$\begin{aligned} T^{\perp}_{\rm ave} & = & \omega_i\left(\omega_i-\zeta\right) \nonumber\\[-5.5pt] \label{eq:Taves}\\[-5.5pt] T^{\parallel}_{\rm ave} & = & \left(2+\omega_i\right)\left(\omega_i-\zeta\right)-2\omega_f \nonumber\end{aligned}$$ for perpendicular and parallel photon polarizations. The same expressions apply to both Feynman diagrams. These terms are identical to those that appear in Eq. (13) of [@BH07]. The differences that arise between the JL and ST states are contained in the terms and in the terms in the differential cross section in Eq. (\[eq:dsigmas\]), which are described for each basis state in Appendixes \[sec:STforms\] and \[sec:JLforms\]. It is instructive to compare these developments with previous work in the nonresonant domain. Then can be set, and the three forms in Eq. (\[eq:dsigmas\]) coalesce to the spin-averaged one: $$\left(\dover{d\sigma }{ d\cos \theta _f } \right)^{\perp , \parallel }_{\rm ave} \; =\; \dover{{\cal F}}{2}\, T^{\perp, \parallel}_{\rm ave} \left\{ \dover{1}{(\omega_i-B)^2} + \dover{1}{(\omega_i+B - \zeta )^2} \right\} \quad . \label{eq:dsigma_zero_width}$$ This can quickly be shown to be equivalent to Eq. (22) of Gonthier [et al.]{} [@Getal00], which was directly derived from the JL formulation of [@DH86]. Our presentation here is an independent derivation, based on the developments of [@Sina96]. A further independent check is provided by the exposition of Herold [@Herold79]. In the special case of , Eqs. (8) and (9) in [@Herold79] can be routinely demonstrated to be identical to Eq. (22) of [@Getal00] after a modicum of algebra, and therefore also to our result in Eq. (\[eq:dsigma\_zero\_width\]). Total cross sections resulting from Eq. (\[eq:dsigma\_zero\_width\]) are explored in Sec. \[sec:tot\_csect\]. The remaining ingredient that needs to be posited is the mathematical form for the spin-averaged cyclotron width . This is taken from Eqs. (13) and (14) of [@BGH05], and can also be found in [@Latal86; @PBMA91; @HL06]. The average rates for cyclotron transitions at nonzero for electrons are scaled (by ) into dimensionless form: $$\Gamma\;\equiv\; \Gamma_{\rm ave} (p_z)\; =\; \dover{\fsc B}{{\cal E}_1} \; I_1(B) \quad , \quad {\cal E}_1\; =\; \sqrt{1 + 2 B + p_z^2}\quad , \label{eq:Gammave_red}$$ with $$I_1(B)\; = \; \int_0^{\Phi} \frac{d\kappa \, e^{-\kappa}}{ \sqrt{(\Phi -\kappa )\, (1/\Phi - \kappa )}}\; \Biggl\lbrack 1 - \frac{\kappa}{2} \biggl( \Phi + \frac{1}{\Phi}\, \biggr)\, \Biggr\rbrack \quad ,\quad \Phi \; =\; \frac{\sqrt{1+2 B}-1}{\sqrt{1+2B}+1} \label{eq:I1B_BGH05}$$ expressing the integration over the angles of radiated cyclotron photons. Note that represents the product precisely at the resonance condition , so that . The integral for can alternatively be expressed as an infinite series of Legendre functions of the second kind; see [@BGH05] for details. This width is mostly needed for the diagram, for which for the intermediate state so that reduces to the specific value listed in Eq. (\[eq:Em\_def\]). The presence of the factor in Eq. (\[eq:Gammave\_red\]) essentially accounts for time dilation when boosting along [B]{} from the electron rest () frame; the Lorentz factor for this boost is simply . When , i.e. in Eq. (\[eq:I1B\_BGH05\]) and in Eq. (\[eq:Gammave\_red\]), the width reduces to , the form widely invoked for nonrelativistic astrophysical applications of magnetic Compton scattering. In the opposite asymptotic extreme, namely, , appropriate to the inner magnetospheric regions of magnetars, the limit of Eq. (\[eq:I1B\_BGH05\]) quickly reveals that . In the fundamental resonance (for ), so that this limit of the width reduces to , independent of . As remarked above, spanning these two regimes, is always realized, underpinning, from the outset, the self-consistent incorporation of the width decay formalism in the complex exponentials for the intermediate electron state. As a concluding discursive offering, it must be emphasized that the calculations offered here do not include channels for pair production in the final state and therefore are strictly valid only below pair creation threshold, i.e., , characterizing a Lorentz invariant under boosts along [B]{}. For much (but not all) of the parameter space considered here, this domain is realized. Since we are restricting considerations to transitions, the kinematic relation for scattering in Eq. (\[eq:kinematics\_photons\]) can be used to demonstrate that is maximized when , realizing a well-defined value: $$\dover{\partial}{\partial\theta_f} \left( \omega_f\sin\theta_f \right) \; =\; 0 \quad\Rightarrow\quad \cos\theta_f \; =\; \dover{\omega_i}{1+\omega_i} \quad\Rightarrow\quad \omega_f\sin\theta_f \; =\; \sqrt{1+2\omega_i} -1\quad . \label{eq:wsinthet_max}$$ Clearly when , this opens up a portion of space for which the scattered photon is above pair creation threshold, i.e., , specifically for the polarization state of this final photon. For resonant scattering at , this applies to supercritical fields. This availability of pair creation channels for the scattering process was summarized in Fig. 7 of [@Getal00]. For the case considered here, it is generally relevant only to scatterings into the polarization, since the threshold for pair creation by mode photons is (in units of ), which usually exceeds the maximum of just derived if . Hence, in summary, when the incident photon possesses an energy in excess of around 2 MeV in the ERF, care must be taken to apply the calculations presented here in scattered angle domains where remains below 2. Fortunately, as will become apparent in the next section (see Fig. \[fig:Angular2\] and associated discussion), this generally corresponds to domains where the peak contribution to the total cross section is realized. In other words, the largest values of that might precipitate pair creation simultaneously reduce the exponential in the factor in Eq. (\[eq:Factor\]), and therefore the differential cross section. A more complete formalism incorporating pair creation channels (e.g., see [@Weise14]), where accessible, is beyond the scope of the present work. This concludes the general elements leading to the assembly of scattering differential cross sections in strong fields; the exposition now turns to specific illustrations of the results of these calculations, for both differential and total cross sections. Results: Characteristics of the Cross Section {#sec:csect_results} ============================================= The focus now turns to the core properties of the differential and total cross sections, and a comparison of the three forms, and to obtaining a comparatively compact analytic approximations to the full ST form in the resonance. Angular distributions {#sec:ang_dist} --------------------- Traditionally, it is the third of these differential forms in Eq. (\[eq:dsigmas\]) that has been used in the literature to take into account the relativistic modifications to the width of the resonance, but not the spin dependence of the width, which emerges because the lifetime of the intermediate ($n=1$) state depends on its spin. Away from the resonance, this dependence becomes effectively immaterial as sums over the intermediate spins generate results that are independent of the choice of electron wave functions: this assertion becomes apparent by simply setting in all the forms in Eq. (\[eq:dsigmas\]). In contrast, averaging over the intermediate state spins behaves more like a harmonic mean near the peak of the resonance, and that is where the differences in the cross sections are most profound, as shall become evident. As argued in the Introduction, this is a domain of importance for computations of Compton scattering in astrophysical models of neutron star magnetospheres. While the differential cross section developed within the ST basis states is formally the correct one, in the following figures, we compare the ST differential cross section to the one using the JL basis states and the traditional one that we refer to as the average cross section. ![The differential cross sections from Eq. (\[eq:dsigmas\]) as a function of the final scattering angle $\theta_f$ for a magnetic field of $B=3$. The upper, middle, and bottom sections display the cross section for ST and JL basis states, and the spin-independent cross section using the average width, respectively. The left, middle, and right sections display the cross sections right below the resonance at $\omega_i/B=0.998$, at the resonance $\omega_i/B=1.$ and right above the resonance at $\omega_i/B=1.002$, respectively. The red dashed curves and the blue dotted curves correspond to perpendicular and parallel polarizations of the scattered photon, respectively. All cross sections are scaled in units of the Thomson cross section.[]{data-label="fig:Angular1"}](Fig2) We display in Fig. \[fig:Angular1\], the angular distributions of the three separate differential cross sections from Eq. (\[eq:dsigmas\]) for a magnetic field of $B=3$ at which the intermediate state spin influences precipitate the largest differences. The incoming photon energies in the ERF are chosen to be in the left wing, at the peak, and in the right wing of the resonance. The red dashed curves correspond to the perpendicular polarization of the final photon, and show similar shapes in all three cross sections, while the blue dotted curves corresponding to the parallel polarization display some small differences in their shapes, especially in the dip near $\theta_f=1$. The dip can be understood by considering where the of Eq. (\[eq:Taves\]) actually goes to zero under the condition $$\cos\theta_f = \frac{\omega_i}{2+\omega_i} \label{eq:TpapaMin}$$ Note that $T^{(m),\parallel}_{\rm ave}$ remains positive on either side of this zero. The JL cross section also displays a minimum in the parallel polarization at a slightly larger angle than in the case of the average cross section. A small minimum is observed in the case of the ST basis states, but the cross section there does not go to zero. In the case of $T^{(m),\perp}_{\rm ave}$, there is no minimum. One striking feature of the ST angular distributions is that at the peak of the resonance, both perpendicular and parallel differential cross sections become identical: there is then no dependence on the polarization states of either incoming or outgoing photons. A derivation of the origin of this property is given in Sec. \[sec:res\_peak\]. However, even very slightly removed from resonance peak, the perpendicular polarization clearly dominates over the parallel polarization. Observe that the differential cross section for both forward scattering ($\theta_f\approx 0$) and backscattering ($\theta_f >\pi /2$) is considerably lower than the peak value at , for all three formulations. The backscattering case corresponds to large values of the argument for the exponential. ![The differential cross sections from Eq. (\[eq:dsigmas\]) as a function of the final scattering angle $\theta_f$ for a magnetic field of $B=3$. In this figure the photon energy is high above the resonance at $\omega_i/B=20$. The upper, middle, and bottom sections display the cross section for ST and JL basis states, and the spin-independent cross section using the average width, respectively. The red dashed curves and the blue dotted curves correspond to perpendicular and parallel polarizations of the scattered photon, respectively, as in Fig. \[fig:Angular1\]. The profound dips in the cross sections are actually true zeros defined by Eq. (\[eq:TpapaMin\]), and they are truncated in the plot for visual clarity.[]{data-label="fig:Angular2"}](Fig3) High above the resonance, the angular distributions begin to manifest an additional minimum that is associated with both polarization modes, and is due to the exponential in the common factor in the cross section: see Eq. (\[eq:Factor\]). In Fig. \[fig:Angular2\], we exhibit the angular distribution at $\omega_i/B=20$ for $B=3$ for each of the three cross sections. The photon energy is now high enough to discern the appearance of a prominent local minimum. This feature approximately corresponds to the exponential achieving a minimum (i.e., is at a local maximum) at $$\cos\theta_f = \frac{\omega_i}{1+\omega_i}\quad , \label{eq:ExpMin}$$ as noted previously in Eq. (29) of [@Getal00]. Observe that this local minimum is offset slightly from the zero for addressed in Eq. (\[eq:TpapaMin\]). Care must be exercised when developing a numerical integration routine, for example for computing the total cross section, to properly take into account these characteristics of the angular distributions. The overall maximum of the differential cross sections arises when the argument of the exponential is as small as possible. Since this illustration is for deep into the Klein-Nishina regime, this occurs when . Straddling this peak, again the forward scattering and backscattering values are greatly reduced in comparison. ![The ratio of the JL and average differential cross sections from Eq. (\[eq:dsigmas\]) to that of the ST cross section as a function of the final scattering angle $\theta_f$ for a magnetic field of $B=3$ at resonance. The upper and bottom sections display the ratio for JL cross section to that of ST and the ratio of the average cross section to that of ST, respectively. The red dashed and blue dotted curves indicate the ratios for perpendicular and parallel polarizations.[]{data-label="fig:Angular3"}](Fig4) The main differences displayed in the JL and averaged cross sections relative to the ST angular distributions occur at the resonance toward the end of the angular distribution near $\theta_f=\pi$ or the backward scattered photons antiparallel to the magnetic field, as seen in Fig. 4. We plot the ratios of the JL and average cross sections to that of the ST distribution for both perpendicular (red dashed curves) and parallel (blue dotted curves) polarizations as a function of the scattered angle $\theta_f$, highlighting the backscattering region $\theta_f >\pi /2$. As the scattered angle approaches $\theta_f=\pi$, these ratios become very large, implying that the JL and spin-averaged cross sections dramatically overestimate the cross section relative to the more correct ST result. This is not as critical a failing as it could be, since near $\theta_f=\pi$, the values of the cross sections are significantly diminished. Total cross sections {#sec:tot_csect} -------------------- For many physics and astrophysical considerations, the angle-integrated total cross section is an informative quantity. Here we perform numerical integrations over the scattered photon angle $\theta_f$ taking into account the local minima and maxima manifested by the JL and average angular distributions, as encapsulated in Eqs. (\[eq:TpapaMin\]) and (\[eq:ExpMin\]) and portrayed in Figs. \[fig:Angular1\] and \[fig:Angular2\]. Analytic results for the total cross section will be considered later in this subsection. For a magnetic field of $B=3$, in Fig. \[fig:Int1\] we display the angle-integrated cross sections in the upper panels, while the ratios of the JL and averaged cross sections are in the lower panels as a function of the incident photon energy in units of the cyclotron energy. The extremely narrow resonance region in the left panel is expanded in the right panel to highlight the structure near the peak of the cyclotron resonance. Observe that the resonance shape is symmetric about the peak, an artifact of the approximation of taking only the leading order dependence in the widths when squaring the resonant denominators, as discussed just prior to Eq. (\[eq:Nterm\]). Relaxing this simplification would introduce a very slight asymmetry of order to the energy profile of the resonance. Another interesting feature of the cross sections is the portion far below the resonance where they flatten out to a constant for [both]{} the ST and JL spin-dependent cases: the origin of this is discussed in Sec. \[sec:low\_freq\]. The choice of the magnetic field was made to illustrate the maximal difference at the peak of the resonance between the ST and JL formulations, and it corresponds to the case of fairly low altitudes in the magnetospheres of magnetars. ![The angle-integrated cross sections (in units of ), averaged over polarization, are displayed in the upper panels with an exploded view near the resonance in the upper right panel. The spin-dependent JL (green solid curve), the ST (blue solid curve), and the spin-averaged (red solid curve) cross sections are displayed in the upper panels. In the bottom panels, the ratios of the JL and average cross sections to that of the ST cross section are displayed. The diagonal dashed black line is the familiar nonrelativistic result in Eq. (\[eq:NonrelAve2\]) that applies when . The horizontal dotted line is the low-frequency spin-dependent anomaly whose asymptotic form is deducible from Eq. (\[eq:dsigma\_lowfreq\_tot\]). The dots represent numerical evaluations of the integral expression for the total cross section in Eq. (\[eq:sigma\_spinave\]) for the left panel, i.e., outside the resonance, and for the approximate cross section in the resonance in Eq. (\[eq:sig\_resonance\_fin\]) for the right-hand panel. []{data-label="fig:Int1"}](Fig5) The cumulative contribution of the aforementioned excesses of the spin-dependent JL and average differential cross sections above that of the ST one becomes evident near the resonance shown in Fig. 5. For this $B=3$ example, the resonant cross section is overestimated by 40% and 20% by the JL and average cross sections, respectively. The doubled-peaked curve for the average cross section at the resonance exhibits a minimum at the resonance due to fairly rapid swings with $\omega$ in the contributions from the perpendicular and parallel polarization scattering modes. This is illustrated in the blown-up in Fig. \[fig:Int2\], which exhibits the dependence through the resonance for the individual polarization modes. As seen before in Fig. \[fig:Angular3\], the angular distribution with the average widths generates a lower contribution from the parallel polarization, thereby producing the minimum at the resonance seen in Fig. \[fig:Int1\]. This is more than compensated for by the excess seen at the peak of the $\perp$ resonance profile, so that the polarization-summed cross section for the spin-averaged case generates the excess over the ST case depicted in Fig. \[fig:Int1\]. In contrast, the JL cross section that includes the spin-dependent widths does not manifest such opposing polarization-dependent variations near the resonance, but it still yields an overestimate of the cross section near the resonance relative to the correct ST form. ![The ratios of JL (green curves) and spin-averaged (red curves), angle-integrated cross sections to that of the ST cross section are displayed in a zoomed-in view near the resonance for the perpendicular (upper panel) and parallel (lower panel) polarizations.[]{data-label="fig:Int2"}](Fig6) The largest differences between the cross sections developed using the ST and the JL basis states in the spin-dependent width formulations occur around $B=3$. To survey the magnetic field dependence of the cross sections, examples for subcritical and highly supercritical fields are depicted in Fig. \[fig:Int3\]. Outside the resonance, all formulations for the cross section converge to the same result, as they should, corresponding to setting in Eq. (\[eq:dsigmas\]). An exception to this is for the low-frequency regime, which will be addressed in Sec. \[sec:low\_freq\]. The importance of spin-dependent effects in the resonance begins to diminish significantly above $B=10$, since then the choice of spin state for the cyclotron decay width becomes immaterial: all cyclotron decay rates approach the spin-averaged one, as previously noted in Fig. 2 of [@BGH05]. It is notable that the average cross section also overpredicts the resonant cross section at low magnetic fields, due primarily to an overestimate of the perpendicular polarization, a nuance that is discussed in Sec. \[sec:res\_peak\] below. ![The angle-integrated cross sections (in units of ), averaged over polarization, are displayed in the upper panels with a zoomed-in view near the resonance in the upper right panel. The spin-dependent JL (green curves), the ST (blue curves), and the spin-averaged (red curves) cross sections are displayed in the upper panels. In the bottom panels, the ratios of the JL and spin-averaged cross sections to that of ST cross section are displayed. The nonrelativistic limit () in Eq. (\[eq:NonrelAve2\]) is shown in the left panel as a black dashed line. As in Fig. \[fig:Int1\], the horizontal dotted line is the low-frequency spin-dependent anomaly with : see Eq. (\[eq:dsigma\_lowfreq\_tot\]). The dots represent numerical evaluations of the integral expression for the total cross section in Eq. (\[eq:sigma\_spinave\]) outside the resonance.[]{data-label="fig:Int3"}](Fig7) Away from the resonance, in general, the spin-dependent influences are minimal, and spin-averaged formalism is usually sufficient: in such domains, both the ST and JL forms collapse to the spin-averaged form in Eq. (\[eq:dsigmas\]), i.e., to that in Eq. (\[eq:dsigma\_zero\_width\]). A notable exception arises at very low frequencies , and this is discussed in the following subsection. The spin-averaged expression for the differential cross section in Eq. (\[eq:dsigma\_zero\_width\]) can be integrated over using the protocol developed in BWG11. This method changes the variables of the integration to conveniently render the integrals more compact. The first step is to convert the integration to one in terms of the variable , defined in Eq. (\[eq:kinematics\_photons\]). This represents what would be the ratio of the final to initial ERF photon energies in conventional, nonmagnetic Compton scattering. The integration limits become . The kinematic relation in Eq. (\[eq:kinematics\_photons\]) can be rearranged to generate the identity $$2\omega_i r^2\sin^2\theta_f \; \equiv\; Q(r,\, \omega_i )\; =\; \dover{2}{\omega_i}\, (1-r)\, \Bigl[ (2\omega_i +1)r-1\Bigr]\quad . \label{eq:Qvar_def}$$ It turns out that a more convenient variable for expressing the angular integration is $$\phi \;\equiv\; \dover{\omega^2_f\sin ^2\theta_f}{2\omega_i} \; =\; \dover{1-\sqrt{1-Q}}{1+\sqrt{1-Q}}\quad . \label{eq:phivar_def}$$ This then encapsulates the angular dependence of the argument of the exponential in the factor. This change of variables amounts to the integration mapping $$2 \int_{-1}^1 d\cos\theta_f \, {\cal F} \;\to\; \dover{3\sigt}{4\omega_i} \int_0^{\Phi} \dover{e^{-\omega_i \phi /B}\, d\phi}{\sqrt{1-2\phi z+\phi^2}} \quad ,\quad z\; = 1 + \dover{1}{\omega_i}\quad . \label{eq:dtheta_f_to_dphi_map}$$ The dependence of the integrand is contained in the factors in square brackets in Eq. (\[eq:dsigmas\]), and it must be remembered that the factor of 2 appearing here accounts for the two terms present in each of these factors in Eq. (\[eq:dsigmas\]). One subtlety is that there are two branches of that map over to the same interval for the integration. Being obtained by inverting Eq. (\[eq:phivar\_def\]) for , and solving the resulting quadratic for , these are described by $$r_{\pm} \; =\; \dover{z(1+\phi) \pm \sqrt{1-2\phi z+\phi^2}}{(1+\phi)(1+z)} \quad ,\quad z\; = 1 + \dover{1}{\omega_i}\quad . \label{eq:r_branches}$$ These two branches are summed over, simplifying the algebraic complexity of the and in Eq. (\[eq:Taves\]) somewhat, and the resulting integrals span the range $$0\; \leq\;\phi\;\leq\; \Phi \;\equiv\; \dover{\sqrt{1+2\omega_i}-1}{\sqrt{1+2\omega_i}+1} \; =\; z - \sqrt{z^2-1}\quad . \label{eq:phi_range}$$ Since the terms in the resonant denominators can be neglected, these manipulations lead to the compact forms for the total cross section: $$\begin{aligned} \sigma_{\perp} & \approx & \dover{3\sigt}{8} \int_{0}^{\Phi} \dover{ e^{-\omega_i \phi /B}\, d\phi }{\sqrt{1-2\phi z+\phi^2}} \nonumber\\[0pt] & & \quad \times \left\{ \left\lbrack \dover{1}{1+2\omega_i}\, \dover{1}{(\Delta_1)^2} + \dover{1}{\Delta_2} \right\rbrack g^{\perp}(z,\, \phi ) - \dover{2 B}{(\Delta_2)^2} \left( 1-2\phi z+\phi^2 \right) \right\} \nonumber\\[-5.5pt] \label{eq:sigma_spindep}\\[-5.5pt] \sigma_{\parallel} & \approx & \dover{3\sigt}{8} \int_{0}^{\Phi} \dover{ e^{-\omega_i \phi /B}\, d\phi }{\sqrt{1-2\phi z+\phi^2}} \nonumber\\[0pt] & & \quad \times \left\{ \left\lbrack \dover{1}{1+2\omega_i}\, \dover{1}{(\Delta_1)^2} + \dover{1}{\Delta_2} \right\rbrack g^{\parallel}(z,\, \phi ) - \dover{2 (B+2 \phi )}{(\Delta_2)^2} \left( 1-2\phi z+\phi^2 \right) \right\} \nonumber\end{aligned}$$ for the polarized results away from the cyclotron fundamental. Here for the diagram, captures the denominator of the second Feynman diagram, and $$\begin{aligned} g^{\perp}(z,\, \phi ) & = & z - \phi \nonumber\\[0pt] g^{\parallel}(z,\, \phi ) & = & z + (1-2z^2) \phi \label{eq:g_integrand_def}\\[0pt] g(z,\, \phi ) & = & \dover{g^{\perp}+g^{\parallel}}{2} \; =\; z(1-\phi z ) \nonumber\end{aligned}$$ The polarization-summed result is $$\begin{aligned} \sigma \; =\; \sigma_{\perp} + \sigma_{\parallel} &\approx & \dover{3\sigt}{4} \int_{0}^{\Phi} \dover{ e^{-\omega_i \phi /B}\, d\phi }{\sqrt{1-2\phi z+\phi^2}} \nonumber\\[-5.5pt] \label{eq:sigma_spinave}\\[-5.5pt] & & \quad \times \left\{ \left\lbrack \dover{1}{1+2\omega_i}\, \dover{1}{ (\Delta_1)^2} + \dover{1}{\Delta_2} \right\rbrack g(z,\, \phi ) - \dover{2 (B+\phi )}{(\Delta_2)^2} \left( 1-2\phi z+\phi^2 \right) \right\} \nonumber\end{aligned}$$ In the magnetic Thomson limit, , when also and Klein-Nishina corrections are not sampled, , and the integrations simplify with the exponential collapsing to unity. The terms are then negligible, and becomes independent of . The integrals are now almost trivial, yielding $$\sigma \;\approx\; \dover{\sigt}{2} \left[ \frac{\omega_i^2 }{\left(\omega_i-B\right)^2} + \frac{\omega_i^2 }{\left(\omega_i +B\right)^2 } \right] \quad , \quad B\;\ll\; 1\quad . \label{eq:sigma_mag_Thomson}$$ This matches the total cross section deduced from Eq. (16) of [@Herold79], for either initial polarization state, when the latter is specialized to the case of photons incident along the magnetic field. Finally, note that the integrals for the cross sections in Eqs. (\[eq:sigma\_spindep\]) and (\[eq:sigma\_spinave\]) can be expressed analytically in terms of an infinite series of Legendre functions of the second kind. Such a development is outlined in Appendix \[sec:csect\_analytics\]. To provide context for these results, it is instructive to illustrate the contribution of excited states for the final electron at frequencies . Since both ST and JL formulations coalesce to the spin-averaged one, and the cyclotron width at the fundamental can be set to zero, this can be done by integrating Eqs. (11)–(14) in [@Getal00] over the scattered photon angle . The resulting total cross sections, summed over all values with , are displayed for field strengths in Fig. \[fig:tot\_csect\_KN\], using the JL codes developed in [@Getal00]. Therein it becomes evident that for highly subcritical fields, the contributions only become significant well above the cyclotron fundamental. As the field rises, when , then the contribution still dominates in the resonance and a bit above, but not at higher energies: see [@Getal00], Fig. 2, for depictions of the cases . Since the resonant domain is so important for many neutron star applications, this plot illustrates the motivation for confining our resonant study here to just the channel. ![Total angle-integrated cross sections (in units of ), averaged over initial photon polarizations, for the case where the width in the resonance is set to zero, , and the ST and JL formulations are identical to the spin-averaged one. The solid colored curves are computed using the JL formulation in [@Getal00] for all permitted electron excitation quantum numbers . The black dashed curves constitute the contribution only, i.e., integrations over Eq. (\[eq:dsigma\_zero\_width\]) when summing modes, highlighting its dominance below and through the cyclotron fundamental. The red long-dashed curve is the field-free Klein-Nishina cross section. The and cases very closely approximate the Klein-Nishina result when , the example less so.[]{data-label="fig:tot_csect_KN"}](Fig8.eps) We note that recently [@Weise14] has criticized the computations of [@Getal00] that are summed over all values, claiming them to be erroneous and too large by a simple factor . If such an error had been present, then because of the equivalence of Eq. (\[eq:dsigma\_zero\_width\]) to independent analyses of magnetic Compton scattering, the works of [@DH86; @Herold79; @BAM86] would thereby all be called into question. Such an issue is best probed when and , i.e., deep in the Klein-Nishina regime. Then in the asymptotic limit of low field strengths, , the correct magnetic Compton cross section for a complete summation over all accessible values (i.e., excitation states of the final electron) should approach the well-known field-free Klein-Nishina result, for any value of , and therefore for . This is in fact borne out for the case in Fig. 2 of [@Getal00], and for and in Fig. \[fig:tot\_csect\_KN\] here. These reproduce the Klein-Nishina cross section with impressive precision when . We thereby conclude that the formulation in this paper and those of [@Herold79; @DH86; @BAM86; @Getal00] are all correct, and in agreement for the specialization. It should be noted that in the supercritical field regime, the Landau level energy separation is never vastly inferior to [both]{} and for . Therefore, discretization influences are then always prevalent in the magnetic Compton process, so the magnetic cross section will not reduce exactly to the field-free Klein-Nishina form even for large ; such a property is indeed evident for the case in Fig. \[fig:tot\_csect\_KN\], and also at higher field strengths in Fig. 2 of [@Getal00]. Scattering at low frequencies {#sec:low_freq} ----------------------------- One expects to recover the nonrelativistic, magnetic Thomson cross section in the limit as $B\rightarrow 0$. Here we explore this regime, and also highlight the character far below the resonance (i.e., for ) for arbitrary field strengths. Consider the cross section using the spin-averaged width, the third form in Eq. (\[eq:dsigmas\]). For and far below the resonance, , we have and $\omega_f \rightarrow \omega_i$, so that this cross section goes to the nonrelativistic form in [@Herold79]. Retaining the polarization dependence, the differential forms for the cross section, valid for , can be expressed as $$\begin{aligned} \left(\dover{d\sigma }{ d\cos \theta _f } \right)^{\perp}_{\rm ave} & \approx & \frac{ 3\sigt }{16} \left[ \frac{\omega_i^2 }{\left(\omega_i-B\right)^2} + \frac{\omega_i^2 }{\left(\omega_i +B\right)^2 } \right] \quad , \nonumber\\[-5.5pt] \label{eq:NonrelAve}\\[-5.5pt] \left(\dover{d\sigma }{ d\cos \theta _f } \right)^{\parallel}_{\rm ave} & \approx & \frac{ 3\sigt }{16} \left[ \frac{\omega_i^2 }{\left(\omega_i-B\right)^2} + \frac{\omega_i^2 }{\left(\omega_i +B\right)^2 } \right] \cos^2\theta_f\quad ,\nonumber\end{aligned}$$ These results can be deduced with the aid of Eq. (\[eq:Tave\_lowfreq\]), together with the fact that when . Summing over polarizations, we then have the limiting form $$\left(\frac{d\sigma }{ d\cos \theta _f } \right)_{\rm ave} \; \approx\; \frac{ 3\sigt }{8} \frac{\omega_i^2 }{B^2} \left[ 1 + \cos^2\theta_f \right] \quad \Rightarrow \quad \sigma \;\approx\; \sigt \frac{\omega_i^2 }{B^2} \quad \hbox{for}\quad \omega_i \;\ll\; \min \{1,\, B \} \quad . \label{eq:NonrelAve2}$$ This asymptotic expression for is indicated as a black dashed line in both Figs. \[fig:Int1\] and \[fig:Int3\], and is nicely reproduced by the full spin-averaged numerical evaluations. The dependence at low frequencies appears explicitly in a classical description of magnetic Thomson scattering (e.g. [@CLR71; @GS73]; see also Chapter 4 of Mészáros [@Mesz92]), as does the factor that is the hallmark of dipole radiation mechanisms. Classically, its origin is in Larmor formalism for accelerating charges, when the electron that is constrained by the magnetic field is driven at the frequency of the incoming photon that propagates along [B]{}. The Fourier transform of the photon’s slowly oscillating electric field generates the defining contribution to the radiative power for circular polarization “eigenmodes” that are appropriate for photons moving along the field lines. The cyclotron frequency then scales the acceleration that precipitates the “emission” of a scattered photon: see Sec. 4.1 of [@Mesz92] for pedagogical details. In this description, the effective duration () of the interaction far exceeds the gyroperiod, , of the electron, so that the electron’s gyrational response is only an adiabatic influence. The same dependence on the incoming photon frequency emerges in nonrelativistic quantum mechanical derivations [@CLR71; @Herold79]. Therein it derives from the Fourier transforms encapsulated in the $S$-matrix elements, but again only for the “circular polarization” case of photons initially moving along [B]{}. Specifically, the restrictions imposed by the scattering kinematics in the elastic limit of interplay with the complex exponential plane wave portions of the various electron wave functions and photon states to yield matrix elements proportional when . The low-frequency behavior extends to arbitrary incoming photon angles for and scatterings [@Herold79], a property that is evinced for nonmagnetic Thomson interactions in quantum mechanics (e.g. see Chapter 11 of [@JR80]) due to the orthogonality of the initial and final polarization vectors. This also applies to transitions when the field is present. However, more vector phase space is available for scatterings even when . This generates a significant frequency-independent contribution [@Herold79] for the mode as that dominates all other modes of scattering well below the cyclotron frequency. The consequent disparity between the total cross section for the and polarization states of the incoming photon plays a crucial role in defining the spatial and spectral structure of atmospheres [@Ozel01; @HoLai03; @SPW09] of neutron stars that are permeated by outflowing x-ray emission. Now consider spin-dependent ST and JL formulations. If the same character applied to cross sections that incorporate the spin-dependent widths, one would anticipate that the terms would simply cancel, and we would recover the nonrelativistic form. However, it does not: the denominators of the JL and ST cross sections do not follow the same pattern due to the asymmetry of the spin factors in the cyclotron decay widths for the resonance. In the limit of , the spin factors possess the behavior $$\begin{aligned} \xi_+^{\rm ST} \;\approx\; B & \quad , \quad & \xi_- ^{\rm ST} \;\approx\; 2 \quad ,\nonumber\\[-5.5pt] \label{eq:sigLims}\\[-5.5pt] \xi_+^{\rm JL} \;\approx\; B & \quad , \quad & \xi_- ^{\rm JL} \;\approx\; 2 \quad ;\nonumber\end{aligned}$$ see Eq. (\[eq:xi\_pm\_def\]). This inherent spin asymmetry persists for fields , albeit declining with increasing , and eventually it becomes very small in highly supercritical fields. Regardless of the strength of the magnetic field, for the Thomson regime where , the low-frequency behavior of the spin-averaged terms in the numerators for both diagrams is described by $$T_\mathrm{ave}^{(m),\perp} \;\approx\; \omega_i^2 \quad ,\quad T_\mathrm{ave}^{(m),\parallel} \;\approx\; \omega_i^2\cos^2 \theta_f\quad . \label{eq:Tave_lowfreq}$$ The derivation of equivalent results for the spin-dependent terms is somewhat more involved, but the results condense into forms of comparable simplicity: $$T_\mathrm{spin}^{(m),\perp} \;\approx\; \dover{\left(\epsilon_\perp^2-1\right)^2}{2\epsilon^3_\perp} \quad ,\quad T_\mathrm{spin}^{(m),\parallel} \;\approx\; \dover{\left(\epsilon_\perp^2-1\right)^2}{2\epsilon^3_\perp} \cos^2 \theta_f\quad , \label{eq:Tspin_lowfreq}$$ for . These asymptotic forms apply to both ST and JL formulations, and can be derived using the various results in Appendixes B and C; they do, however, require the additional restriction that . Remembering that and when , the low-frequency limits of the polarization-dependent differential cross sections with either JL or ST spin-dependent widths possess the forms $$\left(\dover{d\sigma }{ d\cos \theta _f } \right)^{\perp, \parallel} \; \approx\; \dover{3 \sigt }{8} \sum\limits_{m = 1}^2 \left[ \frac{2\, T_{\rm ave}^{(m),\perp, \parallel}}{(\epsilon^2_\perp -1)^2} + \dover{ (\xi_-^2 - \xi_+^2) \epsilon_{\perp}^2\, \Gamma^2\, T_{\rm spin}^{(m),\perp , \parallel} } {(\epsilon^2_\perp -1)^4} \right] \label{eq:dsigma_lowfreq}$$ Using Eq. (\[eq:xi\_pm\_def\]), for either set of basis states, when . Inserting Eqs. (\[eq:Tave\_lowfreq\]) and (\[eq:Tspin\_lowfreq\]) and summing over the polarization cases, the general result for the low-frequency form of the scattering cross section is $$\dover{d\sigma }{ d\cos \theta _f } \; \approx\; \dover{3 \sigt }{2} \left[ \omega_i^2 + \dover{\Gamma^2}{\epsilon^2_\perp} \right] \dover{1 + \cos^2\theta_f }{(\epsilon^2_\perp -1)^2} \quad ,\quad \omega_i\;\ll\; \min\{ 1,\, B\} \quad . \label{eq:dsigma_lowfreq_fin}$$ This approximation applies for both subcritical and supercritical fields. The contribution comes from the scattering mode, while the mode constitutes the remainder. The total cross section is simply obtained: $$\sigma \; \approx\; \dover{\sigt }{B^2} \left[ \omega_i^2 + \dover{\Gamma^2}{1+2B} \right] \quad ,\quad \omega_i\;\ll\; \min\{ 1,\, B\} \quad . \label{eq:dsigma_lowfreq_tot}$$ Observe that the or portion results from a partial cancellation between a positive spin-up contribution and a slightly smaller negative term that comes from the spin-down case for the virtual electron. As a result, the differential cross section involving the spin-dependent widths will yield a constant term in the numerator that is proportional to , and this becomes dominant as is extremely small. This anomalous character arises when (i.e., when for the magnetic Thomson case of ), which is only relevant for very small photon frequencies far below the resonance. Such a domain, where when (or for ), is unlikely to play any significant role in astrophysical models. In practice, at such low frequencies, contributions from the scattering mode for small but finite photon incidence angles will yield cross sections [@Herold79; @DH86; @Sina96] that dominate the ones resulting from this specialization here. Note that quantum mechanically, the origin of this behavior is from the inclusion of the spin-dependent widths in the complex exponentials appearing in the wave function \[see Eq. (\[eq:spinor\_coeffs\]) for general forms\] for the intermediate electron state. This propagates through the Fourier transforms incorporated in the scattering matrix elements, specifically the temporal integrations that generate overall energy conservation, so that the quantum “fuzziness” of energies of the excited virtual electron modifies the overall kinematics from cases where the finite lifetime of the intermediate state is not treated. Moreover, this effect is not observed in the spin-averaged calculation, since there is exact cancellation of the pertinent contributions for spin-up and spin-down cases for the virtual electron. Scattering at the peak of the resonance {#sec:res_peak} --------------------------------------- It is instructive to focus on the scattering cross section at the resonance, where the first Feynman diagram contribution dominates: we do so in this and the subsequent subsection. This is a parameter regime that is obviously of great import for resonant Compton scattering invocations in astrophysical models [@BH07; @BWG11; @Belob13; @FT07; @ZTNR11; @NTZ08]. Right at the peak of the cyclotron resonance, the following special values are realized: $$\omega _i \; =\; B \quad ,\quad \epsilon _{m=1}^2 \; =\; 0 \quad ,\quad {\cal E}_{m=1} \; =\; 1+B \quad . \label{eq:ResCon}\\[-1.5pt]$$ Summing over the spin states, the differential cross section can then be expressed as $$\frac{{d\sigma _{{\rm{res}}}^{\perp, \parallel } }}{{d\cos \theta _f }} \; =\; \frac{{3\sigma _{\rm{T}} }}{8}\frac{{\omega _f^2 e^{ - \omega _f^2 \sin ^2 \theta _f /2B} }}{{ B\left( {2B - \omega _f - \zeta } \right)}} \dover{Z^{\perp, \parallel }}{ \left( {1 + B} \right)^2 \Gamma ^2 }\quad . \label{eq:ResCr}$$ Here, the factor that encapsulates the spin dependence of the cross section is $$Z^{\perp, \parallel } \; =\; \frac{{\left( {\xi _ + ^2 + \xi _ - ^2 } \right) T_{{\rm{ave}}}^{(1),\perp, \parallel } + \left( {\xi _ - ^2 - \xi _ + ^2 } \right) T_{{\rm{spin}}}^{(1),\perp, \parallel } }}{{\xi _ + ^2 \xi _ - ^2 }} \quad ; \label{eq:ResZ}$$ this is applicable to all three forms captured in Eq. (\[eq:dsigmas\]), provided is adopted for the spin-averaged case. Remember that , with functional forms given in Eq. (\[eq:Taves\]). The presence of the factor implies that the effective resonant cross section scales as , when integrated over the resonance profile, i.e., it is of the order of times factors that depend on the field strength. It is this integral that dictates the approximate strength of resonant scattering in determining how fast electrons and photons exchange energy in this process. The scattering at therefore masquerades as an effective cyclotron decay, i.e., it is first order in the fine-structure constant. Hence, the significance of resonant Compton interactions for astrophysical settings can be approximately as probable as cyclotron emission. Specializing to the ST basis states, we find that the cross section for perpendicular and parallel polarizations are identical, and the factors assume the form $$Z^{ST, \perp} \; =\; Z^{ST,\parallel} % = 2B\left( {1 + B} \right)\eta- \omega _f - B \; =\; B(1+2 B) - \omega_f \Bigl\{ 1 + 2B(1+B) \, [1-\cos\theta_f] \Bigr\}\quad . \label{eq:ResZST}$$ The differential cross sections for both perpendicular and parallel polarizations then are identical to the single form $$\dover{d\sigma_{\rm{res}}^{ST,\perp ,\parallel } }{d\cos \theta _f } \; =\; \dover{3\sigma _{\rm{T}} }{8} \dover{\omega _f^2 e^{ - \omega _f^2 \sin ^2 \theta _f /2B}}{B\left( {1 + B} \right)^2\Gamma^2} \;\dover{B(1+2 B) - \omega_f \{ 1 + 2B(1+B) \, [1-\cos\theta_f] \} \vsp }{ 2B - \omega_f \{ 1 + B \, [1-\cos\theta_f] \} \vsp } \quad . \label{eq:ResCSST}$$ Since, throughout the paper, the incident photons are assumed to propagate along the magnetic field, the differential cross section is insensitive to the choice of initial photon linear polarization. Yet, the ST cross section is also independent of the linear polarization of the outgoing photon [right at the peak of the resonance]{} (), regardless of its angle of emergence. This exceptional character does not extend to differential cross sections that employ either the spin-averaged or the spin-dependent JL widths: as seen in Fig. \[fig:Angular1\], the perpendicular mode contributes more to the cross section than the parallel mode in both the spin-averaged and the spin-dependent JL cases. Such contrasting behavior is highlighted more incisively in Fig. \[fig:Int\_Bdep\], where the ratios of the JL and spin-averaged cases to the ST one, evaluated exactly at the resonance peak (left panel), are illustrated as a function of the field strength, and for the two final polarization cases. This cross section ratio plot captures the main idea of this paper: that treating the resonant scattering interaction correctly using the Sokolov and Ternov eigenstates of the Dirac equation introduces modifications to in the range of around 10% — 60% for a wide domain of fields, , relative to the spin-averaged formalism that has traditionally been employed in the literature. Moreover, the substantial differences appearing between the JL and ST cross sections at the resonance peak clearly indicate that it is insufficient to merely introduce spin-dependent formalism using JL basis states for the intermediate electron; advancing to the ST formalism that is the centerpiece of this paper is requisite for more precise implementation in astrophysical models. In the limit of , the low behavior of is characterized in Eq. (\[eq:Tave\_lowfreq\]). In contrast, the low dependence of differs from Eq. (\[eq:Tspin\_lowfreq\]), due to contributions from the restriction. In addition, scattering kinematics dictates for , and the exponential in Eq. (\[eq:ResCSST\]) is approximately unity. Finally, the spin factors possess limits as summarized in Eq. (\[eq:sigLims\]). The upshot is that all cross section ratios approach unity for highly subcritical fields, except for the JL mode case, which elicits a larger cross section because of the interplay between the and terms and the widths. The behavior of at the resonance for in Eq. (\[eq:ResZST\]) requires retaining terms to second order in , as given by the expression $$\omega_f \approx B - \frac{B^2}{2}\left( 1 - \cos\theta_f \right)^2 \label{eq:ResOmf}$$ In this limit for either JL or ST basis states, the terms in Eq. (\[eq:ResZST\]) become, to lowest order in , $$Z^{\perp,\parallel} \; =\; \frac{B^2}{2}\left(1+\cos\theta_f\right)^2 \label{eq:ResZSTx}$$ The differential cross section in Eq. (\[eq:ResCSST\]) then reduces to $$\dover{d\sigma_{\rm{res}}^{\perp ,\parallel } }{d\cos \theta _f } \; \approx \; \frac{ 3 \sigma_\mathrm{T} B^2}{ 16 \Gamma^2} \left( 1+\cos\theta_f \right)^2\quad , \label{eq:ResCSSTax}$$ neglecting terms of the order of , and this integrates to give $$\sigma_{\rm{res}}^{\perp ,\parallel } \; \approx \; \frac{ \sigma_\mathrm{T} B^2}{ 2 \Gamma^2} \; = \; \frac{ 9 \sigma_\mathrm{T} }{ 8 \alpha^2 B^2 } \label{eq:ResCSSTin}$$ using from Eq. (\[eq:Gammave\_red\]). On the other hand, in this limit, the scattering cross sections with the spin-averaged widths for perpendicular and parallel polarizations are not equivalent, and the pertinent terms are $$Z^{Ave,\perp} \; = \; 2 B^2 \quad ,\quad Z^{Ave,\parallel} \; = \; 2 B^2 \cos^2\theta_f \quad , \label{eq:ResZav}$$ instead of Eq. (\[eq:ResZSTx\]). Integrating the differential cross section with these $Z$ terms implemented yields the following total cross sections: $$\begin{split} \sigma^{Ave,\perp} \; &= \; \frac{ 3 \sigma_\mathrm{T} B^2}{2\Gamma^2} = \frac{27 \sigma_\mathrm{T}}{ 8 \alpha^2 B^2 } \\ \sigma^{Ave,\parallel} \; &= \; \frac{ \sigma_\mathrm{T} B^2}{2\Gamma^2} = \frac{9 \sigma_\mathrm{T}}{ 8 \alpha^2 B^2 } \label{eq:ResCSav} \end{split}$$ Clearly, the total cross section for the spin-averaged case for the parallel polarization mode is the same as for both the ST and JL formalisms. Yet, the total spin-averaged cross section for the perpendicular polarization is a factor of 3 larger than the ST/JL result in Eq. (\[eq:ResCSSTin\]). The limiting forms of these cross sections exactly at the resonance are illustrated in the left panel in Fig. \[fig:Int\_Bdep\]. ![The ratios of JL (green curves) and average (red curves), angle-integrated cross sections to that of the ST cross section, evaluated right at the peak of the cyclotron resonance, i.e., at (left panel) and somewhat off peak (right panel), i.e., in the wings of the resonance. These are displayed as functions of the magnetic field strength, and for the two outgoing photon polarizations: perpendicular (solid curves) and parallel (dashed) modes. All ratios approach unity asymptotically as since the spin-dependent widths satisfy in this regime. In the left panel only, at the peak of the resonance the ST cross section that forms the benchmark for the ratios is independent of polarization.[]{data-label="fig:Int_Bdep"}](Fig9) For fields , the spin factors approach unity for both ST and JL formalisms. It then follows that , and the cross section becomes approximately the same for all three formulations. This behavior is evident in Fig. \[fig:resonance\_phasespace\], which provides a contour plot of the JL/ST and average/ST ratios near the resonance, for arbitrary field strengths. This graphic offers a comprehensive illustration of the importance and scope of spin-dependent influences in the resonance and its wings. Finally, observe that this ST polarization symmetry is broken when the scattering moves off the peak of the resonance (), a domain illustrated in the right panel of Fig. \[fig:Int\_Bdep\]. Moreover, the most noticeable feature of moving the incident frequency into the wings of the cyclotron resonance is the dramatic reduction in differences between the ST, JL, and average cross sections, concomitant with the decline of the spin-dependent influences highlighted in this exposition. Away from the resonance peak, all ratios asymptotically approach unity in the limits and . Note also that polarization dependence reemerges at the resonance peak when the incident photons do not move parallel to [B]{}, a case not explicitly examined in this paper, but that will form the focus of a future study. ![Continuous contour plots of the ratios (linear scale) of JL (left panels) and average (right panels) angle-integrated cross sections, to that of the ST cross section, evaluated over ranges of frequencies that span the cyclotron resonance, i.e., around . These are displayed as functions of the magnetic field strength, averaging over the two outgoing photon polarizations. The upper and lower panels adopt different scales, the upper ones providing bracketing of the resonance pinned at the width , which depends on , and the lower ones providing an absolute scale independent of . All ratios approach unity asymptotically as since the spin-dependent widths satisfy in this regime.[]{data-label="fig:resonance_phasespace"}](Fig10) Approximate cross section in the resonance {#sec:csect_res_approx} ------------------------------------------ It is by now evident that outside the resonance, the Compton cross section formalism is degenerate between the JL and ST approaches when summed over electron spin states, a domain that is well studied with forms presented elsewhere and in this paper that are useful for various applications. With the enhancement here of including spin-dependent effects in the cyclotron resonance for scatterings, it is desirable to put forward useful analytic expressions that can readily be incorporated in astrophysical models. In our recent study of the rates of resonant Compton cooling of relativistic electrons in the neutron star magnetosphere [@BWG11], we briefly introduced just such an approximate resonance cross section using the ST basis states to describe the impact of spin-dependent resonance widths. In this section, we develop, in fuller details, an approximate expression for the scattering cross section near the resonance, using the ST spin-dependent width. The starting point is Eq. (\[eq:dsigmas\]). One need only consider the first Feynman diagram that contributes to the resonance, which is given by $$\left(\frac{d\sigma }{ d\cos \theta _f } \right)_{\rm res} \; \approx\; {\cal F}\sum_{s=\pm 1} \left[ \frac{\left(T_{\rm ave}^{\perp } + T_{\rm ave}^\parallel\right) + s\left(T_{\rm spin }^{\perp } + T_{\rm spin}^\parallel\right) }{4\left(\omega_i-B\right)^2 + \left(\epsilon_\perp-s\right)^2\left(1+B\right)^2\Gamma^2/\epsilon_\perp^2 } \right] \label{eq:dsigma_res1}$$ where the superscript has been suppressed for the terms. Here, is the spin-average width, and with for spin-up and for spin-down with the sum of the average terms equivalent to used in Eq. (14) of [@BWG11], though in a slightly different form, $$T_{\rm ave} \; =\; T^\perp_{\rm ave}+T^\parallel_{\rm ave} \; = \; 2\Bigl[ \left( 1+\omega_i \right) \left( \omega_i - \zeta \right) - \omega_f \Bigr].$$ Adding together the $T_{\rm spin}$ terms for the perpendicular and parallel polarizations in Eqs. (\[eq:ST\_Tspin\_perp\]) and (\[eq:ST\_Tspin\_par\]) for the first Feynman diagram yields a moderately lengthy numerator. This can be made more compact by eliminating terms that are of higher order in the parameter ; observe that is small relative to within the Lorentz profile of the resonance, for arbitrary field strengths. There is no unique path for such a manipulation, and the result cannot map over precisely to domains outside the resonance. The manipulation path adopted in [@BWG11] leads to $$T_{\rm ave} + s T_{\rm spin} \; \approx\; \frac{\left(\epsilon_\perp-s\right)^2}{2\epsilon^3_\perp} \Bigl[ \left(2\epsilon_\perp + s\right) T_{\rm ave} - s\left(\epsilon_\perp+s\right)^2 \left(\omega_i-\omega_f\right) \Bigr]\quad , \label{eq:Ttot_res_BWG11}$$ an expression contained in Eq. (16) of [@BWG11] that is routinely established using the intermediate result $$T_{\rm spin} \; =\; T^\perp_{\rm spin}+T^\parallel_{\rm spin} \; \approx\; \frac{1}{2\epsilon^3_\perp} \left[ - \left(3\epsilon^2_\perp - 1\right)T_{\rm ave} - 2\left(\epsilon^2_\perp-1\right)\omega_i \left(E_f-1\right) \right] \label{eq:Tspin_res_BWG11}$$ Inserting Eq. (\[eq:Ttot\_res\_BWG11\]) into Eq. (\[eq:dsigma\_res1\]) yields an approximation to the exact differential cross section that is accurate to considerably better than a percent when summed over electron spins. However, isolating the spin contributions, this approximation is precise only to a few percent, with errors compensating when the sum over is performed. To improve the integrity of the approximation, the algebra of the numerator can be expanded modestly to retain all terms of order , eliminating only those of order (there are no terms of higher order in ). This tightens the approximation substantially. The result of this manipulation generates a factor $$\begin{aligned} {\cal N}_s \; = \; 2\epsilon_{\perp}^3 (T_{\rm ave} + s T_{\rm spin}) & = &\left(\epsilon_\perp-s\right)^2 \Bigl\{ \left(2\epsilon_\perp + s\right) T_{\rm ave} - s\left(\epsilon_\perp+s\right)^2 \left(\omega_i-\omega_f\right) \Bigr\} \nonumber\\[-5.5pt] \label{eq:calNs_final}\\[-5.5pt] && - \; 2 s \,\delta\, \bigl(1+\cos\theta_f \bigr) \Bigl[ (1+\omega_i ) \, \zeta - \omega_i (\omega_i + \omega_f) \Bigr] \quad ,\nonumber\end{aligned}$$ the first term of which is equivalent to in [@BWG11]. The polarization-averaged differential cross section spanning the resonance can then be written in a compact form as $$\left(\frac{d\sigma }{ d\cos \theta _f } \right)_{\rm res} \;\approx\; \frac{ {\cal F} }{2\epsilon^3_\perp} \sum_{s=\pm 1} \dover{{\cal N}_s }{{\cal D}_s} \quad ,\quad {\cal D}_s \; =\; 4\left(\omega_i-B\right)^2 + \left(\epsilon_\perp-s\right)^2\left(1+B\right)^2 \, \dover{\Gamma^2}{\epsilon_\perp^2} \quad . \label{eq:dsig_resonance}$$ For [each]{} spin case , this result is numerically accurate to a precision of better than % across the resonance Lorentz profile, for fields in the range , with only a slight degradation of the approximation at highly subcritical and supercritical fields. In particular, for all fields, the precision is improved from % in the wings of the resonance to better than % when , where the core of the resonance peak is sampled. Observe that the do not depend on ; only the and do. At the peak of the resonance, where and the correction term in Eq. (\[eq:calNs\_final\]) does not contribute (nor does its absent counterpart), the sum over spins becomes almost trivial, yielding $$\left(\frac{d\sigma }{ d\cos \theta _f } \right)_{\rm peak} \;\approx\; \frac{ 2 {\cal F} }{(1+B)^2\Gamma^2} \Bigl\{ T_{\rm ave} - (\omega_i - \omega_f) \Bigr\} \quad . \label{eq:dsig_peak}$$ This is equivalent to twice Eq. (\[eq:ResCSST\]), i.e., the sum of identical polarization-dependent results for the resonance peak. The analogous polarization-dependent forms of the differential cross section can be developed in a similar manner, replacing by and , where $${\cal N}^{\perp}_s \; = \; \left(\epsilon_\perp-s\right)^2 \biggl[ \left(2\epsilon_\perp+s\right)T^\perp_{\rm ave} - s \left( \epsilon_{\perp} + s \right)^2 \Bigl\{ \zeta - (\omega_i - \omega_f)/2 \Bigr\} \biggr] - 2 s\, \delta \Bigl[ (1+2\omega_i ) \, \zeta - 2 \omega_i^2 \Bigr] \quad , \label{eq:calNperp_final}$$ with , and $$\begin{aligned} {\cal N}^{\parallel}_s & = & \left(\epsilon_\perp-s\right)^2 \biggl[ \left(2\epsilon_\perp+s\right)T^\parallel_{\rm ave} - s \left( \epsilon_{\perp} + s \right)^2 \Bigl\{ - \zeta + 3 (\omega_i - \omega_f)/2 \Bigr\} \biggr] \nonumber\\[-6.0pt] \label{eq:calNpar_final}\\[-6.0pt] & & \qquad - 2s\, \delta \omega_i \Bigl[ - \zeta + (\omega_i - \omega_f) \Bigr] -2 s\, \delta \cos\theta_f \Bigl[ (1+\omega_i ) \, \zeta - \omega_i (\omega_i + \omega_f) \Bigr] \quad .\nonumber\end{aligned}$$ Observe that . Individually, these polarized approximations are of the same order of accuracy as the combination of Eqs. (\[eq:calNs\_final\]) and (\[eq:dsig\_resonance\]). These forms for , and , when inserted into Eq. (\[eq:dsig\_resonance\]), constitute an extremely useful set of approximations for the magnetic Compton differential cross section in the resonance. They apply specifically to the ST formulation, and provide a concise toolkit for incorporating spin-dependent resonant Compton formalism into astrophysical models. Comparatively compact approximate forms for the total cross section in the resonance can be developed using the protocol detailed in [@BWG11] for resonant Compton cooling rates, and summarized in Sec. \[sec:tot\_csect\]. Specifically, one replaces the integration over by one over the variable that is defined in Eq. (\[eq:phivar\_def\]). There are two branches to the inversion of the quadratic relation between these two variables, encapsulated in Eq. (\[eq:r\_branches\]). This change of variables effects the correspondence $$\int_{-1}^1 d(\cos\theta_f) \, {\cal F} \, \sum_{s=\pm 1}\dover{{\cal N}_s }{{\cal D}_s} \;\to\; \dover{3\sigt}{8\omega_i} \int_{0}^{\Phi} \dover{e^{-\omega_i \phi /B} \, d\phi}{\sqrt{1-2\phi z+\phi^2}} \, \sum_{r_{\pm}} \sum_{s=\pm 1} \dover{{\cal N}_s }{{\cal D}_s} \quad . \label{eq:correspondence}$$ There are two different forms of integrals present in the resulting cross section. The first is $${\cal I} \; =\; \int_{0}^{\Phi} \dover{(1-\phi z )\, e^{-\omega_i \phi /B} }{\sqrt{1-2\phi z+\phi^2}} \, d\phi\quad , \label{eq:calI_def}$$ which appears in Eq. (\[eq:calI\_def\]) in the Sec. \[sec:tot\_csect\] formalism. This integral contributes to the leading order terms that are dominant when . For the sum over polarizations, the approximate cross section can be expressed as $$\sigma_{\rm res} \;\approx\; \frac{3\sigt}{8} \, \dover{\omega_i}{\epsilon^3_\perp} \sum_{s=\pm 1} \dover{{\cal I}_s + s\delta \hat{\cal J}}{{\cal D}_s} \quad , \label{eq:sig_resonance}$$ where for , the integrals can be cast in the form $${\cal I}_s \; =\; (\epsilon_\perp -s)^2\, \biggl\{ 2\epsilon_\perp + \dover{s\delta}{1+2\omega_i} \biggr\} \, {\cal I} \quad , %\; \approx\; 2 \biggl\{ \epsilon_\perp \, (\epsilon_\perp -s)^2 \quad , % + \delta\, \dover{ s (1+ \omega_i) - \epsilon_{\perp}}{1+2\omega_i} \biggr\} % \, {\cal I} \quad , \label{eq:calI_s_def}$$ using Eq. (\[eq:calI\_def\]). There are also residual terms () that involve a more complicated integrand that includes the quadratic factor in the denominator. This introduces an analytic complexity that is largely avoidable with appropriate approximation and simplification. The pathological nature of the integrand can be eliminated by replacing the factor in Eq. (\[eq:calNs\_final\]) by its angle-integrated average. We find that $$\Bigl\langle \cos \theta_f \Bigr\rangle_{\rm peak} \;\approx\; \dover{1+2\omega_i}{2 (1+\omega_i)} \label{eq:cos_thetaf_ave}$$ is an approximation numerically accurate to better than 1.5% when computing the average at the peak of the resonance, i.e. using the differential cross section in Eq. (\[eq:dsig\_peak\]). Employing such an approximation in a small (of order ) term is both tolerable and expedient. Substituting this in Eq. (\[eq:calNs\_final\]), the resulting form is obtained from Eq. (\[eq:sig\_resonance\]) via the replacement $$\hat{\cal J} \;\to\; \dover{3+4\omega_i}{\omega_i (1+\omega_i)}\, {\cal J} \quad ,\quad {\cal J} \; =\; \int_{0}^{\Phi} \dover{e^{-\omega_i \phi /B}\, d\phi }{\sqrt{1-2\phi z+\phi^2}} \quad . \label{eq:calJ_def}$$ This introduces the second integral appearing in the resonance cross section. Collecting results, we now have the final form for the approximate total cross section, $$\sigma_{\rm res} \;\approx\; \frac{3\sigt}{8} \, \dover{\omega_i}{\epsilon^3_\perp} \sum_{s=\pm 1} \dover{1}{{\cal D}_s} \left\{ 2\epsilon_\perp (\epsilon_\perp -s)^2\, {\cal I} + s\delta \, \left\lbrack \dover{(\epsilon_\perp -s)^2}{1+2\omega_i} \, {\cal I} + \dover{3+4\omega_i}{1+\omega_i} {\cal J} \right\rbrack \right\} \quad . \label{eq:sig_resonance_fin}$$ The precision of this approximation relative to exact numerical integrations of the full ST differential cross section is better than around 0.3% for at the peak and within a few percent of the wings of the resonance, and it is considerably better than this tolerance outside the interval . It can be applied in the energy range spanning the resonance. At the resonance peak, where , the leading order term in Eq. (\[eq:sig\_resonance\_fin\]) is simply reproduced by direct manipulation of the integral of Eq. (\[eq:dsig\_peak\]). The same manipulations can be applied to the polarization version of the approximate differential cross section in the resonance, encapsulated via the numerator factor in Eq. (\[eq:calNperp\_final\]). Using the identity , the result is $$\sigma_{\rm res}^{\perp} \;\approx\; \frac{3\sigt}{8} \, \dover{\omega_i}{\epsilon^3_\perp} \sum_{s=\pm 1} \dover{1}{{\cal D}_s} \Biggl\{ \epsilon_\perp (\epsilon_\perp -s)^2 \, {\cal I} - s\, \dover{1+2\omega_i}{1+\omega_i}\, \Bigl({\cal J} - {\cal I}\Bigr) + s\delta \left\lbrack \dover{8\omega_i^2+\omega_i-1}{1+2\omega_i} \, {\cal I} + 6\, {\cal J} \right\rbrack \Biggr\} \quad . \label{eq:sig_resonance_perp_fin}$$ It is then a simple matter to subtract this from Eq. (\[eq:sig\_resonance\_fin\]) to generate the equivalent result for . Numerical evaluation of the and integrals can be facilitated by two algorithms. The first is to employ the class of integrals $${\cal I}_{\nu}(z,\, p)\; =\; \int_0^{\Phi (z)} e^{-p\, \phi} \, \left(1-2 z\phi + \phi^2\right)^{\nu/2}\, d\phi\quad , \label{eq:calInu_def}$$ defined in [@BWG11], then we can simply write and for . Techniques for the series evaluation of the integrals are outlined in Appendix B of [@BWG11]. Alternatively, we can define another related class of integrals $${\cal H}_n(z,\, p)\; =\; \int_0^{\Phi (z)} \dover{\phi^n \, e^{-p\, \phi} \, d\phi}{\sqrt{1-2 z\phi + \phi^2} } \; =\; \sum_{k=0}^{\infty} \dover{ (-p)^k}{k!} \, Q_{k+n}(z)\quad , \label{eq:calHnu_def}$$ where are Legendre functions of the second kind (defined in 8.703 of [@GR80]), and the series equivalence is established by changing variables and using manipulations along the lines of those employed in [@BGH05]. Then one can use $${\cal I} \; = \; {\cal H}_0(z,\, p) - z{\cal H}_1(z,\, p) \quad ,\quad {\cal J} \; =\; {\cal H}_0(z,\, p) \label{eq:calI_calJ_ident}$$ and the Legendre series to efficiently compute the integrals. This second alternative appears to be the more expedient algorithm. Discussion: The Influence of Vacuum Dispersion {#sec:disperse} ============================================== The presentation here has restricted considerations throughout to nondispersive situations where photons move at speed , i.e., . In material media, plasma, and also in the presence of strong large-scale electromagnetic fields, this is only an approximation: dispersion arises and can potentially offer significant modifications to QED mechanisms. Plasma dispersion can be neglected in neutron star magnetospheres, since the density of charges is sufficiently low that the plasma frequency is in the radio-to-infrared band of frequencies, so that x rays and gamma rays propagate essentially in a nondispersive manner: the refractive index induced by the plasma scales roughly as . The situation is very different for vacuum dispersion, and so that it will form the focus of this discussion. It is instructive to assess when corrections to the photon scattering dynamics due to vacuum dispersion or birefrengence effects become important. It has been understood for decades that the magnetized vacuum is dispersive, so photons travel at phase speeds differing from ; these speeds differ for propagation parallel and oblique to the field because of the anisotropy of the polarization tensor . The dispersion relation necessarily attains the form $$\dover{\omega^2}{c^2} \; =\; k_z^2 + {\cal F}(k_{\perp}^2)\quad , \label{eq:dispersion}$$ where the restriction expresses the property that dispersion is zero for propagation along [B]{}. The dispersion arises because spontaneous photon conversion (absorption) processes are permitted in QED in the presence of an external electromagnetic field. The leading order contribution in strong magnetic fields to dispersion is magnetic pair creation, , so one naturally anticipates that dispersion can become significant in supercritical fields , and is of the order of , the fine-structure constant. Since pair threshold is never exceeded for photon propagation along the field, such photons must travel dispersion-free, with a refractive index identical to unity. The polarization tensor and refractive index for the magnetized vacuum could, in principal, be obtained from the pair creation rate via the optical theorem. However, the standard path of choice is to directly compute the polarization tensor by some technique, and often this employs the effective Lagrangian or Schwinger proper-time approach [@Adler71; @TE75; @HH97]. The refractive index can be expressed in the approximate form $$n_{\perp,\,\parallel} \; =\; 1 + \dover{\fsc}{4\pi} \, \sin^2\theta \, N_{\perp,\,\parallel} \bigl( \omega\sin\theta ,\, B \bigr)\quad , \label{eq:refr_index_n}$$ where the functions are relatively manageable double integrals that are dimensionless. Here is assumed; dispersion accessing pair channels will be discussed shortly. As in the rest of the paper, the magnetic field is expressed here in units of the Schwinger field . When the photons propagate at a nonzero angle to the magnetic field, the two polarization modes propagate with different speeds, and the magnetized vacuum is birefringent. In the regime of photon energies well below pair creation threshold (practically, this is ), these integrals distill down to a single integral. For low and high field regimes, the resulting integral can be evaluated analytically (e.g., see Appendix D.5 of [@Dittrich00]). Accordingly, in the subcritical domain, the refractive indices possess the asymptotic forms given in Eq. (46) of [@Adler71] or Eq. (9) of [@TE75]: $$\begin{aligned} n_{\perp} & \approx & 1 + \dover{2\fsc}{45\pi}\, B^2 \sin^2\theta \quad ,\nonumber\\[-5.5pt] \label{eq:n_vac_lowB}\\[-5.5pt] n_{\parallel} & \approx & 1 + \dover{7\fsc}{90\pi}\, B^2 \sin^2\theta \quad ,\quad B\;\ll\; 1\quad .\nonumber\end{aligned}$$ Observe that the labeling convention that [@Adler71] employed was reversed from that used here and elsewhere (e.g. [@TE75]): again, here we ascribe the subscripts according to the orientation of a photon’s [electric]{} field vector relative to its momentum [k]{} and the large-scale field [B]{}. The convention Adler [@Adler71] adopted was defined by the photon’s magnetic field vector orientation. The other low-frequency asymptotic limit of Eq. (\[eq:refr\_index\_n\]) is for , but with small enough that . The appropriate forms for the refractive index can be deduced from Eq. (38) of [@TE75] or Eq. (2.97) of [@Dittrich00]: $$\begin{aligned} \label{eq:disper} n_{\perp} & \approx & 1 + \dover{\fsc}{6\pi}\, \sin^2\theta \quad ,\nonumber\\[-5.5pt] \label{eq:n_vac_highB}\\[-5.5pt] n_{\parallel} & \approx & 1 + \dover{\fsc}{6\pi}\, B\, \sin^2\theta \quad ,\quad B\;\gg\; 1\quad .\nonumber\end{aligned}$$ These limiting forms need to be modified when . For example, when but , Eq. (10) of [@TE75] illustrates that the refractive index is slightly [less than]{} unity so that the eigenmode phase speeds exceed . Notwithstanding, Eqs. (\[eq:n\_vac\_lowB\]) and (\[eq:n\_vac\_highB\]) serve to illustrate the general character of the refractive index of the magnetized vacuum for a large portion of parameter space below pair creation threshold, the domain of relevance to this presentation. It is immediately apparent that vacuum dispersion and birefringence both disappear for photon propagation along the magnetic field, , the restriction in this paper for the incoming photons in the ERF. In addition, the mode always possesses a refractive index very close to unity, with . In contrast, these asymptotic results indicate that the mode can realize significant departures of from unity when , provided that . This domain is largely of academic interest, because to date, the surface polar magnetic fields of magnetars [@Kouv98; @Kouv99] have only been deduced to have values , so one can safely assume that in the magnetospheres of magnetars and normal pulsars, the principal objects of interest for the application of the Compton scattering developments presented in this paper. The above asymptotic formulas apply to domains well below pair creation threshold. When the threshold is reached or exceeded, , the mathematical pathology of the refractive index, i.e., of the functions in Eq. (\[eq:refr\_index\_n\]) is more complicated than is presented by [@Adler71; @TE75]. Precise treatment of the pair resonances contributing to the polarization tensor is then necessary [@Shabad75; @SU84], and it leads to interesting refractive effects in light propagation in curved field morphologies [@SU84; @SU82]: Shabad and Usov observed that light can be captured and channeled by the magnetic field. Such subtleties deserve consideration when rises to and above the pair threshold; however, this is a domain that [a priori]{} requires modification of the external lines in the Feynman diagrams due to the availability of pair channels. Such complexity for a focused and minor portion of kinematic phase space is beyond the scope of the scattering analysis here. To cast further insight into the role of vacuum dispersion for calculations of magnetic Compton scattering, observe that it provides a purely kinematic modification to the differential cross section; see, for example, the plasma dispersion context in the study of [@CLR71]. The dispersion relations for the photons involved in the scatterings are for the incoming photon moving at precisely along the field, and for the scattered photon, where represents either or , depending on its polarization state. The vector relations for the momentum components of the eigenmodes in dispersive cases are provided in Eq. (46) of [@Adler71], expressed in terms of the refractive index . The mathematical development of the cross section proceeds as outlined in Sec. \[sec:csect\_formalism\], invoking the substitution throughout, thereby describing the altered phase velocity of the final photon. This preserves the explicit wavenumber or momentum dependence in the complex exponentials for the spatial integrals in Eqs. (\[eq:S1\_T1\_orig\]) and (\[eq:S2\_T2\_orig\]), and the frequency or energy dependence in the denominators of Eqs. (\[eq:T1\_orig\]) and (\[eq:T2\_orig\]); these denominators arise, of course, from the temporal integrations. However, the relationship between and modifies the scattering kinematic relation in Eq. (\[eq:kinematics\_photons\]) slightly. Since the value of in Eq. (\[eq:kinematics\_electrons\]) is now replaced by , the effect of this modification is simply to replace by in Eq. (\[eq:kinematics\_photons\]). Then the permitted range of scattering angles is , i.e., outside the Čerenkov cones. As this alteration is propagated through the algebraic reduction of the cross section, it is quickly observed that changes to the factors outside the summations and also in the numerators of Eqs. (\[eq:Diff2\]) and (\[eq:Fs\_def\]) are merely affected by the substitution . The same is true for the factor and the and terms in the numerators of Eq. (\[eq:Dspin\]). In addition, the introduction of dispersive corrections slightly modifies the energy of the intermediate state for the Feynman diagram, so that in Eq. (\[eq:Em\_def\]): it is the diagram that receives the largest, albeit small, dispersive corrections. In contrast, the cyclotron resonance is precisely at , and there the cross section is dominated by the diagram, with the energy and momentum of the nondispersive incoming photon being the controlling parameter. Accordingly, including vacuum dispersion influences does not change the resonant frequency, and for , does not significantly broaden the cyclotron resonance beyond that incurred by the non-dispersive cyclotron widths described in the body of this paper — for pulsars and magnetars, vacuum dispersion thereby provides corrections to the magnitude of the scattering cross section of a few percent at most \[see Eq. \[eq:disper\]\] when , and generally much smaller. For this reason, with its concomitant mathematical complexity, the influence of such dispersion is neglected throughout the analysis of this paper, following the precedent set by numerous expositions on Compton scattering in strong magnetic fields. Such a protocol of applying a small dispersion approximation in fields has also been adopted by many authors treating other processes such as pair creation and cyclotron/synchrotron emission, where it yields practically useful results. An exception arises for photon splitting, as Adler [@Adler71] considered, where dispersion opens up new polarization channels that are otherwise forbidden in the limit of zero dispersion: this provides a special situation where it is crucial to consider dispersion effects in the magnetized vacuum. Conclusion ========== This paper has offered QED formalism and new computational developments of Compton scattering in strong magnetic fields, for the specific case of ground-state–ground-state transitions in the electron rest frame, and when photons are incident parallel to the magnetic field. The analyses are extremely relevant to the study of strongly magnetized neutron stars. The calculations treat the very important cyclotron resonance regime, incorporating spin-dependent decay rates for the intermediate excited electron state. Inclusion of the finite lifetimes for the ephemeral states is for physical consistency and thereby generates a convergent scattering cross section at the cyclotron energy, . Correct treatment of such decays in the resonance is required, since the transition rates depend sensitively upon the choice of the wave functions for the virtual electrons. The historical convention in magnetic Compton scattering analyses has been to employ JL [@JL49] wave functions. These fail to preserve spin configurations under Lorentz boosts along [B]{} (e.g., see [@BGH05]), which is problematic for their invocation for scattering in the cyclotron resonance, since the intermediate electron state possesses nonzero momentum parallel to the field. The appropriate choice for QED scattering analyses is instead the ST [@ST68] electron-positron symmetric eigenfunctions of the magnetic Dirac equation, which are simultaneously eigenvectors of the spin operator , where . These have gained more widespread usage in the last two decades, and yield correct, self-consistent determinations of the scattering cross section when incorporating spin influences in the cyclotron resonance. The paper develops general $S$-matrix scattering formalism through much of Sec. \[sec:csect\_formalism\], for the specific restriction that the scattered photon lies below the threshold for the pair creation process. The exposition imposes the approximation , which is exact for photons incident along the field. For the scattered photons, this serves as a good approximation because the refractive index is generally small for fields below around , as discussed at length in Sec. \[sec:disperse\]. The zero-dispersion approximation for the final photon is adopted following the precedent in many studies of strong-field QED processes, and it facilitates mathematical expediency. Here we have detailed the spin-dependent ST analysis at length and derived useful compact expressions for the differential and total cross sections for the first time. We have also developed the analytics in parallel for cases for JL wave function choices, highlighting the differences that arise between using them and the ST eigenstates. Away from the resonance, if , the two approaches are approximately identical, since spin-dependent contributions from the decay of virtual excited states are minuscule in these frequency domains. In the resonance, it is found that the largest spin-dependent modifications generally occur at field strengths close to 3 times the quantum critical field Gauss; such fields are found in the inner magnetospheres near the surfaces of magnetars, the highly magnetized class of neutron stars. Polarization-dependent angular distributions are developed and compared for three cases (spin-dependent ST, spin-dependent JL, and spin-averaged cyclotron decays of the intermediate state). When the incident photon propagates along the field, the cross sections depend only on the linear polarization state of the outgoing photon, and so are tagged (extraordinary mode) and (ordinary mode). Principal forms for the differential cross sections are listed in Eq. (\[eq:dsigmas\]), combined with several constituent equations. It is found that precisely at the resonance, the correct ST formalism is independent of the photon polarization, unlike the cross sections developed with the JL spin-dependent width, or with the spin-averaged width. However, this uniquely occurs exactly at the peak of the resonance, and not in its wings. The spin-dependent influences are largest between $\omega_i/B = 0.99$ and $1.01$. For example, when $B=3 B_\mathrm{cr}$, for polarization-averaged considerations, we find that the JL spin-dependent width formulation and the average width determination of the cross sections overestimate the resonant cross section relative to the ST form by around 40% and 25%, respectively (see Figs. \[fig:Int1\] and \[fig:Int2\]). To facilitate broader usage of our results, we have derived a compact approximate expression for the ST differential Compton cross section in the resonance in Eqs. (\[eq:calNs\_final\]) and (\[eq:dsig\_resonance\]), a version of which has already been employed in the resonant Compton cooling study of [@BWG11] pertinent to magnetar x-ray emission. Polarized equivalents are also supplied using Eqs. (\[eq:calNperp\_final\]) and (\[eq:calNpar\_final\]). Analytic integrations of these approximate forms in the resonance have also been performed, yielding the useful and compact approximate total cross section result in Eq. (\[eq:sig\_resonance\_fin\]). Neutron star modelers will find this form and its polarized equivalents useful in magnetospheric opacity determinations. Above and below the resonance, the angular distributions exhibit a strong minimum, particularly for the polarization mode; in such domains, considerable care must be exercised when performing numerical integrations. For this reason, analytic integrals for the total cross section are provided in Eqs. (\[eq:sigma\_spindep\]) and (\[eq:sigma\_spinave\]), which apply both above and below the resonance. Being valid away from the resonance, these are applicable to all three of the formalisms studied in this paper: ST, JL and spin-averaged. They cannot be used in the low-frequency domain of because of the following anomalous character. The polarization-averaged cross section exhibits an interesting low-frequency behavior at (i.e., when ), where it becomes independent of frequency and establishes the constant value . The origin of this asymptotic dependence is the presence of spin-dependent decay widths in the complex exponentials for the intermediate virtual electron states. This low-frequency limit provides profound departures from the dependence evinced in both classical and nonrelativistic quantum formulations of magnetic Compton scattering. This feature disappears when spin-averaged widths are employed. While an interesting pathological result, such as a constant low-frequency scattering cross section, is unlikely to play a significant role in Compton upscattering invocations for neutron star magnetospheres, because it will be dominated by contributions from even very small incident photon angles with respect to [B]{}. The results presented in this paper are readily applied in astrophysical contexts, principally for models of radiation emission in strongly magnetized neutron stars. In particular, Compton scattering by relativistic electrons speeding along magnetic field lines is the leading candidate for the generation of high energy x-ray tails observed in magnetars. Such an interaction must take place in their magnetospheres, where the magnetic field strengths approach or exceed the quantum critical field $B_\mathrm{cr}$, depending on the altitude of electron-photon collisions. As the Compton process achieves its highest efficiency when the scattering is resonant, the motivation for formulating a correct description of the cross section that incorporates the spin-dependent widths is apparent. To create photons up to 100 keV using surface x rays below 10 keV in energy requires ultrarelativistic electrons, if only single scatterings of photons are invoked. This largely underpins our focus here on the dominant ground-state–ground state scatterings with photons that are incident parallel to [B]{}: in the ERF, the angular distribution of the low energy (target) x rays is Lorentz contracted to a narrow cone collimated along the local field line. As ultrarelativistic electrons cool [@BWG11] in such Compton collisions with x rays emanating from the surface, they eventually enter a mildly relativistic domain. Then, in the ERF, photons scatter at significant angles of incidence relative to [B]{}, for which many harmonics of the cyclotron resonance appear, and the treatment of the cross section becomes more involved mathematically; this regime will be the focus of our future work on the magnetic Compton interaction within the framework of Sokolov and Ternov formalism. We are grateful to Alice Harding for a thorough reading of the paper and for providing numerous useful suggestions for refining the presentation. We thank the referee for helpful comments and questions that led to the improvement of the manuscript. We are also grateful for the generous support of Michigan Space Grant Consortium, the National Science Foundation (Grants No. AST-0607651, No. AST-1009725, No. AST-1009731, and No. PHY/DMR-1004811), and the NASA Astrophysics Theory Program through Grants No. NNX06AI32G, No. NNX09AQ71G, and No. NNX10AC59A. Wave Functions and Photon Polarizations {#sec:wfunc_pol} ======================================= Solutions to the Dirac equation for relativistic magnetic Compton scattering result in a coupled pair of scalar functions $\chi_n(x)$ and $\chi_{n-1}(x)$, which, following the notation in Appendix 1 of [@DB80], are given by the expression $$\chi_n({\bf x}) = { i^n \over{ L\left(\lambda^2\pi\right)^{1/4}\left(2^n n!\right)^{1/2} } } \exp\left[-{1\over{2\lambda^2}}\left(x-a\right)^2\right]H_n\left({{x-a}\over\lambda}\right) \exp\left(-{iay\over\lambda^2}\right)\exp\left(ipz\right)$$ where $L$ is the length of the system as described in Eq. (\[eq:cross\_sect\_form\]) with $a = - \lambda^2 p_y$ and $\lambda=1/\sqrt{B}$. $H_n(x)$ represent Hermite polynomials and $p$ is the momentum of the charge. The wave functions of the electron and positron are constructed from these scalar functions together with the respective ST and JL wave-function coefficients $$\begin{split} \psi^{(-)}_n(x)= &\left(\begin{array}{c} C_1\chi_{n-1} \\ C_2\chi_n \\ C_3\chi_{n-1} \\ C_4\chi_n \end{array} \right) \exp\left(-iE_nt\right) \\ \psi^{(+)}_n(x)= & \left(\begin{array}{c} C_1\chi_{n-1} \\ C_2\chi_n \\ C_3\chi_{n-1} \\ C_4\chi_n \end{array} \right) \exp\left(+iE_nt\right) \label{eq:spinor_coeffs} \end{split}$$ where “+" and “–" refer to the positron and electron and the coefficients $C_j$ are defined for ST and JL basis states in Appendixes \[sec:STforms\] and \[sec:JLforms\]. Both ST and JL wave functions share the same scalar functions $\chi_n$ differing only in their coefficients obtained from the equations in tabular form,(\[eq:STcoe\]) and (\[eq:JLcoe\]). Here $\psi_n(x)$ is the general form of the four-vector wave function that is used in Eq. (\[eq:Sfi\_form\]) and embraces the electron and positron spinor states $u^{(s)}_n(x)$ and $v^{(s)}_n(x)$ incorporated into Eqs. (\[eq:S1\_T1\_orig\]) and (\[eq:S2\_T2\_orig\]), where $s=\pm $ for spin parallel or antiparallel to the external magnetic field $B$, $E_n=\sqrt{1+p^2+2nB}$ is the charge’s energy in the Landau state $n$, and $t$ is time. The photon vector $$\vec k \; = \; \omega \begin{pmatrix} \sin\theta\cos\phi,\, \sin\theta\sin\phi, \, \cos\theta \end{pmatrix}$$ is used to define two photon polarization modes described in magnetic fields with $B$ along the $z$ axis implementing the appropriate two polarization vectors for the nondispersive vacuum, $$\begin{split} \vec{\varepsilon}_\perp=\vec{\varepsilon}_1 = & \left( \sin\phi, \, -\cos\phi, \, 0 \right)\\ \vec{\varepsilon}_\parallel=\vec{\varepsilon}_2=& \left( \cos\theta\cos\phi, \, \cos\theta\sin\phi, \, -\sin\theta \right) ,\\ \end{split}$$ as indicated in Eq. (7.3.5) in [@Mel13] and also used in [@Sina96] and [@DB80], which have the properties $$\begin{split} \vec{\varepsilon}_1 =\; & \frac{\vec{k}\times\vec{B}}{\vert\vec{k}\times\vec{B}\vert}\\ \vec{\varepsilon}_2=\; & \frac{\vec{k}\times(\vec{k}\times\vec{B})}{\vert\vec{k}\times(\vec{k}\times\vec{B})\vert} \\ \vec{\varepsilon}_1\cdot\vec{k} =\; &\; \vec{\varepsilon}_2\cdot\vec{k} =\; 0 \\ \vec{\varepsilon}_1\cdot \vec{\varepsilon}_2 =\; &\; 0 \\ \end{split}$$ with the pertinent components $$\begin{split} \varepsilon_{1,\pm} = & \; \mp i \, e^{\pm\, i\,\phi};\quad \varepsilon_{1,z} = 0 \\ \varepsilon_{2,\pm} = & \; \cos\theta\, e^{\pm\, i\, \phi};\quad \varepsilon_{2,z} = -\sin\theta \\ \end{split}$$ allowing for the development of the polarization-dependent cross sections expressed in Eq. (\[eq:Dspin\]). Development of the Matrix Elements {#sec:Matrix} ================================== In this appendix, the development of the $S$-matrix elements in Eq. (\[eq:S-matrix2\]) and the vertex functions in Eqs. (\[eq:S1\_T1\_orig\]) and (\[eq:S2\_T2\_orig\]) in the lead up to the formulation of the differential cross section in Eq. (\[eq:Diff2\]) is outlined. The matrix elements associated with the terms in brackets in Eqs. (\[eq:S1\_T1\_orig\]) and (\[eq:S2\_T2\_orig\]) have the general form $$\int d^3x\, e^{i{\bf k}\;\cdot\; {\bf x}}\;u^{\dagger (t)}_\ell({\bf x})\; M \;u^{(s)}_n\quad , \quad \hbox{where}\quad M\; =\; \begin{pmatrix} 0,0,\varepsilon_z,\varepsilon_- \\ 0,0,\varepsilon_+ ,-\varepsilon_z \\ \varepsilon_z, \varepsilon_-, 0,0 \\ \varepsilon_+, -\varepsilon_z , 0 , 0 \\ \end{pmatrix} \label{eq:vertex_Mmatrix}$$ is the photon polarization matrix, and the $t$ and $s$ labels denote lepton spin states. As an example, taking the first integral in the product of Eq. (\[eq:S1\_T1\_orig\]) and inserting the final and intermediate wave functions, and the polarization matrix for the first term in brackets of Eq. (\[eq:S1\_T1\_orig\]), we obtain the following expression: $$\begin{split}\label{eq:IDcoe1} \int d^3x & e^{-i{\bf k}_f\cdot{\bf x}} \begin{pmatrix} C_{1,\ell}\chi_{\ell-1}^\dagger, C_{2,\ell}\chi_\ell^\dagger, C_{3,\ell}\chi_{\ell-1}^\dagger, C_{4,\ell}\chi_\ell^\dagger \end{pmatrix} \cdot M \cdot\begin{pmatrix} C_{1,n}\chi_{n-1}\\ C_{2,n}\chi_n \\ C_{3,n}\chi_{n-1}\\ C_{4,n}\chi_n \\ \end{pmatrix}\; = \\ \int d^3x & e^{-i{\bf k}_f\cdot{\bf x}} \begin{pmatrix} \chi_{\ell-1}^\dagger\chi_{n-1}, \chi_\ell^\dagger \chi_n, \chi_{\ell-1}^\dagger\chi_n, \chi_\ell^\dagger\chi_{n-1} \end{pmatrix}\cdot \begin{pmatrix} \varepsilon_z \left[ C_{1,\ell} C_{3,n}+ C_{3,\ell} C_{1,n} \right] \\ - \varepsilon_z \left[ C_{2,\ell}C_{4,n} + C_{4,\ell} C_{2,n} \right] \\ \varepsilon_- \left[ C_{1,\ell} C_{4,n} + C_{3,\ell} C_{2,n} \right] \\ \varepsilon_+ \left[ C_{2,\ell} C_{3,n}+ C_{4,\ell} C_{1,n} \right] \\ \end{pmatrix} \;\; , \end{split}$$ where the spin dependence of the intermediate state is within the coefficients of the wave functions for either an electron or positron, which are kept general for now, i.e., apply to either JL or ST basis states. In this case the $k_f$ is the momentum of the final photon, and $\chi_\ell$ and $\chi_n$ are the final electron and intermediate lepton. The integrals $$\begin{split} \int & d{^3}x e^{i{\bf k}\cdot{\bf x}}\; \chi^\dagger_\ell(\vec x, q, b)\; \chi_{m}(\vec x, p, a) = \\ & \left( \frac{2\pi \lambar }{L}\right)^2\; e^{-k_\perp^2/4B}\; e^{i k_x a}\; e^{-i k_x k_y/4B}\;e^{i({m-\ell})\phi}\;\delta\left( B(b-a)+k_y\right)\;\delta\left(p-q+k_z\right)\;\Lambda_{{m,\ell}}(k_\perp) \end{split}$$ are standard integrals in the literature, for example, derived in Appendix 1 of [@DB80] and also rederived in Appendix D in [@Sina96] where $q$ and $p$ are the momenta of the leptons and $b$ and $a$ are the corresponding values of the $x$ coordinate of the orbit center. Each of these integrals has associated with it a $\Lambda_{\ell, n}(k_\perp)$ function defined here as $$\Lambda_{\ell, n}(k_\perp) \; =\; (-1)^{\ell+S}\sqrt{\frac{S!}{G!}}\left(\frac{k_\perp}{\sqrt{2 B}}\right)^{G-S} L^{G-S}_S\left( \frac{k^2_\perp}{2B}\right) \label{eq:lamb}$$ where $L$ is an associated Laguerre polynomial, $S\equiv\min(\ell,n)$, and $G\equiv\max(\ell,n)$. We have also pulled out of the definition of the $\Lambda_{\ell, n}(k_\perp)$ function the $e^{-k^2_\perp/4B}$ that is included in the definition used in Eq. (D.36) of [@Sina96]. The term is now part of the factor outside of the summation over intermediate states and their spin. The $\Lambda_{\ell, n}(k_\perp)$ terms here are more similar to the ones defined in Eq. (9) of [@DH86]. The $\Lambda_{\ell,m}$ functions have the following important relations $$\begin{split} \Lambda_{\ell,m}(k_\perp ) \; = \; & (-1)^{\ell+m}\; \Lambda_{m,\ell}(k_\perp ) \\ \Lambda_{\ell,m}(0) \; = \; & \delta_{\ell,m} \\ \Lambda_{0,0}(k_\perp ) \; =\; & 1 \\ \Lambda_{0,j}(k_\perp ) \;=\; & \frac{k_\perp }{\sqrt{ 2 j B }}\ \Lambda_{0,j-1}(k_\perp ) \end{split} \label{eq:Lambs}$$ The integral in Eq. (\[eq:IDcoe1\]) becomes $$\begin{split} \int dx^3 e^{-i{\bf k}_f\cdot{\bf x}}\; u^{\dagger}_\ell({\bf x})\; & M_f\; u_n({\bf x}) \; =\; \left(\frac{2\pi \lambar}{L}\right)^2\; e^{-k^2_{\perp,f}/4B}\; e^{-ik_{x,f} a_n}\; e^{-ik_{x,f} k_{y,f}/2B}\\ &\; \delta \Bigl[ -k_{y,f}-B(a_n-a_\ell)\Bigr] \;\delta\left(-k_{z,f}-p_\ell+p_n\right)\;e^{i(n-\ell)\phi_f}\; D^{f,n}_{u,l}\left(k_{\perp,f}\right) \end{split} \label{eq:Dcoe}$$ where the identity $e^{i k_{x,f} a_n } \; = \; e^{i k_{z,f} a_\ell}\; e^{i B k_{x,f} k_{y,f}}$ has been used, leading to the definition of a new complex vertex function $$\begin{split} D^{\ell,n}_{u,k}\left(k_{\perp,f}\right) = & \begin{pmatrix} \Lambda_{\ell-1,n-1}^f, \Lambda_{\ell,n}^f, \Lambda_{\ell-1,n}^f, \Lambda_{\ell,n-1}^f \end{pmatrix}\cdot \begin{pmatrix} \varepsilon_z \left[ C_{1,\ell} C_{3,n}+ C_{3,\ell} C_{1,n} \right] \\ - \varepsilon_z \left[ C_{2,\ell}C_{4,n} + C_{4,\ell} C_{2,n} \right] \\ \varepsilon_- e^{i\phi_f} \left[ C_{1,\ell} C_{4,n} + C_{3,\ell} C_{2,n} \right] \\ \varepsilon_+ e^{-i\phi_f} \left[ C_{2,\ell} C_{3,n}+ C_{4,\ell} C_{1,n} \right] \\ \end{pmatrix}\end{split}$$ and where the subscript $k$ denotes either an electron $u$ or a positron $v$, and the $f$ in the $\Lambda_{\ell,m}^f$ functions refers to the final scattered photon. The extra phase factors $e^{\pm i \phi_f}$ associated with the integrals containing $\chi_{\ell-1,n}$ and $\chi_{\ell,n-1}$ are brought into the $D$ term in order to have the same factor in front of the $D$ term for all four integrals in Eq. (\[eq:IDcoe1\]). Following a similar protocol, the second term in brackets in Eq. (\[eq:S1\_T1\_orig\]) leads to an integral of the form $$\begin{split} \int dx^3 e^{+i{\bf k}_i\cdot{\bf x}}\; u^{\dagger}_n({\bf x})\; & M_i\; u_j({\bf x}) \; =\; \left(\frac{2\pi \lambar}{L}\right)^2\; e^{-k^2_{\perp,i}/4B}\; e^{+i k_{x,i} a_j}\; e^{-ik_{x,i} k_{y,i}/2B}\\ &\; \delta \Bigl[k_{y,i}-B(a_j-a_n)\Bigr]\;\delta\left(k_{z,i}-p_n+p_j\right)\;e^{i(j-n)\phi_j}\; H^{n,j}_{u,l}\left(k_{\perp,i}\right) \end{split}$$ where $$\begin{split} H^{n,j}_{u,k}\left(k_{\perp,i}\right) = & \begin{pmatrix} \Lambda_{j-1,n-1}^i, \Lambda_{j,n}^i, \Lambda_{j,n-1}^i, \Lambda_{j-1,n}^i \end{pmatrix}\cdot \begin{pmatrix} \varepsilon_z \left[ C_{1,n} C_{3,j}+ C_{3,n} C_{1,j} \right] \\ - \varepsilon_z \left[ C_{2,n}C_{4,j} + C_{4,n} C_{2,j} \right] \\ \varepsilon_- e^{ i \phi_i} \left[ C_{1,n} C_{4,j} + C_{3,n} C_{2,j} \right] \\ \varepsilon_+ e^{- i \phi_i}\left[ C_{2,n} C_{3,j}+ C_{4,n} C_{1,j} \right] \\ \end{pmatrix}\end{split}$$ For the first Feynman diagram $\ell=f$ in the $D$ term and $j=i$ in the $H$ term, while for the second Feynman diagram terms in Eq. (\[eq:S2\_T2\_orig\]), the changes $D^{f,n}_{u,l} \rightarrow D^{n,i}_{u,l}$ and $H^{n,i}_{u,l} \rightarrow H^{f,n}_{u,l}$ take place in the indices. However, $D$ and $H$ remain functions of $k_f$ and $k_i$, respectively. We can insert into the $D$ and $H$ terms the polarization components for each of the linear polarizations discussed in Appendix \[sec:wfunc\_pol\]. For the special case of ground-state–ground-state transitions with the initial electron at rest, the nonzero wave-function coefficients for the initial and final electrons are $C_{2,i}=1$, $C_{2,f}$ and $C_{4,f}$, which significantly simplifies the $D$ and $H$ terms to the following perpendicular polarization (for the final photon) forms: $$\begin{split} D^{f,n,\perp}_{u,k}\left(k_{\perp,f}\right) \; =\; & - i \left[ C_{2,f} C_{3,n}+ C_{4,f} C_{1,n} \right] \Lambda_{f,n-1}^f \\ D^{n,i,\perp}_{u,k}\left(k_{\perp,f}\right) \; =\; & i C_{3,n} \Lambda_{n-1,i}^f \\ H^{n,i,\perp}_{u,k}\left(k_{\perp,i}\right) \; =\; & i C_{3,n} \Lambda_{i,n-1}^i \\ H^{f,n,\perp}_{u,k}\left(k_{\perp,i}\right) \; =\; & - i \left[ C_{2,f} C_{3,n}+ C_{4,f} C_{1,n} \right] \Lambda_{n-1,f}^i \end{split}$$ and the parallel polarization forms $$\begin{split} D^{f,n,\parallel}_{u,k}\left(k_{\perp,f}\right) \; =\; & \sin\theta_f \left[ C_{2,f}C_{4,n} + C_{4,f} C_{2,n} \right] \Lambda_{f,n}^f + \cos\theta_f \left[ C_{2,f} C_{3,n}+ C_{4,f} C_{1,n} \right] \Lambda_{f,n-1}^f \\ D^{n,i,\parallel}_{u,k}\left(k_{\perp,f}\right) \; =\; & \sin\theta_f C_{4,n} \Lambda_{n,i}^f + \cos\theta_f C_{3,n} \Lambda_{n-1,i}^f \\ H^{n,i,\parallel}_{u,k}\left(k_{\perp,i}\right) \; =\; & \sin\theta_i C_{4,n} \Lambda_{i,n}^i + \cos\theta_i C_{3,n} \Lambda_{i,n-1}^i \\ H^{f,n,\parallel}_{u,k}\left(k_{\perp,i}\right) \; =\; & \sin\theta_i \left[ C_{2,f}C_{4,n} + C_{4,f} C_{2,n} \right] \Lambda_{n,f}^i + \cos\theta_i \left[ C_{2,f} C_{3,n}+ C_{4,f} C_{1,n} \right] \Lambda_{n-1,f}^i \ . \end{split} \label{eq:DandH}$$ We can now construct the integrals in Eq. (\[eq:S-matrix2\]) with the sum of the $T$ terms defined in Eqs. (\[eq:T1\_orig\]) and (\[eq:T2\_orig\]) providing the contributions from both Feynman diagrams, which can be crafted into the form $$\begin{split}\label{eq:Fshere} B \int da_n & \int dp_n \left\{ T^{(1)}_n + T^{(2)}_n \right\} = \\ & \times \left(\frac{2\pi \lambar }{L}\right)^4 e^{-\left(k^2_{\perp,i}+k^2_{\perp,f}\right)/4B} \delta \Bigl[ k_{y,i}-k_{y,f}-B(a_j-a_f)\Bigr] \, \delta\left(k_{z,i} -k_{z,f}-p_f\right)\\ & \times\frac{1}{ \sqrt{E_f}}\; { e^{i \left(k_{x,i}-k_{x,f}\right)\left(a_j +a_f\right)/2} \; e^{-i \ell(\phi_f+\phi_i)/2} } \left\{ F^{(1)}_{n,s}\; e^{i \Phi} + F^{(2)}_{n,s}\; e^{ -i \Phi} \right\} \end{split}$$ where we have defined the $F$ terms used in Eq. (\[eq:Sfi\_squared\]), $$\begin{split} F^{(1)}_s \; = &\; \sqrt{E_f}\; \left[\frac{ D^{f,n}_{u,u}\left(k_{\perp,f}\right) \; H^{n,i}_{u,u}\left(k_{\perp,i}\right) }{1+\omega_i-E_n+i\Gamma^s/2} + \frac{ D^{f,n}_{u,v}\left(k_{\perp,f}\right) \; H^{n,i}_{u,v}\left(k_{\perp,i}\right) }{1+\omega_i+E_n-i\Gamma^s/2}\right]\\ F^{(2)}_s \; = &\; \sqrt{E_f}\; \left[ \frac{ H^{f,n}_{u,u}\left(k_{\perp,i}\right) \; D^{n,i}_{u,u}\left(k_{\perp,f}\right) }{1-\omega_f-E_n+i\Gamma^s/2} + \frac{ H^{f,n}_{u,v}\left(k_{\perp,i}\right) \; D^{n,i}_{u,v}\left(k_{\perp,f}\right) }{1-\omega_f+E_n-i\Gamma^s/2} \right] \ .\\ \end{split}$$ The factor $\sqrt{E_f}$ is brought into the definition of the $F$ terms \[resulting in the appearance of a $1/\sqrt{E_f}$ factor in Eq. (\[eq:Fshere\])\], as it will cancel when the coefficients of the final electron wave function are introduced. This factor stems from the numerator associated with the integral of the delta function expressing the conservation of energy, discussed below in Eq. (\[eq:EnDel\]). The phase factor $\Phi$ is given by the expression $$\begin{split}\label{eq:PhaseFac} \Phi \; = \; & k_{\perp,i}\, k_{\perp,f} \sin(\phi_i-\phi_f) / 2B { - (\ell-2n)(\phi_f - \phi_i)/2 } \end{split}$$ which is similar to the one presented in Eq. (7) of [@DH86]. Slight differences from the presentation of [@DH86] exist in this development because here we have implemented the particular photon polarizations such that the entire phase dependence is in the $\Phi$ term. Including these into the scattering matrix in Eq. (\[eq:S-matrix2\]) yields $$\begin{split} S_{fi} \; & =\; \frac{- i\; (2 \pi)^4 \, \fsc\; }{ \sqrt {\omega _i \omega _f } }\left(\frac{\lambar}{L}\right)^5 \, \delta \left( {1 + \omega _i - E_{\ell} - \omega _f } \right) \\ &\times e^{-\left(k^2_{\perp,i}+k^2_{\perp,f}\right)/4B}\; \delta\left(k_{y,i}-k_{y,f}-B(a_j-a_f)\right)\; \delta\left(k_{z,i} -k_{z,f}-p_f\right)\\ &\times { e^{i \left(k_{x,i}-k_{x,f}\right)\left(a_j +a_f\right)/2} \; e^{-i \ell(\phi_f+\phi_i)/2}}\; \frac{1}{ \sqrt{E_f}}\; \sum\limits_{n = 0}^\infty \sum\limits_{s = \pm } \left\{ F^{(1)}_{n,s}\; e^{i \Phi} + F^{(2)}_{n,s}\; e^{ -i \Phi} \right\} \label{eq:S-matrix2B} \end{split}$$ where is the fine-structure constant. The modulus squared of the $S$-matrix can now be performed and is given by the expression $$\begin{split} \left\| S_{fi}\right\|^2 \; & =\; \frac{ (2 \pi)^5\,\fsc^2}{{\omega _i\; \omega _f }}\left(\frac{\lambar}{L}\right)^7\frac{cT}{L} \, \delta \left( {1 + \omega _i - E_{\ell} - \omega _f } \right) \\ & \times e^{-\left(k^2_{\perp,i}+k^2_{\perp,f}\right)/2B}\; \delta\left(k_{y,i}-k_{y,f}-B(a_i-a_f)\right)\; \delta\left(k_{z,i} -k_{z,f}-p_f\right)\\ & \times \frac{1}{E_f} \left\| \sum\limits_{n = 0}^\infty \sum\limits_{s = \pm } \Bigl\{ F^{(1)}_{n,s}\; e^{i \Phi} + F^{(2)}_{n,s}\; e^{ -i \Phi} \Bigr\} \right\|^2 \label{eq:S-matrix2D} \end{split}$$ Observe that the standard protocol for squaring the three delta functions in the above equation introduces a $1/2\pi$ factor for each. Setting $\beta_i=0$, the differential cross section in the rest frame of the initial electron is inferred from the expression in Eq. (\[eq:cross\_sect\_form\]), $$\frac{d\sigma}{d\Omega_f} \; = \; \lambar^2 \int \frac{\left\|S_{fi}\right\|^2}{\lambar^2 c T}\frac{L^6\; k_f^2\; dk_f}{(2\pi\lambar)^3}\frac{L\;p_f}{2\pi\lambar}\frac{L\;B\;da_f}{2\pi\lambar} \ .$$ The integral over $k_f$ or $\omega_f$ includes the delta function associated with the conservation of energy, $$\int \delta \left( {1 + \omega _i - E_{\ell} - \omega _f } \right) d\omega_f \; = \; \frac{1}{1-\beta_f\cos\theta_f}\; = \; \frac{E_f}{E_f-p_f\cos\theta_f} \ . \label{eq:EnDel}$$ Performing the integral in Eq. (\[eq:S-matrix2D\]) and implementing kinematic relations, we arrive at the expression for the differential cross section $$\frac{d\sigma}{d\Omega_f} \; = \; \frac{\fsc^2\;\lambar^2\; \omega_f^2\; e^{-\left(\omega_i^2\sin^2\theta_i+\omega_f^2\sin^2\theta_f\right)/2B}}{\omega_i\left[2\omega_i-\omega_f-\omega_i\omega_f\left(1 -\cos\theta_i\cos\theta_f\right)\right]} \left\| \sum\limits_{n = 0}^\infty \sum\limits_{s = \pm } \Bigl[ F^{(1)}_{n,s}\; e^{i \Phi} + F^{(2)}_{n,s}\; e^{ -i \Phi} \Bigr] \right\|^2 \ . \label{eq:DiffB2}$$ The $\phi_f$ dependence is only in the phase terms $\Phi$, and the spin dependence is only in the $F$ terms; therefore, we can set $\phi_i=0$ and integrate over $\phi_f$. The integration of the cross term in the modulus squared leads to a Bessel function $J_{n+k}\left(k_{\perp,i} k_{\perp,f}/B\right)$ , and with $\fsc^2\,\lambar^2=3\, \sigt /\,8\pi $, we have arrived at the expression for the general differential cross section in the form $$\begin{split}\label{eq:DiffB3} & \frac{d\sigma}{d\cos\theta_f} \; = \; \frac{3\,\sigt}{4} \frac{\omega_f^2\; e^{-\left(\omega_i^2\sin^2\theta_i+\omega_f^2\sin^2\theta_f\right)/2B}}{\omega_i\left[2\omega_i-\omega_f-\omega_i\omega_f\left(1 -\cos\theta_i\cos\theta_f\right)\right]} \\ & \ \ \ \times \sum\limits_{s = \pm } \sum\limits_{n = 0}^\infty \left\{ \left\|F_{n,s}^{(1)}\right\|^2 + \left\| F_{n,s}^{(2)} \right\|^2 +\sum\limits_{j = 0}^\infty \left[ F_{n,s}^{(1)} F_{j,s}^{(2)^\dagger} + F_{n,s}^{(2)} F_{j,s}^{(1)^\dagger} \right] J_{n+j}\left(\frac{k_{\perp,i}k_{\perp,f}}{B}\right)\right\} \end{split}$$ reproducing the more general differential cross section in Eq. (3.24) of [@Sina96]. For the main focus of this study, we set $\theta_i=0$ and $k_{\perp,i}=0$. As a result, the cross term in Eq. (\[eq:DiffB3\]) no longer contributes as $J_n(0)=0$. In addition, only $n=1$ contributes to the summation over intermediate states due to the $\Lambda_{\ell,m}^i (0) = \delta_{\ell,m}$ as indicated previously in Eq. (\[eq:Lambs\]). We then have the expression for the differential cross section in the compact form of Eq. (\[eq:Diff2\]), $$\begin{split}\label{eq:DiffB4} \frac{d\sigma}{d\cos\theta_f} \; = \; & \frac{3\sigt}{4} \frac{\omega_f^2\; e^{-\omega_f^2\sin^2\theta_f/2B}}{\omega_i\left[2\omega_i-\omega_f-\omega_i\omega_f\left(1 -\cos\theta_f\right)\right]} \sum\limits_{s = \pm } \left\{ \left\|F_{n=1,s}^{(1)}\right\|^2 + \left\| F_{n=1,s}^{(2)} \right\|^2 \right\} \ . \\ \end{split}$$ The $F$ terms $$\begin{split} F^{(1)}_{n=1,s} \; = &\; \frac{ S^{(1),s}_{u} }{1+\omega_i-E_n+i\Gamma^s/2} + \frac{ S^{(1),s}_{v} }{1+\omega_i+E_n-i\Gamma^s/2}\\ F^{(2)}_{n=1,s} \; = &\; \frac{ S^{(2),s}_{u} }{1-\omega_f-E_n+i\Gamma^s/2} + \frac{ S^{(2),s}_{v} }{1-\omega_f+E_n-i\Gamma^s/2}\\ \end{split}$$ are related to the $T$ terms in Eqs. (\[eq:T1\_orig\]) and (\[eq:T2\_orig\]), which have been integrated over the phase factor $\phi_f$. The $S$ terms are products of the $D$ and $H$ vertex functions, $$\begin{split} S^{(1),s}_{{\cal P}_i,{\cal P}_f,k}\; = & \; \sqrt{E_f} \; D^{f,n,{\cal P}_f}_{u,k}\left(k_{\perp,f}\right)\; H^{n,i,{\cal P}_i}_{u,k}\left(0\right) \\ S^{(2),s}_{{\cal P}_i,{\cal P}_f,k}\; = & \; \sqrt{E_f} \; H^{f,n,{\cal P}_i}_{u,k}\left(0\right)\; D^{n,i,{\cal P}_f}_{u,k}\left(k_{\perp,f}\right) \ , \\ \end{split}$$ for the first and second Feynman diagrams. The ${\cal P}_i$ and ${\cal P}_f$ are the polarization of the incident and final photons, and the $k$ is the virtual lepton, either $u$ or $v$, for the electron and positron. Implementing the definitions of the $D$ and $H$ terms above, we can define the necessary set of required $S$ terms, $$\begin{split} S^{(m),s}_{\perp,\perp,k} \; = \; & \left[ C_{2,f} I_{3,3}^{s,k}+ C_{4,f} I_{1,3}^{s,k} \right] \\ S^{(1),s}_{\perp,\parallel,k} \; = \; & i \left\{ \left[ C_{2,f}I_{4,3}^{s,k} + C_{4,f} I_{2,3}^{s,k} \right] \frac{\omega_f\sin^2\theta_f}{\sqrt{2B}} + \left[ C_{2,f} I_{3,3}^{s,k}+ C_{4,f} I_{1,3}^{s,k} \right] \cos\theta_f \right\} \\ S^{(2),s}_{\perp,\parallel,k} \; = \; & -i \left\{ - \left[ C_{2,f}I_{4,3}^{s,k} + C_{4,f} I_{1,4}^{s,k} \right] \frac{\omega_f\sin^2\theta_f}{\sqrt{2B}} + \left[ C_{2,f} I_{3,3}^{s,k}+ C_{4,f} I_{1,3}^{s,k} \right] \cos\theta_f \right\}\\ S^{(m),s}_{\parallel,\perp,k} \; = \; & - i S^{(m),s}_{\perp,\perp,k} \\ S^{(1),s}_{\parallel,\parallel,k} \; = \; & - i S^{(1),s}_{\perp,\parallel,k} \\ S^{(2),s}_{\parallel,\parallel,k} \; = \; & -i S^{(2),s}_{\perp,\parallel,k} \\ \end{split}$$ where we have used $I_{j,m}^{s,k} = C_{j,n} C_{m,n}$ terms as products of the coefficients of the intermediate state that are dependent on the spin of the leptonic state and are specified in the Appendixes \[sec:STforms\] and \[sec:JLforms\] for ST and JL basis states, respectively. Given that incident photons are along the magnetic field lines, the cross section is determined by the polarization of the final photon and is independent of the incident polarization. We can then drop the specification of the incident polarization in the $S$ terms. Inserting the coefficients of the final electron in the $S$ terms leads to the expression $$S_{ \perp ,k}^{(m), s } \; =\; f \left[ \left( {E_f + 1} \right)I_{3,3}^{ s ,k} - p_f I_{1,3}^{ s ,k} \right] \quad , \label{eq:Sperp}$$ where $$f \; =\; \dover{1}{\sqrt {2\left( {E_f + 1} \right)} } \ . \label{eq:fnorm}$$ In the normalization factor in the denominator of the above equation, the $\sqrt{E_f}$ has been canceled by the $E_f$ coming from the $\beta_f$ in Eq. (\[eq:EnDel\]). The $S$ terms with parallel polarization are $$S_{\|\|,k}^{(m), s } \; =\; i \left\{ (-1)^{m+1} S_{ \perp ,k}^{(1),s} \cos \theta _f + f{{\left[ {\left( {E_f + 1} \right)I_{4,3}^{ s ,k} - p_f I_{2,3}^{ s ,k} } \right] }} \frac{{\omega _f \sin ^2 \theta _f }}{{\sqrt {2B} }}\right\} \quad , \label{eq:Spara}$$ where the index $k$ refers to the electron $u$ and positron $v$ in the intermediate state, $s$ is the spin state, and $m$ refers to the Feynman diagram. The wave-function coefficients for the initial and final states within the context of ultrarelativistic scattering are identical in both JL and ST spinors. The total energy $E_f$ and parallel momentum $p_f$ have the following kinematic relations: $$\begin{aligned} E_f & = & 1 + \omega _i - \omega _f \nonumber\\[-5.5pt] \label{eq:Kine}\\[-5.5pt] p_f & = & \omega _i - \omega _f \cos \theta _f \nonumber\end{aligned}$$ as the final electron is in the ground state. The $S_{ \perp ,k}^{(m), s }$ terms are real, while the $ S_{\|\|,k}^{(m), s }$ terms are imaginary and are used to develop the $N$ terms described in Eq. (\[eq:Nterms\]). Sokolov and Ternov Spinors {#sec:STforms} ========================== In this appendix, the development of the and contributions to the numerators in Eq. (\[eq:dsigmas\]) is outlined for the case of Sokolov and Ternov formalism. The coefficients of the ST spinors for electron and positron states and can be found in [@ST68] as well as in Appendix B of [@Sina96]. We have adapted their presentations to generate the following compact notation for the coefficients outside the spatial Hermite functions and temporal exponentials in Eq. (\[eq:spinor\_coeffs\]) below: $$\label{eq:STcoe} \begin{array}{{20}c} {} &\vline & C_1 & C_2 & C_3 & C_4 \\ \hline {u^ + } &\vline & {f_2 } & { - f_1 } & {f_4 } & {f_3 } \\ {u^ - } &\vline & {f_1 } & {f_2 } & {f_3 } & { - f_4 } \\ {v^ + } &\vline & { - f_4 } & { - f_3 } & {f_2 } & { - f_1 } \\ {v^ - } &\vline & {f_3 } & { - f_4 } & { - f_1 } & { - f_2 } \\ \end{array}$$ where $$\label{eq:fs} \begin{split} f_1 &= g_n^{ST}\sqrt {2B}\, p_m \\ f_2 &= g_n^{ST}\left( {\epsilon_\perp + 1} \right)\left( {\cal E}_m + \epsilon_\perp \right) \\ f_3 &= g_n^{ST}\sqrt {2B} \left( {\cal E}_m + \epsilon_\perp \right) \\ f_4 &= g_n^{ST}\left( {\epsilon_\perp + 1} \right)p_m \\ \end{split}$$ for , and the common normalization factor is $$g_n^{ST} \; =\; \frac{1}{{\sqrt {4{\cal E}_m \epsilon_\perp \left( \epsilon_\perp + 1 \right) \left( {\cal E}_m + \epsilon_\perp \right)} }}$$ The and , which are related by , have the kinematic definitions in Eqs. (\[eq:Em\_def\]) and (\[eq:pm\_def\]), namely $$\begin{aligned} {\cal E}_{m=1} \; =\; \sqrt {\omega _i ^2 + \epsilon _ \perp ^2 } & \quad , \quad & {\cal E}_{m=2} \; =\; \sqrt {\omega _f ^2 \cos ^2 \theta _f + \epsilon _ \perp ^2 } \nonumber\\[-5.5pt] \label{eq:EM_pm_def}\\[-5.5pt] p_{m=1} \; =\; \omega _i & \quad ,\quad & p_{m=2} \; =\; - \omega _f \cos \theta _f \quad .\nonumber\end{aligned}$$ Here the concern is primarily with the intermediate state, with only contributing to the specific ground-state–ground-state scattering involving the resonance at the cyclotron fundamental. Generally the terms in Eqs. (\[eq:Sperp\]) and (\[eq:Spara\]) are defined as products of coefficients of the form $$\label{eq:Iterms} I^{s,k}_{i,j}= C^{s,k}_{i}C^{s,k}_{j}$$ that involve different combinations of the electron and positron spinors. For each terms there are four possibilities, $s=+$, $s=-$, $k=u$ and $k=v$, that capture these combinations. The following five sets of terms are required to define the terms in Eqs. (\[eq:Sperp\]) and (\[eq:Spara\]): $$\label{eq:STIterms} \begin{array}{{20}c} {} &\vline & c_{\rm 1} I_{3,3}^{s ,k} & c_{\rm 1}I_{1,3}^{s ,k}&c_{\rm 2} I_{2,3}^{s ,k} &c_{\rm 2} I_{4,3}^{s ,k}& c_{\rm 2}I_{1,4}^{s ,k} \\ \hline {u^ + } &\vline & {\left( {\epsilon _ \perp + 1} \right)\left( {E_m - \epsilon _ \perp } \right)} \;\; & \;\; {\left( {\epsilon _ \perp + 1} \right)p_m} \;\; & \;\; { - \left( {E_m - \epsilon _ \perp } \right)} \;\; & \;\; p_m \;\; & \;\; {\left( {E_m + \epsilon _ \perp } \right)} \\ {u^ - } &\vline & {\left( {\epsilon _ \perp - 1} \right)\left( {E_m + \epsilon _ \perp } \right)} \;\; & \;\; {\left( {\epsilon _ \perp - 1} \right)p_m} \;\; & \;\; {\left( {E_m + \epsilon _ \perp } \right)} \;\; & \;\; { - p_m} \;\; & \;\; { - \left( {E_m - \epsilon _ \perp } \right)} \\ {v^ + } &\vline & {\left( {\epsilon _ \perp + 1} \right)\left( {E_m + \epsilon _ \perp } \right)} \;\; & \;\; { - \left( {\epsilon _ \perp + 1} \right)p_m} \;\; & \;\; { - \left( {E_m + \epsilon _ \perp } \right)} \;\; & \;\; { - p_m} \;\; & \;\; {\left( {E_m - \epsilon _ \perp } \right)} \\ {v^ - } &\vline & {\left( {\epsilon _ \perp - 1} \right)\left( {E_m - \epsilon _ \perp } \right)} \;\; & \;\; { - \left( {\epsilon _ \perp - 1} \right)p_m} \;\; & \;\; {\left( {E_m - \epsilon _ \perp } \right)} \;\; & \;\; p_m \;\; & \;\; { - \left( {E_m + \epsilon _ \perp } \right)} \;\; , \\ \end{array}$$ where and . As indicated, the columns for the $I$ terms correspond to $k=u$ electron and $k=v$ positron states, while the first and third rows for the $s=+$ case correspond to spin-up or parallel to the external $B$ field, and the second and fourth rows for $s=-$ correspond to spin-down or antiparallel to $B$. Inserting these terms into Eqs. (\[eq:Sperp\]) and (\[eq:Spara\]) with the definitions of the terms in Eqs. (\[eq:Nterm\]), one obtains $$\begin{aligned} N_ + ^{ST,(m), \perp } & =& \frac{f}{2\epsilon_\perp} \Bigl[ {\left( { - 1} \right)^{m + 1} \left( {\epsilon _ \perp + 1} \right)\omega _i q_- - 2B\left( {E_f + 1} \right)} \Bigr] \nonumber \\[-5.5pt] \label{eq:STNperp}\\[-5.5pt] N_ - ^{ST,(m), \perp } &=& \frac{f}{2\epsilon_\perp} \Bigl[ {\left( { - 1} \right)^{m + 1} \left( {\epsilon _ \perp - 1} \right)\omega _i q_- + 2B\left( {E_f + 1} \right)} \Bigr] \quad ,\nonumber\end{aligned}$$ and for parallel polarization, $N$ terms $$\begin{aligned} N_ + ^{ST,(1),\parallel } &=& N_ + ^{ST,(1), \perp } \cos \theta _f + \frac{f}{2\epsilon_\perp} \Bigl[ {\omega _i q_+ - \left( {\epsilon _ \perp - 1} \right)p_f } \Bigr] \omega _f \sin ^2 \theta _f \nonumber \\[2.0pt] N_ - ^{ST,(1),\parallel } &=& N_ - ^{ST,(1), \perp } \cos \theta _f - \frac{f}{2\epsilon_\perp} \Bigl[ {\omega _i q_+ + \left( {\epsilon _ \perp + 1} \right)p_f } \Bigr] \omega _f \sin ^2 \theta _f \nonumber \\[-6.5pt] \label{eq:STNpara}\\[-6.5pt] N_ + ^{ST,(2),\parallel } &=& N_ + ^{ST,(2), \perp } \cos \theta _f + \frac{f}{2\epsilon_\perp} \Bigl[ {\omega _i q_- + \left( {\epsilon _ \perp - 1} \right)p_f } \Bigr] \omega _f \sin ^2 \theta _f \nonumber\\[2.0pt] N_ - ^{ST,(2),\parallel } &=& N_ - ^{ST,(2), \perp } \cos \theta _f - \frac{f}{2\epsilon_\perp} \Bigl[ {\omega _i q_- - \left( {\epsilon _ \perp + 1} \right)p_f } \Bigr] \omega _f \sin ^2 \theta _f \;\; , \nonumber\end{aligned}$$ where $$f\; =\; \dover{1}{\sqrt{2 (E_f+1)}} \quad \hbox{and}\quad q_{\pm} \; =\; E_f+1 \pm p_f\quad . \label{eq:fac}$$ These forms can be reduced with the aid of some useful relations: $$\begin{aligned} p_f \omega_f \sin^2 \theta_f & = & \left( \omega_i - \omega_f \right)q_- \left( 1+\cos\theta_f\right) \nonumber\\ q_+\omega_f\sin^2\theta_f & = & \left(E_f-1+p_f\right)q_-\left(1+\cos\theta_f \right) \label{eq:useful_reln}\\ \omega_f\sin^2\theta_f & = & -\left(E_f-1-p_f\right)\left(1+\cos\theta_f\right) \quad .\nonumber\end{aligned}$$ The result is that the terms assume the following forms: $$\begin{aligned} N^{ST,(m)\perp}_{\pm} & = & \left(\epsilon_\perp\pm1\right)\alpha^\perp \mp \beta^\perp \nonumber\\ N^{ST,(1),\parallel}_{\pm} & = & \left(\epsilon_\perp \mp 1\right)\alpha^{\parallel} \pm \beta^{(1),\parallel} \label{eq:NSTs} \\ N^{ST,(2),\parallel}_{\pm} & = & \left(\epsilon_\perp \mp 1\right)\alpha^{\parallel} \pm \beta^{(2),\parallel} \nonumber\end{aligned}$$ where $$\begin{aligned} \alpha^\perp & = &\left(-1\right)^{1+m} \frac{f}{2\epsilon_\perp} \, \omega_i q_{-} \nonumber\\[-7.5pt] \label{eq:STalphas}\\[-7.5pt] \alpha^\parallel & = & \frac{f}{2\epsilon_\perp} \left(\omega_f-p_f\right) q_{-} \nonumber\end{aligned}$$ and $$\begin{aligned} \beta^\perp & = & \frac{f}{\epsilon_\perp}\, B\left(E_f+1\right) \nonumber \\ \beta^{(1),\parallel} & = & \frac{f}{\epsilon_\perp} \, \Bigl[ \left(E_f+1\right)\left(\omega_i-B\right)\cos\theta_f+\omega_i p_f \Bigr] \label{eq:STbetas} \\ \beta^{(2),\parallel} & = & \frac{f}{\epsilon_\perp} \, \Bigl[ \left(E_f+1\right)\left(\omega_i+B-\zeta\right)\cos\theta_f+\omega_i p_f\left(E_f-p_f\right) \Bigr] \quad .\nonumber\end{aligned}$$ From the definitions in Eqs. (\[eq:Tave\]) and (\[eq:Tspin\]), the and terms with these expressions can be described as $$\begin{aligned} T^{\perp,\parallel}_{\rm ave} & = & \left( 2\epsilon_\perp\alpha^{\perp,\parallel}\right)^2 \nonumber\\[2pt] T^{(m),\perp}_{\rm spin} & = & \frac{1}{2\epsilon^3_\perp} \left[ \left(\epsilon^2_\perp+1\right) T^{\perp}_{\rm ave} - 8\epsilon^2_\perp\left(\epsilon^2_\perp+1\right)\alpha^\perp\beta^\perp + 4\epsilon^2_\perp\left(\beta^\perp\right)^2 \right] \\ T^{(m),\parallel}_{\rm spin} & = & - \, \frac{1}{2\epsilon^3_\perp} \left[ \left(3\epsilon^2_\perp-1\right) T^{\parallel}_{\rm ave} - 8\epsilon^2_\perp\left(\epsilon^2_\perp-1\right)\alpha^\parallel\beta^{(m),\parallel} - 4\epsilon^2_\perp\left(\beta^{(m),\parallel} \right)^2 \right] \;\; . \nonumber \end{aligned}$$ The forms then simply reproduce those listed in Eq. (\[eq:Taves\]), namely, and . The corresponding forms are not as simple, yet they are still manageable: $$T^{(m),\perp}_{\rm spin} \; =\; \dover{1}{2\epsilon^3_\perp} \left\{ \left(\epsilon^2_\perp+1\right)T^{\perp}_{\rm ave} + \left(-1\right)^m \left(\epsilon^4_\perp-1\right) \left(2\omega_i -\zeta\right) +\frac{1}{2}\left(\epsilon^2_\perp-1\right)^2 \left(E_f+1\right) \right\} \label{eq:ST_Tspin_perp}$$ and $$\begin{aligned} T^{(1),\parallel}_{\rm spin} & = & \dover{1}{2\epsilon^3_\perp} \biggl\{ - \left(3\epsilon^2_\perp-1\right)T^{\parallel}_{\rm ave} + 2 \omega_i\left(\epsilon^2_\perp-1\right) \Bigl[ \zeta-2 (E_f - 1) \Bigr] +2\omega^2_i\left(E_f-1\right) \nonumber\\ && \qquad + 2\left(\omega_i-B\right)\cos\theta_f \Bigl[ \left(\epsilon^2_\perp-1\right) \left(2\omega_f\cos\theta_f-\zeta\right)\nonumber\\ && \qquad\qquad\qquad +\left(\omega_i-B\right)\left(E_f+1\right)\cos\theta_f +2 p_f \omega_i \Bigr] \biggr\} \nonumber\\[-5.5pt] \label{eq:ST_Tspin_par}\\[-6.5pt] T^{(2),\parallel}_{\rm spin} & = & \dover{1}{2\epsilon^3_\perp} \biggl\{ -\left(3\epsilon^2_\perp-1\right)T^\parallel_{\rm ave} - 2 \left(\epsilon^2_\perp-1\right)\left(2+\omega_i\right)\zeta\nonumber\\ && \qquad\qquad + \left(E_f-1\right)\left(\omega_i-\zeta\right) \left[ 4\left(\epsilon^2_\perp-1\right)\left(1+\omega_i\right) + \left(\omega_i-\zeta\right)\right] \nonumber\\ && \qquad + 2 \left(\omega_i+B-\zeta\right)\cos\theta_f \Bigl[ \left(\epsilon^2_\perp-1\right)\left(2\omega_f\cos\theta_f-\zeta\right) \nonumber\\ && \qquad\qquad\qquad + \left(\omega_i+B-\zeta\right)\left(E_f+1\right)\cos\theta_f - 2 p_f \left(\omega_i-\zeta\right) \Bigr] \biggr\}\nonumber\end{aligned}$$ With these and terms so defined, the differential cross section using the ST basis states can be routinely obtained using Eq. (\[eq:dsigmas\]). Johnson and Lippmann Spinors {#sec:JLforms} ============================ In this section, we present the coefficient of the particle wave functions in the JL basis followed by the development of the $S$ and $N$ terms required for the $T_{\rm ave}$ and $T_{\rm spin}$ of Eqs. (\[eq:Tave\]) and (\[eq:Tspin\]) to compare the spin-dependent results with those in the ST basis. We obtain the JL coefficients from Appendix J of [@Sina96]. However, we use the notation developed in [@HD91]. As previously indicated, the coefficients of the initial and final electron states are equivalent in both JL and ST basis states, and we only require those of the intermediate state where we assume $n=1$ for our ground-state–ground-state scattering. Our presentation follows the one used in Appendix B. \[eq:JLcoe\] [\[20]{}c]{} && [C\_1 / ]{} & [C\_2 / ]{} & [C\_3 / ]{} & [C\_4 / ]{}\ && 1 & 0 & P & N\ [u\^ - ]{} && 0 & 1 & N & [ - P]{}\ [v\^ + ]{} && [ - P]{} & [ - N]{} & 1 & 0\ [v\^ - ]{} && [ - N]{} & P & 0 & 1\ where $m$ refers to the Feynman diagram with \[eq:Js\] [c]{} P\ N\ f\_m\^[JL]{} =\ with $p_m$ and $E_m$ being defined by the previous kinematic relations of the intermediate state in Eqs. (\[eq:Em\_def\]) and (\[eq:pm\_def\]). We can obtain the product of the pairs of coefficients required for the development of the $N$ terms using Eq. (\[eq:Iterms\]) to get the following five sets of $I$ terms \[eq:JLIterms\] [\[20]{}c]{} && I\_[3,3]{}\^[s ,k]{} & I\_[1,3]{}\^[s ,k]{} & I\_[2,3]{}\^[s ,k]{} & I\_[4,3]{}\^[s ,k]{} & I\_[1,4]{}\^[s ,k]{}\ && [P\^2 ]{} & P & 0 & [NP]{} & N\ [u\^ - ]{} && [N\^2 ]{} & 0 & N & [ - NP]{} & 0\ [v\^ + ]{} && 1 & [ - P]{} & [ - N]{} & 0 & 0\ [v\^ - ]{} && 0 & 0 & 0 & 0 & [ - N]{}\ where $I_{i,j}^{s ,k} = \left(\frac{2E_m}{E_m+1}\right)C^{s,k}_i C^{s,k}_j$. The $I$ terms are required to define the $S$ terms for the perpendicular photon polarization in Eq. (\[eq:Sperp\]) and parallel photon polarization in Eq. (\[eq:Spara\]), which are then used to develop the necessary $N$ terms using Eq. (\[eq:Nterm\]). However, it might be instructive to develop the $F$ terms of Eq. (\[eq:Fs\_def\]) in order to compare to the work of [@HD91]. Therefore, we obtain for both photon polarizations \[eq:JLFterms\] F\_ \^[(m), + ]{} =& D\^[u, + ]{} ( [P - Q]{} )P + D\^[v, + ]{} ( [1 + PQ]{} )\ F\_ \^[(m), - ]{} =& D\^[u, - ]{} N\^2\ F\_\^[(1), + ]{} =& D\^[u, + ]{} { [( [P - Q]{} )P\_f + PN]{} }\ & + D\^[v, + ]{} { [( [1 + QP]{} )\_f + QN]{} }\ F\_\^[(1), - ]{} =& D\^[u, - ]{} { [N\^2 \_f - ( [P + Q]{} )N]{} }\ F\_\^[(2), + ]{} =& D\^[u, + ]{} { [( [P - Q]{} )P\_f + ( [P - Q]{} )N]{} }\ &+ D\^[v, + ]{} ( [1 + QP]{} )\_f\ F\_\^[(2), - ]{} =& D\^[u, - ]{} { [N\^2 \_f + QN]{} }\ where $$\begin{aligned} \label{eq:Dterms} D^{u, \pm } & =& \frac{{\sqrt {E_f + 1} \left( {E_m + 1} \right)}}{{2\sqrt 2 E_m \left[ {\omega _m - E_m + \frac{{i\Gamma ^ \pm }}{2}} \right]}} \nonumber \\ D^{v, \pm } &=& \frac{{\sqrt {E_f + 1} \left( {E_m + 1} \right)}}{{2\sqrt 2 E_m \left[ {\omega _m + E_m - \frac{{i\Gamma ^ \pm }}{2}} \right]}} \end{aligned}$$ Here we have included the spin-dependent widths. However, the form is equivalent to the $F$ terms in the Appendix of [@HD91] after correcting for a couple of typos, where the $F$ terms are the missing $\sqrt{E_f+1}$ multiplicative factor and the occurrence of the term with a single $N$ should be $N^2$. Using the $I$ terms we can develop the $N$ terms for JL basis states in the perpendicular photon polarization with the form \[eq:JLNpepe\] [c]{} N\_ + \^[(m), ]{} = (-1)\^[1+m]{}f\_iq\_- - 2Bf( E\_f + 1 )[L]{}\_o\ N\_ - \^[(m), ]{} = 2Bf( [E\_f + 1]{} )[L]{}\_o\ where $$\begin{split} {\cal L}_o &=\frac{\omega_m+E_m}{2E_m\left(E_m+1\right)} \\ \end{split}$$ and for parallel photon polarizations \[eq:Npapa\] N\_ + \^[(1),]{} &= N\_ + \^[(1), ]{} \_f + f{ \_o-p\_f } \_f \^2 \_f\ N\_ - \^[(1),]{} &= N\_ - \^[(1), ]{} \_f - f { ( [E\_f + 1]{} )p\_m + p\_f ( [E\_m + 1]{} ) } \_f \^2 \_f [L]{}\_o\ N\_ + \^[(2),]{} &= N\_ + \^[(2), ]{} \_f - f{ ( [E\_f + 1]{} )p\_m - p\_f ( [E\_m + 1]{} ) } \_f \^2 \_f [L]{}\_o\ N\_ - \^[(2),]{} &= N\_ - \^[(2), ]{} \_f + f{ \_o + p\_f} \_f \^2 \_f\ with $f$ having the same definition as in the ST terms in Appendix \[sec:STforms\]. Bringing the terms together, the $N$ terms can be rewritten in the form $$\begin{split}\label{eq:JLNpapa} N_ + ^{(1),\parallel } &= f \left[ \begin{array}{c} \omega_i q_-\cos \theta _f -p_f \omega _f \sin ^2 \theta _f \\ + \left\{ \begin{array}{c} \left( E_f + 1 \right) \left[ p_m \omega _f \sin ^2 \theta _f -2 B \cos \theta _f \right] \\ + \left( E_m + 1 \right)p_f \omega _f \sin ^2 \theta _f \end{array}\right\} {\cal L}_o \\ \end{array} \right] \\ N_ - ^{(1),\parallel } &= -f\left[ \begin{array}{c} \left( E_f + 1 \right) \left[ p_m \omega _f \sin ^2 \theta _f - 2B \cos \theta _f \right] \\ + \left( E_m + 1 \right) p_f \omega _f \sin ^2 \theta _f \end{array} \right] {\cal L}_o \\ N_ + ^{(2),\parallel } &= - f\left[ \begin{array}{c} \omega_i q_- \cos \theta _f \\ + \left\{ \begin{array}{c} \left( E_f + 1 \right) \left[ p_m\omega _f \sin ^2 \theta _f + 2B \cos \theta _f \right] \\ - \left( E_m + 1 \right) p_f \omega _f \sin ^2 \theta _f \end{array} \right\} {\cal L}_o \\ \end{array} \right] \\ N_ - ^{(2),\parallel } &= f \left[ \begin{array}{c} p_f \omega _f \sin ^2 \theta _f \\ + \left\{ \begin{array}{c} \left( E_f + 1 \right) \left[ p_m\omega _f \sin ^2 \theta _f + 2B \cos \theta _f \right] \\ - \left( E_m + 1 \right) p_f \omega _f \sin ^2 \theta _f \end{array} \right\} {\cal L}_o \end{array} \right] \\ \end{split}$$ Here the forms are lacking some of the symmetry that is present in the ST forms in Appendix \[sec:STforms\]. It is a little more expedient to develop the form of the $T^{JL}_{\rm spin}$ using an alternative with the form $$T^{JL}_{\rm spin}=\left(N_+ + N_-\right) \left(N_+ - N_-\right)+\frac{\xi^{JL}_{\rm fac}}{4}\left[ \left(N_+ + N_-\right)^2-\left(N_+ - N_-\right)^2\right]$$ The JL forms can be used to detail the $T^{JL}_{\rm spin}$ terms by introducing some temporary variables $$\begin{split} N^{(m),\perp}_+ + N^{(m),\perp}_- &= \left(-1\right)^{m+1} \alpha^\perp \\ N^{(m),\perp}_+ - N^{(m),\perp}_- &= \left(-1\right)^{m+1} \alpha^\perp -2\beta^\perp \\ N^{(1),\parallel}_+ + N^{(1),\parallel}_- &= \alpha^\parallel \\ N^{(1),\parallel}_+ - N^{(1),\parallel}_- &= \alpha^\parallel + 2\beta^\parallel \\ N^{(2),\parallel}_+ + N^{(2),\parallel}_- &= -\alpha^\parallel \\ N^{(2),\parallel}_+ - N^{(2),\parallel}_- &= \alpha^\parallel - 2\left(\alpha_1 + \beta^\parallel \right) \\ \end{split}$$ Unfortunately, the $N$ terms for the case of parallel polarization does not reflect the symmetry in the case of perpendicular polarization; the $N$ term of the parallel polarizations for the second Feynman diagram pick up an additional term $\alpha_1$. The spin factors that determine the spin-dependent widths $\Gamma_\pm$ are defined in [@BGH05] for JL states and are given by the expressions $$\label{eq:SigsJL}\begin{split} \xi _ \pm ^{JL} &= 1 \mp \frac{{E_m + \epsilon _ \perp ^2 }}{{\epsilon _ \perp ^2 \left( {E_m + 1} \right)}} \\ \xi _{{\rm{fac}}}^{JL} &= - \frac{{2\left( {E_m + \epsilon _ \perp ^2 } \right)}}{{\epsilon _ \perp ^2 \left( {E_m + 1} \right)}} \\ {\cal L}_1 &= \frac{{\left( {E_m + \epsilon _ \perp ^2 } \right)}}{{\epsilon _ \perp ^2 \left( {E_m + 1} \right)}} \\ \end{split}$$ Therefore, $\xi^{JL}_{\rm fac} = -2 {\cal L}_1$, which is a bit unfortunate, but this will help us to compactify the terms. The ratio of $\xi_{\pm}^{JL}/\xi_{\pm}^{ST}$ is displayed in Fig. 2 of [@BGH05], where a ratio of 2 occurs for spin-up at lower $B$ fields while a ratio of 1 occurs for spin-down case. However, at high $B \sim 100$ to $1000$, the spin factors become identical and the effects of the spin states on the widths vanish. These variables can aid in the development of the $T^{JL}_{\rm spin}$ terms as defined previously by Eq. (\[eq:Tspin\]), $$\begin{array}{c}\label{eq:JLTspins} T^{(m),\perp}_{\rm spin} =T^\perp_{\rm ave} - \left(-1\right)^{m+1}2\left(1+{\cal L}_1\right) \alpha^\perp\beta^\perp + 2\left(\beta^\perp\right)^2{\cal L}_1 \\ T^{(1),\parallel}_{\rm spin}=T^\parallel_{\rm ave} + 2\left(1+{\cal L}_1\right)\alpha^\parallel\beta^\parallel+ 2\left( \beta^\parallel\right)^2{\cal L}_1 \\ T^{(2),\parallel}_{\rm spin}=-T^\parallel_{\rm ave} + 2\left(1+{\cal L}_1\right) \alpha^\parallel \left( \alpha_1+\beta^\parallel\right) +2 \left(\alpha_1+\beta^\parallel\right)^2{\cal L}_1 \\ \end{array}$$ From Eqs. (\[eq:JLNpepe\]) and (\[eq:JLNpapa\]), we can express the temporary variables in the following terms: $$\begin{split} \alpha^\perp &= f\omega_iq_- \\ \beta^\perp &= 2Bf\left( {E_f + 1} \right){\cal L}_o \\ \alpha^\parallel &= f \left(\omega_f-p_f\right) q_- \\ \alpha_1 &= f\omega_i \cos \theta _f q_-\\\ \beta^\parallel &= f \left\{ \left( E_f + 1 \right) \gamma_1 + (-1)^{m+1} p_f \gamma_2 \right\} {\cal L}_o \ . \\ \end{split}$$ where $$\begin{split} \gamma_1 &= p_m \omega _f \sin ^2 \theta _f - (-1)^{m+1}2B \cos \theta _f \\ \gamma_2 &= \left( E_m + 1 \right) \omega_f\sin^2\theta_f \\ \end{split}$$ Applying these variables to the $T_{\rm spin}$ terms in Eq. (\[eq:JLTspins\]), we obtain the expressions $$\label{eq:JLTpepefin} T^{(m),\perp}_{\rm spin} = T^\perp_{\rm ave} - 2 B {\cal L}_o\left[ \left(-1\right)^{m+1} \left(2\omega_i-\zeta\right) \left( 1 +{\cal L}_1\right) - 2 B \left(E_f+1\right) {\cal L}_1 {\cal L}_o\right] \ , \\$$ $$\begin{aligned} \begin{split} T^{(1),\parallel}_{\rm spin}&=\left\{ \begin{array}{c} T^\parallel_{\rm ave} + \left(1+{\cal L}_1\right) \left(\omega_f-p_f\right)\left[\gamma_1q_-+\gamma_2\omega_f\left(1-\cos\theta_f\right)\right]{\cal L}_o \\ + \left[\left(E_f+1\right)\left(\gamma^2_1+\gamma^2_2\right)+2\gamma_2\left( p_f\gamma_1-\gamma_2\right) \right] {\cal L}_1 {\cal L}^2_o \ , \\ \end{array} \right\} \end{split} \end{aligned}$$ $$\begin{aligned} \begin{split} T^{(2),\parallel}_{\rm spin}&=\left\{ \begin{array}{c} -T^\parallel_{\rm ave} \\ + \left(1-{\cal L}_1\right) \left[ \begin{array}{c} 2\left[\omega^2_i-\left(1+\omega_i\right)\zeta\right]\cos\theta_f \\ \left(\omega_f-p_f\right)\left[\gamma_1q_- -\gamma_2\omega_f\left(1-\cos\theta_f\right)\right]{\cal L}_o \\ \end{array} \right]\\ + \left[ \begin{array}{c} 2\left(\omega_i-\zeta\right)\omega_i\cos^2\theta_f \\ +2 \left[\gamma_1\left( 2\omega_i - \zeta\right) - \gamma_2 \zeta \right] \cos\theta_f {\cal L}_o \\ + \left\{\left(E_f+1\right)\left(\gamma^2_1+\gamma^2_2\right)-2\gamma_2\left[ p_f\gamma_1+\gamma_2\right]\right\}{\cal L}^2_o \\ \end{array} \right] {\cal L}_1 \ . \\ \end{array}\right\} \end{split}\end{aligned}$$ Series Development of Total Cross Sections {#sec:csect_analytics} ========================================== This appendix summarizes the protocol of deriving Legendre series expressions for the polarization-summed cross section in Eq. (\[eq:sigma\_spinave\]) that are applicable outside the cyclotron resonance. The starting point is $$\sigma \;\approx \; \dover{3\sigt}{4} \int_{0}^{\Phi} \dover{ e^{-\omega_i \phi /B}\, d\phi }{\sqrt{1-2\phi z+\phi^2}} \; \left\{ \left\lbrack \dover{1}{1+2\omega_i}\, \dover{1}{ (\Delta_1)^2} + \dover{1}{\Delta_2} \right\rbrack z(1-\phi z ) - \dover{2 (B+\phi )}{(\Delta_2)^2} \left( 1-2\phi z+\phi^2 \right) \right\} \;\; . \label{eq:sigma_spinave_appE}$$ Again, here for the diagram, and corresponds to the denominator of the diagram. The ensuing analysis is made more compact by defining the expressions $$\Upsilon_1 \; =\; \dover{\omega_i^2}{(\omega_i - B)^2} \quad\hbox{and}\quad \Upsilon_2 \; =\; \dover{\omega_i^2}{(\omega_i + B) \, (\omega_i + B + 2\omega_iB)} \label{eq:Upsilon_12_def}$$ to represent and the limit of , respectively. The most involved portion is the term. To manipulate the total cross section, first form a partial fractions decomposition $$\dover{2 (B+\phi )}{(\Delta_2)^2} \; =\; \dover{1}{B}\, \left\{ \dover{1}{\Delta_2} + \dover{ (zB+1)^2+B^2}{(\Delta_2)^2} \right\} \quad . \label{eq:part_frac_Delta2}$$ Then integrate the residual portion by parts using $$2B \int_{0}^{\Phi} \dover{ e^{-\omega_i \phi /B}\, d\phi }{ (\Delta_2)^2} \, \sqrt{1-2\phi z+\phi^2} \; = \; -\Upsilon_2 + \int_{0}^{\Phi} \dover{ e^{-\omega_i \phi /B}\, d\phi }{\sqrt{1-2\phi z+\phi^2}} \dover{1}{ \Delta_2} \biggl[ (z-\phi) + \dover{\omega_i}{B} \left( 1-2\phi z+\phi^2 \right) \biggr] \label{eq:integ_parts_ident}$$ In expressing the integrations, we will again make use of the class of integrals $${\cal H}_n(z,\, p)\; =\; \int_0^{\Phi (z)} \dover{\phi^n \, e^{-p\, \phi} \, d\phi}{\sqrt{1-2 z\phi + \phi^2} } \; =\; \sum_{k=0}^{\infty} \dover{ (-p)^k}{k!} \, Q_{k+n}(z)\quad \label{eq:calHnu_def_appE}$$ in Eq. (\[eq:calHnu\_def\]) that was employed in developing the cross section in the resonance. Here, is a Legendre function of the second kind, defined in 8.703 of [@GR80]. This evaluates the terms nicely. We extend this to treat the pieces by defining $${\cal G}_n(z,\, p, \, \tau)\; =\; \int_0^{\Phi (z)} \dover{\phi^n \, e^{-p\, \phi} \, d\phi}{\sqrt{1-2 z\phi + \phi^2} } \, \dover{1}{\tau -\phi} \;\equiv\; \int_0^{\infty} d\mu\, \int_0^{\Phi (z)} \dover{\phi^n \, e^{-p\, \phi - \mu (\tau -\phi)} \, d\phi}{\sqrt{1-2 z\phi + \phi^2} }\quad . \label{eq:calGnu_def}$$ To make the algebra more compact, we use the definition $$\tau \; =\; \dover{(z^2-1) B^2 + 2 B z + 1}{2B} \; =\; \dover{1}{2B \Upsilon_2}\quad , \label{eq:tau_def}$$ so that . Then we can use the definition of to our advantage: $${\cal G}_n(z,\, p, \, \tau)\; =\; \int_0^{\infty} d\mu\, e^{-\mu\tau} \int_0^{\Phi (z)} \dover{\phi^n \, e^{- (p-\mu )\, \phi} \, d\phi}{\sqrt{1-2 z\phi + \phi^2} } \;\equiv\; \int_0^{\infty} d\mu\, e^{-\mu\tau} \sum_{k=0}^{\infty} \dover{ (\mu -p)^k}{k!} \, Q_{k+n}(z)\quad , \label{eq:calGnu_ident}$$ recognizing that the series identity for is valid for both positive and negative . Reversing the order of summation and integration, the terms of the series now are the integrals $$\int_0^{\infty} (\mu -p)^k\, e^{-\mu\tau} \, d\mu \; =\; \dover{e^{-p\tau}}{\tau^{k+1}} \int_{-p\tau}^{\infty} x^k\, e^{-x}\, dx \; =\; \dover{e^{-p\tau}}{\tau^{k+1}} \, \Gamma( k+1,\, -p\tau )\quad , \label{eq:calGnu_terms}$$ employing the integral representation of the incomplete Gamma function. The right-hand side of Eq. (\[eq:calGnu\_terms\]) distills down to a finite series of terms using 8.352.2 of [@GR80]. It follows that $${\cal G}_n(z,\, p, \,\tau )\; =\; e^{-p\tau}\sum_{k=0}^{\infty} \dover{\Gamma (k+1,\, -p\tau )}{\tau^{k+1}\, k!} \, Q_{k+n}(z) \;\equiv\; \sum_{k=0}^{\infty} \dover{Q_{k+n}(z)}{\tau^{k+1}} \, \sum_{m=0}^k \dover{(-p\tau )^m}{m!} \quad . \label{eq:calGnu_alt}$$ The numerical facility of computing this series representation is only marginally more demanding than computing that for . Assembling these pieces, the integration by parts identity in Eq. (\[eq:integ\_parts\_ident\]) can be recast as $$\int_{0}^{\Phi} \dover{ e^{-\omega_i \phi /B}\, d\phi }{ (\Delta_2)^2} \, \sqrt{1-2\phi z+\phi^2} \; =\; - \dover{\Upsilon_2}{2B} + \dover{1}{4B^2} \left\{ z {\cal G}_0 - {\cal G}_1 + \dover{\omega_i}{B} \Bigl( {\cal G}_0 - 2 z {\cal G}_1 + {\cal G}_2 \Bigr) \right\}\quad . \label{eq:integ_parts_ident2}$$ This can then be combined with all the other terms to yield an expression for the total polarization-summed cross section away from the resonance: $$\begin{aligned} \sigma &\approx & \dover{3\sigt}{4} \biggl\{ \dover{z \Upsilon_1}{1+2\omega_i}\, \Bigl[ {\cal H}_0 - z {\cal H}_1 \Bigr] + \dover{z}{2B} \Bigl[ {\cal G}_0 - z {\cal G}_1 \Bigr] - \dover{1}{2B^2} \Bigl( {\cal G}_0 - 2 z {\cal G}_1 + {\cal G}_2 \Bigr) \nonumber\\[-5.5pt] \label{eq:sigma_spinave_app}\\[-5.5pt] & - & \dover{ (zB+1)^2+B^2}{4B^3} \left[ - 2 B\Upsilon_2 + z {\cal G}_0 - {\cal G}_1 + \dover{\omega_i}{B} \Bigl( {\cal G}_0 - 2 z {\cal G}_1 + {\cal G}_2 \Bigr) \right] \Biggr\} \nonumber\end{aligned}$$ This amounts to an efficient computation using the two series representations for and . 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--- author: - 'Cédric Lorcé[^1]' title: 'The nucleon spin decomposition: news and experimental implications' --- Introduction ============ The question of how to decompose the proton spin into measurable quark/gluon and spin/orbital angular momentum (OAM) contributions has been revived half a decade ago by Chen et al. [@Chen:2008ag; @Chen:2009mr]. They challenged the textbook knowledge [@Jauch; @Berestetskii; @Jaffe:1989jz; @Ji:1996ek] by providing a gauge-invariant decomposition of the gluon angular momentum into spin and OAM contributions. This work triggered many new developments and reopened old controversies about the physical relevance and the measurability of the different contributions. For a recent and detailed review of the topic, see Ref. [@Leader:2013jra]. In this proceeding, we briefly summarize the situation and stress the experimental implications. In section \[sec2\], we present the four main kinds of proton spin decompositions. In section \[sec3\], we sketch the Chen et al. approach and discuss its uniqueness issue. In section \[sec4\], we argue that this problem is basically solved by the theoretical framework used to describe actual experiments and establish the link between the missing pieces, namely the OAM, with parton distributions. Finally, we conclude with section \[sec5\]. The proton spin decompositions {#sec2} ============================== There are essentially four kinds of proton spin decompositions into quark/gluon and spin/OAM contributions [@Lorce:2012rr; @Leader:2013jra], referred to as the Jaffe-Manohar [@Jaffe:1989jz], Chen et al. [@Chen:2008ag; @Chen:2009mr], Ji [@Ji:1996ek] and Wakamatsu [@Wakamatsu:2010qj; @Wakamatsu:2010cb] decompositions, and given by $$\begin{aligned} {\boldsymbol{J}}_\text{QCD}&={\boldsymbol{S}}^q_\text{JM}+{\boldsymbol{L}}^q_\text{JM}+{\boldsymbol{S}}^g_\text{JM}+{\boldsymbol{L}}^g_\text{JM},\\ &={\boldsymbol{S}}^q_\text{Chen}+{\boldsymbol{L}}^q_\text{Chen}+{\boldsymbol{S}}^g_\text{Chen}+{\boldsymbol{L}}^g_\text{Chen},\\ &={\boldsymbol{S}}^q_\text{Ji}+{\boldsymbol{L}}^q_\text{Ji}+{\boldsymbol{J}}^g_\text{Ji},\\ &={\boldsymbol{S}}^q_\text{Wak}+{\boldsymbol{L}}^q_\text{Wak}+{\boldsymbol{S}}^g_\text{Wak}+{\boldsymbol{L}}^g_\text{Wak}.\end{aligned}$$ The common piece to all these decompositions is the quark spin contribution $${\boldsymbol{S}}^q_\text{JM}={\boldsymbol{S}}^q_\text{Ji}={\boldsymbol{S}}^q_\text{Chen}={\boldsymbol{S}}^q_\text{Wak}=\int{\mathrm{d}}^3x\,\psi^\dag\tfrac{1}{2}{\boldsymbol{\Sigma}}\psi.$$ They however differ in the definition of the quark/gluon OAM $$\begin{aligned} {\boldsymbol{L}}^q_\text{JM}&=\int{\mathrm{d}}^3x\,\psi^\dag({\boldsymbol{x}}\times\tfrac{1}{i}{\boldsymbol{\nabla}})\psi,&{\boldsymbol{L}}^g_\text{JM}&=\int{\mathrm{d}}^3x\,E^{ai}({\boldsymbol{x}}\times{\boldsymbol{\nabla}}) A^{ai},\\ {\boldsymbol{L}}^q_\text{Chen}&=\int{\mathrm{d}}^3x\,\psi^\dag({\boldsymbol{x}}\times i{\boldsymbol{D}}_{\text{pure}})\psi,&{\boldsymbol{L}}^g_\text{Chen}&=-\int{\mathrm{d}}^3x\,E^{ai}({\boldsymbol{x}}\times{\boldsymbol{\mathcal D}}^{ab}_{\text{pure}}) A^{bi}_{\text{phys}},\\ {\boldsymbol{L}}^q_\text{Ji}&={\boldsymbol{L}}^q_\text{Wak}=\int{\mathrm{d}}^3x\,\psi^\dag({\boldsymbol{x}}\times i{\boldsymbol{D}})\psi,&{\boldsymbol{L}}^g_\text{Wak}&={\boldsymbol{L}}^g_\text{Chen}-\int{\mathrm{d}}^3x\,({\boldsymbol{\mathcal D}}\cdot{\boldsymbol{E}})^a\,{\boldsymbol{x}}\times{\boldsymbol{A}}^a_{\text{phys}},\end{aligned}$$ and the gluon spin $$\begin{aligned} {\boldsymbol{S}}^g_\text{JM}&=\int{\mathrm{d}}^3x\,{\boldsymbol{E}}^a\times{\boldsymbol{A}}^a,\\ {\boldsymbol{S}}^g_\text{Chen}&={\boldsymbol{S}}^g_\text{Wak}=\int{\mathrm{d}}^3x\,{\boldsymbol{E}}^a\times{\boldsymbol{A}}^a_{\text{phys}}.\end{aligned}$$ Except for the quark spin piece, the other terms in the Jaffe-Manohar decomposition are not gauge invariant. Chen et al. remedied this by splitting of the gauge potential ${\boldsymbol{A}}={\boldsymbol{A}}_{\text{pure}}+{\boldsymbol{A}}_{\text{phys}}$, and formally replacing ordinary derivatives by pure-gauge covariant derivatives $-{\boldsymbol{\nabla}}\mapsto{\boldsymbol{D}}_{\text{pure}}=-{\boldsymbol{\nabla}}-ig{\boldsymbol{A}}_{\text{pure}}$ and explicit occurrences of the gauge field by the physical part ${\boldsymbol{A}}\mapsto {\boldsymbol{A}}_{\text{phys}}$. The difference between the Chen et al. and Wakamatsu decompositions is in the attribution of the so-called potential OAM to either quarks or gluons $${\boldsymbol{L}}_\text{pot}=-\int{\mathrm{d}}^3x\,({\boldsymbol{\mathcal D}}\cdot{\boldsymbol{E}})^a\,{\boldsymbol{x}}\times{\boldsymbol{A}}^a_{\text{phys}}={\boldsymbol{L}}^g_\text{Wak}-{\boldsymbol{L}}^g_\text{Chen}={\boldsymbol{L}}^q_\text{Chen}-{\boldsymbol{L}}^q_\text{Wak},$$ where the QCD equation of motion $({\boldsymbol{\mathcal D}}\cdot{\boldsymbol{E}})^a=g\psi^\dag t^a\psi$ has been used in the last equality. Finally, the difference between the Ji and Wakamatsu decompositions is that the Ji decomposition does not provide any splitting of the total gluon angular momentum into spin and OAM contributions $${\boldsymbol{J}}^g_\text{Ji}={\boldsymbol{S}}^g_\text{Wak}+{\boldsymbol{L}}^g_\text{Wak}=\int{\mathrm{d}}^3x\,{\boldsymbol{x}}\times({\boldsymbol{E}}^a\times{\boldsymbol{B}}^a).$$ The Chen et al. approach {#sec3} ========================== Both the Chen et al. and Wakamatsu decompositions are based on a splitting of the gauge potential into “pure-gauge” and “physical” terms [@Chen:2008ag; @Chen:2009mr; @Lorce:2012rr; @Wakamatsu:2010qj; @Wakamatsu:2010cb] $$\label{decomposition} A_\mu(x)=A^{\text{pure}}_\mu(x)+A^{\text{phys}}_\mu(x).$$ By definition, the pure-gauge term does not contribute to the field strength $$\label{cond1} F^{\text{pure}}_{\mu\nu}=\partial_\mu A^{\text{pure}}_\nu-\partial_\nu A^{\text{pure}}_\mu-ig[A^{\text{pure}}_\mu,A^{\text{pure}}_\nu]=0$$ and changes under gauge transformation as follows $$\label{cond2} A^{\text{pure}}_\mu(x)\mapsto \tilde A^{\text{pure}}_\mu(x)=U(x)[A^{\text{pure}}_\mu(x)+\tfrac{i}{g}\partial_\mu]U^{-1}(x).$$ The physical term is responsible for the field strength $$F_{\mu\nu}=\mathcal D^{\text{pure}}_\mu A^{\text{phys}}_\nu-\mathcal D^{\text{pure}}_\nu A^{\text{phys}}_\mu-ig[A^{\text{phys}}_\mu,A^{\text{phys}}_\nu],$$ and transforms like the latter $$A^{\text{phys}}_\mu(x)\mapsto \tilde A^{\text{phys}}_\mu(x)=U(x)A^{\text{phys}}_\mu(x)U^{-1}(x).$$ This approach is very similar to the background field method [@Lorce:2013bja] and is essentially equivalent to the gauge-invariant approach based on Dirac variables [@Chen:2012vg; @Lorce:2013gxa]. The principal issue with it is that the splitting into pure-gauge and physical fields is not unique. Indeed the following alternative fields $$\bar A^{\text{pure}}_\mu(x)=A^{\text{pure}}_\mu(x)+B_\mu(x),\qquad\bar A^{\text{phys}}_\mu(x)=A^{\text{phys}}_\mu(x)-B_\mu(x),$$ satisfy the defining conditions and , provided that $B_\mu(x)$ transforms in a suitable way under gauge transformations. The transformation $\phi(x)\mapsto\bar\phi(x)$ is referred to as the Stueckelberg transformation [@Lorce:2012rr; @Stoilov:2010pv]. Since the pure-gauge term plays essentially the role of a background field, Stueckelberg dependence corresponds simply to background dependence [@Lorce:2013bja]. It can also be understood from a non-local point of view, where $A_\mu^{\text{pure}}(x)$ and $A_\mu^{\text{phys}}(x)$ appear as particular functionals of $A_\mu(x)$ [@Hatta:2011zs; @Hatta:2011ku; @Lorce:2012ce]. Experimental implications {#sec4} ========================= Because of the Stueckelberg dependence, the Chen et al. and Wakamatsu decompositions are not unique. In practice, one imposes an extra condition to make the splitting well-defined. Opinions diverge about which condition to use and whether physical quantity are allowed or not to be Stueckelberg/background dependent [@Leader:2013jra; @Lorce:2013bja]. As a matter of fact, the proton internal structure is essentially probed in high-energy scattering experiments. The latter can be described within the framework of QCD factorization theorems [@Collins] which involves non-local objects with Wilson lines running essentially along the light-front direction. This specific path for the Wilson lines is equivalent to working with the condition $A^+_{\text{phys}}(x)=0$ [@Hatta:2011zs; @Hatta:2011ku; @Lorce:2012ce]. This means that from an experimental point of view, the latter condition is the most natural one. In particular, it is with this condition that one can interpret the measured quantity $\Delta G$ [@Manohar:1990kr] as the gauge-invariant gluon spin. The OAM is also in principle accessible. The Ji or Wakamatsu OAM can be extracted from leading-twist generalized parton distributions (GPDs) [@Ji:1996ek]. The Chen et al. OAM, which is equivalent to the Jaffe-Manohar OAM considered in the appropriate gauge, is related to transverse-momentum dependent GPDs (GTMDs) [@Lorce:2011kd; @Lorce:2011ni; @Hatta:2011ku]. Unfortunately, it is not known so far how to extract these GTMDs from experiments. Conclusion {#sec5} ========== We briefly discussed the four main kinds of proton spin decompositions. We presented the main features of the Chen et al. approach, where the gauge potential is split into pure-gauge and physical contributions, and noted its similarity with the background field method and the gauge-invariant approach based on Dirac variables. 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--- abstract: 'In the cold inflation scenario, the slow roll inflation and reheating via coherent rapid oscillation, are usually considered as two distinct eras. When the slow roll ends, a rapid oscillation phase begins and the inflaton decays to relativistic particles reheating the Universe. In another model dubbed warm inflation, the rapid oscillation phase is suppressed, and we are left with only a slow roll period during which the reheating occurs. Instead, in this paper, we propose a new picture for inflation in which the slow roll era is suppressed and only the rapid oscillation phase exists. Radiation generation during this era is taken into account, so we have warm inflation with an oscillatory inflaton. To provide enough e-folds, we employ the non-minimal derivative coupling model. We study the cosmological perturbations and compute the temperature at the end of warm oscillatory inflation.' author: - 'Parviz Goodarzi[^1]' - 'H. Mohseni Sadjadi [^2]' title: ' Warm inflation with an oscillatory inflaton in the non-minimal kinetic coupling model' --- Introduction ============ In the standard inflation model, the accelerated expansion and the reheating epochs are two distinct eras [@guth; @inflaton1; @Liddle]. But in the warm inflation, relativistic particles are produced during the slow roll. Therefore, the warm inflation explains the slow roll and onset of the radiation dominated era in a unique framework [@Berera; @Berera2]. Warm inflation is a good model for large scale structure formation, in which the density fluctuations arise from thermal fluctuation [@Berera3; @Berera4]. Various models have been proposed for warm inflation, e.g. tachyon warm inflation, warm inflation in loop quantum cosmology, etc. [@Herrera1; @Herrera2; @Zhang]. Oscillating inflation was first introduced in [@Mukhanov], where it was proposed that the inflation may continue, after the slow roll, during rapid coherent oscillation in the reheating era. An expression for the corresponding number of e-folds was obtained in[@Liddle]. Scalar field oscillation in inflationary model was also pointed out briefly in [@ref], where the decay of scalar fields during their oscillations to inflaton particles was proposed. A brief investigation of the adiabatic perturbation in the oscillatory inflation can be found in [@Taruya]. The formalism used in [@Mukhanov] was extended in [@Lee], by considering a coupling between inflaton and the Ricci scalar curvature. The shape of the potential, required to end the oscillatory inflation, was investigated in [@sami]. The rapid oscillatory phase provides a few e-folds so we cannot ignore the slow roll era in this formalism. Due to small few number of e-folds, a detailed study of the evolution of quantum fluctuations has not been performed. To cure this problem, one can consider nonminimal derivative coupling model. The cosmological aspects of this model have been widely studied in the literature [@sushkov]. The oscillatory inflation in the presence of a non-minimal kinetic coupling was studied in [@sadjadi2] and there was shown that in high-friction regime, the non-minimal coupling increases the e-folds number and so can remedy the problem of the smallness of the number of e-folds arising in [@Mukhanov]. Scalar and tensor perturbations and power spectrum and spectral index for scalar and tensor modes in oscillatory inflation, were derived in [@sadjadi2], in agreement with Planck 2013 data. However, it is not clear from this scenario how reheating occurs or the Universe becomes radiation dominated after the end of inflation. For non-minimal derivative coupling model, the reheating process after the slow roll and warm slow roll inflation are studied in [@good1; @good2] and [@nozari1; @nozari2] respectively. In the present work, inspired by the models mentioned above, we will consider oscillatory inflation in non-minimal derivative coupling model. We will assume that the inflaton decays to the radiation during the oscillation, providing a new scenario: warm oscillatory inflation. Equivalently, this can be viewed as an oscillatory reheating phase which is not preceded by the slow roll. In the second section, we examine conditions for warm oscillatory inflation and study the evolution of energy density of the scalar field and radiation. In the third section, the thermal fluctuation is considered and spectral index and power spectrum are computed. We will consider observational constraints on oscillatory warm inflation parameters by using Planck 2015 data [@planck]. In the fourth section, the temperature at the end of warm inflation is calculated. We will compute tensor perturbation in the fifth section and in the last section, we conclude our results. We use units $\hbar=c=1$ throughout this paper. Oscillatory warm inflation ========================== In this section, based on our previous works [@sadjadi2; @good1; @good2], we will introduce rapid oscillatory inflaton decaying to radiation in non-minimal kinetic coupling model. We start with the action [@Germani1] $$\label{1} S=\int \Big({M_P^2\over 2}R-{1\over 2}\Delta^{\mu \nu}\partial_\mu \varphi \partial_{\nu} \varphi- V(\varphi)\Big) \sqrt{-g}d^4x+S_{int}+S_{r},$$ where $\Delta^{\mu \nu}=g^{\mu \nu}+{1\over M^2}G^{\mu \nu}$, $G^{\mu \nu}=R^{\mu \nu}-{1\over 2}Rg^{\mu \nu}$ is the Einstein tensor, $M$ is a coupling constant with mass dimension, $M_P=2.4\times 10^{18}GeV$ is the reduced Planck mass, $S_{r}$ is the radiation action and $S_{int}$ describes the interaction of the scalar field with radiation. There are not terms containing more than two times derivative, so we have not additional degrees of freedom in this theory. We can calculate energy momentum tensor by variation of action with respect to the metric $$\label{2} T_{\mu\nu}=T^{(\varphi)}_{\mu\nu}+{1\over M^2}\Theta_{\mu\nu}+T^{(r)}_{\mu\nu}.$$ The energy momentum tensor for radiation is $$\label{3} T^{(r)}_{\mu\nu}=(\rho_r+P_r)u_{\mu}u_{\nu}+P_rg_{\mu\nu},$$ where $u^{\mu}$ is the four-velocity of the radiation and $T^{(\varphi)}_{\mu\nu}$ is the minimal coupling counterpart of the energy momentum tensor $$\label{4} T^{(\varphi)}_{\mu\nu}=\nabla_{\mu}\varphi\nabla_{\nu}\varphi-{1\over2}g_{\mu\nu}{(\nabla\varphi)}^2-g_{\mu\nu}V(\varphi).$$ The energy momentum tensor corresponding to the non-minimal coupling term is $$\begin{aligned} \label{5} &&\Theta_{\mu\nu}=-{1\over2}G_{\mu\nu}{(\nabla\varphi)}^2-{1\over2}R\nabla_{\mu}\varphi\nabla_{\nu}\varphi+ R^{\alpha}_{\mu}\nabla_{\alpha}\varphi\nabla_{\nu}\varphi\\ \nonumber &&+R^{\alpha}_{\nu}\nabla_{\alpha}\varphi\nabla_{\mu}\varphi+R_{\mu\alpha\nu\beta}\nabla^{\alpha}\varphi\nabla^{\beta}\varphi +\nabla_{\mu}\nabla^{\alpha}\varphi\nabla_{\nu}\nabla_{\alpha}\varphi\\\nonumber &&-\nabla_{\mu}\nabla^{\nu}\varphi\Box \varphi -{1\over2}g_{\mu\nu}\nabla^{\alpha}\nabla^{\beta}\varphi\nabla_{\alpha}\nabla_{\beta}\varphi+{1\over2}g_{\mu\nu}{(\Box\varphi)}^2\\\nonumber &&-g_{\mu\nu}\nabla_{\alpha}\varphi\nabla_{\beta}\varphi R^{\alpha\beta}.\end{aligned}$$ Energy transfer between the scalar field and radiation is assumed to be $$\label{6} Q_{\mu}=-\Gamma u^{\nu} \partial_{\mu}\varphi \partial_{\nu} \varphi,$$ where $$\label{7} \nabla^{\mu}T^{(r)}_{\mu\nu}=Q^{\nu} \qquad and \qquad \nabla^{\mu}(T^{(\varphi)}_{\mu\nu}+{1\over M^2}\Theta_{\mu\nu})=-Q_{\nu}.$$ The scalar field equation of motion, in Friedmann-Lemaître-Robertson-Walker (FLRW) metric is $$\label{8} (1+{3H^2\over M^2})\ddot{\varphi}+3H(1+{3H^2\over M^2}+{2\dot{H}\over M^2})\dot{\varphi}+V'(\varphi)+\Gamma\dot{\varphi}=0,$$ where $H={\dot{a}\over a}$ is the Hubble parameter, a “dot” is differentiation with respect to cosmic time $t$, prime denotes differentiation with respect to the scalar field $\varphi$. $\Gamma$ is a positive constant, first introduced in [@gamma] as a phenomenological term which describes the decay of $\varphi$ to the radiation during reheating era. This term was vastly used in the subsequent literature studying the inflaton decay in the reheating era (see [@gamma1] and [@kolb] and the references therein), where like our model the inflaton experiences a rapid oscillation phase. In [@gamma2], it is shown that the production of particles during high-frequency regime in reheating era can be expressed by adding a polarization term to the inflaton mass. To do so, a Lagrangian comprising the inflaton field and its interactions with bosonic and fermionic fields was employed. It was shown that the phenomenological term proposed in [@gamma1] can be derived in this context. The precise form of the dissipative term depends on the coupling between inflaton and relativistic particles it decays to, and also on the interactions of relativistic particles. As the nature of inflaton and these relativistic particles are not yet completely known, the precise form of $\Gamma$ is not clear. However one may employ a phenomenological effective field theory or also thermal field theory [@gamma3], to study the effective dependency of $\Gamma$ on temperature and dynamical fields. Thermal effect can also be inserted by including the thermal correction in the equations of motion [@gamma3]. In the period where the inflaton is dominant over relativistic thermal particles, it is safe to approximately take $\Gamma$ as $\Gamma=\Gamma\Big\|_{T=0}$ (like [@gamma]), as explained in [@gamma4]. Similarly, in the framework of slow roll warm inflation, the possibility that $\Gamma$ is a function of $\varphi$ and temperature, was discussed in the literature[@Berera2; @Xiao]. The Friedmann equations are given by $$\begin{aligned} \label{9} H^2&=&{1\over 3M_P^2}(\rho_\varphi+\rho_{r}) \nonumber \\ \dot{H}&=&-{1\over 2M_P^2}(\rho_\varphi+\rho_{r}+P_\varphi+P_{r}).\end{aligned}$$ The energy density and the pressure of inflaton can be expressed as $$\label{10} \rho_\varphi=(1+{9H^2\over M^2}){\dot{\varphi}^2\over 2}+V(\varphi),$$ and $$\label{11} P_\varphi=(1-{3H^2\over M^2}-{2\dot{H}\over M^2}){\dot{\varphi}^2\over 2}-V(\varphi)-{2H\dot{\varphi}\ddot{\varphi}\over M^2},$$ respectively. Energy density of radiation is $\rho_{r}={3\over4}TS$ [@Berera2]. $S$ is the entropy density and $T$ is the temperature. The equation of state parameter for radiation is ${1\over3}$, hence the rate of radiation production is given by $$\label{12} \dot{\rho_{r}}+4H\rho_{r}=\Gamma{\dot{\varphi}}^2.$$ We assume that the potential is even, $ V(-\varphi)=V(\varphi)$, and consider rapid oscillating solution (around $\varphi=0$) to (\[8\]), which in high-friction regime, $H^2\gg M^2$, reduces to $$\label{r1} \ddot{\varphi}+3H\left(1+{2\over 3}{\dot{H}\over H^2}\right)\dot{\varphi}+{M^2V_{,\varphi}\over 3H^2}+{M^2\Gamma\over 3H^2}=0.$$ In our formalism the inflation has a quasi periodic evolution $$\label{r2} \varphi(t)=\phi(t) \cos(\int A(t) dt),$$ with time dependent amplitude $\phi(t)$. The rapid oscillation (or high frequency oscillation) is characterized by $$\label{r3} \left\|{\dot{H}\over H}\right\|\ll A,\,\,\,\left\|{\dot{\phi}\over \phi}\right\|\ll A,\,\,\,\,\left\|{\dot{\rho_{\varphi}}\over \rho_{\varphi}}\right\|\ll A.$$ The existence of such a solution is verified in [@good1]. It is worth to note that for a power law potential $V(\varphi)=\lambda \varphi^q$, (\[r3\]) holds provided that $$\label{r4} \Phi\ll \left(q^2M_P^4 M^2\over \lambda \right)^{1\over q+2},$$ which is opposite to the slow roll condition $\varphi^{q+2}\gg \left(M_P^4M^2\over \lambda\right)$ [@good1]. The period of oscillation is $$\label{13} \tau(t)=2\int_{-\phi}^\phi {d\varphi(t)\over \dot{\varphi}(t)},$$ and the rapid oscillation occurs for $H\ll {1\over \tau}$ and ${\dot{H}\over H}\ll {1\over \tau}$. The inflaton energy density may estimated as $\rho_{\varphi}=V(\phi(t))$. In this epoch $\rho_\varphi$ and $H$ change insignificantly during a period of oscillation in the sense indicated in (\[r3\]). In rapid oscillatory phase, the time average of adiabatic index, defined by $\gamma={\rho_\varphi + P_\varphi\over \rho_\varphi}$ is given by $\gamma=\big<{\rho_\varphi+P_\varphi\over \rho_\varphi}\big>$, where bracket denotes time averaging over one oscillation $$\label{r5} \big<O(t)\big>={\int_t^{t+\tau}O(t')dt'\over \tau}.$$ For a power law potential $$\label{14} V(\varphi)=\lambda\varphi^q,$$ and in high-friction limit $({H^2\over M^2}\gg1)$, the adiabatic index becomes [@sadjadi2] $$\label{15} \gamma\approx{2q\over 3q+6}.$$ By averaging the continuity equation, we obtain [@good1] $$\label{16} <\rho_\varphi\dot{>}+3H\gamma<\rho_\varphi>+{\gamma\Gamma M^2 \over 3H^2}<\rho_\varphi>=0.$$ when the Universe is dominated by $\varphi$-particles, we take $$\label{R1} \Gamma\ll {9H^3\over M^2}.$$ By this assumption the radiation may be still in equilibrium, and besides we can neglect the third term in (\[16\]). But as $H$ decreases, and the radiation production term becomes more relevant, this approximation fails and the third terms in (\[16\]) get the same order of magnitude as the second term at a time $t_{rh}$. Note that at $t_{rh}$ the radiation and inflaton densities have the same order of magnitude $\rho_{\varphi}(t_{rh})\sim \rho_{r}(t_{rh})$ [@good1], [@kolb]. When $t< t_{rh}$, the average of energy density of the scalar field can be approximated as $$\label{17} \big<\rho_\varphi\big>\propto a(t)^{-3\gamma}.$$ By using relation (\[17\]) and the Friedmann equation ($H^2\approx{1\over 3M_P^2}\rho_\varphi$), in the $\varphi$ dominated era, we can easily obtain $$\label{18} a(t)\propto t^{{q+2\over q}}\propto t^{{2\over 3\gamma}}.$$ Therefore the Hubble parameter in the inflaton dominated era can be estimated as $H\approx {2\over3\gamma t}$ . In the rapid oscillation phase and with the power law potential (\[14\]) we can write the amplitude of the oscillation as $$\label{19} \phi(t)\propto a(t)^{-{2\over q+2}}\propto t^{-{2\over q}}.$$ Our formalism is similar to methods used in the papers studying the reheating era after inflation in the minimal case [@kolb]. But in the minimal case, for $\Gamma<<3H$ until $\Gamma\sim H$, where the Universe is dominated by the oscillating inflaton, instead of (\[18\]), we have $a(t)\propto t^{2\over 3}$. In high-friction limit, time averaging over one oscillation gives $$\label{r6} \big<\dot{\varphi}^2(t)\big>={2M^2\over 9H^2(t)}\big<\rho_\varphi(t)-V(\varphi(t))\big>,$$ where we have used that the Hubble parameter changes insignificantly during one period of oscillation. But $$\begin{aligned} \label{r7} \big<\rho_\varphi(t)-V(\varphi(t))\big>&=&{\int_{-\phi(t)}^{\phi(t)}\sqrt{\rho_\varphi-V(\varphi)}d\varphi\over \int_{-\phi(t)}^{\phi(t)} {d\varphi \over \sqrt{\rho_\varphi-V(\varphi)}}}\nonumber \\ &=&\lambda \phi^q(t) {\int_{0}^{1}\sqrt{1-x^q}dx\over \int_{0}^{1} {dx \over \sqrt{1-x^q}}}\nonumber \\ &=&\lambda \phi^q(t){q\over q+2},\end{aligned}$$ therefore $$\label{20} \big<\dot{\varphi}^2\big>\approx\gamma M_P^2 M^2.$$ This relation shows that for non-minimal derivative coupling model and in the rapid oscillation phase, when the Universe is $\varphi$ dominated, $\big<\dot{\varphi}^2\big>$ is approximately a constant. By inserting (\[20\]) into the equation (\[12\]) we obtain $$\label{20.1} \rho_r={3\Gamma \gamma^2 M^2 M_P^2\over (8+3\gamma)}t\bigg[1-({t_{0}\over t})^{(1+{8\over3\gamma})}\bigg],$$ where $t_{0}$ is the time at which $\rho_r=0$. The number of e-folds from a specific time $t_{}\in (t_0,t_{RD})$ in inflation until radiation dominated epoch, is given by $$\label{20.2} \mathcal{N}_I=\int_{t_{}}^{t_{RD}}Hdt\approx\int_{t_{}}^{t_{RD}}{2\over3\gamma t}dt\approx{2\over3\gamma}\ln\bigg({t_{RD}\over t_{}}\bigg),$$ where $t_{RD}$ is the time at which the universe becomes radiation dominated and inflation ceases. At this time $$\label{20.4} \rho_r(T_{RD})\approx\rho_{\varphi}(t_{RD}).$$ We can calculate the temperature at the end of warm inflation by [@Berera] $$\label{20.3} \rho_r(t_{RD})=g_{RD}{\pi^2\over30}T_{RD}^4,$$ where $g_{RD}$ is number of degree of freedom of relativistic particles and $T_{RD}$ is the temperature of radiation at the beginning of radiation dominated era. Cosmological perturbations ========================== In this section, we study the evolution of thermal fluctuation during oscillatory warm inflation. We use the framework used in [@Berera3] and ignore the possible viscosity terms and shear viscous stress [@visineli]. To investigate cosmological perturbations, we split the metric into two components: the background and the perturbations. The background is described by homogeneous and isotropic FLRW metric with oscillatory scalar field and the perturbed sector of the metric determines anisotropy. We assume that the radiation is in thermal equilibrium during warm inflation. The thermal fluctuations arising in warm inflation evolve gradually via cosmological perturbations equations. Until the freeze out time, the thermal noise has not a significant effect on perturbations development [@Berera3]. We consider the evolution equation of the first order cosmological perturbations for a system containing inflaton and radiation. In the longitudinal gauge the metric can be written as [@Weinberg]. $$\label{21} ds^2=-(1+2\Phi)dt^2+a^2(1-2\Psi)\delta_{ij}dx^idx^j.$$ As mentioned before, the energy momentum tensor splits into radiation part $T^{\mu\nu}_r$ and inflaton part $T^{\mu\nu}_{\varphi}$ as $$\label{22} T^{\mu\nu}=T^{\mu\nu}_r+T^{\mu\nu}_{\varphi}.$$ The unperturbed parts of four velocity components of the radiation fluid satisfy $\overline{u}_{ri}=0$ and $\overline{u}_{r0}=-1$. By using normalization condition $g^{\mu\nu}u_{\mu}u_{\nu}=-1$, the perturbed part of the time component of the four velocity becomes $$\label{24} \delta u^0=\delta u_0={h_{00}\over 2}.$$ The space components $\delta u^i$, are independent dynamical variables and $\delta u_i=\partial_i\delta u$ [@Weinberg]. Energy transfer is described by [@Moss] $$\label{25} Q_{\mu}=-\Gamma u^{\nu}\partial_{\mu}\varphi \partial_{\nu}\varphi.$$ We have also $$\label{26} \nabla_{\mu}T^{\mu\nu}_r=Q^{\nu},$$ and $$\label{27} \nabla_{\mu}T^{\mu\nu}_{\varphi}=-Q^{\nu}.$$ (\[25\]) gives $Q_{0}=\Gamma \dot{\varphi}^2$ and the unperturbed equation (\[26\]) becomes $Q_{0}=\dot\rho_r +3H(\rho_r+P_r)$ which is the continuity equation for the radiation field. In the same way Eq.(\[27\]) becomes $-Q_{0}=\dot\rho_{\varphi} +3H(\rho_{\varphi}+P_{\varphi})$. Perturbations to the energy momentum transfer are described by (there is no perturbation for the dissipation factor $\Gamma$ which we have assumed to be a constant) $$\label{28} \delta Q_{0}=-\delta\Gamma\dot{\varphi}^2+\Phi\Gamma\dot{\varphi}^2-2\Gamma\dot{\varphi}\dot{\delta\varphi}$$ and $$\label{29} \delta Q_{i}=-\Gamma\dot{\varphi}\partial_i{\delta\varphi}.$$ The variation of the equation (\[26\]) is $\delta(\nabla_{\mu}T^{\mu\nu}_r)=\delta Q^{\nu}$, so its (0-0) component is $$\label{30} \dot{\delta\rho_r}+4H\delta\rho_r+{4\over3}\rho_r\nabla^2\delta u-4\dot{\Psi}\rho_r= -\Phi\Gamma{\dot\varphi}^2+\delta\Gamma{\dot\varphi}^2+2\Gamma\dot{\delta\varphi}\dot{\varphi}.$$ Similarly, for the $i-th$ component we derive $$\label{31} 4\rho_r\dot{\delta u^i}+4\dot{\rho_r}\delta u^i+20 H\rho_r\delta u^i=-[3\Gamma\dot{\varphi}\partial_i\delta\varphi+\partial_i\delta\rho_r+4\rho_r\partial_i\Phi].$$ The equation of motion for $\delta\varphi$, computed by variation of (\[27\]), is $\delta(\nabla_{\mu}T^{\mu\nu}_{\varphi})=-\delta Q^{\nu}$. The zero component of this equation is $$\begin{aligned} \label{32} (1+{3H^2\over M^2})\ddot{\delta\varphi}+[(1+{3H^2\over M^2}+{2\dot{H}\over M^2})3H+\Gamma]\dot{\delta\varphi} +\delta V'(\varphi)+\dot{\varphi}\delta\Gamma\\\nonumber -(1+{3H^2\over M^2}+{2\dot{H}\over M^2}){\nabla^2\delta\varphi\over a^2}=\\\nonumber -[2V'(\varphi)+3\Gamma\dot{\varphi}-{6H\dot{\varphi}\over M^2}(3H^2+2\dot{H}) -{6H^2\ddot{\varphi}\over M^2}]\Phi\\\nonumber +(1+{9H^2\over M^2})\dot{\varphi}\dot\Phi+{2H\dot{\varphi}\over M^2}{\nabla^2\Phi\over a^2}\\\nonumber +3(1+{9H^2\over M^2}+{2\dot{H}\over M^2}+{2H\ddot{\varphi}\over M^2})\dot{\Psi}+{6H\dot{\varphi}\over M^2}\ddot{\Psi}-{2(\ddot{\varphi}+H\dot{\varphi})\over M^2}{\nabla^2\Psi\over a^2}.\end{aligned}$$ The $0-0$ component of the perturbation of the Einstein equation $G_{\mu\nu}=-8\pi GT_{\mu\nu}$ is $$\begin{aligned} \label{33} -3H\dot{\Psi}-3H^2\Phi+{\nabla^2\Psi\over a^2}=4\pi G\big[-(1+{18H^2\over M^2}){\dot{\varphi}}^2\Phi-{9H{\dot{\varphi}}^2\over M^2}\dot{\Psi} \\\nonumber +{{\dot{\varphi}}^2\over M^2}{\nabla^2\Psi\over a^2}+\acute{V(\varphi)}\delta\varphi+(1+{9H^2\over M^2})\dot\varphi\dot{\delta\varphi} -{2H\dot{\varphi}\over M^2}{\nabla^2{(\delta\varphi)}\over a^2}+\delta\rho_r\big],\end{aligned}$$ and its $i-i$ component is $$\begin{aligned} \label{34} &&(3H^2+2\dot{H})\Phi+H(3\dot{\Psi}+\dot{\Phi})+{\nabla^2(\Phi-\Psi)\over 3a^2}+\ddot{\Psi}=\\\nonumber &&4\pi G[({(3H^2+2\dot{H}){2\dot{\varphi}^2\over M^2}-{\dot{\varphi}}^2+{8H\dot{\varphi}\ddot{\varphi}\over M^2}})\Phi+ {3H{\dot{\varphi}}^2\over M^2}\dot{\Phi} \\\nonumber &&+{{\dot{\varphi}}^2\over M^2}{\nabla^2\Phi\over 3a^2}+({3H{\dot{\varphi}}^2\over M^2}+{2\dot{\varphi}\ddot{\varphi}\over M^2})\dot{\Psi}+{{\dot{\varphi}}^2\over M^2}\ddot{\Psi}+{{\dot{\varphi}}^2\over M^2}{\nabla^2\Psi\over 3a^2} \\\nonumber &&-\acute{V(\varphi)}\delta\varphi-[(-1+{3H^2\over M^2}+{2\dot{H}\over M^2})\dot{\varphi}+{2H\ddot\varphi\over M^2}]\dot{\delta\varphi}\\\nonumber &&-{2H\dot{\varphi}\over M^2}\ddot{\delta\varphi} +{2(\ddot{\varphi}+H\dot{\varphi})\over M^2}{\nabla^2{(\delta\varphi)}\over 3a^2}+\delta P_r].\end{aligned}$$ By using $-H\partial_i\Phi-\partial_i\dot{\Psi}=4\pi G(\rho+P)\partial_i\delta u$, we can obtain (from $(0-i)$ component of the field equation) $$\begin{aligned} \label{35} &&H\Phi+\dot{\Psi}=4\pi G[{3H\dot{\varphi}^2\over M^2}\Phi+{\dot{\varphi}^2\over M^2}\dot{\Psi}+(1+{3H^2\over M^2})\dot{\varphi}\delta\varphi-{2H\dot{\varphi}\over M^2}\dot{\delta\varphi}\\\nonumber &&+(\rho_r+P_r)\delta u].\end{aligned}$$ Using(\[30\]-\[35\]) we can calculate perturbation parameters. Depending on the physical process, e.g. thermal noise, expansion, curvature fluctuations, three separate regimes for the evolution of the scalar field fluctuations may be considered [@Berera3]. But one can generalize this approach, by adding stochastic noise source and viscous terms to cosmological perturbations equations [@visineli]. During inflation the background has two components, oscillatory scalar field and radiation. The energy density of the scalar field decreases due to expansion and radiation generation. Quantities related to the scalar field in the background have oscillatory behaviors. So we replace the background quantities with their average values over oscillation. Also, we consider non-minimal derivative coupling at the high-friction limit. By going to the Fourier space, the spatial parts of perturbational quantities get $e^{ikx}$ where $k$ is the wave number. So $\partial_j\rightarrow ik_j $ and $\nabla^2\rightarrow-k^2 $. Also we define $$\label{37} \delta u=-{a\over k}v e^{ikx}.$$ So (\[30\]) becomes $$\label{38} \dot{\delta\rho_r}+4H\delta\rho_r+{4\over3} ka\rho_r v-4\rho_r\dot{\Phi}=-\Gamma M^2 M_P^2\Phi,$$ and (\[31\]) becomes $$\label{39} 4{a\over k}(\dot{(\rho_r v)}+4H(\rho_r v))=-\delta\rho_r-4\rho_r\Phi.$$ (\[32\]) reduces to $$\begin{aligned} \label{40} &&({3H^2\over M^2})\ddot{\delta\varphi}+[({3H^2\over M^2}+{2\dot{H}\over M^2})3H+\Gamma]\dot{\delta\varphi} +\delta V'(\varphi)=\\\nonumber &&-2V'(\varphi)\Phi+3({9H^2\over M^2}+{2\dot{H}\over M^2})\dot{\Phi}.\end{aligned}$$ From (\[33\]) we have $$\label{41} -3H\dot{\Phi}(1-{3\gamma\over2})-3H^2\Phi(1-3\gamma)={1\over{2M_P^2}}(V'(\varphi)\delta\varphi+\delta\rho_r),$$ and rewrite (\[34\]) as $$\label{42} (3H^2+2\dot{H})\Phi(1-\gamma)+H\dot{\Phi}(4-3\gamma)+\ddot{\Phi}(1-{1\over2}\gamma)={1\over{2M_P^2}}(-V'(\varphi)\delta\varphi+\delta P_r).$$ Note that we have replaced $\dot{\phi}^2$ and $\dot{\phi}$ by their average values i.e. $<\dot{\phi}^2>=\gamma M^2 M_P^2$ and $<\dot{\phi}>=0$. We restrict ourselves to the high-friction regime ${H^2\over M^2}\gg 1$ and the modes satisfying ${k\over a}\ll H$ and the zero-shear gauge $\Phi=\Psi$ [@Berera3] are considered. (\[35\]) may be written as $$\label{43} H\Phi(1-{3\over2}\gamma)+\dot{\Phi}(1-{1\over2}\gamma)=-{2\over{3M_P^2}}{a\over k}(v\rho_r),$$ and the time derivative of (\[35\]) gives $$\label{44} (H\dot{\Phi}+\dot{H}\Phi)(1-{3\over2}\gamma)+\ddot{\Phi}(1-{1\over2}\gamma)=-{2\over{3M_P^2}}{a\over k}(H(v\rho_r)+\dot{(v\rho_r)}).$$ By analyzing the above equations we find $$\label{45} [3H^2(1-{3\over2}\gamma-{1\over3}\gamma)+\dot{H}({2\over3}-{7\over6}\gamma)]\Phi+{5\over6}(4-3\gamma)H\dot{\Phi}+{5\over6}(1-{1\over2}\gamma)\ddot{\Phi}=0.$$ During the rapid oscillation, the Hubble parameter is $H={2\over 3\gamma t}$, therefore (\[45\]) becomes $$\label{46} ({2\over3\gamma})[{2\over\gamma}-{13\over3}+{7\over6}\gamma]{\Phi\over t^2}+{5\over9\gamma}(4-3\gamma){\dot{\Phi}\over t} +{5\over6}(1-{1\over2}\gamma)\ddot{\Phi}=0.$$ This equation has the solution $\Phi\propto t^{\alpha_\pm}$, therefore $$\label{47} ({2\over3\gamma})[{2\over\gamma}-{13\over3}+{7\over6}\gamma]+{5\over9\gamma}(4-3\gamma)\alpha +{5\over6}(1-{1\over2}\gamma)\alpha(\alpha-1)=0.$$ $\alpha's$ are the roots of this quadratic equation. We denote the positive root by $\alpha_+$. From equations (\[41\]) and (\[42\]), we deduce $$\label{48} -{1\over{M_P^2}}V'(\varphi)\delta\varphi=2(3H^2(1-2\gamma)+\dot{H}(1-\gamma))\Phi+(7-{3\over 2}\gamma)H\dot{\Phi}+(1-{1\over2}\gamma)\ddot{\Phi}.$$ It is now possible to use relation $\Phi\propto t^{\alpha_+}$ to obtain $\delta\varphi$ $$\begin{aligned} \label{49} &&-{1\over{M_P^2}}V'(\varphi)\delta\varphi= \nonumber \\ &&\bigg[{4\over3\gamma}({2\over\gamma}-5+\gamma)+{2\over 3\gamma}(7-{15\gamma\over2})\alpha_{+} +(1-{1\over2}\gamma)\alpha_{+}(\alpha_{+}-1)\bigg]{\Phi\over t^2}.\end{aligned}$$ $\delta\varphi$ simplifies to $$\begin{aligned} \label{50} &&\delta\varphi=-C{{M_P^{2+\alpha_{+}}}\over V'(\varphi)}t^{\alpha_{+}-2}\times \nonumber \\ &&\bigg[{4\over3\gamma}({2\over\gamma}-5+\gamma)+{2\over 3\gamma}(7-{15\gamma\over2})\alpha_{+} +(1-{1\over2}\gamma)\alpha_{+}(\alpha_{+}-1)\bigg]\end{aligned}$$ where $C$ is a numerical constant. Thus the density perturbation, from relation (\[50\]), becomes [@dp] $$\begin{aligned} \label{51} &&\delta_H\approx {16\pi\over 5 M_P^{2+\alpha_{+}}} \nonumber \\ &&{V'\delta\varphi\over\bigg[{4\over3\gamma}({2\over\gamma}-5+\gamma)+{2\over 3\gamma}(7-{15\gamma\over2})\alpha_{+} +(1-{1\over2}\gamma)\alpha_{+}(\alpha_{+}-1)\bigg]t^{\alpha_{+}-2}}.\end{aligned}$$ In this relation $\delta\varphi$ is the scalar field fluctuation during the warm inflation, which instead of quantum fluctuation, are generated by thermal fluctuation [@Berera; @gamma3]. Due to the thermal fluctuations, $\varphi$ satisfies the Langevin equation with a stochastic noise source, using which one finds [@nozari1] $$\label{52} \delta\varphi^2={k_FT\over2\pi^2},$$ where $k_F$ is the freeze out scale, containing also terms corresponding to the non-minimal coupling. To compute $k_F$, we must determine when the damping rate of relation (\[42\]) becomes less than the expansion rate $H$. At $t_F$ (freeze out time [@Berera3]), the freeze out wave number $k_F={k\over a(t_F)}$ is given by [@nozari1] $$\label{53} k_F=\sqrt{\Gamma H+3H^2(1+{3H^2\over M^2})}.$$ In the minimal case ${H^2\over M^2}=0$, and (\[53\]) gives the well known result [@gamma3]. $$\label{ref123} \delta\varphi^2={\sqrt{\Gamma H+3H^2}T\over 2\pi^2},$$ which reduces to $\delta\varphi^2={\sqrt{3}H T\over 2\pi^2}$ [@Berera] in weak dissipative regime $\Gamma\ll H$, and to $\delta\varphi^2={\sqrt{\Gamma H}T\over 2\pi^2}$, in the strong dissipative regime $\Gamma\gg H$. For a more detailed discussion about the scalar field fluctuations(\[ref123\]), based on quantum field theory first principles, see [@ref]. In our case, as we are restricted to the the high-friction regime ${H^2\over M^2}\gg 1$ and also use the approximation (\[R1\]) before the radiation dominated era, we have $$\label{R2} \delta\varphi^2={3H^2T\over2 M\pi^2}.$$ Note that our study is restricted to the region $H<\Gamma \lesssim\left({H^2\over 9M^2}\right)H$. By using (\[R2\]), the density perturbation $$\begin{aligned} \label{54} &&\delta_H^2\approx {\left({16\pi\over 5M_P^{2+\alpha_+}}\right)}^2 t^{4-2\alpha_{+}}\times \nonumber \\ &&{V'^2\delta\varphi^2\over\bigg[{4\over3\gamma}({2\over\gamma}-5+\gamma)+{2\over 3\gamma}(7-{15\gamma\over2})\alpha_{+} +(1-{1\over2}\gamma)\alpha_{+}(\alpha_{+}-1)\bigg]^2}\end{aligned}$$ can be rewritten as $$\begin{aligned} \label{55} &&\delta_H^2\approx {\left({128\over 25M_P^{4+2\alpha_+}}\right)}t^{4-2\alpha_{+}}\left({3H^2\over M}\right)T\times \nonumber \\ &&{V'^2\over\bigg[{4\over3\gamma}({2\over\gamma}-5+\gamma)+{2\over 3\gamma}(7-{15\gamma\over2})\alpha_{+} +(1-{1\over2}\gamma)\alpha_{+}(\alpha_{+}-1)\bigg]^2}.\end{aligned}$$ We can now calculate power spectrum from relation $P_s(k_0)={25\over4}\delta_H^2(k_0)$ [@dp]. $k_0$ is a pivot scale. The spectral index for scalar perturbation is $$\label{56} n_s-1={d\ln{\delta_H^2}\over d\ln{k}}.$$ The derivative is taken at horizon crossing $k\approx aH$. The spectral index may be written as $$\label{57} n_s-1={d\ln{\delta_H^2}\over d\ln{(aH)}}=\bigg({1\over H+{\dot{H}\over H}}\bigg){d\ln{\delta_H^2}\over dt}.$$ From $H={2\over3\gamma t}$ we have $$\label{58} n_s-1\approx \bigg({t\over{2\over 3\gamma}-1}\bigg){d\ln{\delta_H^2}\over dt},$$ therefore $$\label{59} n_s-1\approx \bigg({4\over3\gamma}-{5\over2}-2\alpha_{+}\bigg)\bigg({1\over{2\over 3\gamma}-1}\bigg).$$ This relation gives the spectral index as a function of $\gamma$. From Planck 2015 data $n_s=0.9645\pm0.0049$ (68% CL, Planck TT,TE,EE+lowP) $\gamma$ is determined as $\gamma=0.55902\pm0.00016$. Evolution of the Universe and temperature of the warm inflation =============================================================== In this section, by using our previous results, we intend to calculate the temperature of warm inflation as a function of observational parameters for the power law potential (\[14\]) and a constant dissipation coefficient $\Gamma $, in high-friction limit. For this purpose we follow the steps introduced in [@Mielczarek1], and divide the evolution of the Universe from $t_$ (a time at which a pivot scale exited the Hubble radius) in inflation era until now into three parts\ $I-$ from $t_\star$ until the end of oscillatory warm inflation, denoted by $t_{RD}$. in this period energy density of the oscillatory scalar field is dominated.\ $II-$ from $t_{RD}$ until recombination era, denoted by $t_{rec}$.\ $III-$ from $t_{rec}$ until the present time $t_0$.\ Therefore the number of e-folds from horizon crossing until now becomes $$\begin{aligned} \label{60} \mathcal{N}&=&\ln{({a_0\over a_\star})}=\ln{({a_0\over a_{rec}})}+\ln{({a_{rec}\over a_{RD}})}+\ln{({a_{RD}\over a_{\star}})}=\\\nonumber &&\mathcal{N}_{I}+\mathcal{N}_{II}+\mathcal{N}_{III}\end{aligned}$$ Oscillatory warm inflation -------------------------- During the warm oscillatory inflation, the scalar field oscillates and decays into the ultra-relativistic particles. In this period the energy density of oscillatory scalar field is dominated and the Universe expansion is accelerated. The beginning time of radiation dominated era is determined by the condition $\rho_r(t_{RD})\simeq\rho_{\varphi}(t_{RD})$ which gives [@good1; @good2] $$\label{62} {t_{RD}}^3={4(8+3\gamma)\over9\Gamma\gamma^4M^2}.$$ From equations (\[62\]) and (\[20.1\]) we can calculate energy density of radiation at $t_{RD}$ $$\label{63} \rho_r(t=t_{RD})\approx M_P^2\bigg[{12\Gamma^2\gamma^2M^4\over{(8+3\gamma)}^2} \bigg]^{1\over3}.$$ Note that $t_{RD}\sim t_{rh}$, where $t_{rh}$ is defined after (\[R1\]). The temperature of the Universe at the end of oscillatory warm inflation becomes $$\label{64} {T_{RD}}^4\approx {30M_P^2\over\pi^2g_{RD}}\bigg[{12\Gamma^2\gamma^2M^4\over{(8+3\gamma)}^2} \bigg]^{1\over3}.$$ Radiation dominated and recombination eras ------------------------------------------ At the end of the warm inflation the magnitude of radiation energy density equals the energy density of the scalar field. Thereafter the universe enters a radiation dominated era. During this period, the Universe is filled with ultra-relativistic particles which are in thermal equilibrium. In this epoch the Universe undergoes an adiabatic expansion where the entropy per comoving volume is conserved: $dS=0$ [@kolb]. In this era the entropy density, $s=Sa^{-3}$, is [@kolb] $$\label{65} s={2\pi^2\over 45}g T^3.$$ So we have $$\label{66} {a_{rec}\over a_{RD}}={T_{RD}\over T_{rec}}\left({g_{RD}\over g_{rec}}\right)^{1\over 3}.$$ In the recombination era, $g_{rec}$ corresponds to degrees of freedom of photons, hence $g_{rec}=2$. Thus $$\label{67} \mathcal{N}_{II}= \ln\left({T_{RD}\over T_{rec}}\left({g_{RD}\over 2}\right)^{1\over 3}\right).$$ By the expansion of the Universe, the temperature decreases via $T(z)=T(z=0)(1+z)$, where $z$ is the redshift parameter. Hence $T_{rec}$ in terms of $T_{CMB}$ is $$\label{68} T_{rec}=(1+z_{rec})T_{CMB}.$$ We have also $$\label{69} {a_0\over a_{rec}}=(1+z_{rec}).$$ Therefore $$\label{70} \mathcal{N}_{II}+\mathcal{N}_{III}=\ln\left({T_{RD}\over T_{CMB}}\left({g_{RD}\over 2}\right)^{1\over3}\right).$$ Temperature of the warm oscillatory inflation --------------------------------------------- To obtain temperature of the warm inflation we must determine $\mathcal{N}$ in (\[60\]). We take $a_0=1$, so the number of e-folds from the horizon crossing until the present time is $\Delta=\exp(\mathcal{N})$, where $$\label{71} \Delta={1\over a_}={H_\over k_0}\approx {2\over 3\gamma t_ k_0}.$$ By relations (\[71\],\[70\],\[60\]) we can derive $T_{RD}$, $$\label{72} T_{RD}=T_{CMB}{({2\over g_{RD}})}^{1\over3}{2\over3\gamma k_0}\bigg[{4(8+3\gamma)\over9\Gamma\gamma^4M^2} \bigg]^{-{2\over9\gamma}} \times t_{}^{({2\over3\gamma}-1)}.$$ We can remove $\Gamma M^2$ in this relation by (\[64\]) $$\label{72.1} T_{RD}^{(1-{4\over3\gamma})}\approx {2T_{CMB}\over3\gamma k_0}{({2\over g_{RD}})}^{{\gamma-1\over3\gamma}}\bigg[{2\sqrt{5}M_P\over\pi\gamma} \bigg]^{-{2\over3\gamma}}\times t_{}^{({2\over3\gamma}-1)}.$$ By using relation $\mathcal{P}_s(k_0)={25\over4}\delta_H^2(k_0)$ and equation (\[55\]), power spectrum becomes $$\begin{aligned} \label{73} &&\mathcal{P}_s(k_0)\approx \bigg({32\over M_P^{4+2\alpha_{+}}}\bigg) \times \nonumber \\ &&{\left<V'(\varphi_)\right>^2\over\bigg[{4\over3\gamma}({2\over\gamma}-5+\gamma)+{2\over 3\gamma}(7-{15\gamma\over2})\alpha_{+} +(1-{1\over2}\gamma)\alpha_{+}(\alpha_{+}-1)\bigg]^2}\nonumber \\ &&\times t_^{4-2\alpha_{+}}{\sqrt{\Gamma H_+3H_^2(1+{3H_^2\over M^2})}}T_.\end{aligned}$$ In this relation $T_\star$ is the temperature of the universe at the horizon crossing. By relation (\[20.1\]) we can calculate temperature at horizon crossing as a function of $t_$ $$\label{74} T_{}=\bigg[ {90\Gamma\gamma^2M^2M_P^2\over(8+3\gamma)\pi^2g_}\bigg]^{1\over4}t_^{1\over4}.$$ We can remove $\Gamma M^2$ in relation (\[74\]) by (\[64\]) $$\label{75} T_{}=\bigg[ {\pi\gamma g_{RD}^{1\over2}\over2\sqrt{10}M_P}\bigg]^{1\over4} T_{RD}^{3\over2} t_^{1\over4}.$$ We have taken $g_{RD}\sim g_{}$. The time average of the potential derivative may computed as follows $$\begin{aligned} \label{76} <V'>&=&q\lambda{\int_{-\phi}^{\phi}\varphi ^{q-1}{d\varphi \over \dot{\varphi}}\over \int_{-\phi}^{\phi}{d\varphi \over \dot{\varphi}}}\nonumber \\ &=&q\lambda \phi^{n-1}{\int_0^1 {x^{q-1}dx\over \sqrt{1-x^q}}\over \int_0^1 {dx\over \sqrt{1-x^q}}}\nonumber \\ &=&2q\lambda{\Gamma\left({2+q\over 2q}\right)\over \Gamma \left({1\over q}\right)}\phi^{q-1}.\end{aligned}$$ In inflationary regime we have $H^2\approx{1\over 3M_P^2}\rho_\varphi\approx {1\over 3M_P^2}\lambda \phi^q$ and $H^2\approx {4\over 9 \gamma^2 t^2}$, therefore $$\label{76.1} <V'(\varphi_)>=12\lambda{\gamma\over2-3\gamma}{\Gamma\left({1\over 3\gamma}\right)\over \Gamma \left({1\over 3\gamma}-{1\over2}\right)}\left({4M_P^2\over 3\lambda \gamma^2}\right)^{9\gamma-2\over 6\gamma}t_^{2-9\gamma\over 3\gamma}.$$ Thus we can write (\[73\]) as $$\label{77} \mathcal{P}_s(k_0)\approx M_P^{({5\over2}-{4\over3\gamma}-2\alpha_{+})}\lambda^{({2\over3\gamma}-1)}\Gamma^{1\over4}M^{1\over2}g_{RD}^{-{1\over4}}\beta t_^{(-{11\over4}+{4\over3\gamma}-2\alpha_{+})}.$$ $\beta$ is given by $$\label{78} \beta={{2048\over \sqrt{\pi}}\left({90\gamma^2\over 8+3\gamma}\right)^{1\over 4}{\left({4\over 3\gamma^2}\right)}^{9\gamma-2\over3\gamma}\bigg({\Gamma\left({1\over 3\gamma}\right)\over \Gamma \left({1\over3\gamma} -{1\over 2}\right)}\bigg)^2\over(2-3\gamma)^2\bigg[{4\over3\gamma}({2\over\gamma}-5+\gamma)+{2\over 3\gamma}(7-{15\gamma\over2})\alpha_{+} +(1-{1\over2}\gamma)\alpha_{+}(\alpha_{+}-1)\bigg]^2}.$$ From equation (\[77\]), we derive $t_$ as $$\label{78.1} t_=\bigg[{\mathcal{P}_s(k_0)g_{RD}^{1\over4}\over M_P^{({5\over2}-{4\over3\gamma}-2\alpha_{+})}\lambda^{({2\over3\gamma}-1)}M^{1\over 2}\Gamma^{1\over4}\beta}\bigg]^{12\gamma\over16-33\gamma-24\gamma\alpha_+}.$$ By substituting $t_$ from relation (\[78.1\]) into equation (\[72\]), the temperature at the end of warm oscillatory inflation or beginning of the radiation domination is obtained as $$\begin{aligned} \label{79} T_{RD}=T_{CMB}{({2\over g_{RD}})}^{1\over3}{2\over3\gamma k_0}\bigg[{4(8+3\gamma)\over9\Gamma\gamma^4M^2} \bigg]^{-{2\over9\gamma}} \times \\\nonumber \bigg[{\mathcal{P}_s(k_0)g_{RD}^{1\over4}\over M_P^{({5\over2}-{4\over3\gamma}-2\alpha_{+})}\lambda^{({2\over3\gamma}-1)}M^{1\over 2}\Gamma^{1\over4}\beta}\bigg]^{4(2-3\gamma)\over16-33\gamma-24\gamma\alpha_+}.\end{aligned}$$ The number of e-folds during warm oscillatory inflation becomes $$\label{80} \mathcal{N}_I\approx{2\over3\gamma}\ln \left(\left({4(8+3\gamma)\over9\Gamma\gamma^4M^2}\right)^{1\over 3} \left({\mathcal{P}_s(k_0)g_{RD}^{1\over4}\over M_P^{({5\over2}-{4\over3\gamma})-2\alpha_{+}}\lambda^{({2\over3\gamma}-1)}M^{1\over 2}\Gamma^{1\over4}\beta}\right)^{{12\gamma\over-16+33\gamma+24\gamma\alpha_+}}\right).$$ We set $g_{RD}=106.75$, which is the ultra relativistic degrees of freedom at the electroweak energy scale. Also, from Planck 2015 data, at the pivot scale $k_0=0.002Mpc^{-1}$ and in one sigma level, we have $\mathcal{P}_s(k_0)=(2.014\pm0.046)\times10^{-9}$ and $n_s=0.9645\pm0.0049$ $(68\% CL,Planck TT,TE,EE+lowP)$ [@planck]. By using $\gamma=0.55902 $, $M=10^{-16}M_P$, $\lambda=(10^{-8}M_P)^{4-q}$, and $\Gamma=10^{-4}M_P$ in equation (\[79\]) the temperature of the universe at the end of warm inflation and the number of e-folds become $T_{end}\approx3.83\times10^{12}GeV$, and $\mathcal{N}=61.42$ respectively. Tensorial perturbation ====================== In this part, we follow the method used in [@Kaushik] to study tensorial perturbation. The power spectrum for tensorial perturbation is given by [@Kaushik] $$\label{81} {P_{t}(k)}={k^3\over 2\pi^2}\|{v_{k}\over z}\|^2\coth({k\over2T}),$$ where $v_k$ can be calculated from Mukhanov equation [@Germani2] $$\label{82} {d^2v_k\over d\eta^2}+\big(c^2k^2-{1\over z}{d^2z\over d\eta^2}\big)v_k=0.$$ $\eta$ is the conformal time, $c_t$ is the sound speed for tensor mode and $k$ is wave number for mode function $v_k$ [@Germani2] and $z$ is given by $$\label{83} z=a(t)M_p{\sqrt{e_{ij}^\lambda e_{ij}^\lambda }\over2}\sqrt{1-\alpha}.$$ The tensor of polarization is normalized as $e_{ij}^\lambda e_{ij}^{\lambda^\prime}=2\delta_{\lambda\lambda^\prime} $. For our model, with a quasi periodic scalar inflaton background, we have $\alpha={\dot{\varphi}^2\over 2M^2M_p^2}$ and $c$ is given by relation $$\label{84} c^2={1+\alpha\over 1-\alpha}.$$ Therefore $$\label{85} {1\over z}{d^2z\over d\eta^2}=({q\over2}+1)({q\over2}+2)\eta^{-2}.$$ By using this relation, the equation for mode function becomes $$\label{86} {d^2v_k\over d\eta^2}+\big(c^2k^2-({q\over2}+1)({q\over2}+2)\eta^{-2}\big)v_k=0.$$ Solution to this mode equation are the Hankel functions of the first and second kind $$\label{87} v_k(\eta)=\eta^{1\over 2}[C^{(1)}(k)H_\nu^{(1)}(ck\eta)+C^{(2)}(k)H_\nu^{(2)}(ck\eta)].$$ Well within the horizon, the modes satisfy $k\gg aH$, and can be approximated by flat waves. Therefore $$\label{88} v_{k}(\eta)\approx{\sqrt{\pi}\over 2}e^{i(\nu+{1\over 2}){\pi\over 2}}{ (-\eta)}^{1\over 2}H_\nu^{(1)}(-ck\eta).$$ On the other hand, when we want to compute power spectrum, we need to have modes that are outside the horizon. So by taking the limit ${k\over aH}\rightarrow0$, we obtain the asymptotic form of mode function as $$\label{89} v_{k}(\eta)\rightarrow e^{i(\nu+{1\over 2}){\pi \over2 } }2^{(\nu-{3\over2})}{\Gamma(\nu)\over\Gamma({3\over2})}{1\over\sqrt{2ck}}{(-ck\eta)}^{(-\nu+{1\over2})}.$$ By using this relation we can write the power spectrum as $$\label{90} P_t(k)={k^3\over 2\pi^2}{2^{(2\nu-3)}\over\beta^2 a^2}\bigg({\Gamma(\nu)\over\Gamma({3\over2})}\bigg)^2{1\over2c k} {(-ck\eta)}^{(-2\nu+1)}\coth({k\over2T}).$$ In the rapid oscillation epoch $\epsilon={\dot{H}\over H^2}={3\gamma\over 2}$ (see (\[18\])), so we can write the conformal time as $$\label{91} \eta=-{1\over aH}{1\over 1-\epsilon}.$$ At the horizon crossing $c_sk=aH$, we can write (\[90\]) as $$\label{92} P_{t}=A^2_{t}(q)\bigg({H\over M_p}\bigg)^2\coth({k\over2T})\|_{ck=aH},$$ where $$\label{93} A_t(q)={3^{({1\over2})}{2^{(q-{1\over2})}}{\Gamma({3\over2}+{q\over2})}(q+2)^{(-{q+1\over2})}\over\pi\Gamma({3\over2})(q+3)^{({1\over4})}(2q+3)^{({3\over4})}}.$$ The ratio of tensor to scalar spectrum (from relations (\[77\]) and (\[92\])) becomes $$\label{94} r={P_t(k_0)\over \mathcal{P}_s(k_0)}\approx \bigg[{4A^2_{t}(q)g_{RD}^{1\over4}t_^{({4\over3\gamma}-{11\over4}-2\alpha_{+})}\over {9\gamma^2 M_p^{({9\over2}-{4\over3\gamma}-2\alpha_{+})}\lambda^{({2\over3\gamma}-1)}M^{1\over2}\Gamma^{1\over4}}\beta }\bigg]\coth({k\over2T})\bigg\|_{ck=aH}.$$ From relation (\[94\]) by using $\gamma=0.55902 $, $ g_{RD}=106.75$, $M=10^{-16}M_p$ and $\Gamma=10^{-4}M_p$, the ratio of tensor to scaler at the pivot scale $k_*=0.002 Mpc^{-1}$ becomes $r\approx0.081$ which is consistent with planck 2015 data $r_{0.002}<0.10$ $(95\% CL, Planck TT, TE, EE + lowP)$. Conclusion ========== In the standard model of inflation, inflaton begins a coherent rapid oscillation after the slow roll. During this stage, inflaton decays to radiation and reheats the Universe. In this paper, we considered a rapid oscillatory inflaton during inflation era. This scenario does not work in minimal coupling model due to the fewness of e-folds during rapid oscillation. But the non-minimal derivative coupling can remedy this problem in high-friction regime. Therefore we proposed a new model in which inflation and rapid oscillation are unified without considering the slow roll. The number of e-folds was calculated. We investigated cosmological perturbations and the temperature of the Universe was determined as a function of the spectral index. We used a phenomenological approach to describe the interaction term between the inflaton and the radiation, but a precise study about thermal radiation productions must be derived from quantum field theory principles. An attempt in this subject may be found in [@ref]. To complete our study one may consider quantum and thermal corrections to the parameters of the system such as the inflaton mass and its coupling to the radiation. In the slow roll model, the role of theses corrections on the observed spectrum is studied in the literature [@quantum]. Recently in [@be], it was shown that in the warm slow roll inflation, it is possible to sustain the flatness of the potential against the thermal and loop quantum corrections. In the non-minimal derivative coupling model, by power counting analysis, and unitarity constraint which implies $H\ll \Lambda$, where $\Lambda=(H^2M_P)^{1\over 3}$ is the cutoff of the theory, it was shown that for power-law potentials quantum radiation corrections are subleading [@Germani1]. 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--- abstract: '> Dialogue authoring in large games requires not only content creation but the subtlety of its delivery, which can vary from character to character. Manually authoring this dialogue can be tedious, time-consuming, or even altogether infeasible. This paper utilizes a rich narrative representation for modeling dialogue and an expressive natural language generation engine for realizing it, and expands upon a translation tool that bridges the two. We add functionality to the translator to allow direct speech to be modeled by the narrative representation, whereas the original translator supports only narratives told by a third person narrator. We show that we can perform character substitution in dialogues. We implement and evaluate a potential application to dialogue implementation: generating dialogue for games with big, dynamic, or procedurally-generated open worlds. We present a pilot study on human perceptions of the personalities of characters using direct speech, assuming unknown personality types at the time of authoring.' author: - 'Stephanie M. Lukin' - 'James O. Ryan' - \| Marilyn A. Walker\ Natural Language and Dialogue Systems Lab\ University of California, Santa Cruz\ 1156 High Street, Santa Cruz, CA 95060\ [{slukin,joryan,mawalker}@ucsc.edu]{} title: Automating Direct Speech Variations in Stories and Games --- Dialogue authoring in large games requires not only the creation of content, but the subtlety of its delivery which can vary from character to character. Manually authoring this dialogue can be tedious, time-consuming, or even altogether infeasible. The task becomes particularly intractable for games and stories with dynamic open worlds in which character parameters that should produce linguistic variation may change during gameplay or are decided procedurally at runtime. Short of writing all possible variants pertaining to all possible character parameters for all of a game’s dialogue segments, authors working with highly dynamic systems currently have no recourse for producing the extent of content that would be required to account for all linguistically meaningful character states. As such, we find open-world games today filled with stock dialogue segments that are used repetitively by many characters without any linguistic variation, even in game architectures with rich character models that could give an actionable account of how their speech may vary [@klabunde2013greetings]. Indeed, in general, we are building computational systems that, underlyingly, are far more expressive than can be manifested by current authoring practice. These concerns can also be seen in linear games, in which the number of story paths may be limited to reduce authoring time or which may require a large number of authors to create a variety of story paths. Recent work explores the introduction of automatically authored dialogues using expressive natural language generation (NLG) engines, thus allowing for more content creation and the potential of larger story paths [@montfort2014expressing; @lin2011all; @cavazza2005dialogue; @rowe2008archetype]. ![\[est-arch-fig\] [NLG pipeline method of the ES Translator.]{}](sch-pers-arch-cropped-framed-v2){width="3.0in"} [@walker2013using] explore using a dynamic and customizable NLG engine called [personage]{} to generate a variety of character styles and realizations, as one way to help authors to reduce the authorial burden of writing dialogue instead of relying on scriptwriters. [personage]{} is a parameterizable NLG engine grounded in the Big Five personality traits that provides a larger range of pragmatic and stylistic variations of a single utterance than other NLG engines [@MairesseWalker11]. In [personage]{}, narrator’s voice (or style to be conveyed) is controlled by a model that specifies values for different stylistic parameters (such as verbosity, syntactic complexity, and lexical choice). [personage]{} requires hand crafted text plans, limiting not only the expressibility of the generations, but also the domain. Original Fable Dialogic Interpretation of Original Fable ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- A Crow was sitting on a branch of a tree with a piece of cheese in her beak when a Fox observed her and set his wits to work to discover some way of getting the cheese. Coming and standing under the tree he looked up and said, “What a noble bird I see above me! Her beauty is without equal, the hue of her plumage exquisite. If only her voice is as sweet as her looks are fair, she ought without doubt to be Queen of the Birds.” The Crow was hugely flattered by this, and just to show the Fox that she could sing she gave a loud caw. Down came the cheese,of course, and the Fox, snatching it up, said, “You have a voice, madam, I see: what you want is wits.” “It’s a lovely day, I think I will eat my cheese here” the crow said, flying to a branch with a piece of cheese in her beak. A Fox observed her. “I’m going to set my wits to work to discover some way to get the cheese” Coming and standing under the tree he looked up and said, “What a noble bird I see above me! Her beauty is without equal, the hue of her plumage exquisite. If only her voice is as sweet as her looks are fair, she ought without doubt to be Queen of the Birds.” “I am hugely flattered!” said the Crow. “Let me sing for you!” Down came the cheese,of course, and the Fox, snatching it up, said, “You have a voice, madam, I see: what you want is wits."\ [@reed2011step] introduce SpyFeet: a mobile game to encourage physical activity which makes use of dynamic storytelling and interaction. A descendant of [personage]{}, called SpyGen, is its NLG engine. The input to SpyGen is a text plan from Inform7, which acts as the content planner and manager. [@reed2011step] show that this architecture allows any character personality to be used in any game situation. However their approach was not evaluated and it relied on game specific text plans. [@rishes2013generating] created a translator, called the ES-Translator ([est]{}), which bridges a narrative representation produced by the annotation tool Scheherezade, to the representation required by [personage]{}, thus not requiring the creation of text plans. Fig. \[est-arch-fig\] provides a high level view of the architecture of [est]{}, described in more detail below. Scheherazade annotation facilitates the creation of a rich symbolic representation for narrative texts, using a schema known as the [story intention graph]{} or [sig]{} [@ElsonMcKeown10; @elson2012b]. A [sig]{} represents the sequence of story events, as well as providing a rich representation of the intentions, beliefs, and motivations of story characters. The [est]{} takes the [sig]{} as input, and then converts the narrative into a format that [personage]{} can utilize. However, the approach described in [@rishes2013generating] is limited to telling stories from the third person narrator perspective. This paper expands upon the [est]{} to enable annotation of direct speech in Scheherazade, that can then be realized directly as character dialogue. We explore and implement a potential application to producing dialogue in game experiences for large, dynamic, or procedurally-generated open worlds, and present a pilot study on user perceptions of the personalities of story characters who use direct speech. The contributions of this work are: 1) we have can modify a single, underlying representation of narrative to adjust for direct speech and substitute character speaking styles; and 2) that we can perform this modeling on any domain. ES Translator ============= Aesop’s Fable “The Fox and The Crow" (first column in Fig. \[fc-original\]) is used to illustrate the development and the new dialogue expansion of the [est]{}. Annotation Schema ----------------- ![image](pretty-fox-crow-sig.jpg){width="4.0in"} One of the strengths of Scheherazade is that it allows users to annotate a story along several dimensions, starting with the surface form of the story (first column in Fig. \[foxcrow-sig\]) and then proceeding to deeper representations. The first dimension (second column in Fig. \[foxcrow-sig\]) is called the “timeline layer”, in which the story facts are encoded as predicate-argument structures (propositions) and temporally ordered on a timeline. The timeline layer consists of a network of propositional structures, where nodes correspond to lexical items that are linked by thematic relations. Scheherazade adapts information about predicate-argument structures from the VerbNet lexical database [@Kipper06] and uses WordNet [@WordNet] as its noun and adjectives taxonomy. The arcs of the story graph are labeled with discourse relations. Fig. \[gui\] shows a GUI screenshot of assigning propositional structure to the sentence [The crow was sitting on the branch of a tree]{}. This sentence is encoded as two nested propositions [sit(crow)]{} and the prepositional phrase [on(the branch of the tree)]{}. Both actions ([sit]{} and [on]{}) contain references to the story characters and objects ([crow]{} and [branch of the tree]{}) that fill in slots corresponding to semantic roles. Only the timeline layer is utilized for this work at this time. ![\[gui\] GUI view of propositional modeling](sig-fc-bw.png){width="40.00000%"} In the current annotation tool, the phrase [The fox ... said “You have a voice, madam..."]{} can be annotated in Scheherazade by selecting [say]{} from VerbNet and attaching the proposition [the crow has a voice]{} to the verb [say(fox, able-to(sing(crow)))]{}. However, this is realized as [The fox said the crow was able to sang]{} (note: in the single narrator realization, everything is realized in the past tense at this time. When we expand to direct speech in this work, we realize verbs in the future or present tense where appropriate). To generate [The fox said “the crow is able to sing"]{}, we append the modifier “directly" to the verb “say" (or any other verb of communication or cognition, e.g. “think"), then handle it appropriately in the [est]{} rules described in Section \[est-rules\]. Furthermore, to generate [The fox said “you are able to sing"]{}, instead of selecting [crow]{}, an [interlocutor]{} character is created and then annotated as [say(fox, able-to(sing(interlocutor)))]{}. We add new rules to the [est]{} to handle this appropriately. Translation Rules {#est-rules} ----------------- ![image](sig_dsynts-cropped-framed-bw){width="5.0in"} The process of the [est]{} tranformation of the [sig]{} into a format that can be used by [personage]{} is a multi-stage process shown in Fig. \[sig\_dsynts\] [@rishes2013generating]. First, a syntactic tree is constructed from the propositional event structure. Element A in Fig. \[sig\_dsynts\] contains a sentence from the original “The Fox and the Grapes" fable. The Scheherazade API is used to process the fable text together with its [sig]{} encoding and extract actions associated with each timespan of the timeline layer. Element B in Fig. \[sig\_dsynts\] shows a schematic representation of the propositional structures. Each action instantiates a separate tree construction procedure. For each action, we create a verb instance (highlighted nodes of element D in Fig. \[sig\_dsynts\]). Information about the predicate-argument frame that the action invokes (element C in Fig. \[sig\_dsynts\]) is then used to map frame constituents into respective lexico-syntactic classes, for example, characters and objects are mapped into nouns, properties into adjectives and so on. The lexico-syntactic class aggregates all of the information that is necessary for generation of a lexico-syntactic unit in the DSyntS representation used by the [ realpro]{} surface realizer of [personage]{} (element E in Fig. \[sig\_dsynts\]) [@lavoie1997fast]. [@rishes2013generating] define 5 classes corresponding to main parts of speech: noun, verb, adverb, adjective, functional word. Each class has a list of properties such as morphology or relation type that are required by the DSyntS notation for a correct rendering of a category. For example, all classes include a method that parses frame type in the [sig]{} to derive the base lexeme. The methods to derive grammatical features are class-specific. Each lexico-syntactic unit refers to the elements that it governs syntactically thus forming a hierarchical structure. A separate method collects the frame adjuncts as they have a different internal representation in the [sig]{}. At the second stage, the algorithm traverses the syntactic tree in-order and creates an XML node for each lexico-syntactic unit. Class properties are then written to disk, and the resulting file (see element E in Fig. \[sig\_dsynts\]) is processed by the surface realizer to generate text. Dialogue Realization -------------------- The main advantage of [personage]{} is its ability to generate a single utterance in many different voices. Models of narrative style are currently based on the Big Five personality traits [@MairesseWalker11], or are learned from film scripts [@Walkeretal11]. Each type of model (personality trait or film) specifies a set of language cues, one of 67 different parameters, whose value varies with the personality or style to be conveyed. In [@reed2011step], the SpyGen engine was not evaluated. However previous work [@MairesseWalker11] has shown that humans perceive the personality stylistic models in the way that [ personage]{} intended, and [@Walkeretal11] shows that character utterances in a new domain can be recognized by humans as models based on a particular film character. Here we first show that our new architecture as illustrated by Fig. \[est-arch-fig\] and Fig. \[sig\_dsynts\] lets us develop [sig]{}s for any content domain. We first illustrate how we can change domains to a potential game dialogue where the player could have a choice of party members, and show that the [est]{} is capable of such substitutions. Table \[game\] shows different characters saying the same thing in their own style. We use an openness to experience model from the Big 5 [@MairesseWalker11], Marion from [Indiana Jones]{} and Vincent from [ Pulp Fiction]{} from [@lin2011all], and the Otter character model from [@reed2011step]’s Heart of Shadows. --------------------- ------------------------------------------------------------------------------------------------------------------------ Openness (Big Five) “Let’s see... I see, I will fight with you, wouldn’t it? It seems to me that you save me from the dungeon, you know."\ Marion (Indiana Jones) & “Because you save me from the dungeon pal, I will fight with you!"\ Vincent (Pulp Fiction) & “Oh God I will fight with you!"\ Otter (Heart of Shadows) & “Oh gosh I will fight with you because you save me from the dungeon mate!"\ --------------------- ------------------------------------------------------------------------------------------------------------------------ : \[game\]Substituting Characters With the [est]{}, an author could use Scheherazade to encode stock utterances that any character may say, and then have [personage]{} automatically generate stylistic variants of that utterance pertaining to all possible character personalities. This technique would be particularly ripe for games in which character personality is unknown at the time of authoring. In games like this, which include The Sims 3 [@sims3] and Dwarf Fortress [@dwarffortress], personality may be dynamic or procedurally decided at runtime, in which case a character could be assigned any personality from the space of all possible personalities that the game can model. Short of writing variants of each dialogue segment for each of these possible personalities, authors for games like these simply have no recourse for producing enough dialogic content sufficient to cover all linguistically meaningful character states. For this reason, in these games a character’s personality does not affect her dialogue. Indeed, The Sims 3 avoids natural-language dialogue altogether, and Dwarf Fortress, with likely the richest personality modeling ever seen in a game, features stock dialogue segments that are used across all characters, regardless of speaker personality. When producing the [est]{} [@rishes2013generating] focused on a tool that could generate variations of Aesop’s Fables such as [The Fox and the Crow]{} from Drama Bank [@ElsonMcKeown10]. [@rishes2013generating] evaluated the [est]{} with a primary focus on whether the [est]{} produces [correct]{} retellings of the fable. They measure the generation produced by the [est]{} in terms of the string similarity metrics BLEU score and Levenshtein Distance to show that the new realizations are comparable to the original fable. After we add new rules to the [est]{} for handling direct speech and interlocutors, we modified the original [sig]{} representation of the [Fox and the Crow]{} to contain more dialogue in order to evaluate a broader range of character styles, along with the use of direct speech (second column of Fig. \[fc-original\]). This version is annotated using the new direct speech rules, then run through the [est]{} and [ personage]{}. Table \[pers-fig\] shows a subset of parameters, which were used in the three personality models we tested here: the [laid-back]{} model for the fox’s direct speech, the [shy]{} model for the crow’s direct speech, and the [neutral]{} model for the narrator voice. The [laid-back]{} model uses emphasizers, hedges, exclamations, and expletives, whereas the [shy]{} model uses softener hedges, stuttering, and filled pauses. The [neutral]{} model is the simplest model that does not utilize any of the extremes of the [personage]{} parameters. [Model]{} [Parameter]{} [Description]{} [Example]{} --------------- ----------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------- -- -- -- -- [Softener hedges]{} Insert syntactic elements ([sort of]{}, [kind of]{}, [somewhat]{}, [quite]{}, [around]{}, [rather]{}, [I think that]{}, [it seems that]{}, [it seems to me that]{}) to mitigate the strength of a proposition [‘It seems to me that he was hungry’]{} [Stuttering]{} Duplicate parts of a content word [‘The vine hung on the tr-trellis’]{} [Filled pauses]{} Insert syntactic elements expressing hesitancy ([I mean]{}, [err]{}, [mmhm]{}, [like]{}, [you know]{}) [‘Err... the fox jumped’]{} [Emphasizer hedges]{} Insert syntactic elements ([really]{}, [basically]{}, [actually]{}) to strengthen a proposition [‘The fox failed to get the group of grapes, alright?’]{} [Exclamation]{} Insert an exclamation mark [‘The group of grapes hung on the vine!’]{} [Expletives]{} Insert a swear word [‘The fox was damn hungry’]{} : \[pers-fig\] Examples of pragmatic marker insertion parameters from [personage]{} Single Narrator Realization Dialogic Realization ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The crow sat on the branch of the tree. The cheese was in the beak of the crow. The fox observed the crow. The fox tried he discovered he got the cheese. The fox came. The fox stood under the tree. The fox looked toward the crow. The fox said he saw the bird. The fox said the beauty of the bird was incomparable. The fox said the hue of the feather of the bird was exquisite. The fox said if the pleasantness of the voice of the bird was equal to the comeliness of the appearance of the bird the bird undoubtedly was every queen of every bird. The crow felt the fox flattered her. The crow loudly cawed in order for her to showed she was able to sang. The cheese fell. The fox snatched the cheese. The fox said the crow was able to sang. The fox said the crow needed the wits. The crow sat on the tree’s branch. The cheese was in the crow’s pecker. The crow thought “I will eat the cheese on the branch of the tree because the clarity of the sky is so-somewhat beautiful." The fox observed the crow. The fox thought “I will obtain the cheese from the crow’s nib." The fox came. The fox stood under the tree. The fox looked toward the crow. The fox avered “I see you!" The fox alleged ‘your’s beauty is quite incomparable, okay?" The fox alleged ‘your’s feather’s chromaticity is damn exquisite." The fox said “if your’s voice’s pleasantness is equal to your’s visual aspect’s loveliness you undoubtedly are every every birds’s queen!" The crow thought “the fox was so-somewhat flattering." The crow thought “I will demonstrate my’s voice." The crow loudly cawed. The cheese fell. The fox snatched the cheese. The fox said “you are somewhat able to sing, alright?" The fox alleged “you need the wits!"\ We first illustrate a monologic version of “The Fox and The Crow" as produced by the [est]{} in the first column of Table \[fc-est\]. This is our baseline realization. The second column shows the [ets]{}’s realization of the fable encoded in dialogue with the models described above. We run [personage]{} three times, one for each of our [personage]{} models ([laid-back]{}, [shy]{}, and [neutral]{}), then have a script that selects the narrator realization by default, and in the event of a direct speech instance, piece together realizations from the crow or the fox. We are currently exploring modifications to our system that allows multiple personalities to be loaded and assigned to characters so that [personage]{} is only run once and the construction be automated. Utterances are generated in real-time, allowing the underlying [personage]{} model to change at any time, for example, to reflect the mood or tone of the current situation in a game. User Perceptions {#apps} ================ Here we present a pilot study aimed at illustrating how the flexibility of the [est]{} when producing dialogic variations allows us to manipulate the perception of the story characters. We collect user perceptions of the generated dialogues via an experiment on Mechanical Turk in which the personality models used to generate the dialogic version of “The Fox and The Crow" shown in Fig. \[fc-est\] are modified, so that the fox uses the [shy]{} model and the crow uses the [laid-back]{} model. We have three conditions; participants are presented with the dialogic story told 1) only with the [ neutral]{} model; 2) with the crow with [shy]{} and the fox with [laid-back]{}; and 3) with the crow with [laid-back]{} and the fox with [shy]{}. After reading one of these tellings, we ask participants to provide adjectives in free-text describing the characters in the story. Fig.s \[cloud-crow\] and \[cloud-fox\] show word clouds for the adjectives for the crow and the fox respectively. The [shy]{} fox was not seen as very “clever" or “sneaky" whereas the [laid-back]{} and [neutral]{} fox were. However, the [shy]{} fox was described as “wise" and the [laid-back]{} and [neutral]{} were not. There are also more positive words, although of low frequency, describing the [shy]{} fox. We observe that the [laid-back]{} and [neutral]{} crow are perceived more as “naïve" than “gullible", whereas [shy]{} crow was seen more as “gullible" than “naïve". [Neutral]{} crow was seen more as “stupid" and “foolish" than the other two models. [0.3]{} ![image](cloud-crow-neutral){width="\textwidth"} [0.3]{} ![image](cloud-crow-shy){width="\textwidth"} [0.3]{} ![image](cloud-crow-friendly){width="\textwidth"} Table \[crow-eval\] shows the percentage of positive and negative descriptive words defined by the LIWC [@pennebaker2001linguistic]. We observe a difference between the use of positive words for [shy]{} crow and [laid-back]{} or [neutral]{}, with the [shy]{} crow being described with more positive words. We hypothesize that the stuttering and hesitations make the character seem more meek, helpless, and tricked rather than the [laid-back]{} model which is more boisterous and vain. However, there seems to be less variation between the fox polarity. Both the stuttering [shy]{} fox and the boisterous [laid-back]{} fox were seen equally as “cunning" and “smart". This preliminary evaluation shows that there is a perceived difference in character voices. Furthermore, it is easy to change the character models for the [est]{} to portray different characters. Crow Pos Neg Fox Pos Neg ----------- ----- ----- ----------- ----- ----- Neutral 13 29 Neutral 38 4 Shy 28 24 Shy 39 8 Laid-back 10 22 Laid-back 34 8 : \[crow-eval\] Polarity of Adjectives describing the Crow and Fox (% of total words) [0.3]{} ![image](cloud-fox-neutral){width="\textwidth"} [0.3]{} ![image](cloud-fox-shy){width="\textwidth"} [0.3]{} ![image](cloud-fox-friendly){width="\textwidth"} Conclusion ========== In this paper, we build on our previous work on the [est]{} [@rishes2013generating], and explain how it can be used to allow linguistically naïve authors to automatically generate dialogue variants of stock utterances. We describe our extensions to the [est]{} to handle direct speech and interlocutors in dialogue. We experiment with how these dialogue variants can be realized utilizing parameters for characters in dynamic open worlds. [@walker2013using] generate utterances using [personage]{} and require authors to select and edit automatically generated utterances for some scenes. A similar revision method could be applied to the output of the [est]{}. As a potential future direction, we aim to explore the potential of applying this approach to games with expansive open worlds with non-player characters (NPCs) who come from different parts of the world and have varied backgrounds, but currently all speak the same dialogue in the same way. While above we discuss how our method could be used to generate dialogue that varies according to character personality, the [est]{} could also be used to produce dialogue variants corresponding to in-game regional dialects. [personage]{} models are not restricted to the Big Five personality traits, but rather comprise values for 67 parameters, from which models for unique regional dialects could easily be sculpted. Toward this, [@walker2013using] created a story world called [Heart of Shadows]{} and populated it with characters with unique character models. They began to create their own dialect for the realm with custom hedges, but to date the full flexibility of [personage]{} and its 67 parameters has not been fully exploited. Other recent work has made great strides toward richer modeling of social-group membership for virtual characters [@harrell]. Our approach to automatically producing linguistic variation according to such models would greatly enhance the impact of this type of modeling.\ \ [Acknowledgments]{} This research was supported by NSF Creative IT program grant \#IIS-1002921, and a grant from the Nuance Foundation. Adams, T., and Adams, Z. 2006. . Bay 12 Games. Cavazza, M., and Charles, F. 2005. Dialogue generation in character-based interactive storytelling. In [AIIDE]{}, 21–26. Electronic-Arts. 2009. . Elson, D., and McKeown, K. 2010. Automatic attribution of quoted speech in literary narrative. In [Proc. of AAAI]{}. Elson, D. K. 2012. Detecting story analogies from annotations of time, action and agency. In [Proc. of the LREC 2012 Workshop on Computational Models of Narrative]{}. Fellbaum, C. 1998. Wordnet: An electronic lexical database. IT Press. Harrell, D.; Kao, D.; Lim, C.; Lipshin, J.; Sutherland, J.; and Makivic, J. 2014. The chimeria platform: An intelligent narrative system for modeling social identity-related experiences. Kipper, K.; Korhonen, A.; Ryant, N.; and Palmer, M. 2006. Extensive classifications of english verbs. Proc. of the Third Int. Conf. on Language Resources and Evaluation. Klabunde, R. 2013. Greetings generation in video role playing games. 167. Lavoie, B., and Rambow, O. 1997. A fast and portable realizer for text generation systems. In [Proc. of the fifth conference on Applied natural language processing]{}, 265–268. Lin, G. I., and Walker, M. A. 2011. All the world’s a stage: Learning character models from film. In [AIIDE]{}. Mairesse, F., and Walker, M. A. 2011. Controlling user perceptions of linguistic style: Trainable generation of personality traits. 37(3):455–488. Monfort, N.; Stayton, E.; and Campana, A. 2014. Expressing the narrator’s expectations. Pennebaker, J. W.; Francis, M. E.; and Booth, R. J. 2001. Linguistic inquiry and word count: Liwc 2001. Reed, A. A.; Samuel, B.; Sullivan, A.; Grant, R.; Grow, A.; Lazaro, J.; Mahal, J.; Kurniawan, S.; Walker, M. A.; and Wardrip-Fruin, N. 2011. A step towards the future of role-playing games: The spyfeet mobile rpg project. In [AIIDE]{}. Rishes, E.; Lukin, S. M.; Elson, D. K.; and Walker, M. A. 2013. Generating different story tellings from semantic representations of narrative. In [Int. Conf. on Interactive Digital Storytelling, ICIDS’13]{}. Rowe, J. P.; Ha, E. Y.; and Lester, J. C. 2008. Archetype-driven character dialogue generation for interactive narrative. 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--- abstract: 'Subbarrier fusion of the $^{7}Li + ^{12}$C reaction is studied using an antisymmetrized molecular dynamics model (AMD) with an after burner, GEMINI. In AMD, $^{7}Li$ shows an $\alpha + t$ structure at its ground state and it is significantly deformed. Simulations are made near the Coulomb barrier energies, i.e., E$_{cm} = 3 - 8 MeV$. The total fusion cross section of the AMD + GEMINI calculations as a function of incident energy is compared to the experimental results and both are in good agreement at E$_{cm} > 3 MeV$. The cross section for the different residue channels of the AMD + GEMINI at $E_{cm} = 5 MeV$ is also compared to the experimental results.' address: - '$^1$ Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, 730000,China.' - '$^2$ School of Nuclear Science and Technology, University of Science and Technology of China, Hefei, Anhui 230026, China.' - '$^3$ Graduate University of Chinese Academy of Sciences, Beijing, 100049, China.' - '$^4$ Department of Physics, Henan Normal University, Xinxiang, 453007, China.' author: - 'M. Huang$^1$, F. Zhou$^2$, R. Wada$^{1,}$, X. Liu$^{1,3}$, W. Lin$^{1,3}$, M. Zhao$^{1,3}$, J. Wang$^1$, Z. Chen$^1$, C. Ma$^4$, Y. Yang$^{1,3}$, Q. Wang$^1$, J. Ma$^1$, J. Han$^1$, P. Ma$^1$, S. Jin$^{1,3}$, Z. Bai$^{1,3}$, Q. Hu$^{1,3}$, L. Jin$^{1,3}$, J. Chen$^{1,3}$, Y. Su$^{1,3}$ and Y. Li$^{1,3}$' title: Dynamics aspect of subbarrier fusion reaction in light heavy ion systems --- Introduction ============ Availability of radioactive beam facilities has stimulated theoretical and experimental interest in the structure of nuclei far from the stability line. Nuclear fusion reactions near the Coulomb barrier are strongly affected by the structure of the interacting nuclei, especially with weakly bound neutrons [@Canto06]. Some theoretical calculations predict that the fusion cross section is enhanced over well-bound nuclei because of the larger spatial extent of halo nucleons [@Takigawa93]. On the other hand halo nuclei can easily break up in the field of the other nucleus, due to their low binding energies, before the two nuclei come close enough to fuse and carry away available energy. Early calculations of Hussein [et al.]{} [@Hussein92] indicate that the actual fusion cross section decreases significantly at all energies. However recent couple channel calculations of Hagino [et al.]{} [@Hagino00] have concluded that the fusion cross section increases below the Coulomb barrier because of the neutron halo whereas it decreases above the barrier because of weak coupling of the halo nucleons. Experimentally this is still a hot debate because of experimental difficulties. Another interest we propose here is the influence of the cluster structure in the fusion mechanism. The cluster structures of light nuclei have been studied theoretically from stable to those far away from the stability line. Recent calculations, using an antisymmetrized molecular dynamics model(AMD), indicate that light nuclei exhibit variety of distinct cluster structures [@Kaneda_En'yo95; @Kaneda_En'yo95_1; @Kaneda_En'yo99; @Furutachi09]. The cluster structures are predicted even for nuclei with $Z \sim N$ of Li and Be [@Kaneda_En'yo95_1] (where Z and N are the charge and neutron number in a nucleus, respectively). When nuclei with a well-developed cluster structure are involved in fusion reactions near the barrier, it will be reflected on the fusion cross section, especially in the variety of the exit channel distribution or in the fusion residue distribution. The cluster structure effect may be enhanced in the fusion reaction between light systems. In these systems the Coulomb interaction becomes small and the proximity effect between the two nuclei will be enhanced and therefore the structure of the projectile and/or target may reflect the fusion cross section directly, especially as an enhancement of particular incomplete fusion channels. In this paper we present the calculated results in the study of the fusion reactions of the $^7Li + ^{12}C$ system near the Coulomb barrier using AMD simulations. AMD simulations =============== The initial nuclei of $^7Li$ and $^{12}C$ were produced by the AMD code of Ono [et al.]{} in refs. [@Ono02], using the Gogny interaction. The binding energy and root mean square radius of these initial nuclei are compared to the experimental values in Table 1. All calculated values are in good agreement to those of the experimental values, except for the root-mean square radius of $^7Li$ in which the calculated value is about $20\%$ larger than that of the experiment. The more sophisticated calculation in ref. [@Kaneda_En'yo95_1] of the experimental root mean square radius of $^7Li$ is also well reproduced with a distinct $\alpha + t$ structure. [\[5]{}[l]{}]{} $\0\0$nucleus&AMD&AMD&$\m$Exp.&$\m$Exp.$\0\0$&Binding&rms (fm)&$\m$Binding&$\m$rms (fm)$\0\0$&energy (MeV)&&$\m$energy (MeV)&$\m$$^{7}$Li(t,$\alpha$)&40.00 &3.02&$\m$39.24&$\m$2.43$^{12}$C(3$\alpha$)&92.24 &2.53&$\m$92.16&$\m$2.47 ![\[fig1b\]$^{12}C$ initial nucleus with 3$\alpha$ structure.](Fig1_7Li_initialnuclei_XZ_t370_W.eps){width="14pc"} ![\[fig1b\]$^{12}C$ initial nucleus with 3$\alpha$ structure.](Fig1_12C_initialnuclei_XZ_t560_W.eps){width="14pc"} The density plot of these nuclei are also shown in Fig.\[fig1a\] and Fig.\[fig1b\]. Symbols indicate the location of all nucleons. One can see in both figures that nucleons are well clusterized in space. In the $^7Li$ case, two clusters are observed, the larger one corresponds to an $\alpha$ and the other to a triton, and the nucleus is very deformed. In $^{12}C$, $3 \alpha$ clusters are observed, but the nucleus is compact and much more spherical. Using these initial nuclei, $^7Li + ^{12}C$ reactions were simulated at center of mass energies between 3 to 8 MeV. Calculations were performed in the impact parameter range, b, from 0 to $7 fm$. In $ b > 7 fm$, no fusion reactions are observed. In Fig.\[fig:fig2\] the time evolution of the density distributions is shown as typical examples of the complete and incomplete fusion reactions. On the left panel, a complete fusion reaction is observed. In the middle, only the $\alpha$ particle is transferred into the $^{12}C$ nucleus and triton is escaped as a spectator. On the right panel, only triton is absorbed and the $\alpha$ particle becomes a spectator. The latter two cases are mainly observed at larger impact parameters. In each incident energy, a few thousand to ten thousand events are generated, depending on the fusion cross section, proportional to the impact parameter in the given range. ![ Time evolution of the 2D density plots for typical fusion reactions. Impact parameter, incident energy and reaction product at the bottom of the simulations are indicated on the top of each figure. The density plot is made by projecting that of all nucleons on the X-Z plane. The contour lines are plotted on a linear scale. []{data-label="fig:fig2"}](Fig2_7Li12C_density_W_ta_XZ.eps) The AMD calculations were performed up to times ranging from 3000 fm/c at lower energies to 1000 fm/c at higher energy side and clusterized at the end of the calculation, using a coalescence technique in phase space. The coalescence radius, corresponding to 5 fm in the coordinate space, is used at all energies. The Z, A, excitation energy, angular momentum and velocity vector of each cluster were calculated. Even after such a relatively long time, most clusters were in an excited state. In order to compare the simulated results to those of the experiments, the excited fragments were cooled down using the statistical decay code, GEMINI [@Charity88]. In this calculation, the C++ version of GEMINI was used. These events are referred to as the AMD + GEMINI events hereafter, whereas the events without the GEMINI calculation are called the primary events and referred to as the AMD events. The occurrence of the fusion reactions in the AMD + GEMINI events is defined here by the emission of the fragments with $Z > 6$ in a given event. In Fig.\[fig:fig3\] the calculated fusion cross sections, indicated by closed triangles, are compared to those of the experiments (open circles). The experimental data are taken from ref. [@Mukherjee96]. The experimental data are reproduced well within the experimental errors above $E_{cm} > 3 MeV$ in the absolute scale. The absolute cross sections predicted by the AMD simulations were calculated using the number of events generated in the given impact parameter range. At $E_{cm} \le 3 MeV$ the AMD simulation underestimated the fusion cross sections. In this energy range, the tunneling effect through the Coulomb barrier becomes important and in the present AMD formulation, this process is not incorporated. In the figure the formation cross sections of $^{19}F$ in the primary AMD events are also plotted by open square symbols. As discussed below, there are additional $20-30\%$ incomplete fusion contribution in the primary fusion process. ![ Fusion cross section for the $^{7}Li+^{12}C$. Circles represent experimental results and taken from [@Mukherjee96]. Squares are the primary of AMD results filtered by $Z = 9$. Secondary values of the AMD + GEMINI events filtered by $Z > 6 $ are showed as triangles. []{data-label="fig:fig3"}](Fig3_Final_CS_7Li12C_true.eps) ![ Primary major exit channel distribution at different incident energies. []{data-label="fig:fig4"}](Fig4_Show_channel_7Li12C_ta.eps) ![ Final exit channel distribution of the fusion reactions for the $^{7}Li$ + $^{12}C$. The blue histogram indicated the experimental values. The results of AMD+GEMINI are shown by green histograms. pn and pnn channels also include d and t decays. []{data-label="fig:fig5"}](Fig5_CS_exp_AMDgeminicpp_7Li12C_Ecm5_LBcorre.eps) In Fig.\[fig:fig4\], the fusion channel distribution at the primary stage is shown as a probability distribution. Only the top three major channels are plotted. The $^{19}F$ formation and $^{15}N+\alpha$ channel dominate the fusion reaction at all energies. Complete fusion occurs in about $80\%$ of the cases at the lower incident energies and decreases to about $70\%$ at the higher energies. The third channel contribution is from different reactions at different incident energies, but their probabilities are only a few $\%$ at most. In Fig.\[fig:fig5\], the final exit channel distribution of the fusion reaction is plotted from the AMD + GEMINI events and compared with the experimental results of ref. [@Mukherjee96]. The relative cross section of the major decay channels is fairly well reproduced except the $^{15}N+\alpha$ channel. The suppression of this channel in the experimental results is not yet fully understood. Further study is now underway. SUMMARY ======= The fusion cross section of the $^7Li + ^{12}C$ reaction was studied using the AMD and GEMINI codes. The AMD+GEMINI simulation reproduced the experimental total fusion cross sections reasonably well at $E_{cm} > 3 MeV$ but underestimated it below that energy. The relative experimental exit channel distribution, except the $^{15}N + \alpha$ channel, was well reproduced by the AMD+GEMINI simulation. We thank A. Ono for helpful discussions and communications. And thank him and R. Charity for letting us to use their calculation codes. One of us (R. Wada) thanks “the visiting professorship of senior international scientists” of the Chinese Academy of Sciences for the support. This work is supported by National Natural Science Foundation of China (Grant No. 11075189, 11075190 and 11005127) and Directed Program of Innovation Project of the Chinese Academy of Science (Grant No. KJCX2-YW-N44). Z.Chen thanks the “100 Person Project” of the Chinese Academy of Science. And thank the high-performance computing center of College of Physics and Information Engineering, Henan Normal University. References {#references .unnumbered} ========== [9]{} L.F. Canto [et al.]{}, Nucl. Phys. [A424]{} 1 (2006).N. Takigawa [et al.]{}, Phys. Rev. [C47]{} R2470 (1993).M.S. Hussein [et al.]{}, Phys. Rev. [C46]{}, 377 (1992); M.S. Hussein [et al.]{}, Phys. Rev. Lett. [72]{}, 2693 (1994);M.S. Hussein [et al.]{}, Nucl. Phys. [A588]{}, 85c (1995). K. Hagino [et al.]{}, Phys. Rev. [C61]{} 037602 (2000). Y. Kaneda-En’yo et al., PRC52, 628 (1995) Y. Kaneda-En’yo et al., PRC52, 647 (1995) Y. Kanada-En’yo et al., PRC60, 064304 (1999); Naoya Furutachi, Masaaki Kimura, Akinobu Dot$\acute{e}$ and Yoshiko Kanada-En’yo, Prog. Theor. Phys. [122]{}, 865 (2009). A. Ono, S. Hudan, A. Chbihi, J. D. Frankland, Phys. Rev. [C66]{}, 014603 (2002). R. J. Charity [et al.]{}, Nucl. Phys. [A483]{}, 371, 1988. A. Mukherjee [et al.]{}, Nucl. Phys. [A596]{}, (1996) 299.
--- abstract: 'Analytical expressions for the non-relativistic and relativistic Sunyaev-Zel’dovich effect (SZE) are derived by means of suitable convolution integrals. The establishment of these expressions is based on the fact that the SZE disturbed spectrum, at high frequencies, possesses the form of a Laplace transform of the single line distortion profile (structure factor). Implications of this description of the SZE related to light scattering in optically thin plasmas are discussed.' author: - \| A. Sandoval-Villalbazo$^a$ and L.S. García-Colín$^{b,\,c}$\ $^a$ Departamento de Física y Matemáticas, Universidad Iberoamericana\ Lomas de Santa Fe 01210 México D.F., México\ E-Mail: [email protected]\ $^b$ Departamento de Física, Universidad Autónoma Metropolitana\ México D.F., 09340 México\ $^c$ El Colegio Nacional, Centro Histórico 06020\ México D.F., México\ E-Mail: [email protected] title: 'A convolution integral representation of the thermal Sunyaev-Zel’dovich effect' --- Introduction ================ Distortions to the cosmic microwave background radiation (CMBR) spectrum arise from the interaction between the radiation photons and electrons present in large structures such as the hot intracluster gas now known to exist in the universe. Such distortions are only a very small effect changing the brightness of the spectrum by a figure of the order of 0.1 percent. This effect, excluding the proper motion of the cluster, is now called the thermal Sunyaev-Zel’dovich effect [@SZ1] [@SZ2] and its detection is at present a relatively feasible task due to the modern observational techniques available. Its main interest lies on the fact that it provides information to determine important cosmological parameters such as Hubble’s constant and the baryonic density [@rev]-[@Steen1]. The kinetic equation first used to describe the SZE was one derived by Kompaneets back in 1956 [@Peebles] [@Komp]-[@Weymann]. This approach implies a photon diffusion description of the effect that works basically when the electrons present in the hot intracluster gas are non-relativistic. Many authors, including Sunyaev and Zel’dovich themselves, were very reluctant in accepting a diffusive mechanism as the underlying phenomena responsible for the spectrum distortion. Later on, Rephaeli and others [@rel] computed the distorted spectrum by considering Compton scattering off relativistic electrons. Those works show that, at high electron temperatures, the distortion curves are significantly modified. Useful analytic expressions that describe the relativistic SZE can be easily found in the literature [@Sazonov-RevF]. One possible physical picture of the effect is that of an absortion-emission process in which a few photons happen to be captured by electrons in the optically thin gas. An electron moving with a given thermal velocity emits (scatters) a photon with a certain incoming frequency $\nu _{o}$ and outgoing frequency $\nu $. The line breath of this process is readily calculated from kinetic theory taking into account that the media in which the process takes place has a small optical depth directly related to the Compton parameter $y$. When the resulting expression, which will be called structure factor, is convoluted with the incoming flux of photons obtained from Planck’s distribution, one easily obtains the disturbed spectra. For the non-relativistic SZE, a Gaussian structure factor has been successfully established [@yo], but the obtention of a simple analytic relativistic structure factor is a more complicated task [@Birk]. Nevertheless, this work shows that a relativistic structure factor can be obtained with the help of the expressions derived in Refs. [@Sazonov-RevF] and by the use of simple mathematical properties of the convolution integrals that describe the physical processes mentioned above. In both cases, extensive use has been made of the intracluster gas. To present these ideas we have divided the paper as follows. In section II, for reasons of clarity we summarize the ideas leading to the SZ spectrum for a non-relativistic electron gas emphasizing the concept of structure factor. In section III we discuss the relativistic case using an appropriate equation for the structure factor whose full derivation is still pending but leads to results obtained by other authors using much more complicated methods. Section IV is left for concluding remarks. General Background ================== As it was clearly emphasized by the authors of this discovery in their early publications [@SZ1] [@SZ2] as well as by other authors, the distortion in the CMBR spectrum by the interaction of the photons with the electrons in the hot plasma filling the intergalactic space is due to the diffusion of the photons in the plasma which, when colliding with the isotropic distribution of a non-relativistic electron gas, generates a random walk. The kinetic equation used to describe this process was one first derived by Kompaneets back in 1956 [@Komp]-[@Weymann]. For the particular case of interest here, when the electron temperature $Te\sim 10^{8}K$ is much larger than that of the radiation, $T\equiv T_{Rad}\;(2.726\,\,K)$, the kinetic equation reads [@Peebles] $$\frac{dN}{dy}=\nu ^{2}\frac{d^{2}N}{d\nu ^{2}}+4\nu \frac{dN}{d\nu } \label{Uno}$$ Here $N$ is the Bose factor, $N=(e^{x}-1)^{-1}$ , $x=\frac{h\nu }{kT}$, $\nu $ is the frequency, $h$ is Planck’s constant, $k$ Boltzmann’s constant and $ y $ the “Compton parameter” given by, $$y=\frac{k\,Te}{m_{e}c^{2}}\int \sigma _{T}\,n_{e}c\,dt=\frac{k\,Te}{ m_{e}c^{2}}\tau \label{dos}$$ $m_{e}$ being the electron mass, $c$ the velocity of light, $\sigma _{T}$ the Thomson’s scattering cross section and $n_{e}$ the electron number density. The integral in Eq. (\[dos\]) is usually referred to as the “optical depth, $\tau $”, measuring essentially how far in the plasma can a photon travel before being captured (scattered) by an electron. Without stressing the important consequences of Eq. (\[Uno\]) readily available in many books and articles [@Peebles]-[@imp1], we only want to state here, for the sake of future comparison, that since the observed value of $ N_{o}(\nu )$ is almost the same as its equilibrium value $N_{eq}(\nu )$, to a first approximation one can easily show that $$\frac{\delta N}{N}\equiv \frac{N(\nu )-N_{eq}(\nu )}{N_{eq}(\nu )}=y\left[ \frac{x^{2}e^{x}(e^{x}+1)}{(e^{x}-1)^{2}}-\frac{4xe^{x}}{e^{x}-1}\right] \label{tres}$$ implying that, in the Rayleigh-Jeans region $\left( x<<1\right) $, $$\frac{\delta N}{N}\cong -2y \label{cuatro}$$ and in the Wien limit $\left( x\ >>1\right) $ : $$\frac{\delta N}{N}\cong x^{2}y \label{cinco}$$ If we now call $I_{o}(\nu )$ the corresponding radiation flux for frequency $ \nu $, defined as $$I_{o}(\nu )=\frac{c}{4\pi }U_{v}(T) \label{seis}$$ where $U_{v}(T)=\frac{8\pi \,h\nu ^{3}}{c^{3}}N_{eq}(\nu )$ is the energy density for frequency $\nu $ and temperature $T$ , and noticing that $$\frac{\delta T}{T}=\left( \frac{\partial (\ln I(\nu ))}{\partial (\ln T)} \right) \left( \frac{\delta I}{I}\right) \label{siete}$$ we have that the change in the background brightness temperature is given, in the two limits, by $$\frac{\delta T}{T}\cong -2y\;,\;x<<1 \label{ocho}$$ and $$\frac{\delta T}{T}\cong x\,y\;,\;x>>1 \label{nueve}$$ showing a decrease in the low frequency limit and an increase in the high frequency one. Finally, we remind the reader that the curves extracted from Eq. (\[tres\]) for reasonable values of the parameter $``y"$ show a very good agreement with the observational data. As mentioned before, many authors, including Sunyaev and Zel’dovich themselves, were very reluctant in accepting a diffusive mechanism as the underlying phenomena responsible for the spectrum distortion. The difficulties of using diffusion mechanisms to study the migration of photons in turbid media, specially thin media, have been thoroughly underlined in the literature [@z]-[@b]. Several alternatives were discussed in a review article in 1980 [@imp1] and a model was set forth by Sunyaev in the same year [@d] based on the idea that Compton scattering between photons and electrons induce a change in their frequency through the Doppler effect. Why this line of thought has not been pursued, or at least, not widely recognized, is hard to understand. Two years ago, one of us (ASV) [@yo] reconsidered the single scattering approach to study the SZ effect. The central idea in that paper is that in a dilute gas, the scattering law is given by what in statistical physics is known as the * dynamic structure factor, denoted by $G(k,\nu )$ where $k=\frac{2\pi }{\lambda }$, and $\lambda $ is the wavelength. In such a system, this turns out to be proportional to $\exp (-\frac{(\nu - \overline{\nu} )^{2}}{W^{2}})$ where $W$ is the broadening of the spectral line centered at frequency $\nu $ given by $$W=\frac{2}{c}(\frac{2kTe}{m})^{1/2}\nu \label{diez}$$ If one computes the distorted spectrum through the convolution integral $$I\left( \nu \right) =\int_{0}^{\infty }I_{o}\left( \bar{\nu}\right) G(k,\bar{ \nu}-(1-ay)\nu )\,d\bar{\nu} \label{once}$$ where the corresponding frequency shift $\frac{\delta \nu }{\nu }$ has been introduced through Eqs. (\[ocho\]-\[nueve\]) $(a=-2+x)$, one gets a good agreement with the observational data. This result is interesting from, at least, two facts. One, that such a simple procedure is in agreement with the diffusive picture. This poses interesting mathematical questions which will be analyzed elsewhere, specially since the exact solution to Eq. (1) is known (see Eq. (A-8) ref. [@imp1]). The other one arises from the fact that this is what triggered the idea of reanalyzing the SZ effect using elementary arguments of statistical mechanics and constitutes the core of this paper. Indeed, we remind the reader that if an atom in an ideal gas moving say with speed $u_{x}$ in the $x$ direction emits light of frequency $\nu _{o}$ at some initial speed $u_{x}(0)$, the intensity of the spectral line $I\left( \nu \right) $ is given by $$\frac{I\left( \nu \right) }{I_{o}\left( \nu \right) }=\exp \left[ -\frac{ m\,c^{2}}{2kT}\left( \frac{\nu _{o}-\nu }{\nu }\right) ^{2}\right] \label{doce}$$ Eq. (\[doce\]) follows directly from the fact the velocity distribution function in an ideal gas is Maxwellian and that $u_{x}$, $u_{x}(0)$ and $\nu $ are related through the Doppler effect. For the case of a beam of photons of intensity $I_{o}\left( \nu \right) $ incident on a hot electron gas regarded as an ideal gas in equilibrium at a temperature $T_{e}$ the full distorted spectrum may be computed from the convolution integral given by $$I\left( \nu \right) =\frac{1}{\sqrt{\pi }W\left( \nu \right) } \int_{0}^{\infty }I_{o}\left( \bar{\nu}\right) \exp \left[ -\left( \frac{ \bar{\nu}-f(y)\,\nu }{W\left( \nu \right) }\right) ^{2}\right] \,d\bar{\nu} \label{trece}$$ which defines the joint probability of finding an electron scattering a photon with incoming frequency $\bar{\nu}$and outgoing frequency $ \nu $ multiplied by the total number of incoming photons with frequency $ \bar{\nu}$. $\frac{1}{\sqrt{\pi }}W\left( \nu \right) $ is the normalizing factor of the Gaussian function for $f(y)=1$, $W\left( \nu \right) $ is the width of the spectral line at frequency $\nu $ and its squared value follows from Eq.(\[diez\]) $$W^{2}\left( \nu \right) =\frac{4kT_{e}}{m_{e}c^{2}}\tau \,\nu ^{2}=4\,y\,v^{2} \label{catorce}$$ where $\tau $ is the optical depth whose presence in Eq. (\[catorce\]) will be discussed later. The function $f(y)$ multiplying $\nu $ in Eq. (\[trece\]) is given by $f(y)=1+ay$ where $a=-2$ in the Rayleigh-Jeans limit and $a=xy$ in the Wien’s limit, according to Eqs. (8-9) and the fact that $ \frac{\Delta \nu }{\nu }=\frac{\Delta T}{T}$ for photons. Eq.(\[trece\]) is the central object of this paper so it deserves a rather detailed examination. In the first place it is worth noticing that $I_{o}(\nu )$, the incoming flux, is defined in Eq. (\[seis\]). Secondly, it is important to examine the behavior of the full distorted spectrum in both the short and high frequency limits. In the low frequency limit, the Rayleigh-Jeans limit $ \,I_{o}(\nu )$ $=\frac{2kT\nu ^{2}}{c^{2}}$, so that performing the integration with $a=-2$ and noticing that $y$ is a very small number, one arrives at the result $$\frac{\delta I}{I}\equiv \frac{I(\nu )-I_{o}(\nu )}{I_{o}(\nu )}=-2y,\;x<<1 \label{quince}$$ (\[quince\]) is in complete agreement with the value obtained using Eq.( \[tres\]), the photon diffusion equation. In the high frequency limit where $a=xy$ and $I_{o}(\nu )$ $=\frac{2h\nu ^{3}}{c^{2}}e^{-x}$, a slightly more tedious sequence of integrations leads also to a result at grips with the diffusion equation, namely $$\frac{\delta I}{I}=x^{2}y,\;x>>1 \label{dieciseis}$$ Why both asymptotic results, the ones obtained with the diffusion equation and those obtained from Eq. (\[trece\]) agree so well, still puzzles us. At this moment we will simply think of them as a mathematical coincidence. Nevertheless, it should be stressed that in his 1980 paper, Sunyaev [@d] reached rather similar conclusions although with a much more sophisticated method, and less numerical accuracy for the full distortion curves. The distorted spectrum for the CMBR radiation may be easily obtained by numerical integration of Eq. (\[trece\]) once the optical depth is fixed, $ y$ is determined through Eq.(\[dos\]) and $a=-2+x$. The intergalactic gas cloud in clusters of galaxies has an optical depth $\tau \sim 10^{-2}$ [@rel]. From the results thus obtained in the non-relativistic case, one appreciates the rather encouraging agreement between the observational data and the theoretical results obtained with the three methods, the diffusion equation, the structure factor or scattering law approach, and the Doppler effect. This, in our opinion is rather rewarding and some efforts are in progress to prove the mathematical equivalence of the three approaches. From the physical point of view, and for reasons already given by many authors, we believe that the scattering Doppler effect picture does correspond more with reality, specially for reasons that will become clear in the last section. The relativistic case ===================== Following the discussion of the previous section, Eq.(\[trece\]) may be written as: $$I\left( \nu \right) =\int_{0}^{\infty }I_{o}\left( \bar{\nu}\right) G(\bar{ \nu},\nu )\,d\bar{\nu} \label{diecisiete}$$ where the function $G(\bar{\nu},\nu )\,=\frac{1}{\sqrt{\pi }W\left( \nu \right) }\exp \left[ -\left( \frac{\bar{\nu}-f(y)\,\nu }{W\left( \nu \right) }\right) ^{2}\right] $ and $I_{o}\left( \bar{\nu}\right) $ is defined in Eq.( \[seis\]). It is now clear that in Wien’s limit, when $\frac{h\nu }{kT}>>1$ , Eq. (\[diecisiete\]) becomes the Laplace transform of $I_{o}\left( \bar{ \nu}\right) $ with a parameter $s=\frac{h}{kT}$ so that calling $I_{pW}$ the distorted spectrum in that limit, one gets that: $$I_{pW}(\nu )=\frac{2h}{c^{2}}\int_{0}^{\infty }\bar{\nu}^{3}\,G(\bar{\nu} ,\nu )\,e^{-s\bar{\nu}}d\bar{\nu} \label{dieciocho}$$ Comparing Eq. (\[dieciocho\]) with the result obtained by integrating the Kompaneets equation for the change of intensity in Wien’s limit, namely [@SZ2]-[@Peebles] $$\Delta I=y\,\frac{2k^{3}T_{o}^{3}}{h^{2}c^{2}}x^{4}\frac{e^{x}}{\left( e^{x}-1\right) ^{2}}\left[ -4+F\left( x\right) \right] \label{diecinueve}$$ where $F(x)=x\coth (\frac{x}{2})$, a straight forward inversion of the Laplace transform yields that $$G(\bar{\nu},\nu )=\delta (\bar{\nu}-\nu )-4y\frac{\nu ^{4}}{\bar{\nu}^{3}} \delta ^{\prime }(\bar{\nu}-\nu )+y\frac{\nu ^{5}}{\bar{\nu}^{3}}\delta ^{^{\prime \prime }}(\bar{\nu}-\nu ) \label{veinte}$$ In this equation, the primes denote derivatives with respect to $\nu $ . It may be thus regarded as a representation of $G(\bar{\nu},\nu )$ in terms of the Delta function an its derivatives. To illustrate this point we can show that the RJ and Wien’s limits of integral (\[once\]) arise by substituting $G(\bar{\nu},\nu )$ by a shifted delta function (see appendix). In the relativistic case, the form for $G(\bar{\nu},\nu )$ may also be written down taking the relativistic form for the particle’s energy $E=\frac{ m_{o}\,c^{2}}{\sqrt{1-\frac{u^{2}}{c^{2}}}}$ and performing a power series expansion in powers of $\frac{u^{2}}{c^{2}}$. Defining $z=\frac{k\,T_{e}}{ m_{e}\,c^{2}}$, this leads to an integral for $I(\nu )$ which reads as follows: $$I(\nu )=\frac{1}{\sqrt{\pi }W\left( \nu \right) }\int_{0}^{\infty }I_{o}\left( \bar{\nu}\right) \exp \left[ -\frac{\sqrt{1-\frac{\left[ \bar{ \nu}-f(y)\,\nu \right] ^{2}}{2\tau y}}-1}{z}\right] \,d\bar{\nu} \label{veintiuno}$$ Nevertheless, its integration even in the two limits Wien’s and Rayleigh-Jeans has been unsuccesful. However, by direct numerical integration one obtains curves that agree qualitatively well with those obtained by other methods. Thus, in order to verify the generosity of the delta function representation we proceeded in an indirect way. It has been shown in the literature that a good representation of the relativistic SZE up to second order in $y$ reads as [@Sazonov-RevF]: $$\begin{array}{l} \Delta I=y\,\frac{2k^{3}T_{o}^{3}}{h^{2}c^{2}}x^{4}\frac{e^{x}}{\left( e^{x}-1\right) ^{2}}[-4+F\left( x\right) +\frac{y^{2}}{\tau }(-10+\frac{47}{2 }F(x) \\ -\frac{47}{5}F^{2}(x)+\frac{7}{10}F^{3}(x)-\frac{21}{5}H^{2}(x)+\frac{7}{5} F(x)H^{2}(x))] \end{array} \label{veintidos}$$ Here, $H(x)=x\left[ \sinh (x/2)\right] ^{-1}$. Taking the inverse Laplace transform in Wien’s limit of Eq. (\[veintidos\]) one obtains that $$\begin{array}{l} G_{R}(\bar{\nu},\nu )=\delta (\bar{\nu}-\nu )-\left( 4y+\frac{10y^{2}}{\tau } \right) \frac{\nu ^{4}}{\bar{\nu}^{3}}\delta ^{\prime }(\bar{\nu}-\nu )+ \\ \left( y+\frac{47y^{2}}{2\tau }\right) \frac{\nu ^{5}}{\bar{\nu}^{3}}\delta ^{^{\prime \prime }}(\bar{\nu}-\nu )-\frac{42\,y^{2}}{5\tau }\frac{\nu ^{6}}{ \bar{\nu}^{3}}\delta ^{^{\prime \prime \prime }}(\bar{\nu}-\nu )+\frac{ 7\,y^{2}}{10\tau }\frac{\nu ^{7}}{\bar{\nu}^{3}}\delta ^{^{(IV)}}(\bar{\nu} -\nu ) \end{array} \label{veintitres}$$ There is of course the remaining problem, namely to obtain Eq. (\[veintidos\]) from the convolution integral (\[veintiuno\]). This has so far defied our skills, but is purely a mathematical problem. The main point we want to underline here is the possibility of using expressions such as Eqs. ( \[veinte\]) and (\[veintitres\]), in principle obtainable from convolution integrals with structure factors which provide a useful analytical tool that avoids resorting too complicated and laborious methods. In fact, Fig. 1 shows CMBR distortion comparison curves obtained using Eqs. ( \[diecinueve\]) and (\[diecisiete\]) with (\[veinte\]), for several non-relativistic clusters. Fig. 2 shows the corresponding comparison relativistic curves at higher, realistic, electron temperatures using Eqs. ( \[veintidos\]) and (\[diecisiete\]) with (\[veintitres\]). It is clear that, for any practical purpose, the curves are identical. Indeed, one is not able to identify two different plots in each figure. =3.4in =2.6in =3.4in =2.6in Discussion of the results ========================= It is known that, for the case of a beam of photons of intensity $ I_{o}\left( \nu \right) $ incident on a hot electron gas regarded as an ideal gas in equilibrium at a temperature $T_{e}$, the full non-relativistic SZE distorted spectrum may be computed from the convolution integral given in Eq. (\[trece\]), which defines the joint probability of finding an electron scattering a photon with incoming frequency $\bar{\nu}$and outgoing frequency $\nu $ multiplied by the total number of incoming photons with frequency $\bar{\nu}$. $1-2y$ is a function that corrects the outgoing photon frequency due to the thermal effect [@imp1]. $\frac{1}{ \sqrt{\pi }}W\left( \nu \right) $ is the normalizing factor of the Gaussian, $W\left( \nu \right) $ is the width of the spectral line at frequency $\nu $ and its squared value is defined in Eq. (\[catorce\]). Thus, it is clear that an accurate convolution integral between the undistorted Planckian and a regular function* exists and can be obtained from strictly physical arguments in the non-relativistic SZE. It is interesting, however, that the corresponding relativistic expression does not seem to be a simple physical generalization of Eq. (\[trece\]). Yet, Eqs. (\[diecisiete\],\[veintitres\]) give the correct mathematical description of the relativistic distortion by means of Dirac delta functions and its derivatives. Two questions can now be posed. One, of a rather mathematical fashion, states under what conditions a structure factor such as that appearing in Eq. (\[diecisiete\]) can be written in a representation involving Dirac’s delta functions and its derivatives. This subject seems to be related to the mathematical theory of distributions. The second one, of more physical type, concerns with the possible description of the thermal SZE as a light scattering problem in which CMBR photons interact with an electron gas in a thermodynamical limit that allows simple expressions for the structure factors. If this were the case, the thermal relativistic SZE physics would not need of Montecarlo simulations or semi-analytic methods in its convolution integral description. Emphasis should be made on the fact that the main objective pursued in this work is to enhance the physical aspects of the SZE by avoiding complicated numerical techniques in many cases. This is achieved, as shown, by resorting the concept of structure factors, or scattering laws, widely used in statistical physics. APPENDIX The starting point here is Eq.(\[diecisiete\]). In the RJ limit, $I_{oRJ}\,\left( \bar{\nu}\right) =\frac{2kT}{c^{2}}\bar{\nu}^{2}$. Introducing $G(\bar{\nu},\nu )=\delta (\bar{\nu}-\nu (1-ay))$ one obtains $$I_{RJ}(\nu )=\frac{2kT}{c^{2}}\int_{0}^{\infty }\bar{\nu}^{2}\delta (\bar{\nu }-\nu (1-ay))d\bar{\nu}=\frac{2kT}{c^{2}}\nu ^{2}(1-ay)^{2}$$ keeping terms linear in $y$ we find that $$\frac{I_{RJ}(\nu )-I_{oRJ}\,\left( \nu \right) }{I_{oRJ}\,\left( \nu \right) }=-2ay$$ This result is consistent with Eq. (8) setting $a=1$. Now, in the Wien limit, $I_{oW}\,\left( \bar{\nu}\right) =\frac{2h}{c^{2}} \bar{\nu}^{3}e^{-\frac{h\bar{\nu}}{kT}}$, in this case the introduction of $ \delta (\bar{\nu}-\nu (1-ay))$ leads to $$I_{W}(\nu )=\frac{2h}{c^{2}}\int_{0}^{\infty }\bar{\nu}^{3}e^{-\frac{h\bar{ \nu}}{kT}}\delta (\bar{\nu}-\nu (1-ay))d\bar{\nu}=\frac{2h}{c^{2}}\nu ^{3}(1-ay)^{3}e^{-\frac{h\nu }{kT}}e^{\frac{h\nu }{kT}ay}$$Expanding $e^{-\frac{h}{kT}ay}$ and keeping terms up to first order in $y$ we obtain that$$\frac{I_{W}(\nu )-I_{oW}\,\left( \nu \right) }{I_{oW}\,\left( \nu \right) } =-6ay+axy$$ Finally, setting $a=1$ and considering $x>>6$ in the Wien limit, we obtain the desired result, Eq. (\[nueve\]). This work has been supported by CONACyT (Mexico), project 41081-F. [1]{} Y.B. Zel’dovich and R.A. Sunyaev, Astrophys. Space Sci. 4, 301 (1969). R.A. Sunyaev and Ya. B. Zel’dovich. Comm. Astrophys. Space Phys. 4, 173 (1972). P.J.E. Peebles, Physical Cosmology (Princeton Univ. Press, Princeton, 2nd. Ed. 1993). MM\. Joy and J.E. Carlstrom; Science 291, 1715 (2001) and ref. therein. R.A. Sunyaev and Ya. B. Zel’dovich; Ann. Rev. Astrom. Astrophys. 18, 537 (1980) S.H. Hansen, S. Pastor and D.V. Semikoz, Astrophys. J. 573, (2002) L69 \[astro-ph/0205295\] A.S. Kompaneets; JETP. 4, 730 (1957). R. Weymann; Phys. Fluids 8, 2112 (1965) Y. Rephaeli, Astrophys. J. 445, 33 (1995);A.D. Challinor and A.N. Lasenby, Astrophys. J. 499, 1 (1998); N. Itoh, Y. Kohyama, and S. Nozawa, Astrophys. J. 502, 7 (1998); S. Nozawa, N. Itoh, and Y. Kohyama, Astrophys. J. 507, 530 (1998); G. Rybicky, (2002) astro-ph/0208542 A.D. Dolgov, S.H. Hansen, D.V. Semikoz and S. Pastor, Astrophys. J. 554 74 (2001) \[astro-ph/0010412\]. S.Sazonov and R.A Sunyaev (1998) astro-ph/9804125 ;E.S. Battistelli, M. DePetris, L. Lamagna, F. Melchiorri, E. Palladino, G. Savini, A. Cooray, A.Melchiorri, Y. Rephaeli, M. Shimon (2002) astro-ph/0208027 and refrences therein. Submitted to ApJL. A. Sandoval-Villalbazo Physica A 313 456-462 (2002) A. Sandoval-Villalbazo and L.S. García-Colín (2002) \[astro-ph/0207218\]. S.M. Molnar and M. Birkinshaw (1999) \[astro-ph/9903444\]; G. H. Weiss; Physica A 311, 381 (2002). D.J. Durian and J. Durnick, J. Opt. Soc. Am. A14 235 (1997). A.H. Gandibache and G.H. Weiss in Progress in Optics XXXIV; E. Wolf, Ed. (Elsevier, Amsterdam 1995) p. 332. R.A. Sunyaev; Sov. Astron. Lett. 6, 212 (1980).
--- abstract: 'In many practical wireless systems, the Signal-to-Interference-and-Noise Ratio (SINR) that is applicable to a certain transmission, referred to as Channel State Information (CSI), can only be learned after the transmission has taken place and is thereby outdated (delayed). For example, this occurs under intermittent interference. We devise the backward retransmission (BRQ) scheme, which uses the delayed CSIT to send the optimal amount of incremental redundancy (IR). BRQ uses fixed-length packets, fixed-rate $R$ transmission codebook, and operates as Markov block coding, where the correlation between the adjacent packets depends on the amount of IR parity bits. When the delayed CSIT is full and $R$ grows asymptotically, the average throughput of BRQ becomes equal to the value achieved with prior CSIT and a fixed-power transmitter; however, at the expense of increased delay. The second contribution is a method for employing BRQ when a limited number of feedback bits is available to report the delayed CSIT. The main novelty is the idea to assemble multiple feedback opportunities and report multiple SINRs through vector quantization. This challenges the conventional wisdom in ARQ protocols where feedback bits are used to only quantize the CSIT of the immediate previous transmission.' author: - '\' title: 'Delayed Channel State Information: Incremental Redundancy with Backtrack Retransmission' --- Introduction ============ Motivation ---------- Channel State Information at the Transmitter (CSIT) is essential for achieving high spectral efficiency in wireless systems. Ideally, CSIT should be known before the transmission has started, as in that case the transmitter can optimize the parameters of the transmission, such as the power used or the precoding that is applied in case of Multiple Input Multiple Output (MIMO) transmission. In Frequency Division Duplex (FDD) systems, provision of CSIT necessarily happens through a feedback from the receiver. On the other hand, Time Division Duplex (TDD) systems can utilize channel reciprocity and the transmitter can obtain CSIT by measuring a signal received in the opposite direction. Reciprocity loses its utility when there is interference. Consider the scenario on Fig. \[fig:AtoBinterfC\], where $A$ is the transmitter, $B$ is the receiver and $C$ is an interferer that affects $B$. By assuming that $A$ knows the noise power at $B$, channel reciprocity enables $A$ to estimate the Signal-to-Noise Ratio (SNR), denoted $\gamma_{AB}$, at which $B$ receives the signal of $A$. When the interferer $C$ is active, the quantity that is relevant for the transmitter is the Signal-to-Interference-and-Noise Ratio (SINR) and $A$ has no way of knowing it unless $B$ explicitly reports either the Interference-to-Noise Ratio (INR) $\gamma_{CB}$ or the actual SINR. Hence, the notion of CSIT is broader and incorporates knowledge about the exact conditions at which the signal has been received, including the interference. In many practical scenarios, the interference is intermittent and $B$ cannot know whether/which interferer will be active when the actual transmission of $A$ takes place. Furthermore, even if there is no interference and the channel is TDD, there could not have been a reliable way to measure the SNR $\gamma_{AB}$ before the transmission, for example due to the scheduling structure in the system or fast channel dynamics. On the other hand, after $A$ has completed its transmission, $B$ can send feedback to $A$ about what the actual received SNR or SINR had been during that transmission. We refer to this as a posterior CSIT or delayed CSIT, as opposed to prior CSIT, known before the actual transmission. ![An example of communication scenario in which it is not possible to know the conditions at the receiver $B$ before the transmission from $A$ takes place use to intermittent interference from $C$.[]{data-label="fig:AtoBinterfC"}](./images/AtoBinterfC.pdf) The performance of the communication link that operates with posterior CSIT is certainly inferior compared to the link that operates with prior CSIT, as there are certain parameters that cannot be optimally adapted if CSIT is not known before the transmission. If CSIT is not available before the transmission, then the power that $A$ uses for transmission cannot follow the water filling principle and thus leads to irrecoverable loss, since at the time of getting CSIT, the power has already been used. In other words, if the channel gain is below the threshold, then the power is irreversibly lost to a bad channel state. Similarly, the lack of prior CSIT in MIMO transmission prevents the sender to have optimal spatial focusing and power is irreversibly lost in spatial dimensions that are not optimal. As the last example, multiuser diversity is not recoverable when the posterior CSIT is available, because prior CSIT is used to select the user with the best channel conditions and transmit data to her. When the user is selected blindly, the power may be irreversibly lost to a “bad” user. In this paper we introduce backtrack retransmission (BRQ), capable to use the delayed CSIT in a way that approaches the average throughput achieved with prior CSIT and fixed transmission power. A prominent feature of BRQ is that it adapts the data rate using a fixed rate$-R$ codebook at the physical layer and fixed packet length, but only changing the amount of parity bits based on the delayed CSIT. Thus, BRQ operates as Markov block coding, where the correlation between the adjacent packets depends on the amount of IR parity bits. By selecting asymptotically high rate $R$ and when the delayed CSIT is full, the average throughput becomes equal to the one with prior CSIT. However, this happens at the expense of asymptotically increased delay. The extension of BRQ to the case with a finite number of feedback bits gives rise to a novel transmission scheme: instead of sending feedback after each packet, the feedback bits from multiple slots are assembled and, using vector quantization, provide information relevant to a number of packets transmitted in the past. This challenges the conventional wisdom in ARQ protocols, in which the content of the feedback bits is only associated with the transmission immediately preceding those bits. The price paid by the BRQ is in the increased delay. Related Work ------------ The problem of HARQ over block fading channels has been addressed in many works, but two recent works stand out in their relation to the current work, [@Tuninetti11] and [@Szczecinski13], respectively. The scenario considered in [@Tuninetti11] is essentially different from our scenario of interest, as the CSIT in is not delayed, but it arrives through a finite number of bits as a prior CSIT, such that the transmitter can adapt the power. The reference [@Tuninetti11] provides good insights on the role of the feedback as a CSIT quantizer, but the feedback bits are associated only with the immediately preceding transmission and it is not allowed to accumulate feedback bits over consecutive slots, as we do in the quantized scheme in Section \[sec:Quant\_BRQ\]. The scenario considered in [@Szczecinski13] is directly related to the one treated in this paper, as the authors assume that at the time of transmission, the received CSIT is not correlated with the actual conditions on the channel. The IR bits in [@Szczecinski13] are sent in a standalone way, using variable-length packets — the authors of [@Szczecinski13] correctly point out that this could be a problem in multiuser systems and therefore they propose a workaround by grouping multiple transmissions into fixed-length resources. On the other hand, our approach works with fixed-length packets/slots at the physical layer, such that it naturally follows the modus operandi of the contemporary wireless systems, such as e. g. LTE [@LTE]. The method in [@Szczecinski13] uses rate adaptation at the physical layer, while in our case the physical layer parameters are fixed and a single combination of coding/modulation of rate $R$ is applied. Regarding the implementation of IR, both [@Tuninetti11] and [@Szczecinski13] describe it implicitly, through sufficient accumulated mutual information, while our scheme is based on explicit method for creating the IR bits through random binning. The approach in [@Szczecinski13] gives rise to a complex optimization problem for determining the IR bits and the rate, while in our method this is straightforwardly based on the mechanism of random binning. Finally, we extend our method with delayed CSIT to the case with finite-bit feedback, while [@Szczecinski13] treats only the case of full delayed CSIT. We remark here that the issue of delayed CSIT has recently sparkled a significant interest in terms of improvements that can be achieved in terms of Degrees of Freedom at high SNRs [@Tse12] [@Tandon12]. However, that line of work does not deal with HARQ protocols and is directed towards multi-user scenarios, while in our case we consider relatively low SNR and a single link. System Model and Illustrative Examples ====================================== System Model ------------ We consider a single-user channel with block fading and Gaussian noise. From this point on we use the term SNR, as we will not explicitly consider interference and attribute the channel variation to fading, but we keep in mind that the concepts are applicable to SINR. TX transmits to RX in slots, each slot contains a full packet and takes $N$ channel uses. The value of $N$ is fixed, unless stated otherwise, and sufficiently large to offer reliable communication that is optimal in an information-theoretic sense. The complex $N-$dimensional received signal vector in slot $t$ is: $$\label{eq:ReceivedSignal} \mathbf{y}_t=\sqrt{\gamma_t} \mathbf{x}_t+\mathbf{z}_t$$ where $\mathbf{x}_t$ is the transmitted signal normalized to have unit power and $\mathbf{z}_t$ is a complex random vector with unit variance that represents the contribution of the noise and the interference in slot $t$. Hence, the SNR in the $t-$th slot is equal to $\gamma_t$, drawn independently from a probability density $p_{\Gamma}(\cdot)$ and is unknown to TX prior to the transmission. We note that (\[eq:ReceivedSignal\]) assumes a fixed transmission power, which is reasonable considering the fact that we always assume that $\gamma_t$ is unknown when the transmission takes place; more elaboration will follow. TX uses a single code/modulation combination of rate $R$ and thus sends a total of $b=NR$ data bits in the packet. The maximal rate that the channel TX-RX can support in slot $t$ is: $$\label{eq:Cgamma} C(\gamma_t)=\log_2(1+\gamma_t) \qquad \textrm{[bits/c. u.]}$$ If $R \leq C(\gamma)$ then RX receives the packet correctly, otherwise an outage occurs. Upon outage, RX buffers the received signal in order to use it in future decoding attempts. An efficient way to treat the outage is to use incremental redundancy (IR) [@Tuninetti11]: TX sends additional $r<b$ parity bits to RX, which RX can combine with the signal received during the previous $N$ channel uses and thus possibly recover the original packet. If TX uses $M$ channel uses to transmits the $r$ additional bits and eventually the packet is decoded correctly, then the data rate achieved between TX and RX is $R_r=\frac{N}{N+M}R$. Note that in general $M \neq N$. An optimal incremental redundancy would select the retransmitted bits in such a way that if the accumulated mutual information at the receiver becomes sufficient, the signal is decoded [@Tuninetti11] [@Szczecinski13]. The interesting question is how is $R_r$ related to the channel capacity $C(\gamma)$? Two Examples of Incremental Redundancy -------------------------------------- For the first example, TX sends the packet using a slot with $N$ channel uses and learns the SNR $\gamma$ after the slot. The value of $\gamma$ is such that $R>C(\gamma)$ and an outage occurs. For this example we deviate from our model in two ways: (1) we temporarily break away from the fixed slot structure and assume that the retransmission from TX can take $M$ channel uses, where $M \neq N$ (as in [@Szczecinski13]); (2) the SNR during those $M$ channel uses is constant and equal to $\gamma$, such that TX can perfectly choose the transmission rate used to send the IR bits without experiencing outage. By knowing $\gamma$, TX knows that RX has a side information about the transmitted packet that amounts to $NC(\gamma)<NR=b$ bits, such that the minimal amount of parity bits that $A$ should provide to $B$ in order to decode the packet is: $$\label{eq:r_bits} r=N(R-C(\gamma))$$ The operational interpretation of the minimal number of parity bits bears resemblance to random binning, introduced in the Slepian-Wolf problem of distributed source coding [@Kramer07]. In the Slepian-Wolf problem, the amount of information that one node sends depends on the knowledge of the side information that the other node provides to the receiver. In our setting with a retransmission, TX can act as two different senders separated in time: after the first transmission, TX learns from the CSIT the amount of side information available at RX, adjusts the amount of parity bits and sends them to RX. Specifically, TX divides the $2^b$ messages into $2^r$ bins, where the number of bins is adjusted to the received CSIT, and X sends to RX $r$ bits to describe to which bin the previous message belongs to. If RX decodes the $r$ bits correctly, it combines the bin information with the side information present at RX from the first $N$ channel uses and, RX can decode the original $b$ bits correctly almost surely as $N \rightarrow \infty$. The rate at which TX sends the redundancy bits is $R^{\prime}=C(\gamma)$, such that number of channel uses for retransmission is: $$M=\frac{r}{R^{\prime}}=\frac{N(R-C(\gamma))}{C(\gamma)}$$ and the equivalent rate achieved from TX to RX is: $$R_{TX-RX}=\frac{b}{N+M}=\frac{NR}{N+\frac{N(R-C(\gamma))}{C(\gamma)}}=C(\gamma)$$ which is the same as if TX knew the CSIT a priori. We can conclude that, if the channel is constant during the original rate-$R$ transmission and the incremental redundancy transmission, but TX can only learn the CSIT after the first transmission, the achievable rate between TX and RX is: $$R_{AB}=\min \{R, C(\gamma) \}$$ Besides the inability to adapt the power, the penalty for not knowing the prior CSIT can occur due to a low value of $R$, i. e. TX cannot take advantage of the very high SNRs. ![An example of backtrack decoding with posterior CSIT. In the $i-$th slot, $\mathbf{d}_i$ are the new bits and $\mathbf{r}_{i-1}$ are the redundancy bits, used as a bin index to decode the transmission in slot $i-1$.[]{data-label="fig:Backtracking3slots"}](./images/Backtracking3slots.pdf) But a careful reader can object to the consistency of the previous example. First, to send the IR bits, TX uses a prior CSIT, such that the example cannot cover e. g. a scenario with intermittent interferer. Knowing CSIT in advance also opens the possibility for employing power control, as in [@Tuninetti11]. On the other hand, we persist to the assumption that the channel must be unknown at the time of transmission and ask whether we can still recover the rate. Furthermore, the example violates the required fixed-slot structure and a single coding/modulation combination with a fixed rate. Therefore, as a second example assume that TX sends to TX in slots with fixed size of $N$ channel uses. The SNR from slot to slot is uncorrelated and available to TX only after the transmission. Let TX always apply the same transmission rate $R$ and let us observe three consecutive slots with SNRs equal to $\gamma_1, \gamma_2, \gamma_3$. This is illustrated on Fig. \[fig:Backtracking3slots\]. We deliberately select the SNRs to satisfy $$C(\gamma_1), C(\gamma_2) <R \qquad C(\gamma_3)>R$$ such that outage occurs in the first two slots and, eventually, the transmission of TX can be decoded in the third slot. TX transmits $b=NR$ new bits in the first slot, denoted by $\mathbf{d}_1$. For the second slot, TX knows $\gamma_1$ and sends again $b=NR$ data bits, but the message is prepared in the following way: - The first $r_1=N(R-C(\gamma_1))$ bits are parity bits, denoted by $\mathbf{r}_1$, used to describe the bin index that can be combined with the signal from slot 1 to recover the bits $\mathbf{d}_1$. - The remaining $b-r_1=NC(\gamma_1)$ bits, denoted by $\mathbf{d}_2$ are new bits transmitted in the second slot. It must be noted that the parity bits and the new bits are only separable in a digital domain, but not at the physical layer i. e. the whole packet, sent at rate $R$, needs to be decoded correctly and then RX extracts the parity bits and the new bits. The insertion of parity bits creates dependency between two adjacent packet transmissions, such that the scheme effectively applies Markov block coding. Coming back to the example, outage also occurs in the second slot, such that the message in the third slot consists of $r_2=N(R-C(\gamma_2))$ redundancy bits $\mathbf{r}_2$ and $d_3=NC(\gamma_2)$ new bits $\mathbf{d}_3$. Note that, according to the illustration on Fig. \[fig:Backtracking3slots\], in the example $r_1<r_2$, which means that $\gamma_2<\gamma_1$. As $C(\gamma_3)>R$, RX decodes the message slot 3 and recovers $\mathbf{d}_3$ and $\mathbf{r}_2$. It then uses $\mathbf{r}_2$ as incremental redundancy to decode the transmission in slot 2 and recover $\mathbf{d_2}$ and $\mathbf{r}_1$. Finally, RX uses $\mathbf{r}_1$ to decode the transmission from slot 1 and recover $\mathbf{d}_1$. The average rate is $\bar{R}=\frac{b_1+b_2+b_3}{3N}$, which results in $$\bar{R}=\frac{R+C(\gamma_1)+C(\gamma_2)}{3}$$ The average rate that could have been achieved over the three slots if the CSIT were known a priori is: $$\bar{R}_{\mathrm{prior}}=\frac{C(\gamma_3)+C(\gamma_1)+C(\gamma_2)}{3}$$ Again, a loss compared to prior CSIT can occur due to low $R$, but otherwise the data rates that are achievable with prior CSIT can be recovered with delayed CSIT. The second example illustrates the central proposal in this paper, IR with backtrack retransmission (BRQ). We note that power adaptation cannot help under the assumption that the SNR $\gamma_t$ is unknown and uncorrelated with $\gamma_{t-1}$ at the time of transmission and here is a sketch of the argument to support this. Referring to Fig. \[fig:Backtracking3slots\], it should be noted that RX does not have any side information about the bits $\mathbf{r}_1$ and $\mathbf{r}_2$, since they represent the minimal amount of information that needs to be retransmitted. Therefore, the information carried in $\mathbf{r}_1$ and $\mathbf{r}_2$ is new in the same sense that the information in $\mathbf{d}_1,\mathbf{d}_2, \mathbf{d}_3$ is new. Hence, each transmitted packet carries equal amount of $NR$ new information bits and, since the channel is not known in advance and for symmetry reasons, the transmission power should be equal in each slot. Backtrack Retransmission (BRQ) with Full CSIT ============================================= Here we specify the BRQ protocol with full CSIT. TX sends to RX using a codebook of rate $R$ and the minimal SNR required to decode it is: $$\gamma_R=C^{-1}(R)=2^R-1$$ We pick the system at slot $t$ that has SNR of $\gamma_t$, unknown to TX prior to the transmission. At the end of the slot $t-1$, RX sends the value $\gamma_{t-1}$ to the TX, such that when slot $t$ starts, both TX and RX know $\gamma_{t-1}, \gamma_{t-2}, \ldots$ In addition, RX knows $\gamma_t$, as we assume coherent reception. It is assumed that the last slot in which the receiver has successfully decoded the packet is $t-L$, where $L>1$ such that $\gamma_{t-L} \geq \gamma_R$ and $\gamma_{t-l} <\gamma_R$ for $l=2, 3, \ldots L-1$. Operation in slot $t$: - Transmitter side: 1. Receive the value of $\gamma_{t-1}$. 2. If $\gamma_{t-1} \geq \gamma_R$, fetch $NR$ new bits and transmit them at rate $R$. 3. Else $\gamma_{t-1} < \gamma_R$ and create the bin, consisting of $N(R-C(\gamma_{t-1}))$ parity bits and fetch $NC(\gamma_{t-1})$ new bits. Concatenate the parity and the new bits into a $NR-$bit packet and send at rate $R$. - Receiver side: 1. If $\gamma_{t-1} < \gamma_R$, store the received signal $\mathbf{y}_t$ in memory for later use. 2. Else, decode the packet, set $l=0$ and while $l<L$ do the following: 1. From the packet decoded in slot $(t-l)$ extract the parity bits and the new bits. 2. Use the parity bits from slot $(t-l)$ and the stored $\mathbf{y}_{t-l-1}$ to successfully decode the transmission of TX from slot $(t-l-1)$; 3. Set $l=l+1$. 3. Send $\gamma_{t}$ to TX. Note that whenever TX receives feedback $\gamma_{t-1}>\gamma_R$, it is treated as an ACK. In the BRQ protocol, the decoder buffers the received signals until a slot with $\gamma_{t-1}>\gamma_R$ and then decodes all stored transmissions. The stochastic behavior of the protocol can be described by a renewal reward process [@Tuninetti11], in which a renewal occurs in a slot with $\gamma_{t-1}>\gamma_R$. The reward is calculated as the total number of information bits, excluding the parity bits, that RX decodes in the slot at which a renewal occurs. Let $\gamma_{t-L}, \gamma_t \geq \gamma_R$ with $L>0$ and $\gamma_{t-l}<\gamma_R$ for $0<l<L$. Then the reward in slot $t$ is: $$\label{eq:RewardBits} \rho_t=N \left(R+\sum_{l=1}^{L-1} C(\gamma_{t-l}) \right)$$ where $N$ is the number of channel uses per slot. Since $\gamma_{t-l}<\gamma_R$ for $0<l<L$, from step TX-3 of the protocol it follows that the packets transmitted in slot $(t-j)$ for $0 \leq j <L-1$ will contain parity bits and new bits. The number of parity bits sent in slot $(t-j)$ is $N[R-C(\gamma_{t-j-1})]$, such that the number of new bits in slot $(t-j)$ is $NC(\gamma_{t-j-1})$. Summing up the new bits in the slots $t-L+1, t-L+2, \ldots, t$ results in (\[eq:RewardBits\]). Let TX transmit data to RX using a fixed rate $R$, with full CSIT available a posteriori and using BRQ. The SNR in each slot is drawn independently from $p_{\Gamma}(\gamma)$. Then the average rate is: $$\label{eq:AverageRateFullCSIT} \bar{R}=\int_{0}^{\gamma_R} p_{\Gamma}(\gamma) C(\gamma) \mathrm{d}\gamma + R \int_{\gamma_R}^{\infty} p_{\Gamma}(\gamma) \mathrm{d}\gamma$$ Let us consider another system, termed $R-$limited protocol, in which TX knows $\gamma_t$ at the start of the slot $t$ and uses a fixed transmission power. The $R-$limited protocol adapts the rate in the following way: $$R_t= \max \{ C(\gamma_t), R \}$$ i. e. it uses the instantaneous channel rate if $\gamma_t<\gamma_R$ and otherwise rate $R$. The average rate of the $R-$limited protocol is straightforwardly given by (\[eq:AverageRateFullCSIT\]). We observe the performance of the $R-$limited protocol and BRQ with full CSI on the same set of $L+1$ consecutive slots. The SNRs are selected as $\gamma_{t-L}, \gamma_t \geq \gamma_R$ and $\gamma_{t-l}<\gamma_R$ for all $l=1 \ldots L-1$ i. e. these are slots between two successful decode events for the BRQ protocol. From Lemma 1 it follows that the reward collected for slots $t-l$, with $l=0 \ldots L-1$ is identical for BRQ and for the $R-$limited protocol. Since this is valid for any set of SNRs between two decode events in BRQ, the average rate of the BRQ protocol and the $R-$limited protocol are identical, which proves the theorem. An interesting feature of BRQ is that it adapts the rate by using a single transmission codebook, while the $R-$limited protocol needs to apply a different codebook for different $\gamma$. The adaptation in BRQ is done in digital domain, through the adaptive number of parity bits. Therefore, between two renewals, BRQ operates as Markov block coding in which the statistical dependence between the packets in slots $l$ and $l+1$ depends on the posterior CSIT received for slot $l$. However, BRQ is able to recover the rate achievable by the prior CSIT at the expense of increased delay. We define the delay of a bit that is transmitted as a new bit in slot $t$ and decoded in slot $t+L$ to be $L$ slots. We do not define delay for a parity bit. For example, in slot $t$ TX sends $NC(\gamma_{t-1})$ new bits and $N[R-C(\gamma_{t-1})]$ parity bits. If $\gamma_{t+l} < \gamma_R$ for $0 \leq l <L$ and $\gamma_L \geq \gamma_R$, then the delay for the new bits sent in slot $t$ is $L$. On the other hand, the delay for all the bits sent in the $R-$limited adaptation protocol is zero. In order to calculate the average delay, let us define: $$p_R=\int_{\gamma_R}^{\infty} p_{\Gamma}(\gamma) \mathrm{d}\gamma$$ The time that a given bit spends in the system has a geometric distribution and the average delay is given by: $$\bar{\tau}=\frac{p_R}{1-p_R} \qquad \textrm{[slots]}$$ and increases with $R$. BRQ with Quantized CSIT {#sec:Quant_BRQ} ======================= In this section we investigate how to use the idea of backtrack retransmission when a finite number of $F$ bits are available after each slot. We devise a strategy that sacrifices the delay performance in order to efficiently use the feedback bits and enable backtrack retransmissions. Note that if the $F$ bits that follow the $t-$th slot are used to report a quantized value of $\gamma_t$, then this is a scalar quantization. Our approach is to assemble $LF$ bits and jointly report the quantized versions of $\gamma_t, \gamma_{t+1}, \ldots \gamma_{t+L-1}$ after the slot $(t+L-1)$. ![Backtrack retransmission with vector quantization of CSIT. The LF feedback bits in the even (odd) block are used to report the SNRs in the previous odd (even) block. There are $2L$ BRQ processes running in parallel, each associated with one of the $L$ even or odd slots. Two processes are illustrated, associated with the odd slots $1$ and $2$, respectively.[]{data-label="fig:BlockVQ"}](./images/BlockVQ.pdf) The transmission strategy can be specified as follows. We group the transmission slots into blocks of $L$ slots, where $L$ is sufficiently large. We then differentiate between odd and even blocks, such that $(1,i)$ is the $i-$th odd block and $(2,j)$ is the $j-$th even block. The $l-$th slot of an odd (even) block will be referred to as $l-$th odd (even) slot. The odd and the even blocks are interleaved in time, such that the sequence is $(1,1), (2,1), (1,2), (2,2), (3,1), \ldots$, see Fig. \[fig:BlockVQ\]. Let us assume that the communication starts in block $(1,1)$ and TX transmits new data bits in each slot of the block $(1,1)$, such that in total $LNR$ new bits are transmitted. The $LF$ feedback bits of block $(1,1)$ are not used, which is a waste that becomes negligible asymptotically, as the number of observed blocks goes to infinity. RX records the SNRs of each of the $L$ slots during block $(1,1)$. Specifically, we denote by $\gamma_{1,i,l}$ the SNR in the $l-$th slot of the $i-$th odd block. At the end of block $(1,1)$, $B$ performs vector quantization of $\gamma_{1,1,1}, \gamma_{1,1,2}, \cdots \gamma_{1,1,L}$ and uses the $LF$ feedback bits to report two different messages: (a) in which slots $l$ of block $(1,1)$ the decoding was successful, i. e. $\gamma_{1,1,l} \geq \gamma_R$, and (b) the quantized SNRs $\hat{\gamma}_{1,1,1}, \hat{\gamma}_{1,1,2}, \cdots \hat{\gamma}_{1,1,L}$. These bits are transmitted to TX during the $L$ feedback opportunities of the even block $(2,1)$, as denoted on Fig. \[fig:BlockVQ\]. TX recovers the distorted versions of the SNRs. If TX learns that $\gamma_{1,1,l} \geq \gamma_R$, TX sends $NR$ new bits in slot $(1,2,l)$. Otherwise, $\gamma_{1,1,l} < \gamma_R$ and TX prepares $NR_{1,1,l}$ parity bits and $NR-NR_{1,1,l}$ new bits and transmits them during the slot $(1,2,l)$. The choice of $R_{1,1,l}$ depends on the received value $\hat{\gamma}_{1,1,l}$ as well as the fidelity criterion used for quantization, as explained below. During the first even block $(2,1)$, TX sends new bits in all $L$ slots, unrelated to the transmissions in block $(1,1)$. From the description above it can be inferred that TX runs $2L$ instances of BRQ in parallel, for the $L$ even and the $L$ odd slots, respectively. For example, the BRQ process denoted by $(1,,l)$ is associated with the $l-$th odd slot. Let us observe the $l-$the odd slot $(1,,l)$ and assume successful decoding occurs in block $j$ (slot $(1,j,l)$) and the next one in block $j+J$ (slot $(1,j+J,l)$). Then, using backtrack, RX decodes the signals received in slots $(1,j+1,l), (1,j+2,l), \cdots (1,J,l)$. The operation is analogous for the even slots and the BRQ operation within the $l$ even/odd slots proceeds according to the description in the previous section. The key to this operation is how to perform the quantization. By definition, each SNR $\gamma_l$ is non-negative, and for the quantization we put the following fidelity criterion: $$\label{eq:QuantCriterion} \gamma_l \geq \hat{\gamma}_l-d$$ where $d$ is a positive constant distortion value. This is a rather heuristic criterion, while we will briefly address the problem of optimal criterion in Section \[sec:Discussion\]. The motivation for (\[eq:QuantCriterion\]) can be explained as follows. When TX observes $\hat{\gamma}_l$ it knows that this is not the correct value, but it is desirable to know a lower bound on the true $\gamma_l$, such that TX can be sure that the amount of parity bits sent will be equal or larger than what is minimally required to recover the failed transmission in slot $l$. Thus, the number of parity bits that TX sends to RX for recovering the transmission in the $l-$th slot is: $$N(R-C(\hat{\gamma}_l-d)^+)$$ where $(x)^+=\max\{x,0\}$. With a slight abuse of the notation, we denote the next slot of the BRQ process by $(l+1)$, such that the amount of new bits sent in the $(l+1)-$th slot is: $$NC((\hat{\gamma}_l-d)^+)$$ If we look at a single BRQ instance, then we can use the analysis of the previous section, such that the reward in the slot $t$ in which decoding occurs (Lemma 1) can be written as: $$\label{eq:rho_t} \rho_t = N \left(R+\sum_{l=1}^{L-1} C((\hat{\gamma}_{t-l}-d)^+) \right)$$ Let $p(\hat{\gamma}\|\gamma)$ be a conditional probability distribution used for quantization that satisfies the fidelity criterion (\[eq:QuantCriterion\]). Recall that we have designed the feedback to tell to TX in which slots decoding has occurred i. e. $\gamma \geq \gamma_R$, while for the remaining slots TX decides the number of parity bits based on $\hat{\gamma}$. The probability that a transmission is decoded is $p_R$ and the decoding events from slot to slot are independent. For sufficiently large block length $L$, the number of bits required to describe the slots in the block in which decoding occurred is $LH(p_R)$, where $H(\cdot)$ is the entropy function. Hence, the number of bits available for the vector-quantized versions of the SNRs occurring in a block is $L(F-H(p_R))$ or $F-H(p_R)$ bits per slot i. e. SNR value. The probability density function $q_{R}(\gamma)$ that should be used for quantization is different from the original distribution of the SNR $p_{\Gamma}(\gamma)$, as we only need to quantize the values $\gamma < \gamma_R$. Note that $q_{R}(\gamma)$ depends on the choice of $R$. Specifically: $$q_{\Gamma}(\gamma)=\frac{p_{\Gamma}(\gamma)}{1-p_R} \mathbf{I}(\gamma < \gamma_R)$$ where the indicator function $\mathbf{I}(x<y)=1$ if $x<y$ and is $0$ otherwise. We denote by $S_R(d)$ the rate distortion function computed for $q_{R}(\gamma)$ and using the fidelity criterion (\[eq:QuantCriterion\]), we can establish the following relation: $$\label{eq:exp_dist_bound} d=S_R^{-1}(F-H(p_R))$$ Using similar reasoning as in Theorem 1, considering that the $2L$ BRQ instances are statistically equal and over a long period the waste of the unused feedback in the first block disappears, we can state the following: Let TX transmit data to RX using a fixed rate $R$. Let there be $F$ feedback bits per slot and RX assembles the feedback bits of $L$ blocks. Part of the $LF$ bits are used to report in which slots there was a decoding event and the remaining bits are used to report quantized values of the SNRs to TX. Let $p(\hat{\gamma}\|\gamma)$ be a conditional distribution that satisfies the fidelity criterion (\[eq:QuantCriterion\]) and $p_{\Gamma}(\gamma)$ be the density of SNR in each slot. Then the average rate is: $$\label{eq:AverageRateQuantCSIT} \bar{R}=\int_{\gamma=0}^{\gamma_R} \int_{\hat{\gamma}} p(\hat{\gamma}\|\gamma) p_{\Gamma}(\gamma) C((\hat{\gamma}_l-d)^+) \mathrm{d} \hat{\gamma} \mathrm{d} \gamma + R \int_{\gamma_R}^{\infty} p_{\Gamma}(\gamma) \mathrm{d}\gamma$$ where $d=S_R^{-1}(F-H(p_R))$. As an example, we can consider the Rayleigh fading for which $p_{\Gamma}(\gamma)=\frac{1}{\Gamma}e^{-\frac{\gamma}{\Gamma}}$. Quantization of the exponential distribution with fidelity criterion (\[eq:QuantCriterion\]) has been considered in [@Verdu96]. The derivation of the rate distortion function $S_R$ for given $R$ is a problem on its own and not of direct interest in this initial paper. Instead, we use the $F-H(p_R)$ bits per SNR in a suboptimal way, making a vector quantization according to $p_{\Gamma}(\gamma)$. This is clearly suboptimal as RX reports to TX also quantized versions of the SNR values $\gamma > \gamma_R$, which is redundant. Therefore, the distortion computed accordion to the result from [@Verdu96] represent an upper bound on the distortion that can be achieved if we (properly) quantize according to $q_R(\cdot)$: $$\label{eq:UpperBoundDist} d \leq \Gamma \cdot 2^{-(F-H(p_R))}$$ As $R$ increases, $p_R$ and $H(p_R)$ decrease, while $q_R(\cdot)$ becomes better approximated by $p_{\Gamma}(\cdot)$, such that the upper bound in (\[eq:UpperBoundDist\]) becomes tight. Numerical Illustration ====================== We evaluate BRQ by assuming Rayleigh block fading, independent from slot to slot. Fig. \[fig:Rate\_vs\_SINR\] depicts the average rate (average throughput) of various schemes as a function of the mean SNR $\Gamma$. For each $\Gamma$, the average rate is normalized by the optimal average rate that can be achieved if CSIT is known a priori and water filling is applied. Two schemes with posterior CSIT are evaluated, full CSIT and quantized CSIT with $F=1$ bit per SNR, respectively. For each scheme, two different transmission rates are chosen, $R=\log_2(1+2\Gamma)$ and $R=\log_2(1+3\Gamma)$, respectively. Note that by such a choice, we are fixing the decoding probability to $p_R=e^{-2}$ and $p_R=e^{-3}$, respectively. The scheme with quantized CSIT is evaluated by using the rate distortion function of the exponential distribution, such that $d$ is determined according to (\[eq:exp\_dist\_bound\]), which is suboptimal. As a reference, we have also plotted the average rate when full CSIT is known a priori, but no water filling (fixed power) is applied. When full CSIT is available and the transmission rate $R$ is sufficiently high, then the knowledge of posterior CSIT is equally useful as the prior CSIT for average SNRs $\Gamma=10$ and higher. Furthermore, as water filling is more significant at low SNRs, we can see that with $R=\log_2(1+3\Gamma)$ and full CSIT, BRQ tightly approaches the average rate obtainable by water filling for $\Gamma$ equal to $20$ dB or higher. The scheme with quantized CSIT converges to a fixed value as $\Gamma$ increases (proof omitted due to lack of space). The behavior of the average rate for the quantized scheme is better visible from (\[fig:Rate\_vs\_gR\]), where the average SNR is fixed to $\Gamma$, while $R=\log_2(1+\gamma_R)$ increases. The abscissa shows the scalar ratio $\frac{\gamma_R}{\Gamma}$, while the ordinate shows the absolute value of the average rate in bits per channel use \[bit/c.u.\]. We see that the schemes $F=1, F=2$ still grow in the region in which the scheme with $F=8$ is saturated. The reason is that the schemes with low $F$ are additionally affected by the decrease of $H(p_R)$ as $p_R$ grows, which affects the value of $d$, while $H(p_R)$ has a negligible effect for $F=8$. ![Average rates as a function of the average SNR in a Rayleigh fading, normalized with the average rate achievable with prior CSIT and waterfilling. []{data-label="fig:Rate_vs_SINR"}](./images/Rate_vs_SINR.pdf){width="8.3cm"} ![Average rates as a function of the transmission rate $R$, represented through the equivalent SNR $\gamma_R$. The average SNR is fixed to $\Gamma=10$. []{data-label="fig:Rate_vs_gR"}](./images/Rate_vs_gR.pdf){width="8.3cm"} Discussion and Conclusions {#sec:Discussion} ========================== The main objective of this paper is to introduce a class of new transmission schemes based on backtrack retransmission (BRQ) that are useful when the CSIT can only be available after the transmission has taken place. Compared to the existing works on the topic, BRQ offers an elegant way to use the full posterior CSIT, based on adaptive Markov block coding, thereby closely approaching the average throughput achievable with prior CSIT. By extending the ideas of BRQ to the case where only a finite number of feedback bits are available, we have introduced a way to use the feedback bits which significantly departs from the standard way in which these bits are used in HARQ protocols. We devise a scheme in which it is possible to assemble multiple feedback bits and jointly report multiple CSIT values through vector quantization. In this initial work on the topic we have used the asymptotic information-theoretic results, valid when the number of channel uses per slot $N$ and when the number of slots in a block $L$ (used in vector quantization) goes to infinity. When finite values of $N$ and $L$ are considered, then the protocol should be adjusted in order to account for nonzero probability of error in the transmission as well as in the recovery based on the random binning. This analysis will be put in an extended version of this paper. Another interesting aspect is the choice of the distortion criterion. We have chosen $d$ to be constant, but in general $d$ should depends on the SNR value and the optimal $d$ should maximize the average rate. An interesting extension would be to generalize BRQ to multi-user scenarios. On the other hand, considering the practicality of the assumption about the posterior CSIT, the presented method is relevant for the practical wireless systems, such as LTE, based on fixed-size physical resources and significantly affected by intermittent interference from neighboring cells. It is therefore important to explore how BRQ can be implemented with practical coding/modulation schemes. Acknowledgement {#acknowledgement .unnumbered} =============== The author thanks Prof. Osvaldo Simeone from NJIT for his comments and pointing out the relation to Markov block coding; as well as to Kasper F. Trillingsgaard from Aalborg University for his comments and corrections. Part of this work has been performed in the framework of the FP7 project ICT-317669 METIS, which is partly funded by the European Union. The author would like to acknowledge the contributions of his colleagues in METIS, although the views expressed are those of the author and do not necessarily represent the project. [1]{} D. Tuninetti, “[On the benefits of partial channel state information for repetition protocols in block fading channels]{},” IEEE Trans. Inform. Theory, vol. 57, no. 8, pp. 5036–5053, Aug. 2011. L. Szczecinski, S. R. Khosravirad, P. Duhamel, and M. Rahman, “[Rate Allocation and Adaptation for Incremental Redundancy Truncated HARQ]{},” IEEE Trans. Communications, vol. 61, no. 6, pp. 2580–2590, Jun. 2013. 3GPP, “[LTE–Advanced]{},” http://www.3gpp.org/article/lte-advanced. M. A. Maddah-Ali and D. Tse, “[Completely Stale Transmitter Channel State Information is Still Very Useful]{},” IEEE Trans. Inform. Theory, vol. 58, no. 7, pp. 4418–4431, Jul. 2012. R. Tandon, S. Mohajer, H. V. Poor, and S. Shamai, “Feedback and delayed [CSI]{} can be as good as perfect [CSI]{},” in Proc. IEEE International Conference on Communications (ICC), Jun. 2012. G. Kramer, “[Topics in multi-user information theory]{},” vol. 4, no. 4-5, pp. 265–444, 2007. S. Verdu, “The exponential distribution in information theory,” Probl. Inform. Transm., vol. 32, pp. 86–95, Jan.-Mar. 1996.
--- abstract: \| In this paper, we study a diamond-relay channel where the source is connected to $M$ relays through orthogonal links and the relays transmit to the destination over a wireless multiple-access channel in the presence of an eavesdropper. The eavesdropper not only observes the relay transmissions through another multiple-access channel, but also observes a certain number of source-relay links. The legitimate terminals know neither the eavesdropper’s channel state information nor the location of source-relay links revealed to the eavesdropper except the total number of such links. For this wiretapped diamond-relay channel, we establish the optimal secure degrees of freedom. In the achievability part, our proposed scheme uses the source-relay links to transmit a judiciously constructed combination of message symbols, artificial noise symbols as well as fictitious message symbols associated with secure network coding. The relays use a combination of beamforming and interference alignment in their transmission scheme. For the converse part, we take a genie-aided approach assuming that the location of wiretapped links is known. author: - '[^1]' title: 'The Wiretapped Diamond-Relay Channel' --- Introduction ============ Cloud Radio-Access Network (C-RAN) is a promising architecture to meet the demand for higher data rates in next generation wireless networks. In these systems, the base-stations act as relays and are connected via high-speed backhaul links to a cloud network. Encoding and decoding operations happen centrally in the cloud. The study of fundamental information theoretic limits and optimal coding techniques for such systems is a fertile area of research. Motivated by C-RAN, we study a model where the source is connected to $M$ relay terminals using orthogonal links with a fixed capacity. The relays transmit to the destination over a wireless multiple-access channel. Such a setup is known as the diamond-relay network [@ScheinGallager:00; @schein_thesis; @TraskovKramer:07; @KangLiu:11; @BidokhtiKramer:arxiv15]. We study this model in the presence of an eavesdropper who can eavesdrop the orthogonal links from the source to the relays as well as the wireless transmission from the relays. We require that a message be transmitted reliably to the legitimate receiver, while keeping it confidential from the eavesdropper. We adopt the information-theoretic notion of confidentiality, widely used in the literature on the wiretap channel [@Wyner:75; @CsiszarKorner:78; @CheongHellman:78; @KhistiWornell:10; @KhistiWornell:10_2; @relay-3; @LiangPoor:08; @relay-4; @relay-2; @TekinYener:08; @TekinYener:08general; @GoelNegi:08; @perron_thesis; @relay-5; @relay-6; @YassaeeArefGohari:14; @relay-7; @LeeKhisti:15]. We thus refer to our setup as the [[wiretapped diamond-relay channel]{}]{}. While the secrecy capacity of both the scalar Gaussian wiretap channel and its multi-antenna extension can be achieved using Gaussian codebooks [@CheongHellman:78; @KhistiWornell:10; @KhistiWornell:10_2], the optimal schemes in multiuser channels are considerably more intricate. Very recently, Xie and Ulukus [@XieUlukus:14] studied the secure degrees of freedom (d.o.f.) for one-hop Gaussian multiuser channels such as the multiple-access channel and the interference channel. It turns out that interference alignment [@CadambeJafar:08; @MotahariOveisGharanMaddahAliKhandani:14] plays a central role in achieving the optimal secure d.o.f. While coding schemes based on Gaussian codebooks can only achieve zero secure d.o.f., the schemes presented in [@XieUlukus:14] involve transmitting a combination of information and jamming signals at each transmitter and judiciously precoding them such that the noise symbols align at the legitimate receiver(s), yet mask information symbols at the eavesdropper. This approach has been extended for the case with no eavesdropper’s channel state information (CSI) at the legitimate parties in [@XieUlukus:13blind; @MukherjeeXieUlukus:arxiv15]. We note that a combination of jamming and interference alignment is also required in the confidential MIMO broadcast channel with delayed channel state information [@yang2013secrecy] and the compound MIMO wiretap channel [@khisti2013artificial]. The diamond-relay channel that we consider is a two-hop network where the first hop consists of orthogonal links from the source to the relays, while the second hop is a multiple-access channel from the relays to the destination. Such a network is considerably different from one-hop networks as the channel inputs from the relays need not be mutually independent. The source can transmit common/independent message, common noise, or any combination of those to the relays. In [@LeeZhaoKhisti:arxiv15], for the diamond-relay channel where an eavesdropper can wiretap only the multiple-access portion of the channel, it is shown that transmitting common noise over the source-relay links facilitates more efficient jamming for both the case with full CSI and the case with no eavesdropper’s CSI. For example, a key constituent scheme proposed in [@LeeZhaoKhisti:arxiv15] for the case with no eavesdropper’s CSI is computation for jamming (CoJ) where the source transmits a function of information and common noise symbols to two relays in order that the common noise symbols introduced to jam information symbols at the eavesdropper can be canceled at the destination through suitable precoding. When there are more than two relays, this scheme operates in a time-sharing basis in a way that only two source-relay links are utilized in each sub-scheme.[^2] In the present paper, the optimal secure d.o.f. of the wiretapped diamond-relay channel is established for the case with no eavesdropper’s CSI. In our wiretapped diamond-relay channel, the eavesdropper not only observes the relay transmissions through another multiple-access channel, but also observes a certain number of source-relay links. The legitimate terminals do not know which subset of source-relay links are revealed to the eavesdropper, except for the total number of such links. For achievability, we note that it is not straightforward to extend the proposed schemes in [@LeeZhaoKhisti:arxiv15] because their sub-schemes require asymmetric link d.o.f. and hence the amount of wiretapped information varies over time according to the unknown location of wiretapped links. Thus, we first develop symmetric version of the proposed schemes in [@LeeZhaoKhisti:arxiv15] and then combine them with a secure network coding (SNC) [@CaiYeung:02; @CaiYeung:11]-like scheme to account for non-secure source-relay links. While the conventional SNC has been developed for fully wired networks, we incorporate SNC by utilizing the nature of wireless networks in a way that the additional randomness introduced in the source-relay links due to SNC is canceled at the destination by beamforming over the wireless multiple-access channel. To that end, we judiciously choose the generator matrix for SNC. From a technical point of view, the secrecy analysis involves accounting for the observations from the source-relay links as well as the multiple-access wiretap channel and is considerably more involved. Furthermore, the secure d.o.f. is shown to be the same even when the knowledge of the compromised links is available. Indeed our converse is established via this genie-aided approach. The rest of this paper is organized as follows. In Section \[sec:model\], we formally state our model of wiretapped diamond-relay channel. The main result on the secure d.o.f. is presented in Section \[sec:main\]. For the achievability part, our proposed schemes are described at a high-level in Section \[sec:ach\] and are rigorously stated in Appendix \[appendix:achievability\]. The converse part is proved in Section \[sec:conv\]. We conclude this paper in Section \[sec:conclusion\]. The following notation is used throughout the paper. For two integers $i$ and $j$, $[i:j]$ denotes the set $\{i,i+1,\cdots, j\}$. For constants $x_1,\cdots, x_k$ and $S\subseteq [1:k]$, $x_S$ denotes the vector $(x_j: j\in S)$. This notation is naturally extended for vectors and random variables. For a sequence $x(1),x(2),\cdots$ of constants indexed by time, $x^k$ for positive integer $k$ denotes the vector $(x(j): j \in [1:k])$. This notation is naturally generalized for vectors and random variables. $\lfloor\cdot\rfloor$ denotes the floor function. For positive real number $\delta$ and positive integer $Q$, $\mathcal{C}(\delta, Q)$ denotes the PAM constellation $\delta\{- Q, - Q+1,\cdots, 0, \cdots, Q-1, Q\} $ of $(2Q+1)$ points with distance $\delta$ between points. For positive integers $i$ and $j\in [0:i+1]$, $[j]_i$ denotes $j$ if $j\in [1:i]$, $i$ if $j=0$, and 1 if $j=i+1$. Model {#sec:model} ===== Consider a diamond-relay channel that consists of an orthogonal broadcast channel from a source to $M\geq 2$ relays and a Gaussian multiple-access channel from the $M$ relays to a destination. In the broadcast part, the source is connected to each relay through an orthogonal link of capacity $C$. In the multiple-access part, the channel output $Y_1(t)$ at time $t$ is given as $$\begin{aligned} Y_1(t) = \sum_{k = 1}^M h_k(t) X_k(t) + Z_1(t), \label{eqn:y1_M}\end{aligned}$$ in which $X_k(t)$ is the channel input at relay $k$, $h_k(t)$’s are channel fading coefficients, and $Z_1(t)$ is additive Gaussian noise with zero mean and unit variance. The transmit power constraint at relay $k$ is given as $\frac{1}{n}\sum_{t=1}^nX_{k}^2(t)\leq P$, where $n$ denotes the number of channel uses. In this paper, we consider a scenario illustrated in Fig. \[fig:model\], where an eavesdropper wiretaps both the broadcast part and the multiple-access part of the diamond-relay channel. In the broadcast part, the eavesdropper can wiretap $W$ source-relay links. Let $N=M-W$ denote the number of secure source-relay links. We assume that the location of wiretapped links is unknown to the source, relays, and destination. In the multiple-access part, the eavesdropper observes $Y_2(t)$ at time $t$ given as $$\begin{aligned} Y_2(t) = \sum_{k = 1}^M g_k(t) X_k(t) + Z_2(t), \label{eqn:y2_M}\end{aligned}$$ where $g_k(t)$’s are channel fading coefficients and $Z_2(t)$ is additive Gaussian noise with zero mean and unit variance. ![The wiretapped diamond-relay channel. []{data-label="fig:model"}](model.eps "fig:"){width="90mm"}\ We assume a fast fading scenario where $h_k(t)$’s and $g_k(t)$’s are drawn in an i.i.d. fashion over time according to an arbitrary real-valued joint density function $f(h_1, \cdots, h_M, g_1, \cdots, g_M)$ satisfying that (i) all joint and conditional density functions are bounded and (ii) there exists a positive finite number $B$ such that $\frac{1}{B}\leq \|h_k(t)\|, \|g_k(t)\| \leq B$ for all $k\in [1:M]$.[^3] For notational convenience, let $\mathbf{h}(t)=(h_1(t)~\cdots ~h_M(t))$ and $\mathbf{g}(t)=(g_1(t)~\cdots ~g_M(t))$ denote the legitimate channel state information (CSI) and the eavesdropper’s CSI at time $t$, respectively. We assume that the source does not know both the legitimate CSI and the eavesdropper’s CSI and the eavesdropper knows both the CSI’s. The relays and destination are assumed to know only the legitimate CSI.[^4] A $(2^{nR}, n)$ code consists of a message $G\sim \mbox{Unif}[1:2^{nR}]$,[^5] a stochastic encoder at the source that (randomly) maps $G\in [1:2^{nR}]$ to $(S_1^n, \cdots, S_M^n)\in \mathcal{S}_1^n\times \cdots \times \mathcal{S}_M^n$ such that $\frac{1}{n}H(S_k^n)\leq C$ for $k\in [1:M]$, a stochastic encoder at time $t\in[1:n]$ at relay $k\in [1:M]$ that (randomly) maps $(S_k^n, X_k^{t-1}, \mathbf{h}^t)$ to $X_k(t) \in \mathcal{X}_k$, and a decoding function at the destination that (randomly) maps $(Y_1^n, \mathbf{h}^n)$ to $\hat{G} \in [1:2^{nR}]$. The probability of error is given as $P_e^{(n)}=P(\hat{G}\neq G)$. A secrecy rate of $R$ is said to be achievable if there exists a sequence of $(2^{nR},n)$ codes such that $\lim_{n\rightarrow \infty}P_e^{(n)}=0$ and $$\begin{aligned} \lim_{n\rightarrow \infty}\frac{1}{n}I(W;S_{T}^n,Y_2^n\|\mathbf{h}^n, \mathbf{g}^n)=0\label{eqn:sec_cond}\end{aligned}$$ for all $T\subseteq [1:M]$ such that $\|T\|=W$. The secrecy capacity is the supremum of all achievable secrecy rates. We also consider the case where the location of wiretapped links is known to all parties. Let $T$ denote the index set of relays whose links from the source are wiretapped. In this case, an achievable secrecy rate is defined by requiring to be satisfied for the known $T$. In this paper, we are interested in asymptotic behavior of the secrecy capacity when $P$ tends to infinity. We say a d.o.f. tuple $(\alpha, d_s)$ is achievable if a secrecy rate $R$ such that $\lim_{P\rightarrow \infty} \frac{R}{\frac{1}{2}\log P}=d_s$ is achievable when $\lim_{P\rightarrow \infty} \frac{C}{\frac{1}{2}\log P}=\alpha$. A secure d.o.f. $d_s(\alpha)$ is the maximum $d_s$ such that $(\alpha, d_s)$ is achievable. According to the context, $d_s$ denotes $d_s(\alpha)$. Our model guarantees secrecy from a certain level of relay collusion, i.e., from collusion of any set of up to $W$ relays when the location of wiretapped links is unknown and from collusion of any set of relays that have wiretapped links when it is known. Our achievability results hold for constant channel gains as well as slow-fading channels. Furthermore, our achievability results can be generalized for complex channel fading coefficients by applying [@MaddahAli:09 Lemma 7] in our analysis of interference alignment. Main Result {#sec:main} =========== In this section, we state the main result of this paper. Let us first present the secure d.o.f. of the wiretapped diamond-relay channel when $N=M$, which was established in [@LeeZhaoKhisti:arxiv15 Theorem 6]. \[thm:same\] The secure d.o.f. of the wiretapped diamond-relay channel when $N=M$, i.e., the eavesdropper does not wiretap the broadcast part, is equal to $$\begin{aligned} d_{s} &= \min\left\{M\alpha,\frac{M\alpha + M - 1}{M + 1},1\right\}.\end{aligned}$$ For the achievability part of Theorem \[thm:same\], two constituent schemes were proposed in [@LeeZhaoKhisti:arxiv15]. One scheme is a time-shared version of the blind cooperative jamming (BCJ) scheme, where in each sub-scheme, a single relay operates as a source and the other relays operate as helpers according to the BCJ scheme [@XieUlukus:13blind]. The other scheme is a time-shared version of the CoJ scheme, where in each sub-scheme, the source and a pair of relays operate according to the CoJ scheme [@LeeZhaoKhisti:arxiv15 Scheme 5] while the other relays remain idle. For the converse part of Theorem \[thm:same\], a technique was introduced in [@LeeZhaoKhisti:arxiv15] that captures the trade-off between the message rate and the amount of individual randomness injected at each relay. The following theorem establishes the secure d.o.f. of the wiretapped diamond-relay channel for the general case. \[thm:less\] The secure d.o.f. of the wiretapped diamond-relay channel is equal to $$\begin{aligned} d_{s} &= \begin{cases} \min\left\{\alpha, \frac{M-1}{M}\right\}, &N=1\\ \min\left\{N\alpha,\frac{N\alpha + M - 1}{M + 1},1\right\}, &N\geq 2 \end{cases}\end{aligned}$$ for both the cases with and without the knowledge of location of wiretapped links. Theorem \[thm:less\] indicates that the secure d.o.f. is the same for the cases with and without the knowledge of location of wiretapped links. According to Theorem \[thm:less\], the secure d.o.f. is at most $\frac{M-1}{M}$ when the number of secure links is one. When the number of secure links is one and its location is known, a natural strategy would be to send the message over the secure link and send nothing over the wiretapped links. Then, the multiple-access part becomes the wiretap channel with $M-1$ helpers whose secure d.o.f. is shown to be $\frac{M-1}{M}$ in [@XieUlukus:13blind]. When $N\geq 2$, we can interpret Theorem \[thm:less\] in a way that the secure d.o.f. is decreased as if the link d.o.f. $\alpha$ was decreased by a factor of $\frac{N}{M}$. Because only $N$ out of $M$ links are secure, it is intuitive that the information that can be securely sent over the broadcast part is decreased by a factor of $\frac{N}{M}$. In Fig. \[fig:thm2\], the secure d.o.f. of the wiretapped diamond-relay channel is illustrated when $M=3$ and $N=1,2,3$. ![Secure d.o.f. of the wiretapped diamond-relay channel when $M=3$ and $N=1,2,3$. []{data-label="fig:thm2"}](thm2.eps "fig:"){width="120mm"}\ The achievability part of Theorem \[thm:less\] when the location of wiretapped links is known can be easily proved by generalizing the proposed schemes in [@LeeZhaoKhisti:arxiv15]. However, when the location of wiretapped links is unknown, it is not straightforward to extend the proposed schemes in [@LeeZhaoKhisti:arxiv15] because their sub-schemes require asymmetric link d.o.f. and hence the amount of wiretapped information depends on the unknown location of wiretapped links. To resolve this issue, we first propose simultaneous BCJ (S-BCJ) and simultaneous CoJ (S-CoJ) schemes for the case with the knowledge of location of wiretapped links. Then we incorporate SNC on top of those schemes for the case without the knowledge of location of wiretapped links. Our proposed schemes are described at a high-level in Section \[sec:ach\] and rigorously stated in Appendix \[appendix:achievability\]. The converse part of Theorem \[thm:less\] is proved in Section \[sec:conv\]. Achievability {#sec:ach} ============= To prove the achievability part of Theorem \[thm:less\], it suffices to show the achievability of the following corner points: $(\alpha,d_s)=(\frac{M-1}{MN}, \frac{M-1}{M})$ for $N\geq 1$ and $(\alpha,d_s)=(\frac{2}{N}, 1)$ for $N\geq 2$.[^6] In the following, we first propose S-BCJ scheme achieving $(\alpha,d_s)=(\frac{M-1}{M^2}, \frac{M-1}{M})$ and S-CoJ scheme achieving $(\alpha,d_s)=(\frac{2}{M}, 1)$ for the special case of $N=M$. Then, we generalize them for the general case of $N\leq M$ with the knowledge of location of wiretapped links. Subsequently, we incorporate SNC on top of the S-BCJ and S-CoJ schemes for the case of $N\leq M$ without the knowledge of location of wiretapped links. To provide main intuition behind our schemes, let us give a high-level description of our schemes in this section. A detailed description with rigorous analysis is relegated to Appendix \[appendix:achievability\]. In Fig. \[fig:SBCJ\] and Fig. \[fig:SCOJ\], our proposed schemes are illustrated for $M=3$, where rectangles labeled with $V$, $U$, $F$ represent message, noise, and fictitious message symbols, respectively, and a rectangle labeled with $L$ represents a linear combination of fictitious message symbols. A column of symbols with fat side lines implies that those symbols are aligned and occupy the d.o.f. of a single symbol. Special case of $N=M$ --------------------- ### S-BCJ scheme for $N=M$ achieving $(\alpha, d_s)=(\frac{M-1}{M^2}, \frac{M-1}{M})$ The source represents the message of d.o.f. $\frac{M-1}{M}$ as a vector $V=(V_{k,j}: k\in [1:M], j\in [1:M-1])$ of independent message symbols each of d.o.f. $\frac{1}{M^2}$ and sends $V_k=(V_{k,j}:j\in [1:M-1])$ to relay $k$, which requires $\alpha=\frac{M-1}{M^2}$. Then, relay $k$ sends $V_k$ together with $M$ independent noise symbols $(U_{k,j}: j\in [1:M])$ each of d.o.f. $\frac{1}{M^2}$ in a way that (i) for each $j\in [1:M]$, $U_{k,j}$’s for $k\in [1:M]$ are aligned at the destination and (ii) $V_{k,j}$’s can be distinguished by the destination. Since $U_{k,j}$’s are not aligned and occupy a total of 1 d.o.f at the eavesdropper, the message can be shown to be secure. ### S-CoJ scheme for $N=M$ achieving $(\alpha, d_s)=(\frac{2}{M}, 1)$ The source represents the message of d.o.f. 1 as a vector $(V_1, \cdots, V_M)$ of independent message symbols and generates $M$ independent noise symbols $(U_1, \cdots, U_M)$, where each $V_k$ and $U_k$ has a d.o.f. $\frac{1}{M}$. The source sends $(V_k+U_k, U_{[k+1]_ M})$ to relay $k$, which requires $\alpha=\frac{2}{M}$. Then, the relays send what they have received in a way that (i) each of $U_k$’s is beam-formed in the null space of the destination’s channel and (ii) $V_k$’s can be distinguished by the destination. Since $U_k$’s are not aligned and occupy a total of 1 d.o.f at the eavesdropper, the message can be shown to be secure. General case of $N\leq M$ with the knowledge of location of wiretapped links {#subsec:less_w} ---------------------------------------------------------------------------- In this case, we send nothing over the wiretapped links. Without loss of generality, let us assume that the first $N$ links are secure. ### S-BCJ scheme achieving $(\alpha, d_s)=(\frac{M-1}{MN}, \frac{M-1}{M})$ The source represents the message of d.o.f. $\frac{M-1}{M}$ as a vector $V=(V_{k,j}: k\in [1:N], j\in [1:M-1])$ of independent message symbols each of d.o.f. $\frac{1}{MN}$ and sends $V_k=(V_{k,j}:j\in [1:M-1])$ to relay $k\in [1:N]$ and sends nothing to relay $i\in [N+1:M]$, which requires $\alpha=\frac{M-1}{MN}$. Then, relay $k\in [1:M]$ sends what it has received together with $N$ independent noise symbols $(U_{k,j}: j\in [1:N])$ each of d.o.f. $\frac{1}{MN}$ in a way that (i) for each $j\in [1:N]$, $U_{k,j}$’s for $k\in [1:M]$ are aligned at the destination and (ii) $V_{k,j}$’s can be distinguished by the destination. Since $U_{k,j}$’s are not aligned and occupy a total of 1 d.o.f at the eavesdropper, the message can be shown to be secure. ### S-CoJ scheme for $N\geq 2$ achieving $(\alpha, d_s)=(\frac{2}{N}, 1)$ The source represents the message of d.o.f. 1 as a vector $(V_1, \cdots, V_N)$ of independent message symbols and generates $N$ independent noise symbols $(U_1, \cdots, U_N)$, where each $V_k$ and $U_k$ has a d.o.f. $\frac{1}{N}$. The source sends $(V_k+U_k, U_{[k+1]_ N})$ to relay $k\in [1:N]$ and sends nothing to relay $i\in [N+1:M]$, which requires $\alpha=\frac{2}{N}$. Then, the relays send what they have received in a way that (i) each of $U_k$’s is beam-formed in the null space of the destination’s channel and (ii) $V_k$’s can be distinguished by the destination. Since $U_k$’s are not aligned and occupy a total of 1 d.o.f at the eavesdropper, the message can be shown to be secure. General case of $N\leq M$ without the knowledge of location of wiretapped links ------------------------------------------------------------------------------- When the location of wiretapped links is unknown, the information sent over the source-relay links in the S-BCJ and S-CoJ schemes should be masked by additional randomness. To that end, we introduce independent fictitious message symbols whose total d.o.f. is the same as the total d.o.f. of wiretapped links. Then, in the S-BCJ and S-CoJ schemes assuming a certain location of wiretapped links (e.g., the last $W$ links), we add a linear combination of these fictitious message symbols to each element sent from the source to the relays. By doing so, the message can be shown to be secure regardless of the location of wiretapped links, while the fictitious message symbols are beam-formed in the null space of the destination and hence the achievable secure d.o.f. is not affected. This technique of masking the information by adding a linear combination of fictitious message symbols is similar to SNC [@CaiYeung:02; @CaiYeung:11], and hence we call our schemes S-BCJ-SNC and S-CoJ-SNC. However, it should be noted that our schemes assume computation over real numbers to enable the beam-forming of fictitious message symbols over the multiple-access part, while the conventional SNC has been developed for fully wired networks and assumes computation over finite field. ### S-BCJ-SNC scheme achieving $(\alpha, d_s)=(\frac{M-1}{MN}, \frac{M-1}{M})$ As in the S-BCJ scheme when the last $W$ links are wiretapped, the source represents the message of d.o.f. $\frac{M-1}{M}$ as a vector $V=(V_{k,j}: k\in [1:N], j\in [1:M-1])$ of independent message symbols each of d.o.f. $\frac{1}{MN}$. To mask each symbol, we introduce a vector $F=(F_k: k\in [1:W(M-1)])$ of independent fictitious message symbols each of d.o.f. $\frac{1}{MN}$ and generate a vector $L=(L_{k,j}: k\in [1:M], j\in[1:M-1])$ of linear combinations of fictitious message symbols by computing $L=F\Gamma$ for some integer matrix $\Gamma$. Now, the source sends $(V_{k,j}+L_{k,j}:j\in [1:M-1])$ to relay $k\in [1:N]$ and sends $(L_{i,j}:j\in [1:M-1])$ to relay $i\in [N+1:M]$, which requires $\alpha=\frac{M-1}{MN}$. Then, relay $k\in [1:M]$ sends what it has received together with $N$ independent noise symbols $(U_{k,j}: j\in [1:N])$ each of d.o.f. $\frac{1}{MN}$ in a way that (i) for each $j\in [1:N]$, $U_{k,j}$’s for $k\in [1:M]$ are aligned at the destination, (ii) each of $F_k$’s is beam-formed in the null space of the destination’s channel, and (iii) $V_{k,j}$’s can be distinguished by the destination. By judiciously choosing the generator matrix $\Gamma$, it can be shown that the information leakage is zero (in the d.o.f. sense). ### S-CoJ-SNC scheme for $N\geq 2$ achieving $(\alpha, d_s)=(\frac{2}{N}, 1)$ As in the S-CoJ scheme when the last $W$ links are wiretapped, the source represents the message of d.o.f. 1 as a vector $(V_1, \cdots, V_N)$ of independent message symbols and generates $N$ independent noise symbols $(U_1, \cdots, U_N)$, where each $V_k$ and $U_k$ has a d.o.f. $\frac{1}{N}$. Then, for a vector $F=(F_k: k\in [1:2W])$ of independent fictitious message symbols each of d.o.f. $\frac{1}{N}$, a vector $L=(L_{k}: k\in [1:2M])$ of linear combinations of fictitious message symbols is generated by computing $L=F\Gamma$ for some integer matrix $\Gamma$. Now, the source sends $(V_k+U_k+L_{2k-1}, U_{[k+1]_ N}+L_{2k})$ to relay $k\in [1:N]$ and sends $(L_{2i-1}, L_{2i})$ to relay $i\in [N+1:M]$, which requires $\alpha=\frac{2}{N}$. Then, the relays send what they have received in a way that (i) each of $U_k$’s and $F_k$’s is beam-formed in the null space of the destination’s channel and (ii) $V_{k}$’s can be distinguished by the destination. By judiciously choosing the generator matrix $\Gamma$, it can be shown that the information leakage is zero (in the d.o.f. sense). Converse {#sec:conv} ======== It suffices to prove the converse part of Theorem \[thm:less\] for the case with the knowledge of location of wiretapped links. Proof of the converse part of Theorem \[thm:less\] for the case with the knowledge of location of wiretapped links ------------------------------------------------------------------------------------------------------------------ For the Gaussian multiple-access wiretap channel, it is shown in [@MukherjeeXieUlukus:arxiv15 Section 4.2.1] that there is no loss of secure d.o.f. if we consider the following deterministic model with integer-input and integer-output, instead of and : $$\begin{aligned} Y_1(t)=\sum_{k=1}^M \lfloor h_k(t)X_k(t)\rfloor,~ Y_2(t)=\sum_{k=1}^M \lfloor g_k(t)X_k(t)\rfloor \label{eqn:ch_M}\end{aligned}$$ with the constraint $$\begin{aligned} X_k(t)\in \{0,1,\ldots,\lfloor \sqrt{P}\rfloor\}, k=1,\ldots, M. \label{eqn:pw_M}\end{aligned}$$ Likewise, it can be shown that there is no loss of secure d.o.f. in considering the deterministic model and for the multiple-access part of our model.[^7] Hence, in this section, let us assume that the multiple-access part is given as and , instead of and . Without loss of generality, we assume that the first $N$ links are secure, i.e., $T=[N+1:M]$. Furthermore, we assume that $\mathbf{g}^n$ in addition to $\mathbf{h}^n$ is available at the destination, which only possibly increases the secure d.o.f. Hence, $\mathbf{h}^n$ and $\mathbf{g}^n$ are conditioned in every entropy and mutual information terms in this section, but are omitted for notational convenience. In the following, $c_i$’s for $i=1,2,3,\ldots$ are used to denote positive constants that do not depend on $n$ and $P$. First, we obtain $$\begin{aligned} nR &\overset{(a)}\leq I(W;Y_1^n,S^n_{[N+1:M]}) + nc_1\\ &\overset{(b)}{\leq} I(W;Y_1^n,S^n_{[N+1:M]}) - I(W;Y_2^n,S^n_{[N+1:M]}) + nc_2 \label{eqn:public_fano}\\ & \leq I(W;Y_1^n\|Y_2^n,S^n_{[N+1:M]}) + nc_2 \label{eqn:one_dof}\\ & \leq H(Y_1^n\|Y_2^n,S^n_{[N+1:M]})+ nc_2\label{eqn:public_1}\\ & = H(Y_1^n,Y_2^n\|S^n_{[N+1:M]}) - H(Y_2^n\|S^n_{[N+1:M]}) + nc_2 \\ & \leq H(X_{[1:M]}^n,Y_1^n,Y_2^n\|S^n_{[N+1:M]}) - H(Y_2^n\|S^n_{[N+1:M]}) + nc_2\\ &=H(X_{[1:M]}^n\|S^n_{[N+1:M]})+ H(Y_1^n,Y_2^n\|X_{[1:M]}^n,S^n_{[N+1:M]}) - H(Y_2^n\|S^n_{[N+1:M]}) + nc_2\\ &\overset{(c)}{=} H(X_{[1:M]}^n\|S^n_{[N+1:M]}) - H(Y_2^n\|S^n_{[N+1:M]}) + nc_2\label{eqn:same}\\ &\leq H(X_{[1:M]}^n, S^n_{[1:N]}\|S^n_{[N+1:M]]}) - H(Y_2^n\|S^n_{[N+1:M]}) + nc_2\\ &\leq H(S^n_{[1:N]})+ H(X_{[1:M]}^n\|S^n_{[1:M]}) - H(Y_2^n\|S^n_{[N+1:M]}) + nc_2\\ &\leq nNC +\sum_{i=1}^MH(X_i^n\|S^n_i)- H(Y_2^n\|S^n_{[N+1:M]})+nc_2. \label{eqn:cov12}\end{aligned}$$ where $(a)$ is due to the Fano’s inequality, $(b)$ is from the secrecy constraint, and $(c)$ is because the deterministic model is assumed. Next, to bound $H(X_{i}^n\|S^n_{i})$ for $i\in [1:M]$, we start from to obtain $$\begin{aligned} nR &\leq I(W;Y_1^n,S^n_{[N+1:M]}) - I(W;Y_2^n,S^n_{[N+1:M]}) + nc_2\\ &\leq I(W;Y_1^n,S^n_{[N+1:M]}) - I(W; S^n_{[N+1:M]}) + nc_2 \\ &\leq I(W;Y_1^n\|S^n_{[N+1:M]}) + nc_2 \\ &\leq I(S^n_{[1:N]};Y_1^n\|S^n_{[N+1:M]})+ nc_2 \label{eqn:cov_bc}\\ &\leq H(Y_1^n\|S^n_{[N+1:M]})-H(Y_1^n\|S^n_{[1:M]})+ nc_2 \\ &\overset{(a)}{\leq} H(Y_1^n\|S^n_{[N+1:M]})-H(X_i^n\|S^n_i)+ nc_4, \label{eqn:cov22}\end{aligned}$$ where $(a)$ is due to the following chain of inequalites: $$\begin{aligned} H(Y_1^n\|S^n_{[1:M]})&= H\big(\big\{\sum_{i = 1}^M \lfloor h_i(t) X_i(t)\rfloor\big\}_{t=1}^n \|S^n_{[1:M]}\big)\\ &\geq H\big(\big\{\sum_{i = 1}^M \lfloor h_i(t) X_i(t)\rfloor\big\}_{t=1}^n \|S^n_{[1:M]},X_{i^c}^n\big)\\ &= H\big(\big\{ \lfloor h_i(t) X_i(t)\rfloor\big\}_{t=1}^n \|S^n_{[1:M]},X_{i^c}^n\big)\\ &=H( X_i^n, \big\{ \lfloor h_i(t) X_i(t)\rfloor\big\}_{t=1}^n \|S^n_{[1:M]}, X_{i^c}^n)-H(X_i^n\|\big\{ \lfloor h_i(t) X_i(t)\rfloor\big\}_{t=1}^n ,S^n_{[1:M]}, X_{i^c}^n)\\ &=H( X_i^n \|S^n_{[1:M]}, X_{i^c}^n)-H(X_i^n\|\big\{ \lfloor h_i(t) X_i(t)\rfloor\big\}_{t=1}^n ,S^n_{[1:M]}, X_{i^c}^n)\\ &\geq H( X_i^n \|S^n_{[1:M]}, X_{i^c}^n)-\sum_{t=1}^nH(X_i(t)\|\lfloor h_i(t) X_i(t)\rfloor)\\ &\overset{(a)}{\geq} H( X_i^n\|S^n_{[1:M]}, X_{i^c}^n)-nc_3\\ &\overset{(b)}{=} H( X_i^n\|S^n_i)-nc_3,\end{aligned}$$ where $i^c$ denotes $[1:M]\setminus \{i\}$, $(a)$ is from [@MukherjeeXieUlukus:arxiv15 Lemma 2], and $(b)$ is due to the Markov chain[^8] $X_i^n - (\mathbf{h}^n, \mathbf{g}^n, S^n_i) - (X_{i^c}^n, S^n_{i^c})$. Now, by combining and , we have $$\begin{aligned} (M+1)nR&\leq nNC+MH(Y_1^n\|S^n_{[N+1:M]}) -H(Y_2^n\|S^n_{[N+1:M]})+nc_{5} \\ &= nNC+(M-1)H(Y_1^n\|S^n_{[N+1:M]})+H(Y_1^n\|S^n_{[N+1:M]})-H(Y_2^n\|S^n_{[N+1:M]})+nc_{5}. \label{eqn:diff_pre}\end{aligned}$$ Furthermore, $H(Y_1^n\|S^n_{[N+1:M]})-H(Y_2^n\|S^n_{[N+1:M]})$ can be bounded as follows: $$\begin{aligned} &H(Y_1^n\|S^n_{[N+1:M]})-H(Y_2^n\|S^n_{[N+1:M]})\cr &\leq \max_{s^n_{[N+1:M]}}\Big\{H(Y_1^n\|S^n_{[N+1:M]}=s^n_{[N+1:M]}) -H(Y_2^n\|S^n_{[N+1:M]}=s^n_{[N+1:M]})\Big\} \\ &\leq \max_{p(x_1^n, \cdots, x_M^n) \in \mathcal{P}_{X_{[1:M]}^n} }\Big\{H(Y_1^n)-H(Y_2^n)\Big\}\\ &\overset{(a)}\leq n\cdot o(\log P) \label{eqn:diff}\end{aligned}$$ where $\mathcal{P}_{X_{[1:M]}^n} $ denotes the set of all possible distributions of codewords satisfying the power constraint and $(a)$ is from [@DavoodiJafar:14 Section 6]. By substituting to , we obtain $d_s\leq \frac{N\alpha+M-1}{M+1}$. On the other hand, from , we have $d_s\leq \min(N\alpha, 1)$. Therefore, $$\begin{aligned} d_s\leq \min\left\{N\alpha, \frac{N\alpha+ M-1}{M+1}, 1\right\}.\end{aligned}$$ For the special case of $N=1$, to derive a tighter bound, we start from : $$\begin{aligned} nR&\leq H(X_{[1:M]}^n\|S^n_{[2:M]}) - H(Y_2^n\|S^n_{[2:M]}) + nc_2\\ &\leq H(X_{1}^n\|S^n_{[2:M]})+\sum_{i=2}^M H(X_{i}^n\|S^n_{i}) - H(Y_2^n\|S^n_{[2:M]}) + nc_2. \label{eqn:cov_1}\end{aligned}$$ Now, by combining and , we have $$\begin{aligned} MnR&\leq H(X_{1}^n\|S^n_{[2:M]})+ (M-1)H(Y_1^n\|S^n_{[2:M]})-H(Y_2^n\|S^n_{[2:M]})+nc_6 \\ &=H(X_{1}^n\|S^n_{[2:M]})+ (M-2)H(Y_1^n\|S^n_{[2:M]}) +H(Y_1^n\|S^n_{[2:M]})-H(Y_2^n\|S^n_{[2:M]})+nc_6 \\ &\overset{(a)}{\leq} H(X_{1}^n\|S^n_{[2:M]})+ (M-2)H(Y_1^n\|S^n_{[2:M]}) +n\cdot o(\log P)+nc_6\\ &\overset{(b)}\leq H(X_{1}^n\|X^n_{[2:M]})+ (M-2)H(Y_1^n\|S^n_{[2:M]})+n\cdot o(\log P)+nc_6, \label{eqn:cov_3} \end{aligned}$$ where $(a)$ is from similar steps used to obtain and $(b)$ is due to the Markov chain $X_1^n-(S^n_{[2:M]},\mathbf{h}^n, \mathbf{g}^n)-X^n_{[2:M]}$ that can be shown by marginalizing $p(\mathbf{h}^n,\mathbf{g}^n)p(s_1^n, \cdots, s_M^n)\prod_{t=1}^n \prod_{k=1}^M p(x_{k}(t)\|s_k^n, x_k^{t-1}, \mathbf{h}^t)$. To bound $H(X_{1}^n\|X^n_{[2:M]})$, we have $$\begin{aligned} H(Y_1^n)&= H\big(\big\{\sum_{i = 1}^M \lfloor h_i(t) X_i(t)\rfloor\big\}_{t=1}^n \big)\\ &\geq H\big(\big\{\sum_{i = 1}^M \lfloor h_i(t) X_i(t)\rfloor\big\}_{t=1}^n \|X_{[2:M]}^n\big)\\ &= H\big(\big\{ \lfloor h_1(t) X_1(t)\rfloor\big\}_{t=1}^n \|X_{[2:M]}^n\big)\\ &=H( X_1^n, \big\{ \lfloor h_1(t) X_1(t)\rfloor\big\}_{t=1}^n \|X_{[2:M]}^n) -H(X_1^n\|\big\{ \lfloor h_1(t) X_1(t)\rfloor\big\}_{t=1}^n , X_{[2:M]}^n)\\ &=H( X_1^n \| X_{[2:M]}^n) -H(X_1^n\|\big\{ \lfloor h_1(t) X_1(t)\rfloor\big\}_{t=1}^n , X_{[2:M]}^n)\\ &\geq H( X_1^n \| X_{[2:M]}^n)-\sum_{t=1}^nH(X_1(t)\|\lfloor h_1(t) X_1(t)\rfloor)\\ &\overset{(a)}{\geq} H( X_1^n\| X_{[2:M]}^n)-nc_7, \label{eqn:cov_4}\end{aligned}$$ where $(a)$ is from [@MukherjeeXieUlukus:arxiv15 Lemma 2]. By combining and , we obtain $$\begin{aligned} MnR&\leq H(Y_1^n)+ (M-2)H(Y_1^n\|S^n_{[2:M]})+n\cdot o(\log P)+nc_8\\ &\leq (M-1)H(Y_1^n)+n\cdot o(\log P)+nc_8. $$ Hence, $d_s\leq \frac{M-1}{M}$. Therefore, for $N=1$, we have $$\begin{aligned} d_s\leq \min\left\{\alpha, \frac{M-1}{M}\right\},\end{aligned}$$ which concludes the proof. We note that the proof technique used for the special case of $N=1$ can be extended for the general case of $N\geq 1$, but the resultant bound is loose when $N>1$. Conclusion {#sec:conclusion} ========== In this paper, we established the optimal secure d.o.f. of the wiretapped diamond-relay channel where an external eavesdropper not only wiretaps the relay transmissions in the multiple-access part, but also observes a certain number of source-relay links in the broadcast part. The secure d.o.f. was shown to be the same for the cases with and without the knowledge of location of wiretapped links. For the former case, we proposed two constituent jamming schemes. The first scheme is the S-BCJ scheme where the noise symbols are aligned at the destination and the second scheme is the S-CoJ scheme where the noise symbols are beam-formed in the null space of the destination’s channel. For the latter case where the location of wiretapped links is unknown, we combined a SNC-like scheme with S-BCJ and S-CoJ in a way that (i) the fictitious message symbols of SNC mask all the message and artificial noise symbols transmitted over the source-relay links and (ii) the fictitious message symbols can be canceled at the destination via beamforming at the relays. We believe that our results on the wiretapped diamond-relay channel are an important step towards understanding the secrecy capacity of general multi-terminal networks, which remains a largely open problem till date. Proof of the achievability part of Theorem \[thm:less\] {#appendix:achievability} ======================================================= Let us first state two theorems that give us achievable secrecy rates for the wiretapped diamond-relay channel. These theorems are obtained by assuming symbol-wise relay operations and then considering the wiretapped diamond-relay channel as the wiretap channel [@CsiszarKorner:78]. These theorems are proved in the end of this appendix. \[thm:sec\_rate\] For the wiretapped diamond-relay channel with the knowledge of location of wiretapped links, a secrecy rate $R$ is achievable if $$\begin{aligned} R\leq I(V;Y_1\|\mathbf{h})-I(V;Y_2, S_{T}\|\mathbf{h},\mathbf{g})\end{aligned}$$ for some $p(v)p(s_{[1:M]}\|v)\prod_{k\in[1:M]}p(x_k\|s_k,\mathbf{h})$ such that $H(S_k)\leq C$ and $\E[X_k^2]\leq P$ for $k\in [1:M]$. \[thm:sec\_rate\_no\] For the wiretapped diamond-relay channel without the knowledge of location of wiretapped links, a secrecy rate $R$ is achievable if $$\begin{aligned} R\leq I(V;Y_1\|\mathbf{h})-I(V;Y_2, S_T\|\mathbf{h},\mathbf{g})\end{aligned}$$ for all $T\subseteq [1:M]$ such that $\|T\|=W$, for some $p(v)p(s_{[1:M]}\|v)\prod_{k\in[1:M]}p(x_k\|s_k,\mathbf{h})$ such that $H(S_k)\leq C$ and $\E[X_k^2]\leq P$ for $k\in [1:M]$. Next, let us present two lemmas used for the proof. The proofs of these lemmas are relegated to the end of this appendix. The first lemma is a result from Khintchine-Groshev theorem of Diophantine approximation [@MotahariOveisGharanMaddahAliKhandani:14], which plays a key role in the analysis of interference alignment. \[lemma:Fano\] Consider a random vector $A=(A_1,\cdots,A_{\tau})$ where each of $A_k$’s is a random variable distributed over $\mathcal{C}(\delta,\theta Q)$ for some positive real number $\delta$ and positive integers $\theta$, and $Q$. Assume that a receiver observes $Y$ given as follows: $$\begin{aligned} Y=\sum_{k=1}^{\tau}\lambda_k A_k+Z\end{aligned}$$ where $\lambda_1,\cdots, \lambda_{\tau}$ are random variables whose all joint and conditional density functions are bounded[^9] and $Z$ is a Gaussian random variable with zero mean and unit variance. Fix arbitrary $\epsilon>0$ and $\gamma>0$. If we choose $Q=P^{\frac{1-\epsilon}{2(\tau+\epsilon)}}$ and $\delta=\frac{\gamma P^{1/2}}{ Q}$, it follows $$\begin{aligned} H(A\|Y,\lambda_1,\cdots,\lambda_{\tau})\leq o(\log P). \end{aligned}$$ The following lemma is used for constructing linear combinations of fictitious message symbols for the case without the knowledge of location of wiretapped links. \[lemma:mds\] For any positive integers $j$ and $k$ such that $j\leq k$, we can construct a $j\times k$ matrix $\Gamma$ with the following properties: - each element of $\Gamma$ is a non-negative integer smaller than $p$, where $p$ is the smallest prime number greater than or equal to $k$, and - any $j$ columns are linearly independent. Now, we are ready to prove the achievability part of Theorem \[thm:less\]. Note that it is sufficient to show the achievability of $(\alpha, d_s)=(\frac{M-1}{MN}, \frac{M-1}{M})$ for $N\geq 1$ and $(\alpha, d_s)=(\frac{2}{N}, 1)$ for $N\geq 2$. Proof of the achievability part of Theorem \[thm:less\] for the case with the knowledge of location of wiretapped links ----------------------------------------------------------------------------------------------------------------------- Without loss of generality, let us assume that the first $N$ links are secure, i.e., $T=[N+1:M]$. ### S-BCJ scheme achieving $(\alpha, d_s)=(\frac{M-1}{MN}, \frac{M-1}{M})$ Let us apply Theorem \[thm:sec\_rate\] with the following choice of $p(v)p(s_{[1:M]}\|v)\prod_{k\in[1:M]}p(x_k\|s_k,\mathbf{h})$: $$\begin{aligned} V&=(V_{k,j}: k\in [1:N], j\in [1:M-1])\\ S_k&=\begin{cases} (V_{k,j}: j\in [1:M-1]), & k\in [1:N]\\ \emptyset, &k\in [N+1:M] \end{cases}\\ X_k&=\begin{cases} \sum_{j\in [1:M-1]} \mu_{k,j} V_{k,j} + \sum_{j\in [1:N]} \frac{\nu_j}{h_k}U_{k,j}, & k\in [1:N]\\ \sum_{j\in [1:N]} \frac{\nu_j}{h_k}U_{k,j}, & k\in [N+1:M] \end{cases}\end{aligned}$$ where $V_{k,j}$’s and $U_{k,j}$’s are independently generated according to $\mbox{Unif}[\mathcal{C}(\delta,Q)]$ for some positive real number $\delta$ and positive integer $Q$ specified later, and $\mu_{k,j}$’s and $\nu_j$’s are independently and uniformly chosen from the interval $[-B, B]$. We note that $H(S_k)\leq C$ and $\E[X_k^2]\leq P$ for $k\in [1:M]$ are satisfied if $$\begin{aligned} (M-1)\log (2Q+1)&\leq C \label{eqn:link}\\ \tilde{\gamma} \delta Q &\leq \sqrt{P}, \label{eqn:power}\end{aligned}$$ where $\tilde{\gamma}=(M-1+NB)B$. Then, the channel outputs are given as $$\begin{aligned} Y_1&= \sum_{k=1}^N\sum_{j=1}^{M-1} h_k\mu_{k,j} V_{k,j} + \sum_{k=1}^M \sum_{j=1}^N \nu_j U_{k,j} +Z_1 \\ &=\sum_{k=1}^N\sum_{j=1}^{M-1} h_k\mu_{k,j} V_{k,j} + \sum_{j=1}^N \nu_j\left[\sum_{k=1}^M U_{k,j}\right] +Z_1 \label{eqn:y1_less}\\ Y_2&= \sum_{k=1}^N\sum_{j=1}^{M-1} g_k\mu_{k,j} V_{k,j} + \sum_{k=1}^M \sum_{j=1}^N \frac{g_k\nu_j}{h_k}U_{k,j} +Z_2.\end{aligned}$$ Because $S_T=\emptyset$, Theorem \[thm:sec\_rate\] says that the following secrecy rate is achievable: $$\begin{aligned} R\leq I(V;Y_1\|\mathbf{h})-I(V;Y_2\|\mathbf{h},\mathbf{g}).\label{eqn:achie}\end{aligned}$$ To derive a lower bound on the RHS of , let us derive a lower and an upper bounds on the first and the second terms in the RHS of , respectively. We will apply Lemma \[lemma:Fano\] with $\tau \Leftarrow MN$ and hence we choose $Q = P^{\frac{1 - \epsilon}{2(MN + \epsilon)}}$ and $\delta = \frac{ \gamma P^{1/2}}{Q}$ for some $\epsilon>0$ with $\gamma=\tilde{\gamma}^{-1}$ to satisfy the power constraint . Now, the first term in the RHS of is bounded as follows: $$\begin{aligned} I(V;Y_1\|\mathbf{h})&=H(V)-H(V\|Y_1,\mathbf{h})\\ &\overset{(a)}\geq \log(2 Q+1)^{N(M-1)} - o(\log{P})\\ &\geq \frac{(1-\epsilon)N(M-1)}{2(MN+\epsilon)}\log P-o(\log P),\label{eqn:same_first}\end{aligned}$$ where $(a)$ is due to Lemma \[lemma:Fano\] with the substitution of $Y\Leftarrow Y_1$, $\tau\Leftarrow MN$ and $\theta\Leftarrow M$.[^10] Next, we have $$\begin{aligned} I(V;Y_2\|\mathbf{h},\mathbf{g}) &=I(V,U;Y_2\|\mathbf{h},\mathbf{g})-I(U;Y_2\|V,\mathbf{h},\mathbf{g})\\ &=I(V,U;Y_2\|\mathbf{h},\mathbf{g})-H(U)+H(U\|Y_{2,\mathrm{eff}},\mathbf{h},\mathbf{g})\\ &=I(V,U;Y_2\|\mathbf{h},\mathbf{g})-\frac{(1-\epsilon)MN}{2(MN+\epsilon)}\log P+H(U\|Y_{2,\mathrm{eff}},\mathbf{h},\mathbf{g})\\ &\overset{(a)}\leq I(V,U;Y_2\|\mathbf{h},\mathbf{g})-\frac{(1-\epsilon)MN}{2(MN+\epsilon)}\log P+o(\log P)\\ &\leq h(Y_2\|\mathbf{h},\mathbf{g})-h(Z_2)-\frac{(1-\epsilon)MN}{2(MN+\epsilon)}\log P+o(\log P)\\ &\overset{(b)}\leq \frac{1}{2}\log P -\frac{1}{2}\log 2\pi e -\frac{(1-\epsilon)MN}{2(MN+\epsilon)}\log P+o(\log P)\\ &\leq \frac{\epsilon(MN+1)}{2(MN+\epsilon)}\log P+o(\log P),\label{eqn:same_second}\end{aligned}$$ where $U=(U_{k,j}: k\in [1:M], j\in[1:N])$, $Y_{2,\mathrm{eff}}= \sum_{k=1}^M \sum_{j=1}^N \frac{g_k\nu_j}{h_k}U_{k,j} +Z_2$, $(a)$ follows from Lemma \[lemma:Fano\] with $Y\Leftarrow Y_{2,\mathrm{eff}}$, $\tau\Leftarrow MN$ and $\theta\Leftarrow 1$, and $(b)$ is because all channel fading coefficients are assumed to be bounded away from zero and infinity. By choosing $\epsilon$ sufficiently small, we conclude from , , and that $(\alpha, d_s)=(\frac{M-1}{MN}, \frac{M-1}{M})$ is achievable. ### S-CoJ scheme for $N\geq 2$ achieving $(\alpha, d_s)=(\frac{2}{N}, 1)$ Let us apply Theorem \[thm:sec\_rate\] with the following choice of $p(v)p(s_{[1:M]}\|v)\prod_{k\in[1:M]}p(x_k\|s_k,\mathbf{h})$: $$\begin{aligned} V&=(V_{k}: k\in [1:N])\\ S_k&=\begin{cases} (V_{k}+U_k,U_{[k+1]_N}), & k\in [1:N]\\ \emptyset, &k\in [N+1:M] \end{cases}\\ X_k&=\begin{cases} V_k+U_k-\frac{h_{[k+1]_N}}{h_k}U_{[k+1]_{N}}, & k\in [1:N]\\ \emptyset, & k\in [N+1:M] \end{cases}\end{aligned}$$ where $V_{k}$’s and $U_{k}$’s are independently generated according to $\mbox{Unif}[\mathcal{C}(\delta,Q)]$ for some positive real number $\delta$ and positive integer $Q$ specified later. We note that $H(S_k)\leq C$ and $\E[X_k^2]\leq P$ for $k\in [1:M]$ are satisfied if $$\begin{aligned} 2\log (4Q+1)&\leq C \label{eqn:link1}\\ \tilde{\gamma} \delta Q &\leq \sqrt{P}, \label{eqn:power1}\end{aligned}$$ where $\tilde{\gamma}=2+B^2$. Then, the channel outputs are given as follows: $$\begin{aligned} Y_1&= \sum_{k=1}^N h_k V_k +Z_1 \label{eqn:y1_less4}\\ Y_2&= \sum_{k=1}^N g_k V_k + \sum_{k=1}^N \left(g_k-\frac{h_k\cdot g_{[k-1]_N}}{h_{[k-1]_N}}\right)U_k +Z_2.\end{aligned}$$ Because $S_T=\emptyset$, from Theorem \[thm:sec\_rate\], the following secrecy rate is achievable: $$\begin{aligned} R\leq I(V;Y_1\|\mathbf{h})-I(V;Y_2\|\mathbf{h},\mathbf{g}).\label{eqn:achie1}\end{aligned}$$ To derive a lower bound on the RHS of , let us derive a lower and an upper bounds on the first and the second terms in the RHS of , respectively. We will apply Lemma \[lemma:Fano\] with $\tau \Leftarrow N$ and hence we choose $Q = P^{\frac{1 - \epsilon}{2(N + \epsilon)}}$ and $\delta = \frac{ \gamma P^{1/2}}{Q}$ for some $\epsilon>0$ with $\gamma=\tilde{\gamma}^{-1}$ to satisfy the power constraint . Now, the first term in the RHS of is bounded as follows: $$\begin{aligned} I(V;Y_1\|\mathbf{h})&=H(V)-H(V\|Y_1,\mathbf{h})\\ &\overset{(a)}\geq \log(2 Q+1)^{N} - o(\log{P})\\ &\geq \frac{(1-\epsilon)N}{2(N+\epsilon)}\log P-o(\log P),\label{eqn:same_first1}\end{aligned}$$ where $(a)$ is due to Lemma \[lemma:Fano\] with the substitution of $Y\Leftarrow Y_1$, $\tau\Leftarrow N$, $\theta\Leftarrow 1$. Next, by applying similar steps used to derive , we can show $$\begin{aligned} I(V;Y_2\|\mathbf{h},\mathbf{g})&\leq \frac{\epsilon(N+1)}{2(N+\epsilon)}\log P+o(\log P).\label{eqn:same_second1}\end{aligned}$$ By choosing $\epsilon$ sufficiently small, we conclude from , , and that $(\alpha, d_s)=(\frac{2}{N},1)$ is achievable. Proof of the achievability part of Theorem \[thm:less\] for the case without the knowledge of location of wiretapped links -------------------------------------------------------------------------------------------------------------------------- For the case without the knowledge of location of wiretapped links, we superpose a linear combination of fictitious message symbols to each element of $S_k$ in the S-BCJ and S-CoJ schemes. ### S-BCJ-SNC scheme achieving $(\alpha, d_s)=(\frac{M-1}{MN}, \frac{M-1}{M})$ Let $F=(F_k: k\in [1:W(M-1)])$ denote the vector of $W(M-1)$ fictitious message symbols each independently generated according to $\mbox{Unif}[\mathcal{C}(\delta,Q)]$ for some positive real number $\delta$ and positive integer $Q$ to be specified later. To mask each element transmitted over the source-relay links in the S-BCJ scheme, we generate a vector $L=(L_{k,j}: k\in [1:M], j\in [1:M-1])$ of linear combinations of fictitious message symbols by computing $L=F\Gamma$, where $\Gamma$ is an $W(M-1)\times M(M-1)$ matrix that satisfies the properties in Lemma \[lemma:mds\], i.e., each element of $\Gamma$ is a non-negative integer smaller than $p$, where $p$ is the smallest prime number greater than or equal to $M(M-1)$, and every $W(M-1)$ columns of $\Gamma$ are linearly independent. Note that the domain of each element of $L$ is a subset of $\mathcal{C}(\delta, W(M-1)(p-1)Q)$. Now, we apply Theorem \[thm:sec\_rate\_no\] with the following choice of $p(v)p(s_{[1:M]}\|v)\prod_{k\in[1:M]}p(x_k\|s_k,\mathbf{h})$: $$\begin{aligned} V&=(V_{k,j}: k\in [1:M], j\in [1:M-1])\\ S_k&=\begin{cases} (V_{k,j}+L_{k,j}: j\in [1:M-1]), &k\in [1:N] \\ (L_{k,j}: j\in [1:M-1]), &k\in [N+1:M] \\ \end{cases}\end{aligned}$$ $$\begin{aligned} X_k&=\begin{cases} \sum_{j\in [1:M-1]} \mu_{k,j} (V_{k,j}+L_{k,j})+ \sum_{j\in [1:N]} \frac{\nu_j}{h_k}U_{k,j}, &k\in [1:N]\\ \sum_{j\in [1:M-1]} \rho_{k,j}L_{k,j} +\sum_{j\in [1:N]} \frac{\nu_j}{h_k}U_{k,j}, & k\in [N+1:M] \end{cases}\end{aligned}$$ where $V_{k,j}$’s and $U_{k,j}$’s are independently generated according to $\mbox{Unif}[\mathcal{C}(\delta,Q)]$ and $\mu_{k,j}$’s and $\nu_j$’s are independently and uniformly chosen from the interval $[-B, B]$. $\rho_{k,j}$’s are carefully chosen to cancel out the fictitious message symbols at the destination as follows. We first note that because any $W(M-1)$ columns of $\Gamma$ are linearly independent, for $a\in [1:N]$ and $b\in[1:M-1]$, there exists $(\sigma_{a,b\|k,j}: k\in [N+1:M], j\in [1:M-1])$ such that $$\begin{aligned} L_{a,b}=\sum_{k\in[N+1:M]}\sum_{j\in [1:M-1]}\sigma_{a,b\|k,j}L_{k,j}.\end{aligned}$$ To beam-form each of $F_k$’s in the null space of the destination’s channel, $\rho_{k,j}$ for $k\in [N+1:M]$ and $j\in [1:M-1]$ is chosen as $$\begin{aligned} \rho_{k,j}=-\sum_{a\in[1:N]}\sum_{b\in [1:M-1]}\frac{h_a}{h_k}\mu_{a,b}\sigma_{a,b\|k,j}. \end{aligned}$$ Let $\rho_{\max}$ denote the maximum of $\rho_{k,j}$’s. We note that $H(S_k)\leq C$ and $\E[X_k^2]\leq P$ for $k\in [1:M]$ are satisfied if $$\begin{aligned} (M-1)\log (2(W(M-1)(p-1)+1)Q+1)\leq C \label{eqn:link2}\\ \tilde{\gamma}\delta Q \leq \sqrt{P}, \label{eqn:power2}\end{aligned}$$ where $$\begin{aligned} &\tilde{\gamma}=\max\{(M-1)B(W(M-1)(p-1)+1), (M-1)\rho_{\max}W(M-1)(p-1)\}+NB^2.\end{aligned}$$ Then, the channel outputs are given as $$\begin{aligned} Y_1&= \sum_{k=1}^N\sum_{j=1}^{M-1} h_k\mu_{k,j} V_{k,j} + \sum_{k=1}^M \sum_{j=1}^N \nu_j U_{k,j} +Z_1 \\ &=\sum_{k=1}^N\sum_{j=1}^{M-1} h_k\mu_{k,j} V_{k,j} + \sum_{j=1}^N \nu_j\left[\sum_{k=1}^M U_{k,j}\right] +Z_1 \label{eqn:y1_less2} \\ Y_2&= \sum_{k=1}^N\sum_{j=1}^{M-1} g_k\mu_{k,j} V_{k,j} + \sum_{k=1}^M \sum_{j=1}^N \frac{g_k\nu_j}{h_k}U_{k,j} +\sum_{k=1}^{W(M-1)} \chi_k F_k+Z_2,\end{aligned}$$ where $\chi_k$’s are determined from $\mu_{k,j}$’s, $\rho_{k,j}$’s, $\mathbf{h}$, $\mathbf{g}$, and $\Gamma$. From Theorem \[thm:sec\_rate\_no\], the following secrecy rate is achievable: $$\begin{aligned} R\leq I(V;Y_1\|\mathbf{h})-\max I(V;Y_2,S_T\|\mathbf{h},\mathbf{g}),\label{eqn:achie_no}\end{aligned}$$ where the maximization is over all $T\subseteq [1:M]$ such that $\|T\|=W$. To derive a lower bound on the RHS of , let us derive a lower and an upper bounds on the first and the second terms in the RHS of , respectively. We will apply Lemma \[lemma:Fano\] with $\tau \Leftarrow MN$ and hence we choose $Q = P^{\frac{1 - \epsilon}{2(MN + \epsilon)}}$ and $\delta = \frac{ \gamma P^{1/2}}{Q}$ for some $\epsilon>0$ with $\gamma=\tilde{\gamma}^{-1}$ to satisfy the power constraint . Then, by applying the same bounding techniques used to obtain , we can obtain $$\begin{aligned} I(V;Y_1\|\mathbf{h})\geq \frac{(1-\epsilon)N(M-1)}{2(MN+\epsilon)}\log P-o(\log P).\label{eqn:same_first2}\end{aligned}$$ Next, let us derive an upper bound on the second term in the RHS of . We first fix an arbitrary $T\subseteq [1:M]$ such that $\|T\|=W$. Note that $$\begin{aligned} &I(V;Y_2,S_T\|\mathbf{h},\mathbf{g})=I(V;S_T\|\mathbf{h},\mathbf{g})+I(V;Y_2\|S_T,\mathbf{h},\mathbf{g}). \label{eqn:sec_split}\end{aligned}$$ We first bound the first term in the RHS of as follows: $$\begin{aligned} I(V;S_T\|\mathbf{h},\mathbf{g})&=H(S_T)-H(S_T\|V)\\ &\overset{(a)}\leq W(M-1)\log (2(W(M-1)(p-1)+1)Q+1)-H(S_T\|V) \\ &= W(M-1)\log (2(W(M-1)(p-1)+1)Q+1)-H(L_{k,j}: k\in T, j\in [M-1]) \\ &\overset{(b)}=W(M-1)\log (2(W(M-1)(p-1)+1)Q+1)-H(F) \\ &=W(M-1)\log (2(W(M-1)(p-1)+1)Q+1) -W(M-1)\log (2Q+1)\\ &=W(M-1)\log \frac{2(W(M-1)(p-1)+1)Q+1}{2Q+1}\\ &=o(\log P), \label{eqn:sp_f}\end{aligned}$$ where $(a)$ is because $S_T$ consists of $W(M-1)$ elements where the domain of each element is a subset of $\mathcal{C}(\delta, (W(M-1)(p-1)+1)Q)$ and $(b)$ is because any $W(M-1)$ columns of $\Gamma$ are linearly independent. Next, the second term in the RHS of is bounded as follows: $$\begin{aligned} I(V;Y_2\|S_T,\mathbf{h},\mathbf{g}) &=I(V,U,F;Y_2\|S_T,\mathbf{h},\mathbf{g})-I(U,F;Y_2\|V,S_T,\mathbf{h},\mathbf{g})\\ &\leq I(V,U,F;Y_2\|S_T,\mathbf{h},\mathbf{g})-I(U;Y_2\|F,V,S_T,\mathbf{h},\mathbf{g})\\ &\overset{(a)}= I(V,U,F;Y_2\|S_T,\mathbf{h},\mathbf{g})-H(U)+H(U\|Y_{2,\mathrm{eff}},\mathbf{h},\mathbf{g})\\ &= I(V,U,F;Y_2\|S_T,\mathbf{h},\mathbf{g})-\frac{(1-\epsilon)MN}{2(MN+\epsilon)}\log P+H(U\|Y_{2,\mathrm{eff}},\mathbf{h},\mathbf{g})\\ &\overset{(b)}\leq I(V,U,F;Y_2\|S_T,\mathbf{h},\mathbf{g})-\frac{(1-\epsilon)MN}{2(MN+\epsilon)}\log P+o(\log P)\\ &\leq h(Y_2\|\mathbf{h},\mathbf{g})-h(Z_2)-\frac{(1-\epsilon)MN}{2(MN+\epsilon)}\log P+o(\log P)\\ &\leq \frac{1}{2}\log P- \frac{1}{2}\log 2\pi e-\frac{(1-\epsilon)MN}{2(MN+\epsilon)}\log P+o(\log P)\\ &\leq \frac{\epsilon(MN+1)}{2(MN+\epsilon)}\log P+o(\log P)\end{aligned}$$ where $U=(U_{k,j}: k\in [1:M], j\in[1:N])$, $Y_{2,\mathrm{eff}}= \sum_{k=1}^M \sum_{j=1}^N \frac{g_k\nu_j}{h_k}U_{k,j}+Z_2$, $(a)$ is because $U$ is independent from $F, V, S_T, \mathbf{h}$ and $\mathbf{g}$, and $(b)$ follows from Lemma \[lemma:Fano\] with $Y\Leftarrow Y_{2,\mathrm{eff}}$, $\tau\Leftarrow MN$ and $\theta\Leftarrow 1$. Since the above bounds do not depend on the choice of $T$, we obtain $$\begin{aligned} &\max_T I(V;Y_2,S_T\|\mathbf{h},\mathbf{g})\leq \frac{\epsilon(MN+1)}{2(MN+\epsilon)}\log P+o(\log P). \label{eqn:less_second2}\end{aligned}$$ By choosing $\epsilon$ sufficiently small, it follows from , , that $(\alpha, d_s)=(\frac{M-1}{MN},\frac{M-1}{M})$ is achievable. ### S-CoJ-SNC scheme achieving $(\alpha, d_s)=(\frac{2}{N}, 1)$ Let $F=(F_k: k\in [1:2W])$ denote $2W$ fictitious message symbols each independently generated according to $\mbox{Unif}[\mathcal{C}(\delta,Q)]$ for some positive real number $\delta$ and positive integer $Q$ to be specified later. We generate a vector $L=(L_k: k\in [1:2M])$ of linear combinations of fictitious message symbols by evaluating $L=F\Gamma$, where $\Gamma$ is an $2W\times 2M$ matrix that satisfies the properties in Lemma \[lemma:mds\], i.e., i.e., each element of $\Gamma$ is a non-negative integer smaller than $p$, where $p$ is the smallest prime number greater than or equal to $2M$, and every $2W$ columns of $\Gamma$ are linearly independent. Note that the domain of each element of $L$ is a subset of $\mathcal{C}(\delta, 2W(p-1)Q)$. Now, we apply Theorem \[thm:sec\_rate\_no\] with the following choice of $p(v)p(s_{[1:M]}\|v)\prod_{k\in[1:M]}p(x_k\|s_k,\mathbf{h})$: $$\begin{aligned} V&=(V_{k}: k\in [1:N])\\ S_k&=\begin{cases} (V_{k}+U_k+L_{2k-1},U_{[k+1]_N}+L_{2k}), &k\in [1:N] \\ (L_{2k-1}, L_{2k}), &k\in [N+1:M] \end{cases}\\ X_k&=\begin{cases} V_k+U_k+L_{2k-1}-\frac{h_{[k+1]_N}}{h_k}(U_{[k+1]_{N}}+L_{2k}), &k\in [1:N]\\ \rho_{2k-1}L_{2k-1}+\rho_{2k}L_{2k}, &k\in [N+1:M] \end{cases}\end{aligned}$$ where $V_{k}$’s and $U_{k}$’s are independently generated according to $\mbox{Unif}[\mathcal{C}(\delta,Q)]$ and $\rho_{k}$’s are chosen to cancel out the fictitious message symbols at the destination as follows. We first note that because any $2W$ columns of $\Gamma$ are linearly independent, for $k\in [1:2N]$, there exists $(\sigma_{k\|j}: j\in [2N+1:2M])$ such that $$\begin{aligned} L_{k}=\sum_{j\in[2N+1:2M]}\sigma_{k\|j}L_{j}.\end{aligned}$$ To beam-form each of $F_k$’s in the null space of the destination’s channel, $\rho_{2j-1}$ and $\rho_{2j}$ for $j\in [N+1:M]$ are chosen as $$\begin{aligned} \rho_{2j-1}&=-\sum_{k\in[1:N]}\frac{h_k}{h_j}\sigma_{2k-1\|2j-1}-\sum_{k\in[1:N]}\frac{h_{[k+1]_N}}{h_j}\sigma_{2k\|2j-1}\\ \rho_{2j}&=-\sum_{k\in[1:N]}\frac{h_k}{h_j}\sigma_{2k-1\|2j} -\sum_{k\in[1:N]}\frac{h_{[k+1]_N}}{h_j}\sigma_{2k\|2j}.\end{aligned}$$ Let $\rho_{\max}$ denote the maximum of $\rho_k$’s. We note that $H(S_k)\leq C$ and $\E[X_k^2]\leq P$ for $k\in [1:M]$ are satisfied if $$\begin{aligned} 2\log (4(W(p-1)+1)Q+1)\leq C \label{eqn:link3}\\ \tilde{\gamma} \delta Q \leq \sqrt{P}, \label{eqn:power3}\end{aligned}$$ where $$\begin{aligned} \tilde{\gamma}&=\max\{2+2W(p-1)+B^2(1+2W(p-1)),4\rho_{\max}W(p-1)\}.\end{aligned}$$ Then, the channel outputs are given as follows: $$\begin{aligned} Y_1&= \sum_{k=1}^N h_k V_k +Z_1 \label{eqn:y1_less3}\\ Y_2&= \sum_{k=1}^N g_k V_k + \sum_{k=1}^N \left(g_k-\frac{h_k\cdot g_{[k-1]_N}}{h_{[k-1]_N}}\right)U_k +\sum_{k=1}^{2W} \chi_k F_k+Z_2,\end{aligned}$$ where $\chi_k$’s are determined from $\rho_{k}$’s, $\mathbf{h}$, $\mathbf{g}$, and $\Gamma$. From Theorem \[thm:sec\_rate\_no\], the following secrecy rate is achievable: $$\begin{aligned} R\leq I(V;Y_1\|\mathbf{h})-\max I(V;Y_2,S_T\|\mathbf{h},\mathbf{g}),\label{eqn:achie_no2}\end{aligned}$$ where the maximization is over all $T\subseteq [1:M]$ such that $\|T\|=W$. As in the S-CoJ scheme, we will apply Lemma \[lemma:Fano\] with $\tau \Leftarrow N$ and hence we choose $Q = P^{\frac{1 - \epsilon}{2(N + \epsilon)}}$ and $\delta = \frac{ \gamma P^{1/2}}{Q}$ for some $\epsilon>0$ with $\gamma=\tilde{\gamma}^{-1}$ to satisfy the power constraint . Now, by applying the same bounding techniques used to obtain , the first term in the RHS of is bounded as follows: $$\begin{aligned} I(V;Y_1\|\mathbf{h})&\geq \frac{(1-\epsilon)N}{2(N+\epsilon)}\log P-o(\log P).\label{eqn:same_first3}\end{aligned}$$ Next, let us derive an upper bound on the second term in the RHS of . We first fix an arbitrary $T\subseteq [1:M]$ such that $\|T\|=W$. Note that $$\begin{aligned} I(V;Y_2,S_T\|\mathbf{h},\mathbf{g}) &=I(V;S_T\|\mathbf{h},\mathbf{g})+I(V,U,F;Y_2\|S_T,\mathbf{h},\mathbf{g})-I(U,F;Y_2\|V,S_T,\mathbf{h},\mathbf{g}),\label{eqn:sec_split2}\end{aligned}$$ where $U=(U_{k}: k\in [1:N])$. Then, by applying similar bounding techniques used to obtain , the first term in the RHS of can be bounded as follows: $$\begin{aligned} I(V;S_T\|\mathbf{h},\mathbf{g})\leq o(\log P). \label{eqn:f1}\end{aligned}$$ Next, the second term in the RHS of is bounded as follows: $$\begin{aligned} I(V,U,F;Y_2\|S_T,\mathbf{h},\mathbf{g})&\leq h(Y_2\|\mathbf{h},\mathbf{g})-h(Z_2)\\ &\leq \frac{1}{2} \log P + o(\log P). \label{eqn:f2}\end{aligned}$$ Finally, the third term in the RHS of is bounded as follows: $$\begin{aligned} I(U,F;Y_2\|V,S_T,\mathbf{h},\mathbf{g})&=H(U,F\|V,S_T)-H(U,F\|Y_2,V,S_T,\mathbf{h},\mathbf{g})\\ &\overset{(a)}=H(U,F,V)-H(V,S_T)-H(U,F\|Y_2,V,S_T,\mathbf{h},\mathbf{g})\\ &\overset{(b)}\geq H(U)+H(F)-H(S_T) -H(U,F\|Y_2,V,S_T,\mathbf{h},\mathbf{g})\\ &\overset{(c)}\geq H(U)-H(U,F\|Y_2,V,S_T,\mathbf{h},\mathbf{g})-o(\log P)\\ &= H(U)\!-H(U\|Y_{2},V,S_T,\mathbf{h},\mathbf{g}) \!-H(F\|U,Y_{2},V,S_T,\mathbf{h},\mathbf{g})\!-o(\log P)\label{eqn:eav_split2}\end{aligned}$$ where $(a)$ is because $S_T$ is a function of $U, F$, and $V$, $(b)$ is because $U, F, V$ are mutually independent, and $(c)$ is by applying similar steps used to obtain . To bound $H(U\|Y_{2},V,S_T,\mathbf{h},\mathbf{g})$, we note that $S_T$ can be represented as a vector of length $2W$ given as $$\begin{aligned} S_T=V\Lambda_{V,T}+U\Lambda_{U,T}+F\Gamma_{T}\end{aligned}$$ where $\Lambda_{V,T}$ and $\Lambda_{U,T}$ are $N\times 2W$ matrices that are determined from our choice of $S_k$’s and $\Gamma_{T}$ is a $2W\times 2W$ submatrix of $\Gamma$ corresponding to the choice of $T$. Because any $2W$ columns of $\Gamma$ are linearly independent, the inverse matrix $\Gamma_T^{-1}$ of $\Gamma_T$ exists and hence we can represent $F$ as follows: $$\begin{aligned} F=(S_T-V\Lambda_{V,T}-U\Lambda_{V,T})\Gamma_T^{-1}. \label{eqn:F}\end{aligned}$$ Now, by substituting for $F$ in $Y_2$, $Y_2$ can be represented as a function of $V, U, S_T$ and $Z_2$. Hence, the effective channel output $Y_{2,\mathrm{eff}}$, where the contribution from $V$ and $S_T$ are canceled out, is a linear combination of $U_k$’s and $Z_2$. By applying Lemma \[lemma:Fano\] with $Y\Leftarrow Y_{2,\mathrm{eff}}$, $\tau\Leftarrow N$, and $\theta\Leftarrow 1$, it follows that $$\begin{aligned} H(U\|Y_{2},V,S_T,\mathbf{h},\mathbf{g})&=H(U\|Y_{2,\mathrm{eff}},\mathbf{h},\mathbf{g})\\ &\leq o(\log P). \label{eqn:eav_split3}\end{aligned}$$ Furthermore, due to , it follows that $$\begin{aligned} H(F\|U,Y_{2},V,S_T,\mathbf{h},\mathbf{g})=0\label{eqn:eav_split4}\end{aligned}$$ Therefore, we have $$\begin{aligned} I(U,F;Y_2\|V,S_T,\mathbf{h},\mathbf{g}) &\geq H(U)-o(\log P)\\ &=\frac{(1-\epsilon)N}{2(N+\epsilon)}\log P-o(\log P). \label{eqn:f3}\end{aligned}$$ From , , , we obtain $$\begin{aligned} I(V;Y_2,S_T\|\mathbf{h},\mathbf{g})&\leq \frac{\epsilon(N+1)}{2(N+\epsilon)}\log P+o(\log P). \label{eqn:less_second3}\end{aligned}$$ By choosing $\epsilon$ sufficiently small, it follows from , , that $(\alpha, d_s)=(\frac{2}{N},1)$ is achievable. ### Proof of Theorems \[thm:sec\_rate\] and \[thm:sec\_rate\_no\] {#proof-of-theorems-thmsec_rate-and-thmsec_rate_no .unnumbered} Let us restrict relay operations to be symbol-wise, i.e., at time $t\in [1:n]$, relay $k\in[1:M]$ (randomly) maps $(S_k(t), \mathbf{h}(t))$ to $X_k(t)$ according to $p(x_k\|s_k, \mathbf{h})$. Then, for the case with the knowledge of location of wiretapped links, the wiretapped diamond-relay channel can be considered as the wiretap channel with channel input $(S_k: k\in [1:M])$, legitimate channel output $(Y_1,\mathbf{h})$, eavesdropper channel output $(Y_2, S_T,\mathbf{h},\mathbf{g})$, and channel distribution marginalized from $p(\mathbf{h},\mathbf{g})\prod_{k\in[1:M]}p(x_k\|s_k,\mathbf{h})p(y_1,y_2\|x_{[1:M]},\mathbf{h},\mathbf{g})$. Then, Theorem \[thm:sec\_rate\] is immediate from [@CsiszarKorner:78]. For the case without the knowledge of location of wiretapped links, Theorem \[thm:sec\_rate\_no\] can be proved in a similar manner by assuming there are multiple eavesdroppers each of which observes every different $S_T$ such that $\|T\|=W$. ### Proof of Lemma \[lemma:Fano\] {#proof-of-lemma-lemmafano .unnumbered} According to the Khintchine-Groshev theorem of Diophantine approximation [@MotahariOveisGharanMaddahAliKhandani:14], for any $\epsilon>0$ and almost all $\lambda_1,\cdots, \lambda_{\tau}$ except a set of Lebesque measure zero, there exists a constant $k_{\epsilon}$ such that $$\begin{aligned} \|\sum_{k=1}^{\tau}\lambda_k q_k\|>\frac{k_{\epsilon}}{\max_{k}\|q_k\|^{\tau-1+\epsilon}}\end{aligned}$$ holds for all $(q_1,\cdots, q_{\tau}) \neq \mathbf{0}\in \mathbbm{Z}^{\tau}$. Hence, the minimum distance $d_{\min}(\lambda_1,\cdots,\lambda_{\tau})$ between the points in $\{\sum_{k=1}^{\tau}\lambda_k a_k: a_k\in \mathcal{C}(\delta,\theta Q)\}$ is bounded as follows: $$\begin{aligned} d_{\min}(\lambda_1,\cdots,\lambda_{\tau}) \geq \frac{\delta k_{\epsilon}}{(\theta Q)^{\tau-1+ \epsilon}} \label{eqn:d_min}\end{aligned}$$ for almost all $\lambda_1,\cdots, \lambda_{\tau}$ except a set of Lebesque measure zero. Now, when $(\lambda_1,\cdots,\lambda_{\tau})$ is known to the receiver, let $\hat{A}(\lambda_1,\cdots,\lambda_{\tau})$ denote the estimate of $A$ which is chosen as follows: $$\begin{aligned} \hat{A}(\lambda_1,\cdots,\lambda_{\tau})=\operatorname{arg\,min}_{ (a_1,\cdots, a_{\tau}): \atop a_k\in \mathcal{C}(\delta,\theta Q)} \big\|Y-\sum_{k=1}^{\tau}\lambda_k a_k\big\|.\end{aligned}$$ Then, for the choice of $Q=P^{\frac{1-\epsilon}{2(\tau+\epsilon)}}$ and $\delta=\frac{\gamma P^{1/2}}{Q}$, we have $$\begin{aligned} P(\hat{A}(\lambda_1,\cdots,\lambda_{\tau}) \neq A ) &\leq \E\Big[\exp\left(-\frac{d_{\min}^2(\lambda_1,\cdots,\lambda_{\tau})}{8}\right)\Big]\\ &\overset{(a)}\leq \exp\left(-\frac{\delta^2 k_{\epsilon}^2}{8(\theta Q)^{2(\tau-1 + \epsilon) }}\right)\\ &\leq \exp\left(-\frac{\gamma^2 k_{\epsilon}^2P^{\epsilon}}{8\theta ^{2(\tau-1+\epsilon)}}\right), \end{aligned}$$ where $(a)$ is because the probability that $\lambda_1,\cdots, \lambda_{\tau}$ are included in a set of Lebesque measure zero is zero by assumption. According to the Fano’s inequality, it follows that $$\begin{aligned} H(A\|Y,\lambda_1,\cdots,\lambda_{\tau}) &\leq H(A\|\hat{A}(\lambda_1,\cdots,\lambda_{\tau}))\\ &\leq 1 + P(\hat{A}(\lambda_1,\cdots,\lambda_{\tau})\neq A) \log(\|A\| - 1)\\ & \leq 1 + \exp\left(-\frac{\gamma^2k_{\epsilon}^2P^{\epsilon}}{8\theta^{2(\tau-1+\epsilon)}}\right)\log{(2\theta Q )^{\tau}}\\ & = o(\log{P}),\end{aligned}$$ which concludes the proof. ### Proof of Lemma \[lemma:mds\] {#proof-of-lemma-lemmamds .unnumbered} Let us show that the generator matrix $\Gamma$ of a Reed-Solomon code [@ReedSolomon:60] with block length $k$, message length $j$, and alphabet size $p$ satisfies the aforementioned properties. First, because $\Gamma$ is over the prime field $\mbox{GF}(p)$, its element is an integer in the range $0, \cdots, p-1$, and hence the first property is satisfied. 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Khisti are with the Department of Electrical and Computer Engineering, University of Toronto, Toronto, Canada (e-mail: [email protected]; [email protected]). [^2]: Similarly, the other constituent scheme in [@LeeZhaoKhisti:arxiv15] for the case with no eavesdropper’s CSI operates in a time-sharing basis in a way that only a single source-relay link is active in each sub-scheme. [^3]: The former condition implies that the probability that the channel fading coefficients are in a space of Lebesque measure zero is zero. The latter condition is a mild technical condition to avoid degenerate situations and has a vanishing impact on the d.o.f. because by choosing $B$ large enough, the omitted support set can be reduced to a negligible probability. [^4]: Although the relays are assumed to know the global legitimate CSI, i.e., each relay knows all the channel fading coefficients to the destination, our proposed schemes require relays to know only some local legitimate CSI. [^5]: $\mbox{Unif}[S]$ for a set $S$ denotes the uniform distribution over $S$. When $S=[i:j]$, we use $\mbox{Unif}[i:j]$ instead of $\mbox{Unif}[[i:j]]$. [^6]: Note that for $N=1$, $d_s=\min\left\{\alpha, \frac{M-1}{M}\right\}$ can be shown to be achievable by time-sharing between $(\alpha,d_s)=(0,0)$ and $(\alpha,d_s)=(\frac{M-1}{M}, \frac{M-1}{M})$. For $N\geq 2$, $d_s=\min\left\{N\alpha,\frac{N\alpha + M - 1}{M + 1},1\right\}$ can be shown to be achievable by time-sharing among $(\alpha,d_s)=(0,0)$, $(\alpha,d_s)=(\frac{M-1}{MN}, \frac{M-1}{M})$, and $(\alpha,d_s)=(\frac{2}{N}, 1)$. [^7]: We omit a formal proof as it is straightforward from that in [@MukherjeeXieUlukus:arxiv15 Section 4.2.1]. [^8]: We remind that $\mathbf{h}^n$ and $\mathbf{g}^n$ are conditioned in every entropy and mutual information terms in this section. [^9]: This condition implies that a space of Lebesque measure zero cannot carry a nonzero probability. [^10]: $\theta\Leftarrow M$ is because $\sum_{k=1}^M U_{k,j}$ for each $j\in [1:N]$ is distributed over $\mathcal{C}(\delta, MQ)$ and $\mathcal{C}(\delta,Q)$ is a subset of $\mathcal{C}(\delta, MQ)$.
--- author: - 'Lukas Liebel^1^[^1]' - Ksenia Bittner^2^ - Marco Körner^1^ bibliography: - 'bibliography.bib' title: \| A Generalized Multi-Task Learning Approach to\ Stereo DSM Filtering in Urban Areas --- City models and height maps of urban areas serve as a valuable data source for numerous applications, such as disaster management or city planning. While this information is not globally available, it can be substituted by , automatically produced from inexpensive satellite imagery. However, stereo often suffer from noise and blur. Furthermore, they are heavily distorted by vegetation, which is of lesser relevance for most applications. Such basic models can be filtered by , trained on labels derived from and 3D city models, in order to obtain a refined . We propose a modular multi-task learning concept that consolidates existing approaches into a generalized framework. Our encoder-decoder models with shared encoders and multiple task-specific decoders leverage roof type classification as a secondary task and multiple objectives including a conditional adversarial term. The contributing single-objective losses are automatically weighted in the final multi-task loss function based on learned uncertainty estimates. We evaluated the performance of specific instances of this family of network architectures. Our method consistently outperforms the state of the art on common data, both quantitatively and qualitatively, and generalizes well to a new dataset of an independent study area. Acknowledgements {#acknowledgements .unnumbered} ================ The work of Lukas Liebel was funded by the German Federal Ministry of Transport and Digital Infrastructure (BMVI) under reference 16AVF2019A. [^1]: Corresponding Author: [[email protected]]([email protected])
--- abstract: 'Substantial progress has been made in identifying single genetic variants predisposing to common complex diseases. Nonetheless, the genetic etiology of human diseases remains largely unknown. Human complex diseases are likely influenced by the joint effect of a large number of genetic variants instead of a single variant. The joint analysis of multiple genetic variants considering linkage disequilibrium (LD) and potential interactions can further enhance the discovery process, leading to the identification of new disease-susceptibility genetic variants. Motivated by the recent development in spatial statistics, we propose a new statistical model based on the random field theory, referred to as a genetic random field model (GenRF), for joint association analysis with the consideration of possible gene-gene interactions and LD. Using a pseudo-likelihood approach, a GenRF test for the joint association of multiple genetic variants is developed, which has the following advantages: 1. considering complex interactions for improved performance; 2. natural dimension reduction; 3. boosting power in the presence of LD; 4. computationally efficient. Simulation studies are conducted under various scenarios. Compared with a commonly adopted kernel machine approach, SKAT, GenRF shows overall comparable performance and better performance in the presence of complex interactions. The method is further illustrated by an application to the Dallas Heart Study.' address: - - '\University of Michigan' - '\\Michigan State University' author: - - - - title: Modeling and Testing for Joint Association Using a Genetic Random Field Model --- , , Joint association; Random field; High-Dimensional test; Complex interaction; Linkage disequilibrium. Introduction {#s:intro} ============ With the advance of high-throughput technologies, high-dimensional genetic data have been widely used in association studies for the identification of genetic variants contributing to common complex diseases. While a large number of genetic variants have been revealed today to be individually associated with complex diseases, they only explain a small proportion of heritability (Manolio, et al., 2009). On one hand, complex diseases are likely influenced by the joint effect of genetic variants through complex biology pathways, given the fact that genes are the functional sets (Wei, et al., 2013). On the other hand, the multiple testing problem occurs when one considers a set of single locus analyses, which dramatically diminishes the power. Therefore, the joint analysis of a functional set of genetic variants simultaneously can further enhance the discovery process, leading to the identification of new genetic variants associated with complex diseases (Chatterjee, et al. 2006). While the conventional linear or logistic regression models can easily be used for joint association analyses, they are subject to several issues, such as multiple-collinearity, when dealing with a large ensemble of dense genetic markers. The exponentially increased number of parameters also making the methods impractical to model two-way or high-order interactions among a large number of genetic variants (Cordell, 2009; Ritchie, et al., 2001). Several new statistical methods have also been recently developed for joint association analysis. It is worthwhile to note two recently developed methods: the kernel machine based methods (well known as SKAT)(Wu, et al.; 2010; Wu, et al. 2011) and the similarity regression (SIMreg) (Tzeng, et al. 2009). SKAT is developed from a kernel machine or random effect model framework and SIMreg directly models trait similarity as a function of genetic similarity. Both methods significantly reduce the number of regression parameters, making it feasible and computationally efficient to handle high-dimensional variants. In addition, both SKAT and SIMreg use flexible frameworks to exploit and account for linkage disequilibrium (LD) and potential interactions, which further improve the performance of the methods. The two methods have also been extended in several ways. Tzeng, et al. (2011) extended SIMreg to evaluate gene-environment ($G\times E$) interaction and showed the close link between SIMreg and SKAT; Li, et al. (2012) developed another kernel machine based method for gene-gene ($G\times G$) interaction; Maity, et al. (2011) further applied garrote kernel to evaluate a single variant, considering both the main effect and potential interactions with other genetic variants. In this paper, we propose a novel, random field framework for modeling and testing for the joint association of multiple genetic variants. In this method, we view outcomes as stochastic realizations of a random field on a genetic space and propose to use a random field model, referred to as a genetic random field model (GenRF), to model the joint association. This approach is motivated by development in spatial statistics where outcomes can be viewed as stochastic realizations of a random field on a 2-dimensional space (Cressie, 1993). Thus, our approach can be viewed as a generalization of spatial statistics from a 2-dimensional space to a $k$-dimensional space. This random field perspective leads to a very distinctive model from the aforementioned regression-based methods including SKAT and SIMreg; specifically, GenRF regresses the response of one subject on responses of all other subjects, which is not common in many fields other than spatial or time-series statistics. Although the random field framework may be unfamiliar to some statisticians, as we demonstrate later, the method can be understood from the intuitive idea that genetic similarity will lead to trait similarity if variants are associated with the trait. Under the GenRF model, testing for the joint association reduces to a test involving a scalar parameter. A testing procedure for the joint association using the pseudo-likelihood method is developed. The proposed test possesses many appealing features. For example, it is able to exploit LD and interactions between variants to improve power. The method also allows for adjusting weights of rare variants to boosting power. In addition, it is computationally convenient. Our simulation studies and an application to a real data show that the proposed method can achieve comparable or, in the presence of complex interactions, superior performance to SKAT. The remainder of this article is organized as follows. In Section 2 we set up the notation and introduce related background. We describe the proposed genetic random field model in Section 3.1, the relationship with previous work in Section 3.2, and develop a joint association test under the GenRF model on Section 3.3. The performance of the proposed method is evaluated by simulation studies in Section 4. The proposed method is illustrated by an application to the Dallas Heart Study in Section 5, followed by a discussion in Section 6. Notation and Background ======================= Consider a study where $n$ subjects are sequenced in a region of interest. For subject $i, i=1,\ldots,n$, let ${\mbox{\boldmath $G$}}_i$ $( p\times 1) $ denote the genotype for the $p$ variants within the region and $Y_i$ the trait or phenotype that ${\mbox{\boldmath $G$}}_i$ is potentially associated to. Additionally, one may also collects other covariates, denoted by ${\mbox{\boldmath $X$}}_i$, on each subject including, for example, age, gender, and other demographic and environmental factors. We are interested in studying the joint association between variants ${\mbox{\boldmath $G$}}_{i}$ and trait $Y_{i}$, possibly adjusted for the effect of ${\mbox{\boldmath $X$}}_i$. For example, if $Y_i$ is continuous, one might model the relationship between variants and the phenotype by a multivariate linear regression model, given by $$\begin{aligned} Y_i={\mbox{\boldmath $\alpha$}}^T{\mbox{\boldmath $X$}}_i+{\mbox{\boldmath $\theta$}}^T{\mbox{\boldmath $G$}}_i+\epsilon_i,\end{aligned}$$ where ${\mbox{\boldmath $X$}}_i$ includes an intercept; ${\mbox{\boldmath $\alpha$}}$ and ${\mbox{\boldmath $\theta$}}$ are the coefficients correspondingly. Testing for the joint association of ${\mbox{\boldmath $G$}}_i$ with $Y_i$ can be achieved by testing the null hypothesis $H_0: {\mbox{\boldmath $\theta$}}=\bf{0}$, i.e., $(\theta_1=\theta_2=\ldots =\theta_p=0)$. Although simple and intuitive, this approach has several drawbacks; for example, the usual $p$-degree-of-freedom tests are known to have low power (Goeman, et al., 2006) and it is difficult to include complex, high-dimensional interactions among variants in parametric models. To remedy these issues, SKAT models the effect of ${\mbox{\boldmath $G$}}_i$ using a semiparametric linear model, i.e., $$\begin{aligned} Y_i={\mbox{\boldmath $\alpha$}}^T{\mbox{\boldmath $X$}}_i+h({\mbox{\boldmath $G$}}_i)+\epsilon_i, \label{SKAT}\end{aligned}$$ where $h(\cdot)$ is a nonparametric function assumed to lie in a functional space generated by a positive semidefinite kernel function $K(\cdot, \cdot)$; e.g., $K({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)=\sum_{k=1}^p {\mbox{\boldmath $G$}}_{ik}{\mbox{\boldmath $G$}}_{jk}$ would correspond to a linear model. The form of $h(\cdot)$ is not explicitly specified and instead is implicitly determined by the chosen kernel function. Under this model, testing for the joint association is equivalent to testing $H_0: h({\mbox{\boldmath $G$}}_i)=0, i=1,\ldots, n$. This is achieved by viewing $[h({\mbox{\boldmath $G$}}_1), \ldots, h({\mbox{\boldmath $G$}}_n)]$ as a random vector with mean zero and covariance $\tau {\mbox{\boldmath $K$}}$, where ${\mbox{\boldmath $K$}}$ is an $n\times n$ matrix with the$(i,j)$-th element equal to $K({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)$, and then testing $H_0:\tau=0$ by a variance-component score test (Lin, 1997). In summary, SKAT is based on modeling outcomes in a mixed model framework, where the responses $Y_i$’s are assumed positively correlated (or similar) across $i$ due to the random effect corresponding to $h({\mbox{\boldmath $G$}}_i)$. SKAT improves power partly by allowing for more flexible models in $h({\mbox{\boldmath $G$}}_i)$ via the choice of a proper kernel function $K(\cdot, \cdot)$. As explained in Wu, et al. (2011), the kernel function $K({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)$ can be interpreted as a measure for genetic similarity in the region of interest between the $i$-th and $j$-th subjects and a kernel function better capturing the similarity between individuals and the causal variant effects can increase power. Thus, in the variance-component score test of SKAT, it implicitly assesses the genetic similarity across subjects and also how this genetic similarity leads to positively correlated (or similar) $Y_i$’s. Instead of modeling trait similarity by a positive association, the SIMreg of Tzeng, et. al. (2009) explicitly defines a measure for trait similarity and directly regresses the trait similarity between each pair of subjects on genetic similarity. SKAT and SIMreg lead to similar test statistics and an explicit connection between SIMreg and variance component score tests was demonstrated by Tzeng, et al. (2009). We will not discuss the SIMreg test hereafter and focus on comparing our method mainly with SKAT. Method {#s:model} ====== Genetic Random Field Model -------------------------- Our method is also motivated by the general idea that, if the genetic variants are jointly associated with a trait, then the genetic similarity across subjects will contribute to the trait similarity. To put it in another way, if variants are jointly associated with the trait, then the response of a subject would be close to the response of other subjects who share similar genetic and possibly other variables. Based on this key idea, we propose to directly model the response of each subject as a function of all other responses and the contribution of other responses to $Y_i$ is weighted by their genetic similarity. This is in contrast to SKAT which models the relationship of $Y_i$ with ${\mbox{\boldmath $G$}}_i$ for each $i$ as opposed to that of $Y_i$ with all $Y_j$’s for $j\neq i$. For simplicity, we temporarily assume $Y_i$’s are centered (have mean zero) and there are no other adjustment covariates. Specifically, based on the idea discussed above, we model the conditional distribution of $Y_i$ given all other responses as $$Y_i \| {\mbox{\boldmath $Y$}}_{-i} \sim \gamma \sum_{j\neq i}s({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)Y_{j}+\varepsilon_i, \label{GenRF}$$ where ${\mbox{\boldmath $Y$}}_{-i}$ denotes responses for all other subjects except $Y_i$ ; $s({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)$ is known weights, weighting the contribution of $Y_j$ on approximating (or predicting) $Y_i$ via their genetic similarity; $\gamma$ is a non-negative coefficient measuring the magnitude of the overall contribution, further discussed below; and $\varepsilon_i$’s are random errors corresponding to subject $i$. A proper weight function $s({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)$ gives higher value when the two subjects are more similar in terms of their genetic variants and, as discussed below, can be viewed as a measure for proximity of two subjects in a genetic space. The random errors $\varepsilon_i$’s are assumed to be independent and identically distributed with Normal$(0,\zeta^2)$; robustness of the method to distributions other than normal is discussed in Section \[s:GenRFT\]. A main distinction between model (\[GenRF\]) and the usual regression is that (\[GenRF\]) models the conditional distribution of $Y_i$ given responses of other subjects, whereas in the usual regression one models the conditional distribution of a subject’s response given explanatory variables of the same subject. As the usual regression is useful for predicting or approximating the response of a subject using his/her covariates, model (\[GenRF\]) is useful for approximating the response of a subject using responses of other subjects, where the similarity in responses is due to similarity in genetic variants if variants are associated with the response. The coefficient $\gamma$ indicates the magnitude of the trait similarity as a result of genetic similarity. Thus, $\gamma$ can also be interpreted as a measure for the magnitude of the joint association of ${\mbox{\boldmath $G$}}_i$ with $Y_i$. Specifically, if ${\mbox{\boldmath $G$}}_i$ is not associated with $Y_i$, then regardless of how similar subject $i$ is to other subjects in terms of their genetic variants, the response $Y_i$ is independent of all other $Y_j$’s for $j\neq i$; that is, $\gamma=0$. On the contrary, if ${\mbox{\boldmath $G$}}_i$ is strongly associated with $Y_i$, then one may expect the response of $Y_i$ can largely be predicted by responses of subjects having the same or similar genetic variants and a large $\gamma$ indicates a strong joint association. Therefore, we can test the joint association of genetic variants with the trait by testing a null hypothesis involving a single parameter, i.e., $H_0: \gamma=0$. Models like (\[GenRF\]), where responses are regressed on responses themselves, are referred to as auto-regressive models and, although less commonly used in genetic and biomedical studies, are commonly used in spatial statistics and in modeling time-series. In this article, we propose to view the response as a random field on a genetic space, and from this fresh perspective, model (\[GenRF\]) is formally a conditional auto-regressive (CAR) model (Cressie, 1993). A random field is a generalization of the notation of a stochastic process (Adler and Taylor, 2007). Informally, a stochastic process is a set of random variables indexed by integers or real numbers; for example, a continuous time-series $W_t$, $t\in T$, is a stochastic process with an index set $T=\mathcal{R}$. A random field can be defined in more general spaces with the index set being an Euclidean space of dimension greater than one or other spaces. For example, in spatial statistics, crop yields of regions can be viewed as a realization of a random field defined in a two-dimensional space, denoted by $W_{s, t}$ where $s$ and $t$ indicate the (latitudinal and longitudinal) location of a region. Regions that are closer in location have more similar crop yields if spatial correlation exists. Specifically, for our problem, we may view observed responses as realizations of a random field defined in a $p$-dimensional space of the $p$ genetic variants; that is, corresponding to each “location” in the $p$-dimensional genetic space (equivalently each vector value that ${\mbox{\boldmath $G$}}_i$ may take), there is a random response variable associated with it, denoted by $Y_{G_{i1}, \ldots, G_{ip}}$ in a slight abuse of notation. Similarly, responses from locations that are “closer” in the genetic space are expected to be more similar if the genetic association exists. In this sense, our model is a generalization of the auto-regressive model in time series analysis (one dimensional) and spatial statistics (two dimensional). Models like (\[GenRF\]) were firstly studied in the seminar work of Besag (1974) for random fields and we will term our model (\[GenRF\]) as a genetic random field (GenRF) model. As a matter of fact, the GenRF model is closely related to the conditional auto-regressive model in spatial statistics (Cressie, 1993); that is $s({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)$ analogously defines the proximity of neighbor ${\mbox{\boldmath $G$}}_j$ to ${\mbox{\boldmath $G$}}_i$ and $\gamma$ is the counterpart of a spatial dependence parameter. However, we note that the usual tests of spatial dependence, for example, the Cliff-Ord-test (Cliff and Ord, 1972) and the Lagrange Multiplier test (Burridge, 1980), do not apply in our setting to test for the joint association of variants, as discussed in Section 6. We have yet to define a measure for “closeness” in the genetic space. Suppose each component of ${\mbox{\boldmath $G$}}_i$ records the number of minor alleles in a single locus and takes on values $\{0, 1, 2\}$, respectively, corresponding to three possibilities $\{AA, Aa, aa\}$. Then a sensible measure for closeness or similarity is the so called identity-by-state (IBS) (Wu, et al., 2010), defined as $$\begin{aligned} s({\mbox{\boldmath $G$}}_i, {\mbox{\boldmath $G$}}_j)=\sum_{k=1}^p\{2-\|G_{ik}-G_{jk}\|\}.\end{aligned}$$ That is, the IBS measures the number of alleles in the region of interest shared by two individuals; for example, in the single locus case ($p=1$), $s(AA, AA)=2, s(Aa,aa)=1, s(AA,aa)=0$. The overall similarity between two loci sequences are the sum of shared alleles in all loci in the region of interest between two subjects. Other measures for closeness in the genetic space rather than IBS are also possible, e.g., the other kernel functions discussed in Wu, at al. (2011), providing flexibility in our GenRF model. Similar to SKAT, our GenRF model can also incorporate weights to increase the importance of rare variants. Specifically, one can define $s({\mbox{\boldmath $G$}}_i, {\mbox{\boldmath $G$}}_j)=\sum_{k=1}^p w_k\{2-\|G_{ik}-G_{jk}\|\}$, where $w_k$ is a prespecified weight for variant $k$; see Wu, et al., (2011) for more discussions on $w_k$. The above discussion has focused on the situation where no covariate adjustment is required. If adjustment for other factors, for example, environmental factors, is needed, a natural extension of model (\[GenRF\]) is given by $$Y_i \| {\mbox{\boldmath $Y$}}_{-i}, {\mbox{\boldmath $X$}}_i \sim {\mbox{\boldmath $\beta$}}^T{\mbox{\boldmath $X$}}_i + \gamma \sum_{j\neq i}s({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)(Y_{j}-{\mbox{\boldmath $\beta$}}^T{\mbox{\boldmath $X$}}\!_j)+\varepsilon_i. \label{GenRF2}$$ An intercept term is included in ${\mbox{\boldmath $X$}}_i$ and, as a result, in (\[GenRF2\]) $Y_i$’s are not required to be centered. Under this model, testing for the joint association of ${\mbox{\boldmath $G$}}_i$ with $Y_i$ after adjusting for other factors is also equivalent to testing $H_0: \gamma=0$. We will mainly focus on this more general form of the GenRF model in the development of a testing procedure. For simplicity, the matrix form of GenRF model is given by $${\mbox{\boldmath $Y$}}\| {\mbox{\boldmath $Y$}}_-, {\mbox{\boldmath $X$}}= {\mbox{\boldmath $X$}}{\mbox{\boldmath $\beta$}}+ \gamma {\mbox{\boldmath $S$}}({\mbox{\boldmath $Y$}}-{\mbox{\boldmath $X$}}{\mbox{\boldmath $\beta$}})+{\mbox{\boldmath $\varepsilon$}},$$ where ${\mbox{\boldmath $Y$}}$ is $(Y_1,\ldots, Y_n)^T$; ${\mbox{\boldmath $X$}}$ is an $n\times q$ matrix defined as $({\mbox{\boldmath $X$}}_1^T,\ldots, {\mbox{\boldmath $X$}}_n^T)^T$; ${\mbox{\boldmath $\varepsilon$}}\sim$ Normal $(0, \zeta^2 I_{n\times n})$; and ${\mbox{\boldmath $S$}}$ is an $n\times n$ symmetric matrix with zeros on the diagonal and the $(i,j)$-th element $s({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)$ for $i\neq j$, as only the pairs of $\{i\neq j\}$ are involved in the model. Relationship with SKAT ---------------------- We further compare the GenRF model with SKAT. As we have commented previously, SKAT can also be understood in a similar fashion, i.e., if ${\mbox{\boldmath $G$}}_i$ is jointly associated with $Y_i$, then subjects having similar genetic variants have similar positively correlated responses. In SKAT, the similarity in responses is essentially modeled in a mixed model framework where a random effect, say, $h({\mbox{\boldmath $G$}}_i)$, induces positive correlation among responses. Also in SKAT the similarity in genetic variants (or equivalently the kernel function) does not appear explicitly in the assumed model (\[SKAT\]) but is implicitly related to $h({\mbox{\boldmath $G$}}_i)$ according to the kernel machine regression theory. In contrast, our model in (\[GenRF\]) models the similarity in responses via conditional expectations other than correlations and the genetic similarity is incorporated in the model explicitly. It is easy to see that, in SKAT when $h({\mbox{\boldmath $G$}}_i), i=1,\ldots, n$, are treated as following a multivariate normal distribution, model (\[SKAT\]) leads to $$\begin{aligned} {\mbox{\boldmath $Y$}}\|{\mbox{\boldmath $X$}}\sim {\mbox{\boldmath $X$}}{\mbox{\boldmath $\alpha$}}+ {\mbox{\boldmath $u$}}, \,\,\, {\mbox{\boldmath $u$}}\sim N(0,\sigma^2{\mbox{\boldmath $I$}}+\tau^2{\mbox{\boldmath $K$}}),\end{aligned}$$ where ${\mbox{\boldmath $u$}}$ is an $n$-dimensional random column vector; and ${\mbox{\boldmath $I$}}$ is an $n\times n$ identity matrix. In contrast, according to the factorization theorem of Besag (1974), our GenRF model in (\[GenRF2\]) leads to the following joint distribution, i.e., $$\begin{aligned} {\mbox{\boldmath $Y$}}\|{\mbox{\boldmath $X$}}\sim {\mbox{\boldmath $X$}}{\mbox{\boldmath $\beta$}}+{\mbox{\boldmath $v$}}, \,\,\, {\mbox{\boldmath $v$}}\sim N(0,\zeta^2({\mbox{\boldmath $I$}}-\gamma {\mbox{\boldmath $S$}})^{-1}),\end{aligned}$$ where ${\mbox{\boldmath $v$}}$ is an $n$-dimensional random column vector. Note, the coefficient $\gamma$ used for describing the conditional expectation of $Y_i$ given others in model (\[GenRF\]) actually describes the correlations among $Y_i$’s. It is clear that, under the null hypothesis that there is no association between ${\mbox{\boldmath $G$}}_i$ and $Y_i$, i.e., $\tau=0$ in SKAT or $\gamma=0$ in GenRF, the two models are equivalent and $Y_i$’s are uncorrelated as the covariance matrices are diagonal. Moreover, if $\tau>0$ or $\gamma> 0$ in the corresponding model, both SKAT and GenRF state that $Y_i$’s are positively correlated as a result of having similar genetic variants associated with the trait. However, the two models provide different parameterizations of the covariance matrix and consequently the two methods model the magnitude of the joint association differently, albeit both via a scaler parameter. Moreover, though one can adopt the same IBS similarity for both SKAT and GenRF, ${\mbox{\boldmath $K$}}$ and ${\mbox{\boldmath $S$}}$ are still different on the diagonal, where the diagonal elements of ${\mbox{\boldmath $S$}}$ are zeros. The two parameterizations may result in different sensitivity in detecting departures from the null hypothesis. Therefore, one might expect the two models lead to testing procedures with different efficiency in testing the genetic effect. Next we develop a testing procedure based on the proposed GenRF model and the proposed test is compared with SKAT by simulation studies in Section \[s:simulation\]. Genetic Random Field Test {#s:GenRFT} ------------------------- In this subsection, we focus on developing a test for the null hypothesis $H_0: \gamma=0$ based on model (\[GenRF2\]), referred to as the genetic random field test. Model (\[GenRF2\]) states that, given responses from all other subjects and covariates ${\mbox{\boldmath $X$}}_i$, the conditional distribution of $Y_i$ is normal with mean ${\mbox{\boldmath $\beta$}}^T{\mbox{\boldmath $X$}}_i + \gamma \sum_{j\neq i}s({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)(Y_{j}-{\mbox{\boldmath $\beta$}}^T{\mbox{\boldmath $X$}}_j^T)$ and variance $\zeta^2$. We construct the pseudo-likelihood according to Besag (1975) as $$\begin{aligned} L_{pd}=\prod_{i=1}^n \bigg\{\frac{1}{\sqrt{2\pi \zeta^2}}\exp\Big[-\frac{1}{2\zeta^2}\big\{Y_i-{\mbox{\boldmath $\beta$}}^T{\mbox{\boldmath $X$}}_i -\gamma \sum_{j\neq i}s({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)(Y_{j}-{\mbox{\boldmath $\beta$}}^T{\mbox{\boldmath $X$}}_j)\big\}^2\Big]\bigg\},\end{aligned}$$ which is a product of the conditional densities of $Y_i$ across $i$. Also according to Besag (1975), assuming ${\mbox{\boldmath $\beta$}}$ is known, one may estimate $\gamma$ by the maximum pseudo-likelihood method. It is easy to see that the maximum pseudo-likelihood estimator for $\gamma$ can be obtained by minimizing $\sum_{i=1}^n\big\{Y_i-{\mbox{\boldmath $\beta$}}^T{\mbox{\boldmath $X$}}_i -\gamma \sum_{j\neq i}s({\mbox{\boldmath $G$}}_i,{\mbox{\boldmath $G$}}_j)(Y_{j}-{\mbox{\boldmath $\beta$}}^T{\mbox{\boldmath $X$}}_j\big\}^2$, which in matrix notation is equal to $$\begin{aligned} \{({\mbox{\boldmath $I$}}-\gamma {\mbox{\boldmath $S$}})({\mbox{\boldmath $Y$}}-{\mbox{\boldmath $X$}}{\mbox{\boldmath $\beta$}})\}^T({\mbox{\boldmath $I$}}-\gamma {\mbox{\boldmath $S$}})({\mbox{\boldmath $Y$}}-{\mbox{\boldmath $X$}}{\mbox{\boldmath $\beta$}}).\end{aligned}$$ The minimization leads to an estimator for $\gamma$ given by $$\Rightarrow {\widetilde{\gamma}}=\frac{({\mbox{\boldmath $Y$}}-{\mbox{\boldmath $X$}}{\mbox{\boldmath $\beta$}})^T{\mbox{\boldmath $S$}}({\mbox{\boldmath $Y$}}-{\mbox{\boldmath $X$}}{\mbox{\boldmath $\beta$}})}{({\mbox{\boldmath $Y$}}-{\mbox{\boldmath $X$}}{\mbox{\boldmath $\beta$}})^T{\mbox{\boldmath $S$}}^2({\mbox{\boldmath $Y$}}-{\mbox{\boldmath $X$}}{\mbox{\boldmath $\beta$}})}.$$ Intuitively one expects that a large value of ${\widehat{\gamma}}$ would give us evidence to reject the null hypothesis that $\gamma=0$. In practice, ${\mbox{\boldmath $\beta$}}$ in unknown. We propose to replace ${\mbox{\boldmath $\beta$}}$ by its least square estimator ${\widehat{\mbox{\boldmath $\beta$}}}$ under the null hypothesis $H_0: \gamma=0$, i.e., ${\widehat{\mbox{\boldmath $\beta$}}}=({\mbox{\boldmath $X$}}^T{\mbox{\boldmath $X$}})^{-1}{\mbox{\boldmath $X$}}^T{\mbox{\boldmath $Y$}}$. Substitute ${\widehat{\mbox{\boldmath $\beta$}}}$ into the expression for ${\widetilde{\gamma}}$ and straightforward algebra leads to the final test statistic: $${\widehat{\gamma}}=\frac{{\mbox{\boldmath $Y$}}^T{\mbox{\boldmath $B$}}{\mbox{\boldmath $S$}}{\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}}{{\mbox{\boldmath $Y$}}^T{\mbox{\boldmath $B$}}{\mbox{\boldmath $S$}}^2{\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}}, \label{Test}$$ where ${\mbox{\boldmath $B$}}={\mbox{\boldmath $I$}}-{\mbox{\boldmath $X$}}({\mbox{\boldmath $X$}}^T{\mbox{\boldmath $X$}})^{-1}{\mbox{\boldmath $X$}}^T$. Again a large value of ${\widehat{\gamma}}$ would support the rejection of the null hypothesis. We next show how the p-value for testing $\gamma=0$ can be obtained based on the test statistic ${\widehat{\gamma}}$; i.e., we would like to calculate the probability of ${\widehat{\gamma}}$ greater than the observed value of the statistic under the null hypothesis. As ${\mbox{\boldmath $S$}}$ is not a diagonal or block-diagonal matrix, regular asymptotic argument does not apply. Alternatively, we propose the following procedure to find the p-value by using the exact mixture Chi-square distribution. Suppose $\eta$ is the observed value of the test statistic ${\widehat{\gamma}}$. Since ${\mbox{\boldmath $B$}}{\mbox{\boldmath $S$}}^2{\mbox{\boldmath $B$}}$ is positive-definite, we have $$\begin{aligned} P_{H_0}\bigg(\frac{{\mbox{\boldmath $Y$}}^T{\mbox{\boldmath $B$}}{\mbox{\boldmath $S$}}{\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}}{{\mbox{\boldmath $Y$}}^T{\mbox{\boldmath $B$}}{\mbox{\boldmath $S$}}^2{\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}}>\eta \bigg)=P_{H_0}\bigg(({\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}})^T({\mbox{\boldmath $S$}}-\eta {\mbox{\boldmath $S$}}^2){\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}>0\bigg)\end{aligned}$$ As it is assumed that $\varepsilon_i\sim N(0,\zeta^2)$, i.i.d. across $i$, it follows that ${\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}\sim N(0,\zeta^2{\mbox{\boldmath $B$}}^2)$ under the null hypothesis. On the other hand, the statistic ${\widehat{\gamma}}$ in (\[Test\]) is ancillary to $\zeta^2$ because $\zeta^2$ in the numerator and denominator will cancel out. Therefore, the above equation becomes $$\begin{aligned} P_{H_0}\bigg(({\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}})^T({\mbox{\boldmath $S$}}-\eta {\mbox{\boldmath $S$}}^2){\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}>0\bigg)=P\bigg({\mbox{\boldmath $Z$}}^T({\mbox{\boldmath $S$}}-\eta {\mbox{\boldmath $S$}}^2){\mbox{\boldmath $Z$}}>0\bigg), \end{aligned}$$ where ${\mbox{\boldmath $Z$}}$ is an $n\times 1$ random vector following $N(0, {\mbox{\boldmath $B$}}^2)$. Applying standard results on the distribution of quadratic form of normal random variables, we have $$\begin{aligned} {\mbox{\boldmath $Z$}}^T({\mbox{\boldmath $S$}}-\eta {\mbox{\boldmath $S$}}^2){\mbox{\boldmath $Z$}}\sim \sum_i^n \lambda_i \Phi_i, \end{aligned}$$ where $\Phi_i$’s are i.i.d random variables with $ \chi^2_1$ distribution, and $\{\lambda_i\}$ are the eigenvalues of ${\mbox{\boldmath $B$}}({\mbox{\boldmath $S$}}-\eta {\mbox{\boldmath $S$}}^2){\mbox{\boldmath $B$}}$. The final p-value can be obtained by Davies’ exact method (1980) for the weighted summation of independent Chi-square variables, similar to the p-value calculation used in SKAT (Wu, et al., 2011). The proposed test has several appealing properties. First, due to the analytical form of the test statistic, the computational burden is well controlled. Second, as ${\widehat{\gamma}}$ in (\[Test\]) is ancillary to $\zeta^2$, there is no need to plug in a consistent estimator for $\zeta^2$ as SKAT did. Third, similar to SKAT, the proposed method improves power by exploiting linkage disequilibrium and allowing for possible complex interactions among variants. Linkage disequilibrium can cause correlations between variants, especially when we consider nearby loci. Considering similarity in variants can naturally reduce the degree of freedom. In the extreme case where components of ${\mbox{\boldmath $G$}}_i$ are “perfectly correlated”, the similarity argument will consider the whole set as a single variable, whereas the typical linear regression will have $p$-degree of freedom. In addition, genetic variants involved in the disease pathway are more likely to interact with each other than contribute to risk individually, known as the epistatic variants effect. Specifying two-way interactions in a set of loci is a challenging high-dimensional problem and the situation gets even worse in modeling higher order interactions. Since in our GenRF model we do not directly model the relationship of ${\mbox{\boldmath $G$}}_i$ with $Y_i$, the difficulty of modeling complex interactions are circumvented and the interaction effect is naturally included through measuring genetic similarity. Finally, as SKAT, the proposed GenRF test can boost power of testing rare variants by increasing their weights by specifying $w_k$ appropriately for variant $k$. The derivation of the GenRF test given above is built on the normal distribution assumption. Asymptotically, the proposed test is robust to distributions other than normal. Consider $P_{H_0}\Big(({\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}})^T({\mbox{\boldmath $S$}}-\eta {\mbox{\boldmath $S$}}^2){\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}>0\Big)$, where it is now assumed ${\mbox{\boldmath $Y$}}$ follows an arbitrary distribution. The random quantity $({\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}})^T({\mbox{\boldmath $S$}}-\eta {\mbox{\boldmath $S$}}^2){\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}$ is a quadratic form of ${\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}$ (with mean 0) with matrix ${\mbox{\boldmath $A$}}=({\mbox{\boldmath $S$}}-\eta{\mbox{\boldmath $S$}}^2)$. Rotar (1973) proved that under sufficiently weak conditions on matrix ${\mbox{\boldmath $A$}}$ and for large $n$, $P_{H_0}\big(({\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}})^T({\mbox{\boldmath $S$}}-\eta {\mbox{\boldmath $S$}}^2){\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}>0\big)$ is close to $P_{H_0}({\mbox{\boldmath $Z$}}^T({\mbox{\boldmath $S$}}-\eta {\mbox{\boldmath $S$}}^2){\mbox{\boldmath $Z$}}>0)$, where ${\mbox{\boldmath $Z$}}$ follows $N(0,{\mbox{\boldmath $B$}}^2)$ as defined before. In addition, Gotze and Tikhomirov (1999) gave an upper bound on $\sup_x \big\|P_{H_0}\big(({\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}})^T{\mbox{\boldmath $A$}}{\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}<x\big)-P_{H_0}({\mbox{\boldmath $Z$}}^T{\mbox{\boldmath $A$}}{\mbox{\boldmath $Z$}}<x)\big\|$. These properties lead to the natural robustness of the GenRF test as long as ${\mbox{\boldmath $B$}}{\mbox{\boldmath $Y$}}$ has expectation zero under the null hypothesis, which is true since the least squares estimator ${\mbox{\boldmath $X$}}({\mbox{\boldmath $X$}}^T{\mbox{\boldmath $X$}})^{-1}{\mbox{\boldmath $X$}}^T{\mbox{\boldmath $Y$}}$ is unbiased for the mean of ${\mbox{\boldmath $Y$}}$ under the GenRF model when $\gamma=0$. Our simulation studies in Section \[s:simulation\] further illustrate this robustness. We comment that, as the score test in SKAT is of similar quadratic form, one would expect that SKAT may share this property as well. Simulation {#s:simulation} ========== Simulation studies under various scenarios are conducted to evaluate the performance of GenRF test and to compare it with SKAT. In both methods, we adopt the IBS kernel in the corresponding matrix ${\mbox{\boldmath $S$}}$ or ${\mbox{\boldmath $K$}}$ and set $w_k=1$ for each $k$. Therefore, results from the two methods are comparable. Three sets of simulations are conducted to evaluate the performance of GenRF test 1) under different levels of LD, 2) under different levels of interaction effect, and 3) under different distributions of the response variable. In the first set of simulations, data are simulated under different levels of LD effect. For each Monte Carlo data set, genotypes for $p=10$ loci are simulated for $n=100$ subjects. To simulate the LD effect, the haplotype is simulated one by one for each locus with the minor allele frequency 0.3 and the haplotypes of each adjacent pair of alleles are correlated with a correlation coefficient $\rho$ with $\rho=0, 0.1, 0.2, \ldots, 0.9$ respectively for each scenario. Genotypes are then generated by summing up two haplotype vectors. This way, all the loci are positively correlated with others in the loci set. The response variable is generated according to the following model $$\begin{aligned} Y_i=aG_{i5}+\varepsilon_i, \textrm { where} \quad \varepsilon_i \sim N (0,1), \end{aligned}$$ $a=0$ or 0.5. That is, when $a=0.5$, the 5th variant is associated with the trait. In the second set of simulations, data are generated such that complex interaction effect exists. For each Monte Carlo data set, we take $p=10$, $n=100$, the minor allele frequency=0.3, and the LD parameter $\rho=0.4$. Two distributions are considered. In the normal distribution case, the response is related to variants according to the following model, $$Y_i\|{\mbox{\boldmath $G$}}_i=b\sum_{k=1}^9 G_{i1} G_{ik+1}+\varepsilon_i, \textrm { where } \quad \varepsilon_i \sim N (0,1),$$ and $b$ is set to 0, 0.01, 0.02, $\ldots$, 0.1, respectively, in each scenario. In the exponential distribution case, responses are generated as exponential random variables with rate $\lambda=b\sum_{k=1}^9 G_{i1} G_{ik+1}$ with $b$ equal to $0, 0.02,0.04, \ldots$, 0.2, respectively. We see that these models contain only interactions but not main effect of each locus. In the third set of simulations, we further evaluate the robustness of the GenRF test to distributions other than normal. The setup is the same as that in the first set of simulations with $\rho=0.4$ except that $Y_i$ is generated according to generalized linear model with a linear predictor $aG_{i5}$ and the canonical link function from the following distributions: Standard Normal, Exponential, Binary. The coefficient $a$ is set to 0.6, 1.1 and 2.5 respectively for each of the distributions. For Mixture Normal, we generate two normal distributions with mean difference 10 with equal proportions. The coefficient for Mixture Normal is 2.7. For each simulated data set, we test the joint association of variants using three methods: the proposed GenRF test, SKAT and the usual method based on a linear regression model including only main effects of variants. We note that data are not generated according to GenRF models and, therefore, these simulations should not particularly favor the proposed GenRF test, allowing for a fair comparison among methods. Table 1 shows results for the first set of simulations with LD effect ranging from low to high. All the three tests achieve the type I error rate close to the nominal level. When LD effect does not exist or is low, e.g., $\rho<0.5$, the test based on a linear regression model is most powerful as expected. However, when the LD effect is moderate or high, both the GenRF test and SKAT have higher or even substantially higher power than the linear regression based test by borrowing information from other loci. The power of the GenRF test is comparable to or slightly higher than that of SKAT. Table 2 shows results when there are complex interactions between variants but no main effects. In these scenarios, the linear regression method has low power in detecting the joint association. Both GenRF test and SKAT have much larger power. Moreover, the proposed GenRF test has significantly larger power than SKAT in detecting the joint association effect when complex interactions exist, at least in the scenarios considered here. Table 3 shows the robustness of the proposed GenRF test to distributions other than normal. GenRF test achieves the type I error rate close to the nominal level even when the distribution of the response is not normal; the same holds for SKAT. Similar to results in Table 1, the performance of GenRF test is comparable to that of SKAT when there is a single risk locus. [ccccccccccccc]{} Method&&&&\ &&&\ & && No LD & 0.1 & 0.2 & 0.3 & 0.4&0.5&0.6&0.7&0.8&0.9\ GenRF & Power&&0.435&0.446&0.450&0.508&0.487&0.556&0.634&0.676&0.724&0.804\ & Type I&&0.053&0.042&0.047&0.048&0.055&0.050&0.045&0.040&0.049&0.061\ \ SKAT & Power&&0.420&0.446&0.434&0.496&0.476&0.552&0.616&0.660&0.711&0.800\ & Type I &&0.050&0.046&0.048&0.041&0.045&0.046&0.048&0.052&0.046&0.062\ \ Linear & Power&&0.500&0.516&0.500&0.526&0.504&0.496&0.514&0.493&0.500&0.518\ & Type I&&0.056&0.042&0.054&0.050&0.058&0.050&0.043&0.048&0.049&0.060\ \[t:table1\] [ccccccccccccc]{} Method&&&&\ \ &&&&\ & & & $b=$0.01 & 0.02 & 0.03 & 0.04&0.05&0.06&0.07&0.08&0.09&0.10\ \ GenRF & Power&&0.063&0.120&0.176&0.267&0.395&0.539&0.706&0.816&0.853&0.948\ & Type I&&0.050&0.054&0.053&0.042&0.052&0.049&0.056&0.054&0.046&0.050\ \ SKAT & Power&&0.054&0.088&0.122&0.177&0.282&0.392&0.555&0.673&0.737&0.884\ & Type I && 0.043&0.048&0.044&0.046&0.048&0.044&0.052&0.050&0.041&0.052\ \ Linear & Power&&0.050&0.068&0.092&0.132&0.183&0.270&0.391&0.488&0.570&0.764\ & Type I&& 0.047&0.054&0.053&0.042&0.052&0.048&0.050&0.052&0.050&0.056\ \ &&&&\ & & & $b=$0.02 & 0.04 & 0.06 & 0.08&0.10&0.12&0.14&0.16&0.18&0.20\ \ GenRF & Power&&0.075&0.125&0.212&0.353&0.486&0.557&0.636&0.740&0.784&0.808\ & Type I&&0.046&0.053&0.050&0.058&0.052&0.052&0.056&0.040&0.048&0.045\ \ SKAT & Power&&0.056&0.090&0.155&0.238&0.340&0.416&0.484&0.600&0.649&0.678\ & Type I &&0.044&0.052&0.047&0.048&0.048&0.054&0.048&0.045&0.052&0.044\ \ Linear & Power&&0.058&0.062&0.088&0.127&0.182&0.218&0.277&0.364&0.391&0.410\ & Type I&&0.051&0.052&0.052&0.057&0.054&0.048&0.054&0.046&0.050&0.048\ \ \[t:table2\] [ccccccccccccc]{} Method&&\ &&&\ & && Normal & Exponential & Mixture Normal & Binary\ GenRF & Power&&0.725&0.636&0.582&0.646\ & Type I&&0.051&0.052&0.056&0.046\ \ SKAT & Power&&0.726&0.655&0.582&$\ast$\ & Type I &&0.047&0.046&0.046&$\ast$\ \ Linear & Power&&0.728&0.572&0.568&0.559\ & Type I&& 0.054&0.056&0.054&0.050\ \[t:table3\] Application {#s:application} =========== We applied our method to the Dallas Heart Study (Browning et al., 2004; Victor et al., 2004), which was used previously for illustration of association tests (Liu and Leal, 2010; Wu, et al., 2011). The Dallas Heart Study is a population-based, multi-ethnic study where 3551 residents are recruited. For each subject, Lipids and glucose metabolism have been measured. Individuals who have diabetes mellitus, alcohol dependency or have taken lipids lowering drugs are excluded as these factors may confound the interpretation of associations. In this re-sequencing study, 348 sequence variations in the coding regions of the four genes, ANGPTL3, ANGPTL4, ANGPTL5 and ANGPTL6 are discovered. Most of these variants (86%) are rare with the minor allele frequency less than 1%. We assessed the association between ANGPTL gene families and two traits, specifically high-density lipoprotein (HDL) and triglyceride, using the proposed GenRF test and SKAT, both with the IBS kernel. Except for the association testing procedure, our analysis is otherwise similar to the original approach (Romeo, et al., 2007) that has discovered the association between ANGPTL4 gene and the level of HDL and triglyceride. For testing the association between ANGPTL4 and HDL, both GenRF test and SKAT showed comparable and marginal evidence for the association between ANGPTL4 and HDL (p-value: 0.085 and 0.060 respectively); however, the p-values are not significant at the level of 0.05. Both methods gave strong evidence for the ANGPTL4 and triglyceride association and the evidence from SKAT is stronger for this particular association (p-values: 0.011 and $1.65\times10^{-3}$). ANGPTL5 may also be potentially associated with triglyceride (Liu and Leal, 2012; Romeo, et al., 2009). In our analysis, our GenRF test provided marginal evidence to support this association (p-value: 0.071) while SKAT did not (p-value: 0.353). More results are shown in Table 4. Overall, in this application, the proposed GenRF test and SKAT have comparable performance. [cccccccccccc]{} Method&&\ \ &&\ &&\ & & ANGPTL3 & ANGPTL4 & ANGPTL5 & ANGPTL6\ GenRF & &0.8129&0.0851&0.0009$\ast$&0.6803\ SKAT & &0.6395&0.0596&0.0212$\ast$&0.3313\ \ &&\ &&\ & & ANGPTL3 & ANGPTL4 & ANGPTL5 & ANGPTL6\ GenRF & &0.0136$\ast$&0.0105$\ast$&0.0716&0.3024\ SKAT & &0.0020$\ast$&0.0017$\ast$&0.3527&0.6450\ \[t:table3\] Discussion {#s:discuss} ========== In this article, we have proposed a novel framework for modeling and testing for the joint association of genetic variants with a trait from the perspective of viewing the response as a random field on a genetic space. A random field generalizes the concept of a stochastic process and random field models have widespread applications in areas such as imaging analysis, spatial statistics and so on. Specifically, in our genetic random field model, we view that there is a random response variable associated with each value of the $p$-dimensional vector that corresponds to the $p$ variants. An analogy with spatial statistic is helpful to view this vividly, i.e., each vector value of the $p$ variants can be analogously viewed as a “location” in a space in spatial statistics and we term it the $p$-dimensional genetic space. Our GenRF model is closely related to the conditional autoregressive model in spatial statistics and the parameter $\gamma$ for describing the association of variants with a trait is the counterpart of a spatial correlation parameter for quantifying spatial autocorrelation. However, as we have mentioned earlier, regular tests for spatial correlation cannot be applied in our setting to test for the genetic association. The reason is that the matrix ${\mbox{\boldmath $S$}}$ in our GenRF model does not satisfy the regularity condition usually assumed in spatial statistics for deriving the asymptotic distribution, as each subject has infinite neighbors in the genetic space and ${\mbox{\boldmath $S$}}$ is a dense matrix. Although motivated from very different perspectives, the proposed GenRF model shares similar features with SKAT. Both methods are based, explicitly or implicitly, on the idea that, if the variants of interest are associated with a trait, then subjects with similar variants have similar (or positively correlated) responses. The proposed GenRF method models the similarity in responses by an autoregressive model for a random field, whereas in SKAT similarity in responses are described by positive correlations induced by random effects in the framework of a mixed effect model. As discussed in Section 3.2, actually both methods lead to correlated responses under the alternative hypothesis that variants are associated with the trait but provide different parameterizations of the covariance. Based on the GenRF model, a test for genetic associations is developed and this test shares many of the appealing features of SKAT. The proposed GenRF test is based on testing a null hypothesis involving a single scalar parameter, allowing it to exploit LD to improve power. When the LD effect is moderate or high, our simulations show that the GenRF test achieves much higher power than the method based on a linear regression. Similar to SKAT, the GenRF model is flexible enough to allow for complex interaction effects. Our simulations demonstrate that the GenRF test is even much more powerful than SKAT in the presence of complex interaction effects. Moreover, as SKAT, prespecified variant-specific weights can be incorporated into the model and test to boost power for rare variants. Finally, the GenRF test is computationally easy to implement. In summary, the GenRF test is an appealing alternative to SKAT for testing the joint association of variants with a trait. Based on our simulations, it can achieve overall comparable performance to and sometimes even much better performance than SKAT. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Jonathan Cohen for the permission to use the Dallas Heart Study data and Dajiang Liu for preparing the data. The authors also highly appreciate Michael Boehnke’s comprehensive suggestions, and thank for the valuable comments from Xihong Lin, Seunggeun Lee, William Wen, Veronica Berrocal, Laura Scott, Lu Wang, Dajiang Liu, Bhramar Mukherjee and Hui Jiang. Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York. Besag, J. (1974). Spatial Interaction and the Statistical Analysis of Lattice Systems. [Journal of the Royal Statistical Society. Series B]{} [36(2),]{} 192–236. Besag, J. (1975). Statistical Analysis of Non-Lattice Data. [The Statistician]{} [24(3),]{} 179–195. Browning, J. D., Szczepaniak, L. S., Dobbins, R., Nuremberg, P., Horton, J. D., Cohen, J. C., Grundy, S. M., and Hobbs, H. H. (2004). 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[A bijective proof of the hook-length formula\ for standard immaculate tableaux]{} Emma L.L. Gao$^1$ and Arthur L.B. Yang$^{2}$\ Center for Combinatorics, LPMC-TJKLC\ Nankai University, Tianjin 300071, P. R. China\ Email: $^{1}$[[email protected]]{}, $^{2}$[[email protected]]{} Abstract. In this paper, we present a direct bijective proof of the hook-length formula for standard immaculate tableaux, which arose in the study of non-commutative symmetric functions. Our proof is along the spirit of Novelli, Pak and Stoyanovskii’s combinatorial proof of the hook-length formula for standard Young tableaux. AMS Classification 2010: 05E05 Keywords: composition; hook; hook[-]{}length formula; immaculate tableau; standard immaculate tableau. Introduction ============ In the study of non-commutative symmetric functions, Berg, Bergeron, Saliola, Serrano and Zabrocki [@BBSSZ] introduced the notion of immaculate tableaux, which was indexed by compositions of integers. They obtained an amazingly simple product formula to enumerate standard immaculate tableaux, which is analogous to the hook-length formula for standard Young tableaux. Their proof is by induction on the length of the composition. The objective of this paper is to give a direct bijective proof of the hook-length formula for standard immaculate tableaux. The classical hook length formula for standard Young tableaux was first discovered by Frame, Robinson and Thrall [@FRT]. To explore why hooks appear in this formula, many proofs have been published based on different methods. The first step towards this direction was given by Hillman and Grassl [@HG]. They proved a special case of Stanley’s hook-content formula, from which the hook-length formula follows. Later, Greene, Nijenhuis and Wilf [@GNW] found a probabilistic proof using the hook walk which shows clearly the role of hooks. The first bijective proof was given by Franzblau and Zeilberger [@FZ], though it is not so direct. Even though there were so many proofs, none of them was considered satisfactory. Novelli, Pak and Stoyanovskii [@NPS] presented an elegant bijective proof of the hook-length formula, based on the work of Pak and Stoyanovskii [@PS]. Motivated by Novelli, Pak and Stoyanovskii’s combinatorial proof of the classical hook-length formula, it is natural to consider whether a naturally bijective proof exists for the hook-length formula of standard immaculate tableaux, which could clearly illuminate the role of hooks. In this paper, we shall present such a proof. Let us first review some notation and terminology concerning the hook-length formula for standard immaculate tableaux. A composition $\alpha$ of a positive integer $n$, denoted by $\alpha\models n$, is a tuple $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_k)$ of positive integers such that $\sum_{i=1}^k\alpha_i=n$. The entries $\alpha_i$ are called the parts of $\alpha$, and the number of parts is called the length of $\alpha$, denoted by $\ell(\alpha)$. Each composition is associated to a diagram of left-justified array of cells. Given a composition $\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_k)$, the corresponding diagram has $\alpha_i$ cells in the $i$-th row. Here we number the rows from top to bottom and the columns from left to right. The cell in the $i$-th row and $j$-th column is denoted by the pair $(i, j)$. For example, the diagram of the composition $(4,1,2,3)$ is as follows. Following Berg, Bergeron, Saliola, Serrano and Zabrocki [@BBSSZ], we now introduce the definitions of hooks and immaculate tableaux. Given a composition $\alpha$ and a cell $c=(i,j)$ in $\alpha$, the hook of $c$, denoted $H_c$, is defined by $$\begin{aligned} H_c =H_{i,j}= \left\{ \begin{array}{ll} {\{(i',j'):i\leq i'\leq \ell(\alpha), 1\leq j'\leq \alpha_{i'}\}}, &\mbox{if $j=1$;} \\[5pt] {\{(i,j'):j\leq j'\leq \alpha_i\}}, &\mbox{if $j>1$.} \end{array} \right.\end{aligned}$$ Correspondingly, the hook[-]{}length of the cell $c=(i,j)$, denoted by $h_c$, is defined as $$h_c=h_{i,j}=\|H_{i,j}\|.$$ For example, taking the cells $(1,2)$ and $(2,1)$ of the composition $\alpha=(4,1,2,3)$, the hooks $H_{1,2}$ and $H_{2,1}$ are depicted in Figure \[fig: hook example\] as the sets of dotted cells. Clearly, we have $h_{1,2}=3$ and $h_{2,1}=6$. captype[figure]{} \\[\][0,1+3]{}\ $H_{1,2}$ captype[figure]{} \ [4,1,2,3]{} \* [4]{}\ $H_{2,1}$ We proceed to introduce the concept of immaculate tableaux. Given a composition $\alpha \models n$, a tableau of shape $\alpha$ is defined to be an array $T=(T_{ij})$ obtained by filling the diagram of $\alpha$ with positive integers. For the convenience, let $T_{ij}=\infty$ if the cell $(i,j)$ does not belong to the diagram of $\alpha$. We say that $T$ has content $\beta=(\beta_1,\beta_2,\ldots)$ if, for each $i\geq 1$, there are $\beta_i$ entries equal to $i$ in the array. For any $1\leq i\leq \ell(\alpha)$ and $1\leq j\leq \alpha_i$, the entry $T_{ij}$ is said to be [stable]{} in $T$ if either of the following two conditions holds: - $j>1$ and $T_{ij}\leq T_{i,j+1}$; or - $j=1$ and $T_{ij}<T_{i+1,j}$ and $T_{ij}\leq T_{i,j+1}$. An immaculate tableau of shape $\alpha$ is a tableau $T=(T_{ij})$ of shape $\alpha$ such that: - all entries $T_{ij}$ are stable; and - for any $m\geq 2$, if $m$ appears in $T$, so does $m-1$. Condition (ii) implies that the content of an immaculate tableau must be a composition $\beta$ of the form $(\beta_1,\ldots,\beta_{\ell(\beta)})$ whose components are all positive. Given an immaculate tableau $T$ of shape $\alpha$, we say that it is standard if $T$ has content $(1^n)$. Remarkably, Berg, Bergeron, Saliola, Serrano and Zabrocki [@BBSSZ] found that the standard immaculate tableaux can be enumerated by using the above defined hook-lengths of the cells of the indexed composition. Given a composition $\alpha$, denote by $f^\alpha$ the number of standard immaculate tableaux of shape $\alpha$. The hook-length formula for standard immaculate tableaux reads as follows. \[hook\_immaculate\] For any composition $\alpha\models n$, we have $$\begin{aligned} \label{hook_formula} f^\alpha=\frac{n!}{\prod_{(i,j)\in \alpha}h_{i,j}}. \end{aligned}$$ For instance, the composition $(2, 1, 2)$ has hook lengths given by 5& 1\ 3\ 2& 1 From the hook-length formula it follows that the number of standard immaculate tableaux of shape $(2, 1, 2)$ is $$\frac{5!}{5\cdot1\cdot3\cdot2\cdot1}=4.$$ In fact, there are exactly $4$ standard immaculate tableaux of shape $(2, 1, 2)$ as illustrated below. captype[figure]{} 1& 2\ 3\ 4& 5 captype[figure]{} 1& 3\ 2\ 4& 5 captype[figure]{} 1& 4\ 2\ 3& 5 captype[figure]{} 1& 5\ 2\ 3& 4 A bijective proof of Theorem \[hook\_immaculate\] ================================================= The aim of this section is to present a bijective proof of Theorem \[hook\_immaculate\]. Towards this end, we first rewrite (\[hook\_formula\]) as $$\begin{aligned} \label{eq-rewritten} n!=f^\alpha\prod_{(i,j)\in \alpha}h_{i,j}.\end{aligned}$$ Then we need to construct two sets such that their cardinalities are respectively given by the left-hand side and the right-hand side of . Let $X$ be the set of tableaux of shape $\alpha$ and content $(1^n)$, and let $Y$ be the set of $\{(P, J)\}$, where $P$ is a standard immaculate tableau of shape $\alpha$ and $J$ is an array of shape $\alpha$ with $J_{i,j}\in \{1,\ldots, h_{i,j}\}$. We call $J$ a hook tableau. It is easy to see that $$\begin{aligned} \|X\|=n!\qquad \mbox{ and } \qquad \|Y\|=f^\alpha\prod_{(i,j)\in \alpha}h_{i,j}.\end{aligned}$$ There remains to show that there exists a bijection between $X$ and $Y$. For the construction of our bijection, a total order on the cells of $\alpha$ is needed. Following Novelli, Pak and Stoyanovskii, we totally order the cells of the diagram of $\alpha$ by reverse lexicographic order on their coordinates. Precisely, we have $(i,j)\leq (i',j')$ if and only if $j>j'$; or $j=j'$ and $i\geq i'.$ As will be shown below, this order is critical for the construction of our bijection. Label the cells of $\alpha$ in the given order $c_1<c_2<\cdots<c_n.$ For example, Figure \[The total order\] displays the total order for the diagram of shape $(4,1,4,2,1)$. captype[figure]{} c\_[12]{}& c\_7& c\_4 & c\_2\ c\_[11]{}\ c\_[10]{}& c\_6 & c\_3 & c\_1\ c\_9&c\_5\ c\_8 Given a tableau $T$ of shape $\alpha$ and a cell $c$, let $T^{\leq c}$ (resp. $T^{<c}$) denote the partial tableau composed of all cells $b$ of $T$ with $b\leq c$ (resp. $b<c$). For example, $$\begin{aligned} \mbox{if \qquad} T = \begin{ytableau} 11& 5& 8& 9\\ 3\\ 10& 2 & 4 & 12\\ 1&6 \\ 7 \end{ytableau}, \qquad \mbox{then \qquad} T^{\leq c_8} = \begin{ytableau} \none& 8& 5 & 9\\ \none\\ \none& 12 & 2 & 4\\ \none&6 \\ 7 \end{ytableau}.\end{aligned}$$ For the convenience, we say that the partial tableau $T^{\leq c}$ (resp. $T^{<c}$) is standard if all the entries in $T^{\leq c}$ (resp. $T^{<c}$) are stable with respect to the diagram of $\alpha$. We now construct a map $\psi$ from $Y$ to $X$. Given a pair $(P,J)\in Y$, we construct a tableau $T\in X$ in the following way. Without loss of generality, we may assume that $n>1$. Begin with $(P_1,J_1)=(P,J)$. If $(P_k,J_k)$ are defined for $1\leq k<n$, then let $J_{k+1}=J_k$ except for $(J_{k+1})_{ij}=1$ if the cell $c_{n+1-k}$ lies in the $i$-th row and $j$-th column. Suppose that the $(J_{k})_{ij}$-th cell of the hook set $H_{ij}$, reading from left to right and top to bottom, lies in the $i'$-th row and $j'$-th column of the diagram of $\alpha$. The cells $(i,j)$ and $(i',j')$ uniquely determine a path $L$ in the following way. - If $i=i'$, then let $L=\{(i,j), (i,j+1),\ldots,(i,j')\}$; - Suppose that $i\neq i'$. Then by the definition of the hook function, we must have $i<i'$ and $j=1$. In this case, let $L=\{(i,1), (i+1,1),\ldots,(i',1),(i',2),\ldots,(i',j')\}.$ The tableau $P_{k+1}$ is obtained from $P_k$ by a circular right shift of the entries on the path $L$. If the process ends at $(P_n,J_n)$, then all entries of $J_n$ are $1$, and let $T=P_n$. The map $\psi$ is defined by $\psi(P,J)=T$. Note that $P_1^{\leq c_n}=P$ is standard. A moment’s thought shows that the partial tableau $P_k^{\leq c_{n+1-k}}$ is standard for any $k$. For example, for $\alpha=(4,1,4,2,1)$, let [@llc@]{} P = 1& 5& 8& 9\ 2\ 3& 4 & 11& 12\ 6&10\ 7 & and & J = 8& 2& 1& 1\ 3\ 6& 3 & 1 &1\ 1&1\ 1 . We have a sequence of pairs $\{(P_i, J_i)\}_{i=1}^{12}$ as follows, where the entries in the path $L$ are underlined at each step. [@ccc]{} $i$ &$P_i$ & $J_i$\ $1$& & 5& 8& 9\ \ & & & 12\ 6&10\ 7 & 8& 2& 1& 1\ 3\ 6& 3 & 1 &1\ 1&1\ 1 \ \ $2$& 11& 5& 8& 9\ \ & & 4 & 12\ 6&10\ 7 & 1& 2& 1& 1\ 3\ 6& 3 & 1 &1\ 1&1\ 1 \ \ $3$& 11& 5& 8& 9\ 3\ & 2 & 4 & 12\ &\ 7 & 1& 2& 1& 1\ 1\ 6& 3 & 1 &1\ 1&1\ 1 \ \ $4$& 11& 5& 8& 9\ 3\ 10& 2 & 4 & 12\ &6\ 7 & 1& 2& 1& 1\ 1\ 1& 3 & 1 &1\ 1&1\ 1 $\Rightarrow$ [@ccc]{} $i$ &$P_i$ & $J_i$\ $5$& 11& 5& 8& 9\ 3\ 10& 2 & 4 & 12\ 1&6\ & 1& 2& 1& 1\ 1\ 1& 3 & 1 &1\ 1&1\ 1 \ \ $6$& 11& & & 9\ 3\ 10& 2 & 4 & 12\ 1&6\ 7 & 1& 2& 1& 1\ 1\ 1& 3 & 1 &1\ 1&1\ 1 \ \ $7$& 11& 8& 5 & 9\ 3\ 10& & &\ 1&6\ 7 & 1& 1& 1& 1\ 1\ 1& 3 & 1 &1\ 1&1\ 1 \ \ $8$ & 11& 8& 5 & 9\ 3\ 10& 12 & 2 & 4\ 1&\ 7 & 1& 1& 1& 1\ 1\ 1& 1 & 1 &1\ 1&1\ 1 $\Rightarrow$ [@ccc]{} $i$ &$P_i$ & $J_i$\ $9$ & 11& 8& & 9\ 3\ 10& 12 & 2 & 4\ 1&6\ 7 & 1& 1& 1& 1\ 1\ 1& 1 & 1 &1\ 1&1\ 1 \ \ $10$ & 11& 8& 5 & 9\ 3\ 10& 12 & & 4\ 1&6\ 7 & 1& 1& 1& 1\ 1\ 1& 1 & 1 &1\ 1&1\ 1 \ \ $11$ & 11& 8& 5 &\ 3\ 10& 12 & 2 & 4\ 1&6\ 7 & 1& 1& 1& 1\ 1\ 1& 1 & 1 &1\ 1&1\ 1 \ \ $12$ & 11& 8& 5 & 9\ 3\ 10& 12 & 2 & 4\ 1&6\ 7 & 1& 1& 1& 1\ 1\ 1& 1 & 1 &1\ 1&1\ 1 Our main result is as follows. \[thm-main\] The map $\psi$ is a bijection from $Y$ to $X$. To show that $\psi$ is a bijection, it suffices to construct a map $\phi$ from $X$ to $Y$ such that $\phi=\psi^{-1}$. This map $\phi$ is based on a modified jeu de taquin performed on $X$. Suppose that we are given a tableau $T$ of shape $\alpha$ in $X$. For any $1\leq e\leq n$, denote by $(i,j)$ the unique cell in $T$ such that $T_{ij}=e$. To each $e$, we will associate a transformation $\mathrm{jdt}_e(T)$ of $T$ called a modified jeu de taquin slide of $T$ with respect to $e$. If $e$ is stable, then we do nothing. If $e$ is not stable, then there are two cases to consider according to whether $e$ lies in the first column of the diagram of $\alpha$. - If $T_{ij}=e$ for some $1<j\leq\alpha_i$, then interchange $T_{ij}$ and $T_{i,j+1}$; - If $T_{i1}=e$ for some $1\leq i\leq \ell(\alpha)$, then interchange $T_{ij}$ and the smaller one of $\{T_{i,j+1}, T_{i+1,j}\}$. Then look at the stability of $e$, and repeat the same procedure. This will eventually terminate since the entry $e$, if it moves, will move either downwards or rightwards in $T$ at each step. The path of $e$ in $T$ is defined as the set of the cells that $e$ passes through when applying $\mathrm{jdt}_e(T)$. By convention, we also denote the resulting tableau by $\mathrm{jdt}_e(T)$. We use an example to illustrate the modified jeu de taquin algorithm. Taking $e=10$ and $$\begin{aligned} T = \begin{ytableau} 11& 5& 8& 9\\ 3\\ 10& 2 & 4 & 12\\ 1&6 \\ 7 \end{ytableau},\end{aligned}$$ then the process of $\mathrm{jdt}_e(T)$ is as follows, $$\begin{matrix} T & = & & \begin{ytableau} 11& 5& 8& 9\\ 3\\ \textbf{10}& 2 & 4 & 12\\ \underline{1}&6 \\ 7 \end{ytableau} & \rightarrow & %\raisebox{-5pt} {\begin{ytableau} 11& 5& 8& 9\\ 3\\ 1& 2 & 4 & 12\\ \textbf{10}&\underline{6} \\ 7 \end{ytableau}} &\rightarrow & %\raisebox{-15pt} {\begin{ytableau} 11& 5& 8& 9\\ 3\\ 1& 2 & 4 & 12\\ 6&\textbf{10} \\ 7 \end{ytableau} } & = & \mathrm{jdt}_e(T), \end{matrix}$$ where the entry $e$ is in boldface and the integers interchanged with $e$ are underlined. Clearly, the path of $10$ in $T$ is $\{(3,1),(4,1),(4,2)\}$. The following result is evident, and we omit the straightforward details. \[lemma-path\] Given $T\in X$ and $1\leq e\leq n$, let $(i,j)$ be the unique cell in $T$ such that $T_{ij}=e$. If $j>1$, then the path of $e$ in $T$ is of the form $$\begin{aligned} %\label{eq-1} \{(i,j),(i, j+1),\ldots,(i,j+k)\}, \quad \mbox{for some $k\geq 0$.}\end{aligned}$$ If $j=1$, then the path of $e$ in $T$ is of the form $$\begin{aligned} %\label{eq-2} \{(i,1), (i+1, 1),\ldots,(i+k,1),(i+k, 2),\ldots,(i+k,l)\}, \quad \mbox{for some $k\geq 0$ and $l\geq 1$.}\end{aligned}$$ Moreover, the tableau $\mathrm{jdt}_e(T)$ is obtained from $T$ by a circular left shift of the entries on the path of $e$. The next result shows that the modified jeu de taquin slide preserves the stability of most entries in the tableau. \[standard pro\] Given $T\in X$ and $1\leq e\leq n$, denote by $c$ the unique cell in $T$ such that $T_{c}=e$. Then if $T^{<c}$ is standard, so is $\mathrm{jdt}_e(T)^{\leq c}$. Note that $\mathrm{jdt}_e(T)^{\leq c}=T^{\leq c}$ except for the entries of the cells in the path of $e$ in $T$. Suppose that $c$ is the $(i,j)$ cell of $T$. There are two cases to consider. - If $j>1$, then $\mathrm{jdt}_e(T)^{\leq c}$ has no cell in the first column of the diagram of $\alpha$. We only need to show that the entries of $\mathrm{jdt}_e(T)^{\leq c}$ are strictly increasing in each row. By Lemma \[lemma-path\], the path of $e$ is of the form $$\{(i,j),(i, j+1),\ldots,(i,j+k)\}.$$ Therefore, we have $$(\mathrm{jdt}_e(T))_{rs}=\left\{\begin{array}{ll} T_{r,s+1}, & \mbox{for $r=i$ and $j\leq s\leq j+k-1$},\\[3pt] T_{i,j}, & \mbox{for $r=i$ and $s=j+k$},\\[3pt] T_{rs}, & \mbox{otherwise.} \end{array}\right.$$ The entries of $\mathrm{jdt}_e(T)^{\leq c}$ in each row other than the $i$-th row, are identically the same as those of $T^{<c}$, and hence are increasing since $T^{<c}$ is standard. While for the $i$-th row, the form of the path of $e$ implies that $$T_{i,j+1}<T_{i,j+2}<\cdots<T_{i,j+k}<T_{i,j}<T_{i,j+k+1},$$ that is $$\mathrm{jdt}_e(T)_{i,j}<\mathrm{jdt}_e(T)_{i,j+1}<\cdots<\mathrm{jdt}_e(T)_{i,j+k-1}<\mathrm{jdt}_e(T)_{i,j+k}<T_{i,j+k+1}.$$ Thus, the entries in the $i$-th row of $\mathrm{jdt}_e(T)^{\leq c}$ are also increasing, as desired. - If $j=1$, then $\mathrm{jdt}_e(T)^{\leq c}$ must contain a cell in the first column of the diagram of $\alpha$. By Lemma \[lemma-path\], the path of $e$ is of the form $$\begin{aligned} \{(i,1), (i+1, 1),\ldots,(i+k,1),(i+k, 2),\ldots,(i+k,l)\}.\end{aligned}$$ Our proof will be divided into two subcases: - The case of $l=1$. In this case, we have $$(\mathrm{jdt}_e(T))_{rs}=\left\{\begin{array}{ll} T_{r+1,1}, & \mbox{for ${s}=1$ and $i\leq r\leq i+k-1$},\\[3pt] T_{i,1}, & \mbox{for ${s}=1$ and $r=i+k$},\\[3pt] T_{rs}, & \mbox{otherwise.} \end{array}\right.$$ Since $T^{<c}$ is standard, the form of the path of $e$ implies that $$T_{i+1,1}<\cdots<T_{i+k,1}<T_{i,1}<T_{i+k+1,1} <\cdots<T_{{\ell(\alpha)},1},$$ that is, $$(\mathrm{jdt}_e(T))_{i,1}<\cdots<(\mathrm{jdt}_e(T))_{i+k,1}<(\mathrm{jdt}_e(T))_{i+k+1,1}<\cdots<(\mathrm{jdt}_e(T))_{{\ell(\alpha)},1}.$$ This means that the entries of $\mathrm{jdt}_e(T)^{\leq c}$ are strictly increasing in the first column. There remains to show that each row of $\mathrm{jdt}_e(T)^{\leq c}$ are strictly increasing. Since all entries of $T$ in any other than the first column remain fixed when applying $\mathrm{jdt}_e(T)$, it suffices to show that, for $i\leq r\leq i+k$, $$(\mathrm{jdt}_e(T))_{r,1}<(\mathrm{jdt}_e(T))_{r,2}.$$ This is true, since, for $i\leq r\leq i+k-1$, $$(\mathrm{jdt}_e(T))_{r,1}=T_{r+1,1}<T_{r,2}=(\mathrm{jdt}_e(T))_{r,2},$$ and $$(\mathrm{jdt}_e(T))_{i+k,1}=T_{i,1}<T_{i+k,2}=(\mathrm{jdt}_e(T))_{i+k,2},$$ as implied by the form of the path of $e$. Therefore, the entries of $\mathrm{jdt}_e(T)^{\leq c}$ are increasing along each row. - The case of $l>1$. In this case, we have $$(\mathrm{jdt}_e(T))_{rs}=\left\{\begin{array}{ll} T_{r+1,1}, & \mbox{for $s=1$ and $i\leq r\leq i+k-1$},\\[3pt] T_{i+k,2}, & \mbox{for $s=1$ and $r=i+k$},\\[3pt] T_{i+k,s-1}, & \mbox{for $1<s<l-1$ and $r=i+k$},\\[3pt] T_{i,1}, & \mbox{for $s=l$ and $r=i+k$},\\[3pt] T_{rs}, & \mbox{otherwise.} \end{array}\right.$$ The strict increasing property of the $(i+k)$-th row of $\mathrm{jdt}_e(T)^{\leq c}$ can be proved by using similar arguments for case (a). While similar arguments for case (b1) could be used to show the strict increasing property of other rows of $\mathrm{jdt}_e(T)^{\leq c}$, as well as that of the first column of $\mathrm{jdt}_e(T)^{\leq c}$ except for the relation $\mathrm{jdt}_e(T)_{i+k,1}<\mathrm{jdt}_e(T)_{i+k+1,1}$. By the form of the path of $e$, we see that $$\mathrm{jdt}_e(T)_{i+k,1}=T_{i+k,2}<T_{i+k+1,1}=\mathrm{jdt}_e(T)_{i+k+1,1}.$$ This completes the proof of the strict increasing property of the first column of $\mathrm{jdt}_e(T)^{\leq c}$. Combining (a) and (b), we obtain the desired result. We proceed to describe the inverse map $\phi: X\rightarrow Y$. Suppose that $T\in X$ is a tableau of shape $\alpha$ and the cells of $\alpha$ are ordered as $c_1<c_2<\cdots<c_n$. We shall associate with $T$ a pair $(P,J)\in Y$, as follows. Begin with $(T_1,S_1)$, where $T_1=T$ and $S_1$ is the array of shape $\alpha$ with all entries equal to $1$. If $(T_{k}, S_{k})$ are defined for $1\leq k\leq n-1$, then let $T_{k+1} =\mathrm{jdt}_{e_{k+1}}(T_{k})$, where $e_{k+1}$ is the entry of the cell $c_{k+1}$ in $T_{k}$. Suppose that the path of $e_{k+1}$ in $T_{k}$ starts at $c_{k+1}=(i,j)$ and ends at $(i',j')$. Let $S_{k+1}=S_k$ except for the values $$\begin{aligned} \label{eq-sk} (S_{k+1})_{ij}= \left\{ \begin{array}{lcl} j'-j+1, &\text{if} &i'=i; \\ \alpha_i+\cdots+\alpha_{i'-1}+j', &\text{if} &i'>i. \end{array} \right.\end{aligned}$$ Suppose that the process ends at $(T_{n}, S_{n})$. We claim that $(T_{n}, S_{n})\in Y$. We first use induction to show that $T_{n}$ is a standard immaculate tableau, namely $T_n^{\leq c_n}$ is standard. It is clear that $T_1^{\leq c_1}$ is standard. If $T_{k}^{\leq c_{k}}=T_{k}^{<c_{k+1}}$ is standard, then, by Proposition \[standard pro\], the partial tableau $T_{k+1}^{\leq c_{k+1}}=\mathrm{jdt}_{e_{k+1}}(T_{k})^{\leq c_{k+1}}$ is standard. By induction, we see that $T_n^{\leq c_n}$ is standard. We continue to show that $S_n$ is a hook tableau. For the cell $c_{k+1}=(i,j)$ of the diagram of $\alpha$, we have $(S_n)_{ij}=(S_{k+1})_{ij}$ by the construction of $S_n$. Suppose that the path of $e_{k+1}$ in $T_{k}$ starts at $c_{k+1}=(i,j)$ and ends at $(i',j')$. If $i'>i$, then we must have $j=1$ by Lemma \[lemma-path\]. From and the definition of the hook length it immediately follows that $(S_n)_{ij}=(S_{k+1})_{ij}\leq h_{ij}$. Let $(P,J)=(T_n, S_n)$. The map $\phi$ is defined by $\phi(T)=(P,J)$. Now we are able to prove our main result. Proof of Theorem \[thm-main\]. To prove that $\psi$ is a bijection, it suffices to show that $\phi$ is the inverse map of $\psi$. We first prove that $\phi$ is the left inverse of $\psi$, which implies the injectivity of $\psi$. Precisely, if a pair $(P,J)\in Y$ is mapped to $T\in X$ by $\psi$, then we must have $\phi(T)=(P,J)$. By the construction of $\psi$, there exists a sequence of pairs $\{(P_i,J_i)\}_{i=1}^n$ such that $(P,J)=(P_1,J_1)$ and $(P_n,J_n)=(T,J_n)$, where all entries of $J_n$ are $1$. Consider the transformation from $(P_k,J_k)$ to $(P_{k+1},J_{k+1})$. Let $L$ denote the path determined by the cell $c_{n+1-k}$ of the diagram of $\alpha$. Let $e_{n+1-k}$ denote the entry of $P_{k+1}$ at the cell $c_{n+1-k}$. We only need to show that the path of $e_{n+1-k}$ in $P_{k+1}$ coincides with the path $L$. But this is clear since the partial tableaux $P_k^{\leq c_{n+1-k}}$ and $P_{k+1}^{< c_{n+1-k}}$ are standard, and $P_{k+1}$ is obtained from $P_k$ by a circular right shift of the entries on the path $L$. Therefore, we have $\mathrm{jdt}_{e_{n+1-k}}P_{{k+1}}=P_k$, as the tableau $\mathrm{jdt}_{e_{n+1-k}}P_{{k+1}}$ is obtained from $P_{k+1}$ by a circular left shift of the entries on the path $L$. This implies that the pair $(P_{k+1},J_{k+1})$ will be mapped to $(P_k,J_k)$ during the construction of $\phi(T)$. Thus, we have $\phi(T)=(P,J)$. Next, we show that $\phi$ is the right inverse of $\psi$, which implies the surjectivity of $\psi$. Precisely, if $T\in X$ is mapped to a pair $(P,J)\in Y$ by $\phi$, then we must have $\psi(P,J)=T$. By the construction of $\phi$, there exists a sequence of pairs $\{(T_i,S_i)\}_{i=1}^n$ such that $(T_1,S_1)=(T,S_1)$ and $(P,J)=(T_n,S_n)$, where all entries of $S_1$ are $1$. Consider the transformation from $(T_k,S_k)$ to $(T_{k+1},S_{k+1})$. Let $e_{k+1}$ denote the entry of $T_k$ at the cell $c_{k+1}$. Note that $T_{k+1}=\mathrm{jdt}_{e_{k+1}}(T_k)$. By Lemma \[lemma-path\], $T_{k+1}$ is obtained from $T_k$ by a circular left shift of the entries on the path of $e_{k+1}$ in $T_k$. Moreover, the entry of $c_{k+1}$ in $S_{k+1}$ is determined by , and this entry will uniquely determine a path $L$ in $T_{k+1}$ when encountering the pair $(T_{k+1},S_{k+1})$ during the construction of $\psi(P,J)$. According to the construction of the map $\psi$, the path $L$ must coincide with the path of $e_{k+1}$ in $T_k$. This implies that the pair $(T_{k+1},S_{k+1})$ will be transformed to $(T_k,S_k)$ during the construction of $\psi(P,J)$. Therefore, we have $\psi(P,J)=T$. Combining the above two aspects, we complete the proof of the bijectivity of $\psi$. [Acknowledgements.]{} This work was supported by the 973 Project, the PCSIRT Project of the Ministry of Education and the National Science Foundation of China. [1]{} C. Berg, N. Bergeron, F. Saliola, L. Serrano and M. Zabrocki, A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions. Canadian J. Math. 66 (2014), 525–565. J. Frame, G. Robinson and R. Thrall, The hook graphs of the symmetric group. Canadian J. Math. 6 (1954), 316–325. D. S. Franzblau and D. Zeilberger, A bijective proof of the hook-length formula. J. Algorithms 3 (1982), 317–343. C. Greene, A. Nijenhuis and H. Wilf. A probabilistic proof of a formula for the number of Young tableaux of a given shape. Adv. in Math. 31 (1979), 104–109. A. P. Hillman and R. M. Grassl. Reverse plane partitions and tableau hook numbers. J. Combin. Theory Ser. A 21 (1976), 216–221. J. C. Novelli, I. M. Pak and A. V. Stoyanovski$\mathrm{\check{i}}$, A direct bijective proof of the hook-length formula. Discrete Math. Theor. Comput. Sci. 1 (1997), 53–67. I. M. Pak and A. V. Stoyanovski$\mathrm{\check{i}}$, Bijective proof of the hook formula and its analogues. Funct. Anal. Appl. 26 (1992), 216–218.
--- abstract: 'We experimentally demonstrate topological edge states arising from the valley-Hall effect in two-dimensional honeycomb photonic lattices with broken inversion symmetry. We break inversion symmetry by detuning the refractive indices of the two honeycomb sublattices, giving rise to a boron nitride-like band structure. The edge states therefore exist along the domain walls between regions of opposite valley Chern numbers. We probe both the armchair and zig-zag domain walls and show that the former become gapped for any detuning, whereas the latter remain ungapped until a cutoff is reached. The valley-Hall effect provides a new mechanism for the realization of time-reversal invariant photonic topological insulators.' author: - \| Jiho Noh$^{1}$, Sheng Huang$^{2}$, Kevin Chen$^{2}$, and Mikael C. Rechtsman$^{1}$\ [[ $^{1}$Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA\ $^{2}$Department of Electrical and Computer Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA\ ]{}]{} bibliography: - 'reference.bib' title: 'Observation of Photonic Topological Valley-Hall Edge States' --- Photonic topological insulators (PTIs) are dielectric structures that possess topologically protected edge states that are robust to scattering by disorder [@Lu2014; @Haldane2008; @Wang2009; @Carusotto2011; @Hafezi2011; @Fang2012; @Rechtsman2013; @Hafezi2013; @Khanikaev2013; @cheng2016robust; @Wu2015; @Gao2016]. There are two categories of PTIs: those that break time-reversal symmetry [@Wang2009; @Rechtsman2013] and those that preserve it [@Hafezi2013; @Khanikaev2013; @Wu2015]. In PTIs that break time-reversal symmetry, there exist one-way edge states, which ensure their robustness, due to the lack of counter-propagating partners at same frequency. In those that preserve it, there exist counter-propagating edge states that are protected only against certain classes of disorder. However, the latter can be more straightforward to realize because they do not require strong time-reversal breaking. Photonic topological insulators have been of interest due to the possibility of photonic devices that are less sensitive to fabrication disorder. In the valley-Hall effect, broken inversion symmetry in a two-dimensional honeycomb lattice causes opposite Berry curvatures in the two valleys of the band structure [@Xiao2007; @Yao2008], and has been realized in solid-state two-dimensional materials [@Mak2014; @Gorbachev2014; @Sui2015; @Shimazaki2015; @Ju2015]. The valley-Hall effect is time-reversal invariant and has common characteristics with the spin Hall effect [@Kane95], where the two valleys in the band structure are used as ‘pseudo-spin’ degrees of freedom. It was shown theoretically that valley-Hall topological edge states would arise in analogous photonic structures [@Ma2016; @Dong2017; @Chen2016_01; @Chen2016_02; @fefferman2016bifurcations; @fefferman2016edge; @fefferman2017]. In addition, valley-Hall topological edge states have also been recently studied in the context of topological valley transport of sound in sonic crystals [@Lu2017]. Here, we present the experimental observation of photonic topological valley-Hall edge states at domain walls between valley-Hall PTIs of opposite valley Chern numbers. The bulk-edge correspondence ensures the presence of edge states: the change in valley Chern number across the domain wall is associated with the existence of counter-propagating edge states [@Mong2011; @Delplace2011; @Ju2015]. We realize the photonic valley-Hall topological edge states in evanescently-coupled waveguide arrays, i.e., photonic lattices, fabricated using the femtosecond direct laser writing technique [@Szameit2010]. We probe different types of domain walls, namely the armchair and zig-zag edges. We also enter a fully gapped regime, which is not accessible in solid-state two-dimensional materials. The topological protection associated with the valley-Hall effect applies as long as a single valley is populated and does not mix with the other valley. In general, disorder that has only low spatial frequency components (i.e., is sufficiently smooth) will not mix the valleys. ![image](Fig01.eps){width="17.0cm"} We begin by describing our experimental system, which is composed of an array of evanescently-coupled waveguides arranged in a honeycomb lattice geometry. The laser-writing technique allows us to arbitrarily control the refractive index of the waveguides, by varying the average power of the pulse train in the femtosecond direct laser writing procedure. The geometries of lattices having armchair and zig-zag edges at their domain wall are depicted in Fig. \[Fig\_01\](a) and (b), respectively. The interface is between two regions (top and bottom) that are both honeycomb lattices with opposite signs of the on-site energy detuning between the two component sublattices (this breaks inversion symmetry within each given lattice). The detuning is shown in the figure by the different colors (red and green) of the component sublattices. Experimentally, the detuning is carried out by controlling the refractive index of the waveguide at each site. Fig. \[Fig\_01\](c) shows the two-dimensional bulk band structure of the inversion-symmetry-broken honeycomb lattice, clearly showing the two valleys. This is simply an inversion-symmetry-broken variation of the photonic honeycomb lattices [@Peleg2007; @Bahat-Treidel2008; @Plotnik2013], and as in the graphene band structure, two valleys are located at two non-equivalent K and K$^{\prime}$ points in the first Brillouin zone. The valley Chern number is defined as the difference in integrated Berry curvature associated with the two valleys. Since the Berry curvature points in the opposite directions ($+z,-z$) in the two valleys in a given lattice, and the sign is given by that of the inversion-breaking term, it follows that the top and bottom lattices in Fig. \[Fig\_01\](a,b) must have opposite valley Chern numbers and thus have valley-protected edge states. The diffraction of light through the waveguide array is governed by the paraxial wave equation: $$\begin{split} i\partial_{z}\psi(\textbf{\textrm{r}},z) = -\frac{1}{2k_{0}}\nabla^{2}_{\textbf{\textrm{r}}}\psi(\textbf{\textrm{r}},z)-\frac{k_{0}\Delta n(\textbf{\textrm{r}})}{n_{0}}\psi (\textbf{\textrm{r}},z)\\ \equiv H_{cont}\psi(\textbf{\textrm{r}},z), \label{eq_paraxial} \end{split}$$ where $\psi(\textbf{\textrm{r}},z)$ is the envelope function of the electric field $\textrm{\textbf{E}}(\textbf{\textrm{r}},z)=\psi(\textbf{\textrm{r}},z)\exp^{i(k_{0}z-\omega t)}\hat{x}$, $k_{0}=2\pi n_{0}/\lambda$ is the wavenumber within the medium, $\lambda$ is the wavelength of the laser light, $\omega = 2\pi c/\lambda$, and $\nabla^{2}_{\textbf{\textrm{r}}}$ is the Laplacian in the transverse $(x,y)$ plane. $H_{cont}$ is the continuum Hamiltonian for propagation of the wave in the photonic lattice. $\Delta n$ is the refractive index of the waveguide relative to the index of our medium, $n_{0}=1.47$, which acts as an effective potential in the Schrödinger equation, Eq. (\[eq\_paraxial\]). The inversion symmetry of the lattice is broken by having different $\Delta n_{\textrm{A}}$ and $\Delta n_{\textrm{B}}$ for waveguides in sublattices $A$ and $B$, respectively, which is analogous to having different on-site energies $E_{\textrm{A}}$ and $E_{\textrm{B}}$ in the condensed-matter context. Furthermore, we write two additional waveguides, which we call ‘straw waveguides’ (as discussed previously in Ref. [@CleoFTAI]) into which light is injected. The straws are weakly coupled to the lattice, allowing them to act as an external drive that is injecting light into the system without altering the system’s intrinsic modes. Furthermore, varying the refractive index of the straw, $\Delta n_{\textrm{s}}$, allows for the control of the propagation constant (i.e., energy) of the modes being injected into the structure. By analogy with condensed-matter systems, the straw allows us to control the effective ‘Fermi energy’ of the system, only allowing coupling to modes of a given energy, $E$. The emergence of valley-Hall topological edge states is shown by a full-continuum calculation by diagonalizing $H_{cont}$ in Eq. (\[eq\_paraxial\]) of two-dimensional inversion-symmetry-broken honeycomb lattice ribbons. The unit cell is a strip that is periodic in the horizontal direction (with a periodicity given by the lattice constant), but is many unit cells in the vertical direction and includes the domain wall (in fact, it must contain a minimum of two domain walls). The eigenvalues of the Schrödinger operator given in Eq. (1) are the energies of the calculated eigenmodes. Band structures and therefore bandgap sizes can be engineered by sweeping across $\Delta E/c_{0}$, where $\Delta E = E_A-E_B$ is the difference between the on-site energies in the two sublattices and $c_{0}$ is the coupling strength between the nearest-neighbor waveguides. Experimentally, $\Delta E$ can be controlled by varying both $\Delta n_{\textrm{A}}$ and $\Delta n_{\textrm{B}}$, and $c_{0}$ can be increased by decreasing the distance between the nearest-neighbor waveguides, $d$, and increasing $\lambda$; $c_0(\lambda)$ at fixed $d$=$19\,\mu$m for $\lambda$=1650nm and $\lambda$=1450nm are 2.69cm$^{-1}$ and 1.76cm$^{-1}$, respectively (for the remainder of the work, we logically order long wavelength before short wavelength because the bandgap increases with decreasing wavelength). Fig. \[eq\_paraxial\](d) and (e) show band structures when the armchair and zig-zag edges are placed at the domain wall, respectively, where $\lambda$=1650nm, $d$=$19\,\mu$m, $\Delta n_{\textrm{A}}=2.50\times 10^{-3}$, and $\Delta n_{\textrm{B}}=2.90\times 10^{-3}$, and Fig. \[eq\_paraxial\](f,g) show corresponding band structures with same $d$, $\Delta n_{\textrm{A}}$ and $\Delta n_{\textrm{B}}$ but with $\lambda$=1450nm. For both structures with armchair and zig-zag edge domain walls, the bulk bandgap opens immediately as $\Delta E/c_{0}$ becomes nonzero. However, behaviors of the edge states are different for each case: for the structure with the armchair domain wall, the edge bandgap opens immediately after $\Delta E/c_{0}$ becomes nonzero (Fig. \[eq\_paraxial\](d,f)). For the structure with the zig-zag domain wall, there exists edge states at the mid-gap for small $\Delta E/c_{0}$ (Fig. \[eq\_paraxial\](e)), which indicates the edge bandgap would open only at finite $\Delta E/c_{0}$ (Fig. \[eq\_paraxial\](g)). The two edge state bands shown in green and red in Fig. \[eq\_paraxial\](e,g) are localized close to the straw waveguides (in the center of the figure), and far away from them, respectively. Therefore, only the green bands will be physically accessible in the experiment. This difference between the armchair and zig-zag edges arises because the orientation of the armchair termination is such that it mixes the two valleys; since they may scatter between them, this allows for a matrix element for a gap to open even for small $\Delta E/c_0$. However, the zig-zag edge runs parallel to the line that connects the two valleys in $k$-space, implying that the presence of the edge does not connect them, allowing them to remain ungapped. To experimentally observe the emergence of topological edge states, a beam was launched at the input facet through a lens-tipped fiber, which allows us to couple the beam precisely into a selected straw waveguide. Here, the refractive index of the straw waveguides was calibrated to inject light at the mid-gap and significantly below the mid-gap, in different devices. The energies of the straw waveguide modes were calculated by diagonalizing $H_{cont}$ of a single waveguide. In Fig. \[Fig\_02\], we present the observed diffracted light at the output facet of the array for the case of mid-gap driving (red-dashed lines in Fig. \[Fig\_01\](d-g)). Here, the calculated energy of the straw waveguide modes at the mid-gap energy were -$4.39c_{0}$ and -$11.87c_{0}$ for 1650nm and 1450nm, respectively. We plot: (1) the edge intensity ratio, i.e., the ratio of the light intensity along the domain wall ($I_{edge}$) to the total light intensity ($I$), and (2) the penetration ratio, which is the light intensity that penetrates into the structure normalized by that in the straw. First, in the photonic lattice with the armchair domain wall, we observe that most of the light coupled into the straw waveguide stayed in the straw, not coupling into the waveguide array (Fig. \[Fig\_02\] insets). Both measured edge intensity ratio and penetration ratio were relatively very small, which indicate the presence of the bandgap between the edge modes: i.e., no edge states are available to transport light through the array. This experimental result agrees with the full-continuum calculation having red-dashed lines not crossing any edge states in the band structure as shown in Fig. \[Fig\_01\](d,f). On the other hand, from the analogous structure with zig-zag domain wall, we observed a clear excitation of edge states along the domain wall, which becomes more significant as wavelength is increased. This indicates that at $\lambda$=1450nm, the bandgap is fully open so that the straw waveguide mode is not able to couple into the domain wall but as we increase $\lambda$ to make $\Delta E/c_{0}$ subsequently decrease, the bandgap becomes smaller and eventually edge states couple with the straw waveguide mode at mid-gap. Furthermore, the sharp increase in edge intensity ratio and penetration ratio indicates that the edge state is topological and that there exist edge states having mid-gap energy, respectively. This experimentally establishes the presence of valley-Hall edge states at mid-gap for the zig-zag edge, and the lack thereof for the armchair edge, consistent with theoretical predictions described above. ![(a) Measured edge intensity ratio and (b) penetration ratio when we excite modes at mid-gap. Blue and red dots are from zig-zag and armchair edge domain walls, respectively. (inset) Diffracted light measured at the output facet. The waveguide where light is initially injected is marked with yellow dashed circle.[]{data-label="Fig_02"}](Fig02.eps){width="8.0cm"} We further probe the valley-Hall edge states by changing $\Delta n_{\textrm{s}}$ of the straw waveguides, while keeping $\Delta n_{\textrm{A}}$ and $\Delta n_{\textrm{B}}$ the same, such that we excite modes at a different energy (blue-dashed lines in Fig. \[Fig\_01\](d-g)). Here, the calculated energies of the straw waveguide modes were $-4.98 c_{0}$ and $-13.22 c_{0}$ for 1650nm and 1450nm, respectively. For the armchair edge, the energy coincides with edge bands at $\lambda$=1650nm, but not at $\lambda$=1450nm (Fig. \[Fig\_01\](d,f)). Therefore, we observe confinement to the input straw waveguide at $\lambda$=1450nm followed by increased penetration along the domain wall with increasing wavelength and strong penetration by $\lambda$=1650nm. For the zig-zag edge, however, the energy does not coincide with the state along the domain wall boundary depicted in Fig. \[Fig\_01\](b) (and whose dispersion is shown in green in Fig. \[Fig\_01\](e,g)), but rather the confined state that arises on the opposite side of the system when periodic boundary conditions are imposed in the vertical direction (shown in red in Fig. \[Fig\_01\](e,g)). In other words, since the only edge states localized near the straw waveguide are those drawn in green, there is no penetration along the zig-zag edge for this energy. Therefore, no penetration is observed in the entire wavelength range for the zig-zag edge (Fig. \[Fig\_03\]). ![(a) Measured edge intensity ratio and (b) penetration ratio when we excited modes significantly below mid-gap. Blue and red dots are measured zig-zag and armchair edge domain walls, respectively. (inset) Diffracted light measured at the output facet. The waveguide where light is initially injected is marked with yellow dashed circle.[]{data-label="Fig_03"}](Fig03.eps){width="8.0cm"} In order to confirm that the small edge intensity ratio and penetration ratio measured at $\lambda$=1450nm is indeed the consequence of a large edge state bandgap, as opposed to simply weak inter-waveguide coupling, we injected light at the center of the domain wall such that edge states are directly excited (Fig. \[Fig\_04\](a) and (b)) - in other words, we did not attempt to control the energy by using the straw. If the small penetration ratios were the result of weak coupling strength between the nearest-neighbor waveguides, the injected light would be expected to be strongly confined at the center of the waveguide array, where it is initially injected. However, for both waveguide arrays with zig-zag and armchair domain walls, we observed light diffracting along the domain wall and into the bulk. There is significantly more diffraction along the zig-zag edge as compared to the armchair edge because the armchair edge band is nearly flat and the zig-zag edge band is highly dispersive. However, in both cases, there is clear diffraction into the bulk of the structure, as is expected when we do not drive at a fixed energy using the straw. Furthermore, we examine the case where the system has no inversion breaking whatsoever, namely $\Delta n_{\textrm{A}}$=$\Delta n_{\textrm{B}}$=$\Delta n_{\textrm{s}}$. In this case, there is no bandgap and therefore no edge state of any kind. Upon injecting light into the straw waveguide, we observe strong diffraction into the bulk for both structures shown in Fig. \[Fig\_01\](a,b) of the zig-zag and armchair orientation (Fig. \[Fig\_04\](c,d)). Taken together, these results show that the straw waveguide acts as a reliable ‘spectroscopy tool’ for directly observing the presence in the valley-Hall edge states in the wavelength range 1450nm-1650nm. ![(a) Diffracted light measured at the output facet when we inject directly at the center of the zig-zag and (b) the armchair domain walls. (c) Images of diffracted light measured at the output facet when $\Delta n_{\textrm{A}}$=$\Delta n_{\textrm{B}}$=$\Delta n_{\textrm{s}}$ and the straw waveguide mode is initially excited for the zig-zag orientation and (d) the armchair domain walls. The lack of detuning leads to the lack of an edge state. All measurements are carried out at $\lambda$=1450nm. Waveguides where light is initially injected are marked with yellow dashed circle.[]{data-label="Fig_04"}](Fig04.eps){width="8.0cm"} In summary, we have realized the photonic valley-Hall topological edge states in two-dimensional honeycomb photonic lattices with broken inversion symmetry. We have experimentally demonstrated that it is possible to open very large bandgaps and therefore enter a fully gapped regime even for the structure with zig-zag edge domain walls, which was not possible in solid-state two-dimensional materials. Auxiliary straw waveguides placed at either end of the domain walls made it possible to access and excite a desired energy within the bulk bandgap, allowing for a convenient ‘spectroscopy’ tool for the waveguide array energies. Being a time-reversal invariant system, the valley-Hall effect could provide a straightforward route towards realizing photonic topological edge states, particularly in an on-chip platform. Thus, while valley-Hall edge states are not rigorously protected against any class of disorder, they will be protected against disorder that is sufficiently smooth (and thus does not allow inter-valley scattering). The linear, static, and non-magnetic nature of the design will also allow for lower optical loss compared to other approaches to topologically-protected photonic states (for example, magnetic materials are typically very lossy). Furthermore, the photonic valley-Hall effect could provide a natural platform for photonic quantum simulation of topological phenomena, perhaps by coupling the photonic modes to atoms or excitons. During the writing of the manuscript, we became aware of an analogous work in the microwave regime [@Wu2017]. M.C.R. acknowledges the National Science Foundation under award number ECCS-1509546 and the Penn State MRSEC, Center for Nanoscale Science, under award number NSF DMR-1420620 as well as the Alfred P. Sloan Foundation under fellowship number FG-2016-6418. K.P.C. acknowledges the National Science Foundation under award numbers ECCS-1509199 and DMS-1620218.
--- abstract: 'The spectral density of various ensembles of sparse symmetric random matrices is analyzed using the cavity method. We consider two cases: matrices whose associated graphs are locally tree-like, and sparse covariance matrices. We derive a closed set of equations from which the density of eigenvalues can be efficiently calculated. Within this approach, the Wigner semicircle law for Gaussian matrices and the Marcenko-Pastur law for covariance matrices are recovered easily. Our results are compared with numerical diagonalization, finding excellent agreement.' author: - Tim Rogers - Koujin Takeda - Isaac Pérez Castillo - Reimer Kühn bibliography: - 'CavityMethod.bib' title: Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices --- Introduction ============ What started as an approximation to the complex Hamiltonian of heavy nuclei has become a very interesting area of research in its own right. Although the statistical properties of random matrices had been tackled before, it was that treatment by Wigner in nuclear physics during 1950’s which boosted the research of what is currently known as Random Matrix Theory (RMT) [@Mehta1991]. The list of applications of this theory has been expanding ever since, ranging from physics, to computer science and finance. Specifically, applications in physics include nuclear theory, quantum chaos, localization, theory of complex networks, and more (see, for instance, [@Guhr1998] for an extensive review).\ From a theoretical and practical viewpoint one of the central quantities of interest in RMT is the spectral density of an ensemble of random matrices. While some cases have been completely analyzed during the last decades, many others have not been fully explored. Consider, for instance, the ensemble of symmetric random matrices whose entries are independently and identically distributed Gaussian variables. Among many of its properties, it is well-known that its spectral density is given by the Wigner semicircle law [@Wigner; @Mehta1991; @edwardsjones]. Another instance is the ensemble of covariance matrices, whose spectral density is given by the Marcenko-Pastur law [@Marcenko1967]. And the list continues.\ Interestingly, the change of introducing sparsity in such ensembles, i.e. many entries being zero, complicates the mathematical analysis enormously [@dorogovtsev; @rodgersbray; @biroli; @cugliandolo; @nagao]. Lacking more powerful mathematical tools, one must rely on approximative schemes to the spectral density, e.g. the Effective Medium Approximation (EMA), the Single Defect Approximation (SDA) [@biroli; @cugliandolo; @nagao].\ In this work, we tackle the problem of evaluating the spectral density of sparse random matrices by using the cavity method [@MPV; @mezard]. As we will show, this approach may offer new theoretical and practical advantages: from a theoretical point of view, it offers an alternative, and we believe easier, method to (re)derive the spectral density; practically, the resulting cavity equations can be interpreted as a belief-propagation algorithm on single instances, which can be then easily implemented. The resulting spectral density is a clear improvement over those obtained by approximative schemes. A complementary study using the replica method, and emphasizing ensemble aspects has appeared elsewhere [@Reimer].\ This work is organized as follows: in Sec. \[section:cavity\] we first mention how the spectral density can be recast as a problem of interacting particles on a graph. The subsequent problem is then analyzed by the cavity method in two cases: locally-tree like graphs and sparse covariance matrices. We derive cavity equations for large single instances and check that the dense limit gives the correct results. In Sec. \[sect:numerics\] we use the cavity equations as an algorithm to calculate the spectral density and compare these results with numerical diagonalization. The last section is for conclusions. Cavity approach to the spectral density {#section:cavity} ======================================= Consider an ensemble $\mathcal{M}$ of $N\times N$ symmetric matrices. If we denote with $\{\lambda^A_{i}\}_{i=1,\ldots,N}$ as the set of eigenvalues of a given matrix $A\in\mathcal{M}$, its spectral density is defined as follows $$\varrho_{A}(\lambda)=\frac{1}{N}\sum_{i=1}^N\delta(\lambda - \lambda_i^A). \label{eq:spectral}$$ The spectral density of the ensemble, denoted as $\rho(\lambda)$ results from averaging $\varrho_A(\lambda)$ over $\mathcal{M}$.\ As it was shown by Edwards and Jones [@edwardsjones], the density can be rewritten as $$\varrho_{A}(\lambda)=-\lim_{\epsilon\rightarrow 0^+}\frac{2}{\pi N}\text{Im}\left[\frac{\partial}{\partial z }\log\mathcal{Z}_{A}(z)\right]_{ z =\lambda-i\epsilon},$$ where $$\mathcal{Z}_{A}(z)=\int\left[\prod_{i=1}^N\frac{dx_i}{\sqrt{2\pi}}\right]e^{-\frac{1}{2}\sum_{i,j=1}^N x_i(z I -A)_{ij}x_j}. \label{eq:partition}$$ In writing the expression , we have been careless with the Gaussian integrals, so that as they stand they are not generally convergent; we simply follow the prescription as in [@Mezard1999; @Brezin2006], and do not worry unnecessarily about imaginary factors, so that we can introduce a Gibbs-Boltzmann probability distribution of $\bm{x}$, viz. $$\begin{split} P_A(\bm{x})=\frac{1}{\mathcal{Z}_A(z)}e^{- \mathcal{H}_A(\bm{x},z)} \label{eq:Gibbs} \end{split}$$ with $$\mathcal{H}_A(\bm{x},z)=\frac{1}{2}\sum_{(i,j)\in\mathcal{G}_{A}}^Nx_i( z I-A)_{ij}x_j. \label{eq:Hamiltonian}$$ In this way, the spectral density $\varrho_{A}(\lambda)$ is recast into a statistical mechanics problem of $N$ interacting particles $\bm{x}=(x_1,\ldots, x_N)$ on a graph $\mathcal{G}_A$ with effective Hamiltonian . By $\mathcal{G}_{A}$ we refer to a weighted graph with $N$ nodes and edges for each interacting pair $(i,j)$ with weight $A_{ij}$, when $A_{ij}\neq 0$. For later use, we introduce the following notation: the set of neighbors of a node $i$ will be denoted as $\partial i$; for a given subset of nodes $\mathcal{B}$ we define $\bm{x}_{\mathcal{B}}=(x_{\ell_1},\ldots,x_{\ell_{\|\mathcal{B}\|}})$ with $\ell_1,\ldots,\ell_{\|\mathcal{B}\|}\in\mathcal{B}$ and with $\|\mathcal{B}\|$ the number of nodes of $\mathcal{B}$; $k_i=\|\partial i\|$ denotes the number of neighbors of node $i$, while $c=\frac{1}{N}\sum_{i=1}^N k_i$ is the average connectivity.\ Note that within this approach, Eq. for the spectral density $\varrho_A(\lambda)$ can be rewritten as follows: $$\begin{split} \varrho_A(\lambda)&=\lim_{\epsilon\rightarrow 0^+}\frac{1}{\pi N}\sum_{i=1}^N \text{Im}\left[\langle x_i^2\rangle_{z}\right]_{z=\lambda-i\epsilon}\,, \label{eq:secondmoment} \end{split}$$ where $\langle \cdots\rangle_{z}$ denotes average over distribution .\ In previous works [@edwardsjones; @rodgersbray; @biroli; @cugliandolo; @nagao], the averaged spectral density $\rho(\lambda)$ is dealt with by using the replica approach, or in [@Fyodorov1991; @Rodgers1990] using Supersymmetric methods. For general sparse matrices, it was shown in [@rodgersbray; @biroli; @cugliandolo; @nagao; @Fyodorov1991] that the analysis of the resulting equations from either the replica or the Supersymmetric methods was a rather daunting task. To push the analysis further, the authors in [@cugliandolo; @nagao] resorted to a series of approximative schemes, originally introduced in [@biroli]. The simpler of such approximations, the EMA, assumes that all nodes are equivalent and play the same role [@biroli; @cugliandolo; @nagao]. This type of approximation works better the larger the average connectivity $c$ of the graph. However, for low and moderate values of $c$ it fails to provide an accurate description of central part and the tails (see, for instance, [@cugliandolo]) and of the presence of weighted Dirac delta peaks in the spectral density [@golinelli; @bauer]. Other approximations, like the SDA, also fail to give an accurate description of the spectrum.\ To improve our understanding of spectral properties of sparse matrices, we tackle the problem from a different perspective. Instead of considering the averaged spectral density $\rho(\lambda)$, we note, as shown in Eq. , that to calculate $\varrho_A(\lambda)$ we simply need the local marginals $P_{i}(x_i)$ from the Gibbs-Boltzmann distribution $P_{A}(\bm{x})$. The cavity method offers a way to calculate them [^1]. To illustrate this we consider two cases: the ensemble of symmetric locally tree-like sparse matrices, and the ensemble of sparse covariance matrices. Tree-like symmetric matrices ---------------------------- Let us start by analyzing the spectral density of sparse graphs $\mathcal{G}_{A}$ which are tree-like, as the one depicted in Fig. \[fig:one\]. By tree-like we mean short loops are rare. ![Left: Part of a tree-like graph $\mathcal{G}_A$ showing the neighborhood of node $i$. Right: Upon removal of node $i$, on the resulting cavity graph $\mathcal{G}^{(i)}_A$ , the neighboring sites $j$,$k$ and $l$ become uncorrelated.[]{data-label="fig:one"}](Plots/removal.eps){width="250pt"} Due to the tree-like structure we note that for each node $i$, the joint distribution of its neighborhood $P(\bm{x}_{\partial i})$ is correlated mainly through the node $i$. If, instead of the original graph $\mathcal{G}_A$, we consider a system where the node $i$ is removed (see Fig. \[fig:one\]), on the resulting cavity graph $\mathcal{G}^{(i)}_A$ the joint distribution $P^{(i)}(\bm{x}_{\partial i})$ factorizes, i.e. $$P^{(i)}(\bm{x}_{\partial i})=\prod_{\ell\in\partial i}P^{(i)}_\ell(x_\ell)\,.$$ This factorisation, which is exact on trees, is called Bethe approximation. On the cavity graph, the set of cavity distributions $\{P^{(j)}_i(x_i)\}$ obeys simple recursive equations, viz. $$\begin{split} P^{(j)}_i(x_i)&=\frac{e^{-\frac{1}{2} z x_i^2}}{Z_i^{(j)}}\int d \bm{x}_{\partial i\setminus j}e^{x_i\sum_{\ell\in\partial i\setminus j} A_{i\ell}x_\ell} \prod_{\ell\in\partial i\setminus j} P^{(i)}_\ell(x_\ell) \, \label{eq:cavity} \end{split}$$ for all $i=1,\ldots, N$ and for all $j\in \partial i$. Once the cavity distributions are known, the marginal distributions $P_{i}(x_i)$ of the original system $\mathcal{G}_{A}$ are given by $$\begin{split} P_i(x_i)&=\frac{e^{-\frac{1}{2} z x_i^2}}{Z_i}\int d \bm{x}_{\partial i}e^{x_i\sum_{\ell\in\partial i} A_{i\ell}x_\ell} \prod_{\ell\in\partial i} P^{(i)}_\ell(x_\ell) \, \label{eq:real} \end{split}$$ for all $i=1,\ldots, N$. While there is [in general]{} no a-priori reason to expect cavity distributions to be simple, they [are]{} for the present system: the set (\[eq:cavity\]) of equations is clearly self-consistently solved by Gaussian $P_i^{(j)}$s. Hence, upon assuming the cavity distributions to be Gaussian, namely, $$P^{(i)}_\ell(x)=\frac{1}{\sqrt{2\pi\Delta_\ell^{(i)}}}e^{-\frac{1}{2\Delta_\ell ^{(i)}}x^2}$$ the set of equations is transformed into a set of equations for the cavity variances $\Delta_j^{(i)}( z )$, viz. $$\Delta_i^{(j)}( z )=\frac{1}{ z -\sum_{\ell\in\partial i\setminus j}A_{i\ell}^2\Delta_\ell^{(i)}( z )} \label{eq:cavityvar}$$ for all $i=1,\ldots, N$ and for all $j\in \partial i$. Similarly, by Eq. the marginals $P_{i}(x_i)$ are Gaussian with variance $\Delta_i$ related to the cavity variances by $$\Delta_i( z )=\frac{1}{ z -\sum_{\ell\in\partial i}A_{i\ell}^2\Delta_\ell^{(i)}( z )}\,. \label{eq:realvar}$$ Eqs. and comprise the final result. For a given graph $\mathcal{G}_A$, one iterates the cavity Eqs. until convergence is reached. Once the cavity variances are known, the variances $\Delta_i$ are given by Eqs. , from which the spectral density $\varrho_A(\lambda)$ is obtained by $$\varrho_A(\lambda)=\lim_{\epsilon\rightarrow 0^+}\frac{1}{\pi N}\sum_{i=1}^N \text{Im}\left[\Delta_i( z )\right]_{z=\lambda-i\epsilon}\,.$$ It is worth noting that equations equivalent to those in can be derived by the method described in [@Abou1973] and can be related to self-returning random walks [@malioutov; @dorogovtsev].\ Notice also that the set of cavity equations must be solved for complex $z=\lambda-i\epsilon$, so that the cavity variances are in general complex, and then the limit $\epsilon\to 0^{+}$ is performed. Instead, we perform this limit explicitly in the cavity equations. To do so, we separate these equations into their real and imaginary parts and then do explicitly the limit $\epsilon\to0^{+}$ by naïvely assuming that the imaginary part is non-vanishing in such a limit (see discussion below). Denoting $(a_i^{(j)},b_i^{(j)})=[\text{Re}(\Delta_i^{(j)}),\text{Im}(\Delta_i^{(j)})]$, we obtain $$\begin{split} a_i^{(j)}&=\frac{\lambda-h^{(j)}_i(\bm{a})}{\left(\lambda- h^{(j)}_i(\bm{a})\right)^2+\left(h^{(j)}_i(\bm{b})\right)^2}\\ b_i^{(j)}&=\frac{h^{(j)}_i(\bm{b})}{\left( \lambda-h^{(j)}_i(\bm{a})\right)^2+\left(h^{(j)}_i(\bm{b})\right)^2} \label{eq:realIm} \end{split}$$ with $$\begin{split} h^{(j)}_i(\bm{v})&=\sum_{\ell\in \partial i\setminus j}A_{i\ell }^2v^{(i)}_\ell\,. \end{split}$$ Our results are exact, as long as the average connectivity $c$ of the graphs considered remains finite in the limit $N \to \infty$. ### Large $c$ limit: The Wigner Semicircle Law {#large-c-limit-the-wigner-semicircle-law .unnumbered} To assess our approach, note that from the set of equations and we can easily recover the Wigner semicircle law in the large $c$ limit. By this limit we understand that the $k_i\to c$ and $c\to \infty$, and assume that the graph is already “infinitely” large [^2]. To do this large $c$ limit, we take the entries of the matrix $A$ to be $A_{ij}=J_{ij}/\sqrt{c}$, with $J_{ij}(=J_{ji})$ a Gaussian variable with zero mean and variance $J^2$. From Eqs. and , we note that for large $c$, we have that $\Delta_i^{(j)}(z)=\Delta_{i}(z)+\mathcal{O}(c^{-1})$ [^3]. Upon defining $$\Delta=\lim_{c\to\infty}\frac{1}{c}\sum_{\ell\in\partial i}\Delta_\ell,$$ we obtain that $$\lim_{c\to\infty}\sum_{\ell\in\partial i}A_{i\ell}^2\Delta_\ell^{(i)}=\lim_{c\to\infty}\frac{1}{c}\sum_{\ell\in\partial i}J_{i\ell}^2\Delta_\ell^{(i)}=J^2 \Delta\,.$$ Thus, in the large $c$ limit Eq. yields $$\Delta=\frac{1}{z-J^2\Delta}\,,$$ which gives the well-known Wigner semicircle law [@Wigner] $$\begin{split} \rho(\lambda)&=\frac{1}{2\pi J^2}\sqrt{4J^2-\lambda^2}\,. \end{split}$$ Covariance matrices ------------------- Let us consider now matrices $A$ of the type $$A_{ij}=\frac{1}{d}\sum_{\mu=1}^P \xi_{i\mu}\xi_{j\mu}\,, \label{eq:covariance}$$ where ${{\bm{\xi}}}$ is an $N\times P$ matrix with entries $\xi_{i\mu}$. To this matrix we can associate a bipartite graph $\mathcal{G}_{{{\bm{\xi}}}}$ with $N+P$ nodes, divided into two sets indexed by $i=1,\ldots,N$ and $\mu=1,\ldots,P$ (see Fig. \[fig:dual\]). A pair of nodes $(i,\mu)$ is connected if $\xi_{i\mu}\neq 0$. We refer to these nodes as $\bm{x}$-nodes and $\bm{m}$-nodes, respectively. We will consider the bipartite graph $\mathcal{G}_{{{\bm{\xi}}}}$ to be tree-like, i.e. many of the entries $\xi_{i\mu}$ are zero. We also introduce $d=(1/P)\sum_{\mu=1}^P k_\mu$ with $k_\mu=\|\partial \mu\|$, i.e. the average connectivity of the $\bm{m}-$nodes. Clearly, $c=\alpha d$ with $\alpha=P/N$. ![Left: Graph $\mathcal{G}_{A}$ for covariance matrices. Right: Bipartite graph $\mathcal{G}_{{{\bm{\xi}}}}$ for the matrix ${{\bm{\xi}}}$. For sake of clarity, self-interactions in the graph $\mathcal{G}_{A}$ are not drawn.[]{data-label="fig:dual"}](Plots/dual.eps){width="250pt"} In this case it is more convenient to apply the cavity method on the bipartite graph $\mathcal{G}_{{{\bm{\xi}}}}$. To do so we write the effective Hamiltonian as follows $$\begin{split} \mathcal{H}_{A}(\bm{x},z)&= \frac{1}{2}z\sum_{i=1}^N x_i^2 - \frac{1}{2}\sum_{\mu=1}^P m_\mu^2(\bm{x}_{\partial \mu})\,, \end{split}$$ where we have defined the overlaps $$m_\mu(\bm{x}_{\partial \mu}) = \frac{1}{\sqrt{d}}\sum_{i\in \partial \mu} \xi_{i\mu}x_i.$$ Note that, due to our choice for ${{\bm{\xi}}}$ and the relation between the matrices $A$ and ${{\bm{\xi}}}$, the corresponding graph $\mathcal{G}_A$ (see Fig. \[fig:dual\]) is locally clique-like with self-interactions.\ We work our cavity equations in the bipartite graph $\mathcal{G}_{{{\bm{\xi}}}}$. Here, the variables $\bm{x}$ are on the $\bm{x}$-nodes while the variables $\bm{m}=(m_1,\ldots,m_p)$ are on the $\bm{m}$-nodes. Since we have two types of nodes, we apply the cavity method twice: around $\bm{x}$-nodes and around $\bm{m}$-nodes. We define $Q^{(i)}_{\nu}(m_\nu)$ as the cavity distribution of $m_\nu$ in the absence of a node $i$, and $ P^{(\mu)}_i(x_i)$ is the cavity distribution of $x_i$ in the absence of node $\mu$. We find the following set of equations for these cavity distributions $$\begin{split} P^{(\mu)}_i(x_i)&=\frac{e^{-\frac{1}{2}z x_i^2}}{Z_i^{(\mu)}}\int d\bm{m}_{\partial i\setminus \mu}\, e^{ \frac{1}{2}\sum_{\nu\in\partial i\setminus \mu} \left(m_\nu+\frac{1}{\sqrt{d}}\xi_{i\nu}x_i\right)^2}\\ &\times\prod_{\nu\in\partial i\setminus \mu} Q^{(i)}_\nu(m_\nu)\,, \label{eq:cavitycovar1} \end{split}$$ for all $i=1,\ldots, N$ and $\mu\in\partial i$. Also $$\begin{split} Q^{(i)}_{\nu}(m_\nu)&=\frac{1}{Z_\nu^{(i)}}\int d \bm{x}_{\partial \nu\setminus i}\, \delta\left( m_\nu-\frac{1}{\sqrt{d}}\sum_{\ell \in\partial \nu\setminus i}\xi_{\ell\nu}x_\ell \right)\\ &\times \prod_{\ell \in\partial \nu\setminus i} P^{(\nu)}_\ell(x_\ell)\,, \label{eq:cavitycovar2} \end{split}$$ for all $\nu=1,\ldots, P$ and $i\in\partial \nu$. Obviously, for the marginal distributions $P_i(x_i)$ we obtain $$\begin{split} P_i(x_i)&=\frac{e^{-\frac{1}{2}z x_i^2}}{Z_i}\int d\bm{m}_{\partial i}\, e^{ \frac{1}{2}\sum_{\nu\in\partial i} \left(m_\nu+\frac{1}{\sqrt{d}}\xi_{i\nu}x_i\right)^2}\\ &\times\prod_{\nu\in\partial i} Q^{(i)}_\nu(m_\nu)\,, \label{eq:realcovar} \end{split}$$ for all $i=1,\ldots, N$. As before, we see from the set of equations and that the Gaussian measure is a fixed point. Thus, by taking $P^{(\mu)}_i(x_i)$ and $Q^{(i)}_\mu(m_\mu)$ to be Gaussian distributions with zero mean and variances $\Delta_i^{(\mu)}$ and $\Gamma_\mu^{(i)}$, respectively, we obtain the following set of equations for the cavity variances $$\left\{\begin{split} \Delta_i^{(\mu)}(z) &= \frac{1}{z - \frac{1}{d}\sum_{\nu\in\partial i\setminus \mu}\xi_{i\nu}^2\frac{1}{1-\Gamma_\nu^{(i)}(z)}}\\ \Gamma_\nu^{(i)}(z) &= \frac{1}{d}\sum_{\ell \in\partial \nu\setminus i}\xi_{\ell\nu}^2\Delta_\ell^{(\nu)}(z)\,. \end{split}\right. \label{eq:Pastur1}$$ Similarly, if we denote with $\Delta_i$ the variance of the marginal $P_i(x_i)$ on the original graph $\mathcal{G}_{{{\bm{\xi}}}}$ we obtain $$\Delta_i(z) = \frac{1}{z - \frac{1}{d}\sum_{\nu\in\partial i}\xi_{i\nu}^2\frac{1}{1-\Gamma_\nu^{(i)}(z)}}\,. \label{eq:Pastur2}$$ ### Large $c$ limit: The Marcenko-Pastur law {#large-c-limit-the-marcenko-pastur-law .unnumbered} For the sake of simplicity we take the non-zero entries of the matrix ${{\bm{\xi}}}$ to have values $\pm 1$, so that $\xi_{i\mu}^2=1$ in Eqs. and . Let us consider the spectral density in the large $c$ limit of the bipartite graph $\mathcal{G}_{{{\bm{\xi}}}}$. By this limit we mean $k_\mu\to d$, $k_i\to c$ and $d,c\to \infty$ while $\alpha$ remains finite. As before, the difference between cavity variances and variances is $\mathcal{O}(c^{-1})$. If we define $$\begin{split} \Delta&=\lim_{d\to\infty} \frac{1}{d}\sum_{\ell \in\partial \nu}\Delta_\ell=\lim_{d\to\infty} \frac{1}{d}\sum_{\ell =1}^d\Delta_\ell\,, \end{split}$$ from Eq. we obtain $$\begin{split} \Delta = \frac{1}{z - \alpha\frac{1}{1-\Delta}}\,. \end{split}$$ Upon solving this equation for $\textrm{Im}(\Delta)$ we obtain the Marcenko-Pastur law [@Marcenko1967] of dense covariance matrices $$\begin{split} \rho( \lambda ) &= \frac{1}{2\pi\lambda}\sqrt{-\lambda^2 + 2\lambda(\alpha+1)+(\alpha-1)^2}\\ &\hspace{10mm} + C_0(1-\alpha) \delta( \lambda )\,. \end{split}$$ with $C_0=1$ for $\alpha\leq 1$ and $C_0=0$ for $\alpha>1$. A slightly different expression is found by Nakanishi and Takayama [@takayama], where the difference comes from not considering the diagonal terms. This could also be implemented fairly straightforwardly to obtain the spectral density as in [@takayama]. Numerical Results and Comparison {#sect:numerics} ================================ For general sparse matrices, we solve the cavity equations numerically and compare the results with exact numerical diagonalization. We consider again the two cases of locally tree-like and sparse covariance matrices. Tree-like symmetric matrices ---------------------------- To test our cavity equations, we choose Poissonian graphs $\mathcal{G}_{A}$ where each entry $A_{ij}$ of the $N\times N$ matrix $A$ is drawn independently from $$P(A_{ij})= \frac{c}{N}\pi(A_{ij})+ \left(1-\frac{c}{N}\right)\delta(A_{ij})$$ with $c$ the average connectivity, and $\pi(x)$ is the distribution of non-zero edge weights. For the distribution of weights we study two cases: bimodal distribution, i.e. $$\pi(A_{ij})=\frac{1}{2}\delta(A_{ij}-1)+\frac{1}{2}\delta(A_{ij}+1)$$ and Gaussian distribution with zero mean and variance $1/c$.\ For the purpose of fairly comparing later with exact numerical diagonalization, we have analyzed the cavity equations for rather small matrices. However, we have checked that the convergence of these equations is generally fairly fast and we are able to evaluate the spectral density of very large matrices in reasonable time. In both, the bimodal and the Gaussian cases, we generated matrices with $N=1000$. For each matrix we run our cavity Eqs. until convergence is reached and then obtain the spectral density from Eqs. . The result is averaged over 1000 samples. For such sizes we have also calculated the spectral density by exact numerical diagonalization and averaged over 1000 samples.\ ![\[fig:B3\] Spectral density of Poissonian graphs with bimodal edge weights and average connectivity $c=3$. Red Square markers are the results of numerical diagonalization with $N=1000$, averaged over $1000$ samples. Blue circles are the results of the cavity approach with $N=1000$, averaged over $1000$ samples. The dashed line corresponds to the SDA and dotted line is the EMA. The inset shows the tail of the spectral density.](Plots/B3.eps){width="230pt"} The numerical results from the cavity approach and exact numerical diagonalization for the bimodal case is plotted in Fig. \[fig:B3\] for average connectivity $c=3$. We have also compared our results with the spectral density obtained by EMA and SDA (see [@cugliandolo; @biroli; @dorogovtsev] for details about the approximations). As we can see, our results are a clear improvement over the EMA and SDA results, as they are in excellent agreement with numerical diagonalization. Even the tail of the spectrum, usually not obtained with these approximation, (see inset of Fig. \[fig:B3\]) is well reproduced by our approach.\ It is well known that the spectrum of these type of ensembles contains a dense collection of Dirac delta peaks [@golinelli; @bauer], which are not fully captured by the previous approximations. Without a prior analysis, one wonders how the cavity equations can be used to obtain such contributions. A practical way out is to reconsider the limit $\epsilon\to 0$, by leaving a small value of $\epsilon$ in the cavity equations, which implies approximating Dirac deltas by Lorentzian peaks. ![\[fig:B3\_v2\] Comparison of cavity equations for $\epsilon=0$ (blue circles) and $\epsilon=0.005$ (Continuous red line), for Poissonian graphs with bimodal edge weights and average connectivity $c=3$ ($N=1000$ and average over $1000$ samples). The inset shows the Dirac delta structure in the central region.](Plots/spectrum_C3_N1000_MC1000_Naveg1000_epsilon_0.005_BIMODAL.eps){width="230pt"} In Fig. \[fig:B3\_v2\] we have rerun the set of eqs and with a small value of $\epsilon$. The Dirac delta contributions, whose exact positions within the spectral density is discussed in [@golinelli], are now clearly visible. A more detailed study on this issue within the context of localisation can be found in [@Reimer].\ In Fig. \[fig:G\] we plot the results of both numerical diagonalization and the cavity method when $\pi(x)$ is a Gaussian distribution with zero mean and variance $1/c$. Once again, our results are in excellent agreement with the numerical simulations.\ ![\[fig:G\] Spectral density of Poissonian graphs with Gaussian edge weights and average connectivity $c=4$. Red Square markers are for the results of numerical diagonalization with $N=1000$, averaged over $1000$ samples. Blue circles are results of the cavity approach with $N=1000$ and average over $1000$ samples.](Plots/G.eps){width="230pt"} Covariance matrices ------------------- ![Spectral density of covariance matrices with $N=4000$, $d=12$, $\alpha=0.3$. Average over 1000 samples. Red Square markers are for the results of numerical diagonalization. Blue circles are results of the cavity approach. The dashed line corresponds to the SEMA.[]{data-label="fig:H"}](Plots/H.eps){width="230pt"} We have also analyzed numerically in the case of sparse covariance matrices. Here the entries $\xi_{i\mu}$ of the $N\times P$ matrix ${{\bm{\xi}}}$, are drawn according to the distribution $$P(\xi_{i\mu})=\frac{d}{N}\pi(\xi_{i\mu})+\left(1-\frac{d}{N}\right)\delta(\xi_{i\mu})\, ,$$ where $\pi(\xi_{i\mu})$ is a bimodal distribution $$\pi(\xi_{i\mu})=\frac{1}{2}\delta(\xi_{i\mu}+1)+\frac{1}{2}\delta(\xi_{i\mu}-1)$$ In Fig. \[fig:H\], we compare the results of direct diagonalization, the cavity method, and the symmetric effective medium approximation (SEMA), introduced in [@nagao]; here we make the same choice of parameters. The inset figure shows detail of the tail region of the plot, where the difference between the SEMA and the other results can be clearly seen. Conclusions =========== In this work, we have re-examined the spectral density of ensembles of sparse random symmetric matrices. By following Edwards and Jones [@edwardsjones], we have mapped the problem into an interacting system of particles on a sparse graph, which was then analyzed by the cavity approach. Within this framework, we have derived cavity equations on single instances and used them to calculate the spectral densities of sparse symmetric matrices. Our results are in good agreement with numerical diagonalization and are a clear improvement to previous works based on approximative schemes. We have also shown that, to account for the Dirac delta contribution to the spectrum, one may approximate Dirac delta peaks by Lorentzians, by leaving a small value for $\epsilon$ [@Reimer].\ It is well known that cavity and replica methods are equivalent (see for instance [@mezard] for diluted spin glasses), so one may wonder in which aspects our work differs from the ones presented in [@rodgersbray; @biroli; @cugliandolo; @nagao]. Generally, for interacting diluted systems with continuous dynamical variables one expects an infinite number of cavity fields to parametrize the cavity distributions. The authors in [@rodgersbray; @biroli; @cugliandolo; @nagao] decided to tackle such a daunting task by resorting to approximations. In this work, we simply realize that, for the problem at hand, the cavity distributions are Gaussian, so that the problem can be solved exactly by self-consitently determining the variances of these distributions.\ In future studies we expect to extend the method presented here to the analysis of more general aspects of random matrices. KT thanks the hospitality of the Disordered Systems Group, at the department of Mathematics, King’s College. He is supported by Grand-in-aid from MEXT/JSPS, Japan (No.18079006) and Program for Promoting Internationalization of University Education, MEXT, Japan (Support for Learning Overseas Advanced Practices in Research). The authors thank M Mézard and Y Kabashima for discussions and G Parisi for discussions and for pointing out earlier work on the subject. IPC also thanks ACC Coolen for his work during the initial stages of this Guzai project. [^1]: This approach have been used in [@Ciliberti2004] within the context of Anderson localisation. [^2]: Alternatively, one could naïvely take $c\to N$ and then $N\to\infty$. In the complete, or fully connected, graph ($c=N$) the cavity equations are still valid, but the reason for the decorrelation is statistical rather than topological. [^3]: This is the usual derivation of TAP equations from the cavity equations. In this case the difference between cavity fields and effective fields does not produce an Onsager reaction term.
--- abstract: 'In high energy heavy-ion collisions, the degrees of freedom at the very early stage can be effectively represented by strong classical gluonic fields within the Color Glass Condensate framework. As the system expands, the strong gluonic fields eventually become weak such that an equivalent description using the gluonic particle degrees of freedom starts to become valid. We revisit the spectrum of these gluonic particles by solving the classical Yang-Mills equations semi-analytically with the solutions having the form of power series expansions in the proper time. We propose a different formula for the gluon spectrum which is consistent with energy density during the whole time evolution. We find that the chromo-electric fields have larger contributions to the gluon spectrum than the chromo-magnetic fields do. Furthermore, the large momentum modes take less time to reach the weak-field regime while smaller momentum modes take more time. The resulting functional form of the gluon spectrum is exponential in nature and the spectrum is close to a thermal distrubtion with effective temperatures around $0.6$ to $0.9\, Q_s$ late in the Glasma evolution. The sensitiveness of the gluon spectrum to the infrared and the ultraviolet cut-offs are discussed.' author: - Ming Li bibliography: - 'boost\_invariant\_gluon\_spectrum.bib' title: 'Revisiting the Gluon Spectrum in the Boost-Invariant Glasma from a Semi-Analytic Approach' --- Introduction ============ In high energy heavy-ion collisions, the time evolution of the produced quark-gluon plasma has been successfully described by relativistic hydrodynamic models [@Kolb2004]. One of the prerequisites for hydrodynamics to be applicable is the local thermal equilibrium assumption. Comparisons with experimental data indicate that hydrodynamics starts very early in the collisions. This early thermalization has been a challenging theoretical problem which is still under active research and debate. Recently, an effective kinetic theory in the weak coupling regime was applied to bridge the early Glasma stage and the hydrodynamics stage [@Kurkela:2015qoa]. One of the inputs in this approach is the initial phase space distribution of the gluons which is usually parameterized as either a step function [@Kurkela:2014tea; @Blaizot:2017lht] or a Gaussian form [@Kurkela:2015qoa; @Kurkela:2014tea; @Tanji:2017suk]. On the other hand, the gluon distribution at late time in the Glasma evolution has been extensively investigated by numerically solving the boost-invariant classical Yang-Mills equations [@Krasnitz:2000gz; @Krasnitz:2001qu; @Lappi:2003bi; @Krasnitz:2003jw; @Blaizot:2010kh]. Incorporating the rapidity dependence [@Lappi:2011ju] has also been explored. In these numerically simulations, the gluon distribution in the weak field regime is fitted to be a Bose-Einstein distribution for lower momentum modes and a power law form for higher momentum modes. It would be interesting to reexamine the gluon spectrum in the boost-invariant Glasma from a different approach, which will be the topic of this paper. We focus on the simplest boost-invariant classical Yang-Mills equations and the evolution of the Glasma during the very early time $\tau \lesssim 1.0\,\rm{fm/c}$. For important physics originating from violating the assumption of boost-invariance, such as Glasma instabilities and possible pressure isotropization induced, we refer the readers to [@Romatschke:2005pm; @Romatschke:2006nk; @Fukushima:2007ja; @Fujii:2008dd; @Fukushima:2011nq; @Berges:2012cj; @Dusling:2012ig; @Gelis:2013rba; @Ipp:2017lho; @Ruggieri:2017ioa]. There is also the recently found universal self-similar gluon distribution at extremely large proper time in simulating the 3+1D classical Yang-Mills equations assuming an initially ($\tau\sim 1/Q_s$) overpopulated and anisotropic gluon distribution [@Berges:2013eia; @Berges:2013fga; @Berges:2013lsa; @Berges:2014bba; @Berges:2015ixa].\ The paper is organized as follows. In section II, we propose a different formula for the gluon spectrum in the boost-invariant Glasma and discuss its relation with the conventional formula used in the literature. Section III is devoted to the actual computations of the gluon spectrum using a power series expansion method. We work in the leading $Q^2$ approximation and show contributions from the chromo-electric fields and the chromo-magnetic fields explicitly. Results are given in Section IV and comparisons with results from numerical simulations are given. The Appendix includes main computational steps and expressions. Formula for the Gluon Spectrum ============================== In the Color Glass Condensate (CGC) framework, particularly the McLerran-Venugapolan model [@McLerran:1993ni; @McLerran:1993ka] applied to the high energy heavy-ion collisions, describing the very early stages of the collisions is equivalent to solving the classical Yang-Mills equations with appropriate initial conditions [@Kovner:1995ja; @Kovner:1995ts]. In general, solving the full 3+1D classical Yang-Mills equations is needed to obtain both transverse dynamics and longitudinal dynamics. For the study of the gluon spectrum, we focus on the boost-invariant situation to be aligned with the previous numerical simulations. The classical Yang-Mills equations in the Fock-Schwinger gauge $(A^{\tau} =0)$ under the assumption of boost-invariance are $$\label{eoms} \begin{split} &\frac{1}{\tau}\frac{\partial}{\partial \tau}\frac{1}{\tau}\frac{\partial}{\partial \tau} \tau^2 A^{\eta} - [D^i,[D^i,A^{\eta}]] =0 \, , \\ &\frac{1}{\tau}\frac{\partial}{\partial \tau} \tau \frac{\partial}{\partial \tau} A_{\perp}^i -ig\tau^2 [A^{\eta},[D^i,A^{\eta}]] - [D^j,F^{ji}] =0 \, ,\\ \end{split}$$ supplemented by the constraint equation $$\label{constraint} ig\tau [A^{\eta},\frac{\partial}{\partial \tau} A^{\eta}] - \frac{1}{\tau} [D^i,\frac{\partial}{\partial \tau} A^i_{\perp}] =0 \,.$$ The constraint equation comes from the equation of motion related to the $A^{\tau}$ component after we choose the Fock-Schwinger gauge. The Yang-Mills equations are written in the Milne coordinates $(\tau, x, y,\eta)$ with the proper time $\tau=\sqrt{t^2-z^2}$ and the pseudorapidity $\eta = \frac{1}{2}\ln\frac{t+z}{t-z}$. The non-Abelian vector potentials $A^{\eta}(\tau,\mathbf{x}_{\perp})$ and $A^i_{\perp}(\tau,\mathbf{x}_{\perp})$ $(i= x, y)$ are independent of the pseudorapidity $\eta$ due to the assumption of boost-invariance; they are matrices in the $SU(3)$ color group space. The covariant derivative is $D^i=\partial^i - igA^i_{\perp}$ and the field strength tensor is $F^{ij}=\partial^i A^j_{\perp} -\partial^j A^i_{\perp} -ig[A^i_{\perp}, A^j_{\perp}] $. The initial conditions [@Kovner:1995ja; @Gyulassy:1997vt] for the equations of motion are $$\label{ics} \begin{split} &A^i_{\perp}(\tau=0,\mathbf{x}_{\perp}) = A_1^i(\mathbf{x}_{\perp}) + A^i_2(\mathbf{x}_{\perp}) \, , \\ &A^{\eta}(\tau=0,\mathbf{x}_{\perp}) = -\frac{ig}{2}[A_1^i(\mathbf{x}_{\perp}),A_2^i(\mathbf{x}_{\perp}] \, , \\ &\frac{\partial}{\partial\tau} A^i_{\perp}(\tau=0,\mathbf{x}_{\perp}) =0 ,\quad \frac{\partial}{\partial\tau} A^{\eta} (\tau=0,\mathbf{x}_{\perp}) =0 \,. \end{split}$$ Here $A^i_1(\mathbf{x}_{\perp})$ and $A^i_2(\mathbf{x}_{\perp})$ are the pure gauge fields produced by the two colliding nuclei individually until the collision. Once the non-Abelian gauge potentials $A^{\eta}$ and $A^i_{\perp}$ are solved, physical quantities like the energy-momentum tensor can be computed accordingly. The energy-momentum tensor is defined as $ T^{\mu\nu} = F^{\mu\lambda} F^{\nu}_{\,\,\lambda} + \frac{1}{4} g^{\mu\nu} F^{\kappa\lambda}F_{\kappa\lambda}$ with the general field strength tensor $F_{\mu\nu} =\partial_{\mu} A_{\nu} -\partial_{\nu} A_{\mu} -ig[A_{\mu}, A_{\nu}]$. Tracing over color indexes is understood in the definition of the energy-momentum tensor. The energy-momentum tensor thus defined is local in space-time and gauge-invariant. Among the various components of the energy-momentum tensor, the energy density play a crucial role in the definition of the gluon spectrum. $$\label{energydensity} \varepsilon(x) \equiv T^{00} (x)= \frac{1}{2}( \vec{E}^2(x)+\vec{B}^2(x)).$$ The contributions from the chromo-electric field $\vec{E}$ and the chromo-magnetic field $\vec{B}$ are related to the field strength tensor by $$\label{energydensitycomponents} \begin{split} &E^zE^z = \frac{1}{\tau^2} F_{\tau\eta}F_{\tau\eta}\, ,\\ &E_{\perp}^iE_{\perp}^i = \cosh^2\eta F_{i\tau}F_{i\tau} -\frac{1}{\tau}\sinh 2\eta F_{i\tau} F_{i\eta} \\ &\qquad\qquad+ \frac{1}{\tau^2}\sinh^2\eta F_{i\eta}F_{i\eta}\, ,\\ &B^zB^z = \frac{1}{2}F_{kl}F_{kl}\,,\\ &B_{\perp}^iB_{\perp}^i = \sinh^2\eta F_{i\tau}F_{i\tau} -\frac{1}{\tau} \sinh 2\eta F_{i\tau}F_{i\eta}\\ &\qquad\qquad + \frac{1}{\tau^2}\cosh^2\eta F_{i\eta}F_{i\eta}\, .\\ \end{split}$$ where the field strength tensor has subscripts in terms of the Milne coordinates, $F_{mn}$ with $m,n = (\tau, x,y,\eta)$. The gluon spectrum $dN/d^2\mathbf{k}_{\perp} dy$, which is the number of gluons per unit two dimensional transverse momentum and per unit rapidity, is constructed by requiring it be consistent with the local energy density in reproducing the total energy $$\label{requirement0} \begin{split} E_{\rm{tot}}(\tau) &=\int d^2\mathbf{k}_{\perp} dy\, \omega(\mathbf{k}_{\perp},y,\tau)\, \frac{dN}{d^2\mathbf{k}_{\perp} dy}(\tau)\, ,\\ &=\int d^2\mathbf{x}_{\perp} d\eta\, \tau \cosh\eta \,\varepsilon(\mathbf{x}_{\perp},\eta,\tau) \, . \end{split}$$ Here $\omega(\mathbf{k}_{\perp},y,\tau)$ is the dispersion relation function that characterizes the gluonic particles in the Glasma which, in principle, should be time-dependent. In the strong-field regime, the dispersion relation function can be highly nontrivial due to the strong coherence among the gluonic particles. Also, it is not unambiguous whether it is legitimate to define a quasiparticle dispersion relation in the strong-field regime. However, once entering the weak-field regime when particles approximately decohere, the dispersion relation is approximately time-independent and it makes sense to talk about the dispersion relation for the quasiparticles. Unfortunately, there are no a prior derivations for the dispersion relation. For the discussions in this paper, we choose the dispersion relation of free massless particles $\omega(\mathbf{k}_{\perp},y,\tau) = \omega(\mathbf{k}_{\perp}) = k_{\perp}$ for the boost-invariant situation as in [@Krasnitz:2000gz; @Krasnitz:2001qu; @Lappi:2003bi; @Krasnitz:2003jw; @Blaizot:2010kh] while keeping in mind that the problem of choosing dispersion relations is still not rigorously resolved. With the boost-invariance assumption, $dy=d\eta$ and we focus on the central rapidity region $\eta=0$. The requirement becomes $$\label{requirement} \frac{1}{\tau} \int d^2\mathbf{k}_{\perp} k_{\perp} \frac{dN}{d^2\mathbf{k}_{\perp} dy}(\tau) =\int d^2\mathbf{x}_{\perp} \,\varepsilon(\mathbf{x}_{\perp},\tau)\, .$$ The $1/\tau$ factor is purely geometric in nature as it originates from the usage of the Milne coordinates $(\tau, x, y, \eta)$. With the help of the Fourier transformations, one can easily verify that the following expression for the gluon spectrum satisfies the requirement . $$\label{newgluonspectrum} \begin{split} \frac{dN}{d^2\mathbf{k}_{\perp}dy} = \frac{1}{2(2\pi)^2}\frac{1}{k_{\perp}}\bigg\{&\Big[\tau F_{i\tau}(\tau,\mathbf{k}_{\perp})F_{i\tau}(\tau,-\mathbf{k}_{\perp}) + \frac{1}{\tau} F_{\tau\eta}(\tau,\mathbf{k}_{\perp})F_{\tau\eta}(\tau,-\mathbf{k}_{\perp})\Big]\\ &+\left[\frac{\tau}{2}F_{ij}(\tau,\mathbf{k}_{\perp})F_{ij}(\tau,-\mathbf{k}_{\perp}) + \frac{1}{\tau} F_{i\eta}(\tau,\mathbf{k}_{\perp})F_{i\eta}(\tau,-\mathbf{k}_{\perp}) \right]\bigg\}\, .\\ \end{split}$$ The terms in the first square bracket of equation represents contributions from the chromo-electric fields while the terms in the second square bracket represents the contributions from the chromo-magnetic fields, see Eq.. The formula is consistent with the energy density during the whole time evolution. Similar expressions have been used in [@Fujii:2008km] where the dispersion relation is chosen to be $\omega(\mathbf{k}_{\perp}) =\sqrt{k_{\perp}^2 + m^2}$ with an arbitrary effective mass $m$ included. On the other hand, the formula differs from those used in the literature [@Krasnitz:2000gz; @Krasnitz:2001qu; @Lappi:2003bi; @Krasnitz:2003jw; @Blaizot:2010kh] in the chromo-magnetic part where formula contains the full non-Abelian features while the conventional expressions are Abelian in nature. One of the advantages of the formula over the conventional expression is that one can follow the whole time evolution of the Glasma and tell when the strong fields becomes weak mode-by-mode in which self-interactions of gluons become less important compared to the kinetic terms. In addition, formula has gauge-invariant meaning as it is related to the gauge-invariant local energy density, while in [@Krasnitz:2000gz; @Krasnitz:2001qu; @Lappi:2003bi; @Krasnitz:2003jw; @Blaizot:2010kh] the expression for the gluon spectrum is explicitly gauge dependent and the additional Coulomb gauge $\partial_i A^i = 0$ has to be imposed. Finally, the expression puts the contributions of the chromo-magnetic part and chromo-electric part on an equal footing and makes their comparison meaningful. Computing the Gluon Spectrum ============================ To compute the gluon spectrum , one first needs to solve the classical Yang-Mills equations . We follow the semi-analytic approach proposed in [@Fries:2006pv; @Chen:2015wia] where the gauge potential $A^{\eta}$ and $A^i_{\perp}$ are expressed as power series expansions in the proper time $\tau$. Recursive relations of the gauge potentials $A^{\eta}$ and $A^i_{\perp}$ are deduced so that the solutions can be obtained order by order in the power series expansions. Mathematically, this is a rigorous approach to solving the differential equations involved. However, in practice, it is difficult to compute the higher order terms as the number of terms involved grow enormously as one goes to higher orders. To capture contributions from the higher order terms in the power series expansion, we assume a momentum scale separation $Q^2\gg Q_s^2 \gg m^2$ in [@Li:2016eqr]. As a result, we only retain the leading terms that have the highest powers in $Q^2$ while disregarding the subleading terms involving logarithmics of $Q^2$ . There we introduced an infrared cut-off $m$ and a ultraviolet cut-off $Q$. The ultraviolet cut-off $Q$ is introduced so that particles with transverse momentum larger than $Q$ are not included in the effective classical fields. The infrared cut-off $m$ can be viewed as the $\Lambda_{QCD}$ scale. Moreover, the $Q_s$ is the gluon saturation scale which characterizes the typical transverse momentum of the gluonic particles. This leading $Q^2$ approximation, which includes minimal amounts of non-Abelian effects in the time evolution, is an improvement on the Abelian approximation discussed in [@Chen:2015wia; @Fujii:2008km]. The Abelian approximation takes into account the full non-Abelian initial conditions while ignoring non-linear self-interactions of the gluon fields in their time evolutions [@Kovner:1995ja; @Kovner:1995ts; @Kovchegov:1997ke; @Kovchegov:2005ss].\ The ensuing two steps are: one first computes the following correlation functions and then perform the Fourier transformations with respect to the transverse coordinates, $$\label{fourcorrelationfunctions} \begin{split} & \Big\langle \tau F_{i\tau}(\tau,\mathbf{x}_{\perp}) F_{i\tau}(\tau,\mathbf{y}_{\perp})\Big\rangle\, ,\quad \Big\langle \frac{1}{\tau} F_{\tau\eta}(\tau,\mathbf{x}_{\perp})F_{\tau\eta}(\tau,\mathbf{y}_{\perp}) \Big\rangle\, ,\\ & \Big\langle \frac{\tau}{2} F_{ij}(\tau,\mathbf{x}_{\perp}) F_{ij}(\tau,\mathbf{y}_{\perp})\Big\rangle\, ,\quad \Big\langle \frac{1}{\tau} F_{i\eta}(\tau,\mathbf{x}_{\perp}) F_{i\eta}(\tau,\mathbf{y}_{\perp})\Big\rangle\, . \\ \end{split}$$ The bracket $\langle \ldots \rangle$ indicates averaging over different configurations of the initial color distributions at the end of the computations. We only compute the event-averaged gluon spectrum in this paper. For works related to the event-by-event observables within the semi-analytic approach, we refer the readers to [@Fries:2017fwk]. These four terms in , before averaging over the initial color distributions, are also expressed as power series expansions in the proper time, $$\label{firstterm} \tau F_{i\tau}(\tau,\mathbf{x}_{\perp}) F_{i\tau}(\tau,\mathbf{y}_{\perp})=\sum_{n=2}^{\infty}\sum_{k=1}^{n-1} \frac{k(n-k)}{4^{n-1}[k!(n-k)!]^2} [D_x^j,[D_x^{\{2k-2\}}, B_0(\mathbf{x}_{\perp})]][D_y^j, [D_y^{\{2n-2k-2\}}, B_0(\mathbf{y}_{\perp})]]\tau^{2n-1}\, .$$ $$\label{secondterm} \frac{1}{\tau} F_{\tau\eta}(\tau,\mathbf{x}_{\perp})F_{\tau\eta}(\tau,\mathbf{y}_{\perp})= \sum_{n=0}^{\infty}\sum_{k=0}^{n} \frac{1}{4^n[k!(n-k)!]^2} [D_x^{\{2k\}}, E_0(\mathbf{x}_{\perp})][D_y^{\{2n-2k\}}, E_0(\mathbf{y}_{\perp})] \tau^{2n+1}\, .$$ $$\label{thirdterm} \frac{\tau}{2} F_{ij}(\tau,\mathbf{x}_{\perp}) F_{ij}(\tau,\mathbf{y}_{\perp}) =\sum_{n=0}^{\infty}\sum_{k=0}^{n} \frac{1}{4^n[k!(n-k)!]^2} [D_x^{\{2k\}}, B_0(\mathbf{x}_{\perp})] [D_y^{\{2n-2k\}}, B_0(\mathbf{y}_{\perp})] \tau^{2n+1}\,.$$ $$\label{fourthterm} \frac{1}{\tau} F_{i\eta}(\tau,\mathbf{x}_{\perp}) F_{i\eta}(\tau,\mathbf{y}_{\perp}) = \sum_{n=2}^{\infty}\sum_{k=1}^{n-1}\frac{k(n-k)}{4^{n-1} [k!(n-k)!]^2} [D_x^i,[D_x^{\{2k-2\}}, E_0(\mathbf{x}_{\perp})]][D_y^i, [D_y^{\{2n-2k-2\}}, E_0(\mathbf{y}_{\perp})]]\tau^{2n-1}\, .$$ In obtaining the above expressions, we used the results for the different components of the field strength tensor $F_{i\tau}$, $F_{\tau\eta}$, $F_{ij}$ and $F_{i\eta}$ under the leading $Q^2$ approximation in [@Li:2016eqr]. Note that the equations and are very similar. Their only difference lies in whether the initial ($\tau=0)$ field is the longitudinal chromo-electric field $E_0(\mathbf{x}_{\perp})$ or the longitudinal chromo-magnetic field $B_0(\mathbf{x}_{\perp})$. The same observation applies to the equations and . Let us recall the difference between the initial chromo-electric field and chromo-magnetic field [@Lappi:2006fp; @Chen:2015wia], $$\label{initialEandB} \begin{split} &B_0(\mathbf{x}_{\perp}) = ig\epsilon^{mn}[ A_1^m(\mathbf{x}_{\perp}), A_2^n(\mathbf{x}_{\perp})],\\ & E_0(\mathbf{x}_{\perp}) = ig\delta^{mn}[ A_1^m(\mathbf{x}_{\perp}), A_2^n(\mathbf{x}_{\perp})] \, .\\ \end{split}$$ The initial longitudinal chromo-electric field and the longitudinal chromo-magnetic field are different event-by-event $E_0(\mathbf{x}_{\perp}) \neq B_0(\mathbf{x}_{\perp})$. But they contribute the same to the initial energy density after averaging over all the events $\langle E_0(\mathbf{x}_{\perp})E_0(\mathbf{x}_{\perp})\rangle=\langle B_0(\mathbf{x}_{\perp})B_0(\mathbf{x}_{\perp})\rangle$. The spatial indexes in $\delta^{mn}$ and $\epsilon^{mn}$ will be contracted when averaging over the initial color distributions. In the calculation of the local energy-momentum tensor in [@Li:2016eqr; @Chen:2015wia], similar computational procedures had been encountered. However, in that situation, the limit $\mathbf{r}_{\perp}=\mathbf{x}_{\perp} -\mathbf{y}_{\perp} \rightarrow 0$ was taken while here finite values of the $\mathbf{r}_{\perp}=\mathbf{x}_{\perp} -\mathbf{y}_{\perp}$ have to be retained as Fourier transformations from the coordinates space to the momentum space will be implemented. All the techniques needed have already been discussed in [@Li:2016eqr; @Chen:2015wia]; more details on the correlation functions with finite values of $\mathbf{r}_{\perp}$ are given in the Appendix. We summarize the final results here: $$\label{final_result_Ei} \begin{split} &\mathfrak{E}^i\mathfrak{E}^i\equiv \frac{1}{k_{\perp}}\Big\langle \tau F_{i\tau}(\tau,\mathbf{k}_{\perp})F_{i\tau}(\tau,-\mathbf{k}_{\perp})\Big\rangle\\ &= (\pi R_A^2)(2\varepsilon_0)\Bigg[ \sum_{n=3}^{\infty} (-1)^n \mathcal{C}_2(n,k_{\perp}) (Q\tau)^{2n-1} \left[\ln\frac{Q^2}{m^2}\right]^{-2}\\ &+ \sum_{n=2}^{\infty}(-1)^n \mathcal{C}_1(n,k_{\perp})(Q\tau)^{2n-1}\left[\ln\frac{Q^2}{m^2}\right]^{-1}\Bigg]\, ,\\ \end{split}$$ $$\label{final_result_Bi} \begin{split} &\mathfrak{B}^i\mathfrak{B}^i\equiv \frac{1}{k_{\perp}}\Big\langle\frac{1}{\tau} F_{i\eta}(\tau,\mathbf{k}_{\perp})F_{i\eta}(\tau,-\mathbf{k}_{\perp})\Big\rangle\\ &= (\pi R_A^2)(2\varepsilon_0)\Bigg[ \sum_{n=3}^{\infty} (-1)^n \tilde{\mathcal{C}}_2(n,k_{\perp}) (Q\tau)^{2n-1} \left[\ln\frac{Q^2}{m^2}\right]^{-2} \\ &+ \sum_{n=2}^{\infty} (-1)^n\mathcal{C}_1(n,k_{\perp})(Q\tau)^{2n-1}\left[\ln\frac{Q^2}{m^2}\right]^{-1}\Bigg]\, ,\\ \end{split}$$ $$\label{final_result_Ez} \begin{split} &\mathfrak{E}^z\mathfrak{E}^z \equiv \frac{1}{k_{\perp}} \Big\langle\frac{1}{\tau} F_{\tau\eta}(\tau,\mathbf{k}_{\perp})F_{\tau\eta}(\tau,-\mathbf{k}_{\perp})\Big\rangle\\ &=(\pi R_A^2)(2\varepsilon_0)\Bigg[\sum_{n=2}^{\infty}(-1)^n\mathcal{D}_2(n,k_{\perp}) (Q\tau)^{2n+1}\left[\ln\frac{Q^2}{m^2}\right]^{-2}\\ &+\frac{1}{2}\mathcal{G}_0(k_{\perp})(Q\tau) + \sum_{n=1}^{\infty}(-1)^n\mathcal{D}_1(n,k_{\perp}) (Q\tau)^{2n+1}\left[\ln\frac{Q^2}{m^2}\right]^{-1}\Bigg]\, ,\\ \end{split}$$ $$\label{final_result_Bz} \begin{split} &\mathfrak{B}^z\mathfrak{B}^z \equiv \frac{1}{k_{\perp}} \Big\langle \frac{\tau}{2} F_{ij}(\tau,\mathbf{k}_{\perp})F_{ij}(\tau,-\mathbf{k}_{\perp}) \Big\rangle\\ &=(\pi R_A^2)(2\varepsilon_0)\Bigg[\sum_{n=2}^{\infty}(-1)^n\tilde{\mathcal{D}}_2(n,k_{\perp}) (Q\tau)^{2n+1}\left[\ln\frac{Q^2}{m^2}\right]^{-2}\\ &+ \frac{1}{2}\mathcal{G}_0(k_{\perp})(Q \tau)+ \sum_{n=1}^{\infty}(-1)^n\mathcal{D}_1(n,k_{\perp}) (Q\tau)^{2n+1}\left[\ln\frac{Q^2}{m^2}\right]^{-1}\Bigg]\, .\\ \end{split}$$ We use $\mathfrak{E}^i\mathfrak{E}^i$, $\mathfrak{B}^i\mathfrak{B}^i$, $\mathfrak{E}^z\mathfrak{E}^z$ and $\mathfrak{B}^z\mathfrak{B}^z$ to label the four terms. They are ultimately related to their counterparts in the expression for the energy density Eq. . The $R_A$ is the radius of the colliding nucleus. The initial $(\tau=0)$ energy density $\varepsilon_0$ [@Fries:2006pv; @Chen:2015wia] serves as a normalization factor, $$\varepsilon_0 = 2\pi \frac{N_c}{N_c^2-1} \left(\frac{g^2}{4\pi}\right)^3\mu^2\left[\ln\frac{Q^2}{m^2}\right]^2$$ Here $N_c=3 $ is the number of colors and $g$ is the strong coupling constant which depends on the energy scales. The $\mu$ is an input paramter in the McLerran-Venugopalan model that characterizes the Gaussian width of the color fluctuations from the large-$x$ partons within each nucleus. It depends on the transverse coordinate $\mathbf{x}_{\perp}$ in general while we assume homogeneity of $\mu$ on the transverse plane in our discussions of the Glasma evolution. The initial flows due to the inhomogeneity on the transverse plane are discussed in detail in [@Chen:2013ksa; @Chen:2015wia; @Fries:2017ina]. In addition, $\mu$ is quantitatively related to the gluon saturation scale $Q_s$ [@Lappi:2007ku]. Note that we assume the two colliding nuclei are the same so that the gluon saturation scales are the same, as well as the ultraviolet and the infrared cut-offs. The coefficient functions $\mathcal{C}_1(n,k_{\perp})$, $\mathcal{C}_2(n,k_{\perp})$, $\tilde{\mathcal{C}}_2(n,k_{\perp})$, $\mathcal{G}_0(k_{\perp})$, $\mathcal{D}_1(n,k_{\perp})$, $\mathcal{D}_2(n,k_{\perp})$, $\tilde{\mathcal{D}}_2(n,k_{\perp})$ are given in Appendix \[coefficientfunction\_appendix\]. These coefficient functions depend on the input parameters: the ultraviolet cut-off $Q$, the infrared cut-off $m$ and the gluon saturation scale $Q_s$. As power series expansions in $Q\tau$, $\mathfrak{E}^i\mathfrak{E}^i$ and $\mathfrak{B}^i\mathfrak{B}^i$ have the lowest order $(Q\tau)^1$ while $\mathfrak{E}^z\mathfrak{E}^z$ and $\mathfrak{B}^z\mathfrak{B}^z$ have the lowest order $(Q\tau)^3$. It is not surprising to notice that the expressions of $\mathfrak{E}^i\mathfrak{E}^i$ and $\mathfrak{B}^i\mathfrak{B}^i$ are almost the same except for the minor difference in the coefficient functions $\mathcal{C}_2(n,k_{\perp})$ and $\tilde{\mathcal{C}}_2(n,k_{\perp})$. The same observation applies to the expressions of $\mathfrak{E}^z\mathfrak{E}^z$ and $\mathfrak{B}^z\mathfrak{B}^z$. Mathematically speaking, these differences originate from the difference in the initial longitudinal chromo-electric field $E_0$ and the longitudinal chromo-magnetic field $B_0$, see Eq. . It involves spatial index contraction with either $\delta^{mn}$ or $\epsilon^{mn}$ when averaging over initial color fluctuations. Physically speaking, these minor differences represent non-Abelian effects in the time evolutions that deviate from the Abelian approximation where there exists duality between the $E$-fields and the $B$-fields.\ As power series expansions in $Q\tau$, one would naively expect the convergence radius of these four terms to be $\tau_c \sim 1/Q$, which is around $0.05\,\rm{fm/c}$ for $Q=4.0\,\rm{GeV}$. However, the coefficient functions $\mathcal{C}_1(n,k_{\perp})$, $\mathcal{C}_2(n,k_{\perp})$, $\tilde{\mathcal{C}}_2(n,k_{\perp})$, $\mathcal{D}_1(n,k_{\perp})$, $\mathcal{D}_2(n,k_{\perp})$ and $\tilde{\mathcal{D}}_2(n,k_{\perp})$ decrease very fast as one increases the order $n$ of the power series expansions, see Fig. \[fig:coefficients\]. The fast decrease of these coefficients compensates for the increase of $(Q\tau)^n$ when extending to regions of larger proper time. As a result, the convergence radius is approximately enhanced by ten times to $\tau_c \sim 0.5\, \rm{fm/c}$ . This point becomes apparent in the results shown in the next section. [0.5]{} \[fig:Ccoefficients\] [0.5]{} \[fig:Dcoefficients\] Results and Discussions ======================= The input parameters are chosen to be $Q = 4.0 \,\rm{GeV}$, $m=0.2\,\rm{GeV}$ and $Q_s = 1.2\,\rm{GeV}$ as in [@Li:2016eqr] to satisfy the assumption on the scale separation $Q^2\gg Q_s^2\gg m^2$. The strong coupling constant $g$ is calculated at the momentum scale $Q$. These values will be the benchmark input values for comparisons when varying one of them while keeping the other two fixed. In the numerical computations, we cut the power series expansion to the order of $n=60$. Depending on the proper time window one is interested in, higher order terms in the power series expansion can also be incorporated although the computational time will increase dramatically. Additionally, there is the limit on the convergence radius that prohibits extension to larger values of the proper time $\tau$. This reveals the limitation of the small proper time power series expansion method. Figure \[fig:dndk\_components\] shows the time evolution of the four terms , , and in the gluon spectrum for the momentum mode $k_{\perp}=Q_s$. The contributions from the chromo-electric part $\mathfrak{E}^i\mathfrak{E}^i+\mathfrak{E}^z\mathfrak{E}^z$ is larger than that from the chromo-magnetic part $\mathfrak{B}^i\mathfrak{B}^i+\mathfrak{B}^z\mathfrak{B}^z$ as shown in Fig. \[fig:dndk\_BEcomponents\]. Late in the evolution, the fields become weak so that the non-Abelian self-interacting terms are less important than the kinetic terms. Ideally, if the self-interacting effects could be completely ignored, one has the abelianized theory where there exists duality between the chromo-electric field $\mathbf{E}$ and chromo-magnetic field $\mathbf{B}$. We would have the same contributions to the gluon spectrum from the chromo-electric fields and the chromo-magnetic fields. However, non-Abelian self-interacting effects persist even in the weak field regime. As a result, the initial difference between the chromo-electric field $E_0$ and the chromo-magnetic field $B_0$ is passed on nonlinearly to the late time so that their differences show up even in the event-averaged results as demonstrated by Fig. \[fig:dndk\_BEcomponents\]. Note that although $E_0$ and $B_0$ are different for a single event, after averaging over all the initial color distributions, $\langle E_0E_0 \rangle $ is the same as $\langle B_0B_0 \rangle $, which is also demonstrated by Fig. \[fig:dndk\_BEcomponents\]. Figure \[fig:dndk\_tau\] shows the time evolution of four different momentum modes $k_{\perp}/Q_s =0.8$, $k_{\perp}/Q_s = 1.0$, $k_{\perp}/Q_s = 1.2$ and $k_{\perp}/Q_s=1.5$ from the gluon spectrum. After a short proper time of continuous increasing, they all saturate at constant values. These plateau features are reminiscent of the fact that the energy density $\varepsilon(\tau)$ approximately behaves as $1/\tau$ at late time, which means free streaming. Once reaching the plateau regions, the gluon spectrum is independent of time. This feature is further identified as the criteria that the classical gluon fields switch to the weak field regime from the initial strong field regime. A time independent gluon spectrum thus has physical meaning and can be intepretated as distribution of the particle numbers. Apparently, larger momentum modes reach the weak field regime faster than the smaller momentum modes do as can be seen from Fig. \[fig:dndk\_tau\].\ We show the gluon spectrum and the energy density spectrum at $\tau=0.6\,\rm{fm/c}$ in Fig. \[fig:gluon\_spectrum\_thermal\]. We reorganize the gluon spectrum as the number of gluons per unit transverse area, per unit radian, per unit rapidity and per transverse momentum magnitude $k_{\perp}$, $$n(k_{\perp}) \equiv \frac{dn}{dk_{\perp}} =k_{\perp}\, \frac{dN}{dy d^2\mathbf{k}_{\perp}}\frac{1}{(\pi R_A^2)}\, .$$ The area under the curve $n(k_{\perp})$ represents the total number of gluons per unit area and per unit radian. The energy density spectrum is then defined as $$\label{fittingfunctions2} \varepsilon(k_{\perp}) = k_{\perp} n(k_{\perp}) =k_{\perp}^2\, \frac{dN}{dy d^2\mathbf{k}_{\perp}}\frac{1}{(\pi R_A^2)}\,.$$ The functional form of the gluon spectrum $n(k_{\perp})$ is first fitted using a thermal distribution function (Bose-Einstein distribution) with finite effective mass $M_{\rm{eff}}$ and finite effective temperature $T_{\rm{eff}}$. $$\label{thermalfittingfunction} n(k_{\perp})=a_1\, \left(e^{\sqrt{k_{\perp}^2+M_{\rm{eff}}^2}/T_{\rm{eff}}} -1\right)^{-1}$$ The fitting parameters are $a_1= 2.695\, \rm{GeV}$, $M_{\rm{eff}} = 0.717\,\rm{GeV}$ and $T_{\rm{eff}} = 0.843\,\rm{GeV}$. The effective temperature is roughly $T_{\rm{eff}} \sim 0.7 Q_s$. Apparently, the gluon spectrum is close to but slightly different from the equilibrium Bose-Einstein distribution. [0.5]{} [0.5]{} The deviation from the Bose-Einstein distribution is amplified in the energy density spectrum, Fig. \[fig:dedk\_kt\_thermal\]. We then introduce a modification function $h(k_{\perp})$ in the fitting function. $$\label{nonthermal_function_1} n(k_{\perp})= a_2\, \left(e^{\sqrt{k_{\perp}^2+\tilde{M}_{\rm{eff}}^2}/\tilde{T}_{\rm{eff}}}-1\right)^{-1}\, h(k_{\perp})\, .$$ The modification function is $$h(k_{\perp})=\frac{1+ a_3\sqrt{k_{\perp}^2+\tilde{M}_{\rm{eff}}^2}/\tilde{T}_{\rm{eff}}+a_4\left(\sqrt{k_{\perp}^2+\tilde{M}_{\rm{eff}}^2}/\tilde{T}_{\rm{eff}} \right)^2}{1+ a_5\sqrt{k_{\perp}^2+\tilde{M}_{\rm{eff}}^2}/\tilde{T}_{\rm{eff}}+a_6\left(\sqrt{k_{\perp}^2+\tilde{M}_{\rm{eff}}^2}/\tilde{T}_{\rm{eff}} \right)^2}.$$ The fitting result is shown in Fig. \[fig:gluon\_spectrum\_nonthermal\_1\]. For the nonthermal function , the fitting parameters are $a_2=2.633 \,\rm{GeV}$, $\tilde{M}_{\rm{eff}} = 0.831\,\rm{GeV}$, $\tilde{T}_{\rm{eff}} = 0.937 \,\rm{GeV}$, $a_3 = -1.440 $, $a_4 = 0.623$, $a_5 = -1.520$ and $a_6 =0.692$. Here the effective temperature is roughly $\tilde{T}_{\rm{eff}} \sim 0.8\,Q_s$. Both the gluon spectrum and the energy density are fitted well with the nonthermal function . [0.5]{} [0.5]{} It is interesting that one can use a different nonthermal function that fits the gluon spectrum result as well as . $$\label{nonthermal_function_2} n(k_{\perp}) = a_2(e^{k_{\perp}/\tilde{T}_{\rm{eff}}} -1)^{-1}\, k_{\perp}\, \tilde{h}(k_{\perp}),$$ with $$\tilde{h}(k_{\perp}) = \frac{1+a_3\,k_{\perp} + a_4\,k_{\perp}^2}{1+ a_5\,k_{\perp} + a_6\,k_{\perp}^2}$$ The fitting results are shown in Fig \[fig:gluon\_spectrum\_nonthermal\_2\]. The fitting parameters are $a_2=2.574$, $a_3=-0.569 \,\rm{GeV}^{-1}$, $a_4=0.771\,\rm{GeV}^{-2}$, $a_5=-1.084\,\rm{GeV}^{-1}$, $a_6 = 1.581\,\rm{GeV}^{-2}$ and $\tilde{T}_{\rm{eff}} = 0.776\,\rm{GeV}$. Here the effective temperature is roughly $\tilde{T}_{\rm{eff}} \sim 0.65\,Q_s$. [0.5]{} [0.5]{} $Q_s \, (\rm{GeV})$ 1.0 1.2 1.5 -------------------------------------- -------- -------- -------- $ \tilde{M}_{\rm{eff}}\, (\rm{GeV})$ 0.775 0.831 0.997 $\tilde{T}_{\rm{eff}}\, (\rm{GeV}) $ 0.875 0.937 1.087 $a_2$ 1.658 2.633 4.410 $a_3$ -1.273 -1.440 -1.843 $a_4$ 0.529 0.623 0.837 $a_5$ -1.431 -1.520 -1.906 $a_6$ 0.651 0.692 0.900 : The fitting parameters for the nonthermal fitting function when choosing different values of $Q_s$ while $Q=4.0\,\rm{GeV}$ and $m=0.2\,\rm{GeV}$. []{data-label="table:different_Qs"} In comparison with the first nonthermal fitting function , the second nonthermal fitting function assumes a zero effective mass and the functional form of the modification function is multiplied by an additional $k_{\perp}$. Both fittings give much better results than the thermal fitting function . The different forms of the fitting functions indicate that the main feature of the functional form for the gluon spectrum is exponential. The effective mass term $\tilde{M}_{\rm{eff}}$ is not necessary while the effective temperature $\tilde{T}_{\rm{eff}}$ which is approximately $0.6\, Q_s\sim0.9 \, Q_s$ characterizes the typical momentum for the gluonic modes at the weak field regime of the Glasma evolution. It is worth noting that in [@Krasnitz:2001qu; @Krasnitz:2003jw] the gluon spectrum had already been fitted with the Bose-Einstein distribution function for lower momentum modes. However, the fitted curves were for $dN/dyd^2\mathbf{k}_{\perp}$ in [@Krasnitz:2001qu; @Krasnitz:2003jw] rather than for $k_{\perp}dN/dyd^2\mathbf{k}_{\perp}$ as fitted in the current paper. Also, those higher momentum modes were fitted with a power law function so as to compare with the results from perturbative QCD calculations. In our computations, the momentum modes reside in the range from $m=0.2\,\rm{GeV}$ to $Q=4.0\, \rm{GeV}$ within which descriptions in terms of the classical fields are assumed to be justified. Therefore, momentum modes lower than the scale $m$ or larger than the scale $Q$ should be understood as coming from extrapolations. Higher moments of the gluon distributions beyond the energy density spectrum (first moment of the gluon spectrum) should be able to reveal further deviations from a pure Bose-Einstein distribution. We are content with the energy density spectrum as a second constraint for the fittings and not considering higher moments of the gluon distribution.\ $Q \, (\rm{GeV})$ 3.0 4.0 5.0 -------------------------------------- -------- -------- -------- $ \tilde{M}_{\rm{eff}}\, (\rm{GeV})$ 0.859 0.831 0.693 $\tilde{T}_{\rm{eff}}\, (\rm{GeV}) $ 0.898 0.937 1.016 $a_2$ 2.748 2.633 1.716 $a_3$ -1.381 -1.440 -1.786 $a_4$ 0.533 0.623 1.112 $a_5$ -1.409 -1.520 -1.945 $a_6$ 0.559 0.692 1.217 : The fitting parameters for the nonthermal fitting function when choosing different values of $Q$ while $Q_s = 1.2\,\rm{GeV}$ and $m=0.2\,\rm{GeV}$. []{data-label="table:different_Q"} To compare different results when varying the input parameters, we normalize the gluon spectrum by the total number of gluons per unit area, per unit radian $N= \int dk_{\perp} n(k_{\perp})$. The function $f(k_{\perp})=n(k_{\perp})/N$ therefore has the meaning of probability density. In Fig. \[fig:dndk\_three\_Qs\], the gluon spectrums for three different values of $Q_s$ are presented. Other input parameters are chosen to be the same as the benchmark values. Increasing the values of the gluon saturation scale $Q_s$ can be realized by increasing the collision energys of the colliding nuclei. The gluon saturation scale $Q_s$, which is linearly related to the effective temperature $\tilde{T}_{\rm{eff}}$, characterizes the typical momentum of the gluonic system at the weak field regime of the Glasma evolution. Larger values of $Q_s$ mean smaller weights at the lower momentum while smaller values of $Q_s$ indicate larger weights at lower momentum. Figure \[fig:dndk\_three\_Qs\] is consistent with this qualitative properties. Note that the area under each curve is normalized to be one. The corresponding effective mass $\tilde{M}_{\rm{eff}}$ and the effective temperature $\tilde{T}_{\rm{eff}}$ when fitted with the nonthermal function by changing $Q_s$ are given in Table \[table:different\_Qs\]. Both $\tilde{M}_{\rm{eff}}$ and $\tilde{T}_{\rm{eff}}$ increase as $Q_s$ is increased. The effective temperature $\tilde{T}_{\rm{eff}}$ is roughly $0.6\,Q_s\sim0.9\, Q_s$. Figure \[fig:dndk\_three\_Q\] shows the results when varying the ultraviolet cut-off $Q$. Other input parameters are the same as the benchmark values. As can be seen, the results are barely sensitive to the changes of ultraviolet cut-offs. The fitting parameters when changing the ultraviolet cut-offs are given in Table \[table:different\_Q\]. Figure \[fig:dndk\_three\_m\] shows the results for different values of the infrared cut-off $m$. The differences are noticeable. Smaller values of the $m$ incorporate more lower momentum modes, thus increases the weights in the lower momentum regions. The corresponding fitting parameters when changing the infrared cut-offs are listed in Table \[table:different\_m\]. $m \, (\rm{GeV})$ 0.1 0.2 0.3 -------------------------------------- -------- -------- -------- $ \tilde{M}_{\rm{eff}}\, (\rm{GeV})$ 0.457 0.831 1.254 $\tilde{T}_{\rm{eff}}\, (\rm{GeV}) $ 0.920 0.937 0.954 $a_2$ 2.015 2.633 2.875 $a_3$ -1.657 -1.440 -0.908 $a_4$ 1.516 0.623 0.236 $a_5$ -1.476 -1.520 -0.962 $a_6$ 1.117 0.692 0.270 : The fitting parameters for the nonthermal fitting function when choosing different values of $m$ while $Q=4.0\,\rm{GeV}$ and $Q_s=1.2\,\rm{GeV}$.[]{data-label="table:different_m"} Conclusion And Outlook ====================== In high energy heavy-ion collisions, understanding the complete time evolution of the Glasma state is important to gain insights on the very initial stages of the collisions. For the simplest boost-invariant situation, we reexamined the gluon spectrum from a semi-analytic approach. We proposed a different formula for the gluon spectrum which is closely related to the local energy density studied before. We showed that the gluon spectrum has different contributions from the chromo-electric part and the chromo-magnetic part, which reflects the effects of non-Abelian self-interactions in the weak field regime of the Glasma evolution. All the momentum modes reach their plateau regions after certain times, which is consistent with the free-streaming $(\varepsilon \sim 1/\tau)$ at the late time of the Glasma evolution. However, larger momentum modes take less time to enter the weak field regime while smaller mometum modes take more time. To have a meaningful result for the gluon spectrum, one need to make a proper time cut-off large enough so that most of the momentum modes of the gluon spectrum are not changing with time. We took $\tau = 0.6\,\rm{fm/c}$ and we found that the functional form of the gluon spectrum is nonequilibrium in nature but is close to a thermal distribution with effective temperatures around $0.6\, Q_s \sim0.9\, Q_s$. The gluon spectrum is essentially exponential with modification functions that account for the deviations from the equilibrium. This functional form is different from either the Gaussian distributions or the step functions used in the literature. It would be interesting to see how the system evolves starting from these different forms of the initial gluon spectrum. In addition, the close-to-equilibrium feature of the gluon spectrum may give us some hints on the early thermalization problem. Apparently, the boost-invariant gluon spectrum lacks information about the longitudinal dynamics. It is necessary to go beyond the boost-invariance assumption, especially for the initial conditions, to explore the dependence on the longitudinal momentum for the gluon spectrum. Acknowledgement {#acknowledgement .unnumbered} =============== I would like to thank J. I. Kapusta for encouragements and important discussions. I am grateful to R. J. Fries and G. Chen for many helpful discussions and correspondance. I also thank L. McLerran, B. Schenke, R. Venugopalan, D. Kharzeev, H.-U. Yee and P. Tribedy for discussions. This work was supported by the U. S. Department of Energy grant DE-FG02-87ER40328. Correlation Functions with Finite Range ======================================= The relevant correlation functions involve an auxiliary function $\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})$. A few examples [@Chen:2015wia] are $$\begin{split} &\langle A_a^i(\mathbf{x}_{\perp})A_b^j(\mathbf{y}_{\perp})\rangle \\ = &\nabla_x^i\nabla_y^j\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\mathcal{T}(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\delta_{ab}\, ,\\ \end{split}$$ $$\begin{split} &\langle D^kA^i_a(\mathbf{x}_{\perp})D^lA_b^j(\mathbf{y}_{\perp}) \rangle\\ =& \nabla_x^k\nabla_x^i\nabla^l_y\nabla_y^j\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\mathcal{T}(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\delta_{ab}\, ,\\ \end{split}$$ $$\begin{split} &\langle D^kD^lA^i_a(\mathbf{x}_{\perp})A_b^j(\mathbf{y}_{\perp}) \rangle \\ =& \nabla^l_x\nabla_x^k\nabla_x^i\nabla_y^j\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\mathcal{T}(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\delta_{ab}\, ,\\ \end{split}$$ $$\begin{split} &\langle D^mD^nD^kD^lA^i_a(\mathbf{x}_{\perp})A_b^j(\mathbf{y}_{\perp}) \rangle \\ =& \nabla_x^m\nabla_x^n\nabla^l_x\nabla_x^k\nabla_x^i\nabla_y^j\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\mathcal{T}(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\delta_{ab}\, .\\ \end{split}$$ with $$\begin{split} &\mathcal{T}(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\\ =&\frac{2g^2}{g^4N_c\Gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})}\left\{\rm{exp}\left[\frac{g^4N_c}{2(N_c^2-1)}\Gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\right]-1\right\}\, ,\\ \end{split}$$ and $$\label{gammafunction} \Gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp}) = \Gamma(r) = \frac{\mu}{8\pi} r^2\ln m^2r^2\, .$$ Here $r=\|\mathbf{x}_{\perp}-\mathbf{y}_{\perp}\|$. The main efforts are to calculate the auxiliary function $\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})$ and its higher order derivatives. The $\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})$ is expressed as $$\label{gammafunction} \gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp}) = \mu\int \frac{d^2\vec{k}_{\perp}}{(2\pi)^2} e^{i\mathbf{k}_{\perp}(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})} G(\mathbf{k}_{\perp})G(-\mathbf{k}_{\perp})\, .$$ Here $G(\mathbf{k}_{\perp}) = 1/k^2_{\perp}$ is the momentum space Green function. To get meaningful results, the integral in has to be regularized. In [@Fujii:2008km; @Chen:2015wia] an infrared scale $m$ is introduced to modify the expression of $G(\mathbf{k}_{\perp})$ from $1/k^2_{\perp}$ to $1/(k_{\perp}^2+m^2)$ while the ultraviolet cut-off $\Lambda$ is imposed on the upper integration limit. In this paper, we explicitly impose the infrared cut-off $m$ and the ultraviolet cut-off $Q$ as the momentum integration limits $$\label{gammafunction2} \gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp}) = \mu\int_m^Q \frac{d^2\mathbf{k}_{\perp}}{(2\pi)^2} e^{i\mathbf{k}_{\perp}(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})} \frac{1}{k^4_{\perp}}\, .$$ Taking derivatives on $\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})$ is carried out inside of the integral before the momentum integration $$\label{gammafunctionderivative} \begin{split} \nabla_x^i\nabla_y^j\gamma(\vec{x}_{\perp},\vec{y}_{\perp}) &= \mu\int \frac{d^2\vec{k}_{\perp}}{(2\pi)^2} e^{i\vec{k}_{\perp}(\vec{x}_{\perp}-\vec{y}_{\perp})} \frac{k_{\perp}^ik_{\perp}^j}{k^4_{\perp}}\\ &\simeq\mu\frac{\delta^{ij}}{2} \int \frac{d^2\vec{k}_{\perp}}{(2\pi)^2} e^{i\vec{k}_{\perp}(\vec{x}_{\perp}-\vec{y}_{\perp})} \frac{k_{\perp}^2}{k^4_{\perp}}.\\ \end{split}$$ We assume rotational invariance on the transverse plane in the momentum space and thus only keep the symmetric part of $k^i_{\perp}k^j_{\perp}$ which is $\delta^{ij}k_{\perp}^2/2$. An equivalent approach is to evaluate the integral in first and then take derivatives on the spatial function obtained $$\label{spatialapproach} \begin{split} &\nabla_y^j\nabla_x^i\gamma(r) = -\frac{\partial^2\gamma(r)}{\partial r^j\partial r^i}\\ =&-\delta^{ij}\frac{1}{r}\frac{\partial \gamma(r)}{\partial r}-\frac{r^ir^j}{r^2}\left(\frac{\partial^2\gamma(r)}{\partial r^2} - \frac{1}{r}\frac{\partial \gamma(r)}{\partial r}\right).\\ \end{split}$$ The second approach coincides with the first approach after making the approximation $r^ir^j/r^2\simeq \delta^{ij}/2$ in , which is valid as long as $0\lesssim mr\ll 1$. We will follow the first approach examplified by . Two more examples are $$\begin{split} &\nabla_x^k\nabla_y^l\nabla_x^i\nabla_y^j\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp}) \\ =& \mu\int \frac{d^2\mathbf{k}_{\perp}}{(2\pi)^2} e^{i\mathbf{k}_{\perp}(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})} \frac{k_{\perp}^ik_{\perp}^j k_{\perp}^k k_{\perp}^l}{k^4_{\perp}}\\ =&\mu\frac{\Delta^{ijkl}}{8}\int \frac{d^2\mathbf{k}_{\perp}}{(2\pi)^2} e^{i\mathbf{k}_{\perp}(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})}\, ,\\ \end{split}$$ $$\begin{split} &\nabla_x^m\nabla_y^n\nabla_x^k\nabla_y^l\nabla_x^i\nabla_y^j\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp}) \\ =& \mu\int \frac{d^2\mathbf{k}_{\perp}}{(2\pi)^2} e^{i\mathbf{k}_{\perp}(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})} \frac{k_{\perp}^ik_{\perp}^j k_{\perp}^k k_{\perp}^l k_{\perp}^m k_{\perp}^n}{k^4_{\perp}}\\ =&\mu\frac{\Delta^{ijklmn}}{48}\int \frac{d^2\mathbf{k}_{\perp}}{(2\pi)^2} e^{i\mathbf{k}_{\perp}(\mathbf{x}_{\perp}-\mathbf{y}_{\perp})}k_{\perp}^2.\\ \end{split}$$ The spatial index functions $\Delta^{ijkl}$ and $\Delta^{ijklmn}$ are the sum of all possible products of the Kronecker delta functions $$\begin{split} &\Delta^{ijkl}=\delta^{ij}\delta^{kl}+\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk}\, ,\\ &\Delta^{ijklmn}=\delta^{ij}\Delta^{klmn}+\delta^{ik}\Delta^{jlmn}\\ &\qquad\qquad+\delta^{il}\Delta^{jkmn}+\delta^{im}\Delta^{jkln}+\delta^{in}\Delta^{jklm}\, . \end{split}$$ The general expression for $n\geq 2$ is evaluated as $$\begin{split} &\nabla_x^{i_1}\nabla_y^{i_2}\ldots\nabla_x^{i_{2n-1}}\nabla_y^{i_{2n}}\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\\ =&\frac{\mu}{2\pi}\frac{\Delta^{i_1i_2\ldots i_{2n-1}i_{2n}}}{(2n)!!} \frac{1}{r^{2n-2}} \frac{1}{2}\frac{z^{2n-2}}{n-1}\, {}_1F_2[n-1;1,n;-z^2/4]\Big \vert_{mr}^{Qr}\\ \simeq&\frac{\mu}{4\pi}\frac{\Delta^{i_1i_2\ldots i_{2n-1}i_{2n}}}{(2n)!!} \frac{Q^{2n-2}}{n-1}\, {}_1F_2[n-1;1,n;-(Qr)^2/4]\, .\\ \end{split}$$ In the second equality, we take into account the requirement $0\lesssim mr\ll 1$ so that the contribution from the lower integration limit $mr$ can be ignored. The $n=1$ case is computed separately $$\begin{split} &\nabla_x^i\nabla_y^j\gamma(\mathbf{x}_{\perp},\mathbf{y}_{\perp})\\ \simeq&\frac{\mu}{4\pi}\frac{\delta^{ij}}{2}\bigg[ -\frac{(Qr)^2}{4}\, {}_2F_3[1,1;2,2,2;-(Qr)^2/4] + \ln{\frac{Q^2}{m^2}} \bigg]\, .\\ \end{split}$$ Both expressions involve the Hypergeometric functions ${}_1F_2[a;b,c;z]$ and ${}_2F_3[a,b;c,d; z]$, respectively. Let us summarize the general expressions for the correlation functions that are used in the computation of the gluon spectrum, $$\begin{split} &\langle D^{i_1}D^{i_2}\ldots D^{i_{2n}} A^i_a(\mathbf{x}_{\perp})A^j_b(\mathbf{y}_{\perp})\rangle\\ =&(-1)^n \frac{\mu}{4\pi} \frac{\Delta^{i_1i_2i_3\ldots i_{2n}ij}}{2(n+1)!!} \frac{Q^{2n}}{n} {}_1F_2\left[n;1,n+1;-\frac{(Qr)^2}{4}\right]\\ &\times\mathcal{T}(\mathbf{x}_{\perp},\mathbf{y}_{\perp}) \delta_{ab}\, ,\\ \end{split}$$ $$\label{2pointfunction} \begin{split} &\langle A^i_a(\mathbf{x}_{\perp})A^j_b(\mathbf{y}_{\perp})\rangle\\ =&\frac{\mu}{4\pi}\frac{\delta^{ij}}{2}\bigg[ -\frac{(Qr)^2}{4}\, {}_2F_3[1,1;2,2,2;-(Qr)^2/4] + \ln{\frac{Q^2}{m^2}} \bigg]\\ &\times\mathcal{T}(\mathbf{x}_{\perp},\mathbf{y}_{\perp}) \delta_{ab}\, .\\ \end{split}$$ In the limit $r\rightarrow 0$, the term containing the hypergeometric function in vanishes. With further replacement of $Q\leftrightarrow 1/r$, one recovers the well-known result of the two-point correlation function in the McLerran-Vegnugopalan model [@JalilianMarian:1996xn].\ The Coefficient Functions {#coefficientfunction_appendix} ========================= The coefficient functions $\mathcal{C}_1(n,k_{\perp})$ and $\mathcal{C}_2(n,k_{\perp})$ are $$\mathcal{C}_1(n,k_{\perp}) = \sum_{k=1}^{n-1}\frac{1}{4^n}\frac{2(2n-2k)(2k)}{[k!(n-k)!]^2}\frac{1}{2}\left(\frac{1}{n-1}\right)\mathcal{F}_2(n,k_{\perp})\,,$$ $$\begin{split} \mathcal{C}_2(n,k_{\perp})& = \sum_{k=1}^{n-1}\frac{1}{4^n}\frac{2(2n-2k)(2k)}{[k!(n-k)!]^2}\sum_{\beta=0}^{k-1}\sum_{\alpha=0}^{n-k-1}\sum_{\sigma=0}^{\beta}\sum_{\rho=0}^{\alpha}\binom{n-k-1}{\alpha+\rho}\binom{\alpha+\rho}{2\rho} \binom{k-1}{\beta+\sigma}\binom{\beta+\sigma}{2\sigma}\\ &\frac{1}{2\rho+2\sigma+1}\binom{2\rho+2\sigma+2}{\rho+\sigma+1}\frac{1}{2^2}\frac{1}{\alpha+\beta+1}\frac{1}{n-\alpha-\beta-2}\mathcal{F}_1(n,\alpha,\beta,k_{\perp})\\ +&\sum_{k=2}^{n-1}\frac{1}{4^n}\frac{(2n-2k)(2k)}{[k!(n-k)!]^2}\sum_{\beta=0}^{k-2}\sum_{\alpha=0}^{n-k-1}\sum_{\sigma=0}^{\beta}\sum_{\rho=0}^{\alpha}\binom{n-k-1}{\alpha+\rho}\binom{\alpha+\rho}{2\rho} \binom{k-1}{\beta+\sigma+1}\binom{\beta+\sigma+1}{2\sigma+1}\\ &\frac{1}{2\rho+2\sigma+3}\binom{2\rho+2\sigma+4}{\rho+\sigma+2}\frac{1}{2^3}\frac{1}{\alpha+\beta+1}\frac{1}{n-\alpha-\beta-2}\mathcal{F}_1(n,\alpha,\beta,k_{\perp})\times 2\\ +&\sum_{k=2}^{n-2}\frac{1}{4^n}\frac{(2n-2k)(2k)}{[k!(n-k)!]^2}\sum_{\beta=0}^{k-2}\sum_{\alpha=0}^{n-k-2}\sum_{\sigma=0}^{\beta}\sum_{\rho=0}^{\alpha}\binom{n-k-1}{\alpha+\rho+1}\binom{\alpha+\rho+1}{2\rho+1} \binom{k-1}{\beta+\sigma+1}\binom{\beta+\sigma+1}{2\sigma+1}\\ &\frac{1}{2\rho+2\sigma+3}\binom{2\rho+2\sigma+4}{\rho+\sigma+2}\frac{1}{2^2}\frac{1}{\alpha+\beta+1}\frac{1}{n-\alpha-\beta-2}\mathcal{F}_1(n,\alpha,\beta,k_{\perp})\, .\\ \end{split}$$ The auxilliary functions $\mathcal{F}_1(n,k_{\perp})$ and $\mathcal{F}_2(n,\alpha,\beta,k_{\perp})$ represent the implementation of Fourier transformations $$\begin{split} \mathcal{F}_1(n,\alpha,\beta,k_{\perp}) =& \frac{1}{k_{\perp}Q}\int^{1/m}_0 dr\,(2\pi r) J_0(k_{\perp} r)\, {}_1F_2\left[\alpha+\beta+1;1,\alpha+\beta+2; -\frac{(Qr)^2}{4}\right]\\ &\times {}_1F_2\left[n-\alpha-\beta-2;1,n-\alpha-\beta-1;-\frac{(Qr)^2}{4}\right](\tilde{\mathcal{T}}(r))^2\, ,\\ \end{split}$$ $$\begin{split} \mathcal{F}_2(n,k_{\perp}) =& \frac{1}{k_{\perp}Q}\int^{1/m}_0 dr\,(2\pi r) J_0(k_{\perp} r) {}_1F_2\left[n-1;1,n; -\frac{(Qr)^2}{4}\right]\\ &\times \left(-\frac{(Qr)^2}{4}{}_2F_3\left[1,1;2,2,2;-\frac{(Qr)^2}{4}\right]\left[\ln\frac{Q^2}{m^2}\right]^{-1}+1\right) (\tilde{\mathcal{T}}(r))^2\, .\\ \end{split}$$ The function $\tilde{\mathcal{T}}(r)$ is a rescaled expression of $\mathcal{T}(r)$ so that $\tilde{\mathcal{T}}(r) \rightarrow 1$ as $r\rightarrow 0$, $$\tilde{\mathcal{T}}(r)=\frac{2(N_c^2-1)}{g^4N_c \Gamma(r)}\left\{\rm{exp}\left[\frac{g^4N_c}{2(N_c^2-1)}\Gamma (r))\right]-1\right\}\,.$$ The integration limits for $r$ in the Fourier transformations are chosen to be $0$ and $1/m$ to be consistent with our approximation $0\lesssim mr \ll 1$. The prefactor $1/k$ in the expressions of $\mathcal{F}_1(n,\alpha, \beta, k_{\perp})$ and $\mathcal{F}_2(n, k_{\perp})$ originates from the dispersion relation in Eq. while the prefactor $1/Q$ is due to the additional $1/\tau$ geometrical factor in Eq. when matching the expansions in $Q\tau$. As explained in [@Li:2016eqr], the binomial coefficients in the expression of $\mathcal{C}_2(n,k_{\perp})$ come from distributing multiple covariant derivatives $D_x $ to either the $A_1(\mathbf{x}_{\perp})$ field or $A_2(\mathbf{x}_{\perp})$ field in evaluating the following expressions, (of course, the distributions are also made for the covariant derivative $D_y$ to either the $A_1(\mathbf{y}_{\perp})$ field or the $A_2(\mathbf{y}_{\perp})$ field.) $$\begin{split} &\Big \langle [D_x^j,[D_x^{\{2k-2\}}, [ A_1^m(\mathbf{x}_{\perp}), A_2^n(\mathbf{x}_{\perp})]]][D_y^j, [D_y^{\{2n-2k-2\}}, [ A_1^p(\mathbf{y}_{\perp}), A_2^q(\mathbf{y}_{\perp})]]] \Big \rangle\, ,\\ &\Big\langle [D_x^{\{2k\}}, [ A_1^m(\mathbf{x}_{\perp}), A_2^n(\mathbf{x}_{\perp})]] [D_y^{\{2n-2k\}}, [ A_1^p(\mathbf{y}_{\perp}), A_2^q(\mathbf{y}_{\perp})]]\Big\rangle\, .\\ \end{split}$$ To obtain the coefficient function $\tilde{\mathcal{C}}_2(n,k_{\perp})$, we replace the factors $1/(2\rho+2\sigma+1)$ and $1/(2\rho+2\sigma+3)$ inside the nested summations in the expression of $\mathcal{C}_2(n,k_{\perp})$ with the pure number $1$. These two factors inside the nested summations come from spatial index contractions with $\epsilon^{mn}\epsilon^{pq}$ for the $B_0$ field while they give a pure number 1 if contractions are made with $\delta^{mn}\delta^{pq}$ for $E_0$ field.\ The coefficient functions $\mathcal{D}_1(n,k_{\perp})$ and $\mathcal{D}_2(n,k_{\perp})$ are $$\mathcal{D}_1(n,k_{\perp}) = \sum_{k=0}^{n}\frac{1}{4^n}\frac{(n-k+1)(k+1)}{(n-k)!(n-k+1)!k!(k+1)!}\frac{1}{n}\mathcal{G}_3(n,k_{\perp})\, ,$$ $$\begin{split} \mathcal{D}_2(n,k_{\perp})& = \sum_{k=0}^{n}\frac{1}{4^n}\frac{(n-k+1)(k+1)}{(n-k)!(n-k+1)!k!(k+1)!} \Bigg[\sum_{\alpha=0}^{n-k}\sum_{\beta=0}^{k}\sum_{\rho=0}^{\alpha}\sum_{\sigma=0}^{\beta}\binom{n-k}{\alpha+\rho}\binom{\alpha+\rho}{2\rho}\\ &\binom{k}{\beta+\sigma}\binom{\beta+\sigma}{2\sigma}\binom{2\rho+2\sigma+2}{\rho+\sigma+1}\frac{1}{2^2}\left(\frac{1}{n-\alpha-\beta}\right)\left(\frac{1}{\alpha+\beta}\right)\mathcal{G}_1(n,\alpha,\beta,k_{\perp})\\ + &\sum_{\alpha=0}^{n-k-1}\sum_{\beta=0}^{k-1}\sum_{\rho=0}^{\alpha}\sum_{\sigma=0}^{\beta}\binom{n-k}{\alpha+\rho+1}\binom{\alpha+\rho+1}{2\rho+1} \binom{k}{\beta+\sigma+1}\binom{\beta+\sigma+1}{2\sigma+1}\\ &\binom{2\rho+2\sigma+4}{\rho+\sigma+2}\frac{1}{2^2}\left(\frac{1}{n-\alpha-\beta-1}\right)\left(\frac{1}{\alpha+\beta+1}\right)\mathcal{G}_2(n,\alpha,\beta,k_{\perp}) \Bigg]\, .\\ \end{split}$$ The functions $\mathcal{G}_0(k_{\perp})$, $\mathcal{G}_1(n,\alpha,\beta,k_{\perp})$, $\mathcal{G}_2(n,\alpha,\beta,k_{\perp})$ and $\mathcal{G}_3(n,k_{\perp})$ also represent the implementation of the Fourier transformations, $$\mathcal{G}_0(k_{\perp}) = \frac{1}{k_{\perp}Q}\int_{0}^{1/m}dr\, (2\pi r) J_0(k_{\perp} r)\, \left(-\frac{(Qr)^2}{4} {}_2F_3\left[1,1;2,2,2; -\frac{(Qr)^2}{4}\right]\left[\ln\frac{Q^2}{m^2}\right]^{-1}+1\right)^2 [\tilde{\mathcal{T}}(r)]^2\, ,$$ $$\begin{split} \mathcal{G}_1(n,\alpha,\beta,k_{\perp}) =& \frac{1}{k_{\perp}Q}\int_0^{1/m}dr\, (2\pi r) J_0(k_{\perp} r)\, {}_1F_2\left[\alpha+\beta;1,\alpha+\beta+1;-\frac{(Qr)^2}{4}\right]\\ &\times {}_1F_2\left[n-\alpha-\beta;1,n-\alpha-\beta+1; -\frac{(Qr)^2}{4}\right][\tilde{\mathcal{T}}(r)]^2\, , \end{split}$$ $$\begin{split} \mathcal{G}_2(n,\alpha,\beta,k_{\perp}) =&\frac{1}{k_{\perp} Q} \int_0^{1/m}dr\, (2\pi r) J_0(k_{\perp} r)\, {}_1F_2\left[\alpha+\beta+1;1,\alpha+\beta+2;-\frac{(Qr)^2}{4}\right]\\ &\times {}_1F_2\left[n-\alpha-\beta-1;1,n-\alpha-\beta; -\frac{(Qr)^2}{4}\right] [\tilde{\mathcal{T}}(r)]^2\, , \end{split}$$ $$\begin{split} \mathcal{G}_3(n,k_{\perp}) =& \frac{1}{k_{\perp}Q}\int_0^{1/m}dr\, (2\pi r) J_0(k_{\perp} r)\, {}_1F_2\left[n;1,n+1;-\frac{(Qr)^2}{4}\right]\\ &\times\left(-\frac{(Qr)^2}{4} {}_2F_3\left[1,1;2,2,2; -\frac{(Qr)^2}{4}\right]\left[\ln\frac{Q^2}{m^2}\right]^{-1}+1\right)[\tilde{\mathcal{T}}(r)]^2\,. \end{split}$$ To obtain $\tilde{\mathcal{D}}_2(n,k_{\perp})$ from $\mathcal{D}_2(n,k_{\perp})$, one just need to insert the factor $1/(2\rho+2\sigma+1)$ into the first nested summation of $\mathcal{D}_2(n,k_{\perp})$ and the factor $1/(2\rho+2\sigma+3)$ into the second nested summation of $\mathcal{D}_2(n,k_{\perp})$.\
--- author: - \| \ Camerino University, Italy & The University of Melbourne, Australia\ E-mail: title: 'AdS/CFT as classical to quantum correspondence in a Virtual Extra Dimension' --- According to Witten [@Witten:1998zw], in AdS/CFT “quantum phenomena \[...\] are encoded in classical geometry”, without however involving any explicit quantization condition. In this paper we investigate the origin of the “classical to quantum” correspondences of XD theories. It will be derived in terms: of Klein’s original attempt to derive quantum mechanics from compact eXtra-dimension (XD) theory with Periodic Boundary Conditions (PBCs); of Kaluza’s and Nordström’s XD geometrical description of gauge interactions; of de Broglie’s assumption of spacetime recurrence associated to every particle. This leads to a consistent semiclassical description of elementary particles quantum behaviors defined in [@Dolce:tune; @Dolce:2009ce] and reviewed in [@Dolce:cycle]. Here we summarize results of [@Dolce:AdSCFT]. In the atomistic description characterizing modern physics, every physical system is represented in terms of a set of elementary particles and their local retarded relativistic interactions. QM tells us, through the Planck constant, that a spacetime recurrence of instantaneous periodicity $T^{\mu}=\{T_{t},\vec \lambda_{x}/c\}$ is associated to every elementary particle of four-momentum $\bar p_{\mu}=\{\bar E /c,-\mathbf{\bar{p}}\}$. As noted by de Broglie, the spacetime recurrence of a particle of mass $\bar M$ is fully characterized by the Compton time $T_{\tau} = h / \bar M c^{2}$, the intrinsic periodicity of the proper time $\tau$, or equivalently by the quantum recurrence $s\in (0,\lambda_{s}]$ of the worldline parameter $s= c \tau$, with $\lambda_{s} = c T_{\tau} $ (Compton length). In a generic reference frame the spacetime recurrence resulting from this worldline periodicity is in fact described by the covariant relation $T_\tau \bar M c^2 \equiv T^{\mu} {\bar{p}}_{\mu} c \equiv h $. We have performed a Lorentz transformation $c T_\tau= c \gamma T_{t} - \gamma \vec \beta \cdot \vec \lambda_{x}$, $\bar E(\mathbf{\bar{p}}) = \gamma \bar M c^{2}$ and $\mathbf{\bar{p}} = \gamma \vec {\beta} \bar M c$. This means that the classical-relativistic dynamics of a particle described by its 4-momentum $\bar p_{\mu} =\hbar \bar \omega_{\mu} / c $ can be equivalently encoded in retarded modulations of the corresponding local spacetime periodicity $T_{\mu}$. In undulatory mechanics elementary particles are described in terms of phasors or waves (“periodic phenomena”) in which the spacetime coordinates enter as angular variables. Their periodicities describe the kinematics of the particle though $\hbar$. That is, every system in physics can be consistently described in terms of modulations of elementary spacetime cycles with minimal topologies $\mathbb S^1$. We want to impose the intrinsic periodicity $T^{\mu}$ of elementary particles as a constraint. This represents a semiclassical quantization condition. A particle with intrinsic periodicity is similar to a “particle in a box”. Through discrete Fourier transform the persistent periodicity $T^{\mu}$ directly implies a quantization of the conjugate spectrum $p_{n}^{\mu} = n \bar p^{\mu}$; $n$ is the quantum number associated to the topology $\mathbb S^{1}$. The quantization of the energy spectrum associated to the persistent time periodicity $T_{t}$ is the harmonic spectrum $E_{n} = n \bar E = n h / T_{t}$. A free bosonic particle can be therefore represented as a one dimensional bosonic string $\Phi(x)$ vibrating in compact spacetime dimensions of length $T^{\mu}$ and Periodicity Boundary Conditions (PBCs — denoted by the circle in $\oint$): $${\mathcal{S}}^{\lambda_{s}} = \oint^{T^\mu} d^4 x {\mathcal{L}}(\partial_\mu \Phi(x),\Phi(x)) = \oint^{T'^{\mu} =\Lambda^\mu_\nu T^\nu} d^4 x' {\mathcal{L}}( \partial'_\mu \Phi'(x'),\Phi'(x'))\,.\label{generic:actin:comp4D}$$ As known from string theory or XD theory, PBCs (or combinations of Dirichlet and Neumann BCs) minimizes the action at the boundary so that all the relativistic symmetries of (\[generic:actin:comp4D\]) are preserved. This is a consequence of the fact that relativity fixes the differential structure of spacetime whereas the only requirement for the BCs is to fulfill the variational principle. The expansion in harmonics of a field/string vibrating with persistent periodicity is $\Phi(x) = \sum_{n} \phi_n(x) = \sum_{n} A_{n} \exp[{-\frac{i}{\hbar} p_{n \mu} x^{\mu}}]$. To check the covariance we use a Lorentz transformation $x_\mu \rightarrow x'_\mu = \Lambda_\mu^\nu x_\nu$ as a generic transformation of variables in the free action (\[generic:actin:comp4D\]) so that the transformed boundary of the resulting action yields a solution with transformed periodicity $T^\mu \rightarrow {T'}^\mu = \Lambda^\mu_\nu T^\nu$. This describes the four-momentum $\bar p_\mu \rightarrow {{\bar p}'}_\mu = \Lambda_\mu^\nu \bar p_\nu$ of the free particle in the new frame, according to $\bar p'_{\mu} c T'^{\mu} = h$. That is, $T^{\mu}$ transforms as a contravariant tangent 4-vector with the relativistic constraint induced by the underlying Minkowsky metric $\frac{1}{T^{2}_{\tau}} = \frac{1}{T_{\mu}} \frac{1}{T^{\mu}}$. This is the geometric description in terms of periodicities of the relativistic dispersion relation $\bar M c^{2} = \bar p_{\mu} \bar p^{\mu}$. Thus the harmonic energy spectrum of our system in a generic reference frame is $E_{n} (\bar \mathbf p) = n h / T_{t} (\bar \mathbf p) = n \sqrt{\bar \mathbf p^{2} c^{2} + \bar M^{2} c^{4}}$, which is the energy spectrum prescribed by ordinary second quantization (after normal ordering) for the single mode of periodicity $T(\bar \mathbf p)$ of a free bosonic field. This is a first semiclassical correspondence with ordinary QFT. We also note a dualism to Kaluza-Klein (KK) theories. In the rest frame the proper time periodicity (compact worldline) describes a quantized rest energy spectrum, a mass spectrum, $E_{n}(0)/c^{2} \equiv M_{n} = n \bar M = n h / \lambda_{s} c = n h / T_{\tau} c^{2} $, similar to a KK tower of compactification length $\lambda_{s}$. As also noted by Einstein, a relativistic clock is “phenomenon passing periodically through identical phases”. By assuming intrinsic periodicity every isolated particle can be therefore regarded as a reference clock. As in the Cs atomic clock whose reference “tick” of periods $10^{-10} s$ is fixed by an electronic energy gap, an isolated particle of energy $\bar E$ has regular “ticks” of persistent periodicity $T_{t}$ that can be used to define the unit of time. The so-called internal clock of an electron $T_{\tau} \sim 10^{-21} s$ has been observed in a recent experiment [@2008FoPh...38..659C]. The heavier the mass of the particle, the faster the periodicity ($\bar E \sim 1$ TeV $\rightarrow T_{t} \sim 10^{{-27}} s$ ). In a generic point $x=X$, a relativistic interaction of a particle can be characterized by the local retarded variations of four-momentum w.r.t. the free case $\bar{p}_{\mu}\rightarrow\bar{p}'_{\mu}(X)=e_{\mu}^{a}(x)\|_{x=X}\bar{p}_{a}$. Through $\hbar$, interaction can be equivalently encoded by local retarded modulations of the internal clock of the particle $T^{\mu}\rightarrow T'^{\mu}(X)\sim e_{a}^{\mu}(x)\|_{x=X}T^{a}$, that is by local “stretching” the compactification spacetime dimensions of (\[generic:actin:comp4D\]). Therefore a generic interaction can be equivalently encoded in a locally deformed metric $\eta_{\mu\nu}\rightarrow g_{\mu\nu}(X)=[e_{\mu}^{a}(x)e_{\nu}^{b}(x)]\|_{x=X}\eta_{ab}$. This description can be easily checked by using the local transformation of reference frame $dx_{\mu}\rightarrow dx'_{\mu}(X)=e_{\mu}^{a}(x)\|_{x=X}dx_{a}$ as substitution of variables in the free action (\[generic:actin:comp4D\]), [@Dolce:tune; @Dolce:AdSCFT]. The resulting action with deformed metric $g_{\mu\nu}(X)$ describes a locally modulated solution of periodicity $T'^{\mu}(X)$: we pass from a free solution of persistent type $\phi_{n}(x) \propto \exp[-\frac{i}{\hbar} p_{n \mu} x^{\mu}] $ to the interacting solution of modulated type $\phi'_{n}(x) \propto \exp[-\frac{i}{\hbar} \int^{x_{\mu}} d x'^{\mu} p_{n \mu}(x') ] $. Note also that in our formalism the kinematics of the interaction turns out to be equivalently encoded on the boundary, a la holographic principle. Such a geometrodynamical description of generic interactions is of the same type of General Relativity. In a weak Newtonian interaction, the corresponding energy variation $\bar{E}\rightarrow\bar{E}'\sim\left(1+{GM_{\odot}}/{\|\mathbf{x}\|c^2}\right)\bar{E}$ implies, through $\hbar$, a modulation of time periodicity $T_{t}\rightarrow T_{t}'\sim\left(1-{GM_{\odot}}/{\|\mathbf{x}\|c^2}\right)T_{t}$, redshift and time dilatation. If we also consider the variation of momentum and the corresponding modulation of spatial periodicity, the resulting metric encoding the Newtonian interaction is actually the linearized Schwarzschild metric. Similar to Weyl’s original proposal, gauge interaction can be obtained by considering local variations of flat reference frame $dx^{\mu}(x)\rightarrow dx'^{\mu}\sim dx^{\mu} - e dx^{a} \omega^{\;\mu}_{a}(x)$. Parametrizing by means of a vectorial field $\bar{A}_{\mu}(x) \equiv \omega_{\;\mu}^{a}(x)\bar{p}_{a}$, the resulting interaction scheme is actually $\bar{p}'_{\mu}(x) \sim \bar{p}_{\mu}- e\bar{A}_{\mu}(x)$, see [@Dolce:tune] for more details. Now we summarize the correspondence to ordinary relativistic quantum mechanics. A vibrating string with modulated periodicity is the typical classical system that can be described locally in a Hilbert space. The modulated harmonics of such a string form locally a complete set w.r.t the corresponding local inner product $\left\langle \phi\|\chi \right\rangle $. The harmonics defines locally a Hilbert base $\left\langle x \| \phi_{n}\right\rangle = \phi_{n}(x)$. Thus a modulated vibrating string, generic superposition of harmonics, is represented by a generic Hilbert state $ \left\| \phi \right\rangle = \sum a_{n} \left\| \phi_{n} \right\rangle$. The non-homogeneous Hamiltonian $\mathcal H$ and momentum $\mathcal P_{i}$ operator are introduced as the operators associated to the 4-momentum spectrum of the locally modulate string: $\mathcal P_{\mu}\left \|\phi_{n} \right\rangle = p_{n \mu} \left\|\phi_{n} \right\rangle$, where $\mathcal{P}_\mu = \{\mathcal H, - \mathcal{P}_{i}\}$. From the modulated wave equation, the temporal and spatial evolution of every modulated harmonics satisfies $i \hbar \partial_{\mu} \phi_{n}(x) = p_{n}(x) \phi_{n}(x)$, thus the time evolution of our string with modulated periodicity represented by $\left\langle \phi\|\chi \right\rangle $ is given by the ordinary Schrödinger equation $i \hbar \partial_{t} \left\|\phi \right\rangle = \mathcal H \left\|\phi \right\rangle$. Moreover, since we are assuming intrinsic periodicity, this classical-relativistic theory implicitly contains the ordinary commutation relations of QM. This can be seen by evaluating the expectation value of a total derivative $\partial_{x} F(x)$, and considering that the boundary terms of the integration by parts cancel each other owing the assumption of intrinsic periodicity. For generic Hilbert states we obtain: $i \hbar \partial_{x} F(x) = [F(x),\mathcal{P}]$ and $i \hbar = [x,\mathcal{P}]$ for $F(x)=x$, [@Dolce:cycle; @Dolce:tune; @Dolce:AdSCFT; @Dolce:2009ce]. The correspondence with ordinary relativistic QM can also be seen from the fact that, remarkably, the classical evolution of such a relativistic vibrating string with all its modulated harmonics is described by the ordinary Feynman Path Integral (we are integrating over a sufficiently large number $N$ of spatial periods so that $V_{\mathrm{x}} = N \lambda_{x}$ is bigger than the interaction region) $$\mathcal{Z}=\int_{V_{\mathrm{x}}} {\mathcal{D}\mathrm{x}} \exp[{\frac{i}{\hbar} \mathcal{S}(t_{f},t_{i})}]\,.\label{eq:Feynman:Path:Integral}$$ As usual, the $\mathcal S$ is the classical action of the corresponding interaction scheme, with lagrangian $\mathcal{L} = \mathcal P x - \mathcal H $. This result has a very intuitive justification in the fact that in a cyclic geometry such as that associated to the topology $\mathbb S^{1}$, the classical evolution of $\phi(x)$ from an initial configuration to a final configuration is given by the interference of all the possible classical paths with different windings numbers, without relaxing the classical variational principle. Thus the harmonics of the vibrating string/field $\phi(x)$ can be interpreted semiclassically as quantum excitations [@Dolce:cycle; @Dolce:tune; @Dolce:AdSCFT; @Dolce:2009ce]. In this formalism it is straightforward to note that the cyclic worldline parameter enters into the equations in remarkable analogy with the cyclic XD of the KK theory [@Dolce:AdSCFT]. The solution $\phi$ can be in fact formally derived from a corresponding massless KK field by identifying the cyclic XD with a worldline parameter. For this reason we address the cyclic worldline parameter $s$ as “virtul XD”. If we in fact denote the XD and its compactification length with $s$ and $\lambda_{s}$ in a KK massless theory $dS^{2} = dx_{\mu} d x^{\mu} - d s^{2} \equiv 0$, and we identify the XD with the worldline parameter $s = c \tau$ we obtain our 4D theory $d s^{2} = dx_{\mu} d x^{\mu} $ with cyclic worldline parameter of periodicity $\lambda_{s}$, and thus the spacetime periodicity $T^{\mu}$ by Lorentz transformation. In this case the quantized mass spectrum, the analogous of the KK tower, $M_{n} = n \bar M = n h / \lambda_{s} c$ is directly associated to the periodicity $\lambda_{s}$ of worldline parameter $s$ though discrete Fourier transform, whereas in the KK theory $M_{n}$ is obtained indirectly through the EoMs. That is, by assuming a VXD, the KK modes are virtual in the sense that they are not 4D independent particles of mass $M_{n}$, they are the excitations of the same 4D elementary system. The quantized spectrum of a free boson is such that $p_{n \mu} x^\mu = M_{n} c s$ and thus we have the correspondence $\sum_n e^{-i p_{n \mu} x^\mu/\hbar} \leftrightsquigarrow \sum_n e^{-i M_{n} c s/\hbar} $. Such a collective description of the KK mode is typical of the holographic approach, where however a source field $\phi_{\Sigma}$ can be used as BCs to integrate out the heavy KK modes and achieve an effective description of the XD theory: $ {\mathcal{S}}^{5D}(s_{f},s_{i})\sim\mathcal{S}^{Holo}_{\Phi\|_{\Sigma}=e\phi_{\Sigma}}(s_{f},s_{i})+\mathcal{O}(E^{eff}/\bar{M})\label{VXD:holo:appr} $. This also means that, in analogy with the formalism of Light-Front-Quantization, the KK modes form the base $\left\| \phi_{n} \right\rangle$ of a Hilbert space, the evolution along the XD of a KK field is $i \hbar \partial_{s}\left\| \phi \right\rangle = \mathcal M c \left\| \phi \right\rangle $, where the mass operator $\mathcal M \left\| \phi_{n} \right\rangle = M_{n }\left\| \phi_{n} \right\rangle$ satisfies implicit commutation relations $[\mathcal M, s] = i \hbar$ owing the cyclic behavior of $s$. By means of the duality to XD theories, the generic interaction scheme $\bar{p}'_{\mu}(X)$ can be equivalently encoded in a corresponding deformed VXD metric $G_{MN} =\small{\left(\begin{array}{cc} g_{\mu\nu}& 0 \\ 0 & 1 \end{array}\right)}$ (this description should include dilatons or softwalls in the case of finite VXD). Under this dualism gauge interactions turn out to be encoded in a virtual Kaluza metric. By combining the correspondence between classical cyclic dynamics and relativistic QM [@Dolce:2009ce], the geometrodynamical formulation of interactions as modulation of spacetime periodicity [@Dolce:tune] and the dualism with XD theories in the holographic description, we obtain that the classical configurations of the modulated harmonic modes of $\phi$ in a curved XD background encodes the quantum behavior of the corresponding interaction scheme. This correspondence for interacting particles is summarized by the following relation (with implicit source term [@Dolce:AdSCFT]) $$\int_{V_{\mathbf{x}}} {\mathcal{D}\mathbf{x}} \exp[{ \frac{i}{\hbar}\mathcal{S}'}] \leftrightsquigarrow \exp[{ \frac{i}{\hbar} \mathcal{S}^{Holo}_{\Phi\|_{\Sigma}=e\phi_{\Sigma}}}] \,.\label{VXD:QFT:corr}$$ Indeed, the description of physics in terms of elementary cycles pinpoints, at a semiclassical level, the fundamental correspondence between classical XD geometry and 4D quantum behavior of AdS/CFT, [@Witten:1998zw]. To check this we consider the example of the Quark-Gluon-Plasma (QGP) freeze-out, in which the classical dynamics of the interaction scheme are described by the Bjorken Hydrodynamical Model, [@Magas:2003yp]. During the exponential freeze-out the 4-momentum of the QGP fields decays exponentially with the laboratory time, i.e. with the proper time: $\bar E \rightarrow \bar{E}(s) = e^{-ks/c}\bar{E}$. In terms of QCD thermodynamics, $k$ represents the gradient of Newton’s law of cooling. Thus, in the the massless approximation ($E \simeq c p$), the 4-momentum of QGP during the freeze-out decreases conformally and exponentially $ \bar{p}_{\mu}\rightarrow\bar{p}'_{\mu}(s)\simeq e^{-ks/c}\bar{p}_{\mu}$. Equivalently, through the Planck constant, we have that the spacetime periodicity has an exponential and conformal dilatation $T^{\mu}\rightarrow T'^{\mu}(s)\simeq e^{ks/c}T^{\mu}\,.\label{eq:deform:4period:QGP} $According to our geometrodynamical description of interaction, this modulation of periodicity is therefore encoded in the substitution of variables $dx_{\mu}\rightarrow dx'_{\mu}(s)\simeq e^{-ks}dx_{\mu}\,. $The QGP freeze-out is thus encoded by the warped metric $ds^{2}=e^{-2ks/c}dx_{\mu}dx^{\mu}$. By treating the worldline parameter $s$ as a VXD, the exponential dilatation of the 4-periodicity during the QGP freeze-out of massless fields ($dS^{2}\equiv0$) can be equivalently encoded in the virtual AdS metric $dS^{2}\simeq e^{-2ks/c}dx_{\mu}dx{}^{\mu}-ds^{2}\equiv0 $. The energy of the QGP during the freeze-out is therefore parametrized, though the Planck constant, in terms of the time periodicity $T_{t}(s) = e^{ks/c}/k = h / E(s)$. This is formally the conformal parameter $z(s) \equiv T_{t}(s)$ which in fact describes the inverse of the energy in ordinary AdS/CFT. It varies from the initial state (after the formation in a collider experiment) characterized by small time periodicity $T_{t}^{UV}=\frac{h}{\Lambda}=\frac{e^{ks^{UV}/c}}{k}$, to a state characterized by large time periodicities $T_{t}^{IR}=\frac{h}{\mu}=\frac{e^{ks^{IR}/c}}{k}$. The massless approximation means infinite proper time periodicity, i.e. infinite VXD. Thus the AdS geometry encoding the freeze-out has no boundaries. Indeed, if we consider the propagation of a 5D gauge theory with 5D bulk coupling $g_{5}$ in an infinite VXD, the effective coupling of the corresponding 4D theory behaves logarithmically w.r.t. the infrared scale $g^{2}\simeq\frac{g_{5}^{2}k}{\log\frac{\mu}{\Lambda}} $. This reproduces the quantum behavior of the strong coupling constant as long as we suppose $\frac{{1}}{{k}}\sim\frac{{N_{c}g_{5}^{2}}}{{12\pi^{2}}}$, [@Pomarol:2000hp]. In agreement with the AdS/CFT dictionary, the classical dynamics associated to an infinite VXD actually encodes the quantum behavior of a conformal theory. Indeed this has an intuitive justification in terms of undulatory mechanics and relativistic geometrodynamics. In our description, a massive system is characterized by a finite proper time periodicity $\lambda_{s}$. Thus, if we want to describe a QGP of massive fields we must assume a compact VXD. Through the holographic approach [@ArkaniHamed:2000ds] with small IR scale, the classical configurations on this compact warped VXD is effectively described by $\Pi^{Holo}(q^{2})\sim-\frac{q^{2}}{2kg_{5}^{2}}\log\frac{q^{2}}{\Lambda^{2}}$. This approximately matches the two-point function of QCD and the asymptotic freedom $\frac{1}{e_{eff}^{2}(q)}\simeq\frac{1}{e^{2}}-\frac{{N_{c}}}{{12\pi^{2}}}\log\frac{q}{\Lambda}. $ This quantum behavior has been obtained without imposing any explicit quantization except BCs. We note that in a consistent description of the massive case we must also abandon the conformal behavior between the temporal and spatial components ($T_{t} \neq \lambda_{x} /c$). This means the AdS metric must be consistently deformed, for instance, by introducing dilatons in the metric or “soft-walls”. We know that these geometries reproduces a realistic hadronic spectrum. Similarly to Veneziano’s original idea of strings, in our description the hadrons are indeed energy (quantum) excitations, virtual KK modes, of the same fundamental string vibrating with characteristic compact worldline parameter and deformed spacetime encoding the interaction. Such a geometrodynamical description of the masses is relevant to understand the gauge symmetry breaking and thus of the Higgs mechanism [@Dolce:tune; @Dolce:AdSCFT]. In AdS/CFT quantum behavior \[...\] are encoded in classical geometry [@Witten:1998zw]. We conclude that this central aspect of AdS/CFT has a heuristic semiclassical justification in terms of undulatory mechanics and relativistic geometrodynamics [@Dolce:AdSCFT]. The quantization is equivalently obtained semiclassically by means of PBCs, in analogy with a “particle in a box”, or Light-Front-Quantization. Every quantum particle is represented as a classical string vibrating in spacetime(minimal topology $\mathbb S^{1}$) whose harmonic energy levels are the quantum excitations of the system, as proven by the formal correspondence to ordinary QFT [@Dolce:2009ce]. Such a pure 4D description of elementary particles has an explicit dualism with XD theories. The cyclic worldline parameter of the theory enters into the equations in formal analogy with the XD of a KK theory. The KK modes turn out to encode quantum excitations of the same 4D system. In analogy with general relativity, the local spacetime modulations of periodicity encoding a given interaction scheme (i.e. local variations of four-momentum) can be equivalently described in terms of spacetime geometrodynamics. As show in [@Dolce:tune] such a description also yields ordinary gauge interactions, similarly to original Kaluza’s and Weyl’s proposal. By combining all these correspondences of the dynamics in modulated compact spacetime we have inferred semiclassically that the classical configurations in a deformed VXD geometry reproduces the quantum behavior of a corresponding interaction scheme [@Dolce:AdSCFT]. Though AdS/CFT has a very rich phenomenology only partially investigated here and the validity of our approach is semiclassical, our description confirms the classical to quantum correspondence noted in [@Witten:1998zw], and the application to the QGP freeze-out yields analogies to AdS/QCD. [99]{} Witten E 1998 [Adv. Theor. Math. Phys.]{} [2]{} 505 \[[hep-th/9803131]{}\] Dolce D 2013 [Europhys. Lett.]{} [102]{} 31002 [\[ [arXiv:1305.2802]{}\]](http://arxiv.org/abs/1305.2802) Dolce D 2012 [Ann. Phys.]{} [327]{} 1562 [\[ [arXiv:1110.0315]{} \]](http://arxiv.org/abs/1110.0315) Dolce D 2012 [Ann. Phys.]{} [327]{} 2354 [\[ [arXiv:1110.0316]{} \]](http://arxiv.org/abs/1110.0316) Dolce D 2011 [Found. Phys.]{} [41]{} 178 [\[ [arXiv:0903.3680]{}\]](http://arxiv.org/abs/0903.3680) Catillon P. at.al. 2008 [Found. Phys.]{} [38]{} 659 Magas V K, et.al 2003 [Eur. Phys. J.]{} [C30]{} 255–261 \[[nucl-th/0307017]{}\] Pomarol A 2000 [Phys. Rev. Lett.]{} [85]{} 4004 \[[hep-ph/0005293]{}\] Arkani-Hamed N, Porrati M and Randall L 2001 [JHEP]{} [08]{} 017 \[[hep-th/0012148]{}\]
--- abstract: 'The propagation of ultra-high energy cosmic rays in extragalactic magnetic fields can be diffusive, depending on the strength and properties of the fields. In some cases the propagation time of the particles can be comparable to the age of the universe, causing a suppression in the flux measured on Earth. In this work we use magnetic field distributions from cosmological simulations to assess the existence of a magnetic horizon at energies around 10$^{18}$ eV.' author: - \| [Rafael Alves Batista, G[ü]{}nter Sigl]{}\ II. Institute for Theoretical Physics, University of Hamburg\ Luruper Chaussee, 149, 22761 Hamburg, Germany\ title: 'Magnetic horizons of ultra-high energy cosmic rays' --- Introduction ============ During their propagation ultra-high energy cosmic rays (UHECRs) can be deflected by the intervening cosmic magnetic fields, namely the extragalactic and galactic. The extragalactic magnetic field has different strengths in different regions of the universe. For instance, in the center of clusters of galaxies it is $\sim$10 $\mu$G, with coherence length of the order of 10 kpc. The existence of magnetic fields in the voids is still controversial [@neronov2010], but there are some indications that they can be $\sim$10$^{-15}$-10$^{-12}$ G, with typical coherence lengths of the order of 1 Mpc [@neronov2010]. The propagation of cosmic rays in the extragalactic magnetic fields can be diffusive if the scattering length is much smaller than the distance from the source to the observer. Depending on the magnetic field strength and diffusion length, a significant fraction of these particles can have trajectory lengths comparable to the Hubble radius. In this case, a suppression in the flux of cosmic rays is expected compared to the case in which magnetic fields are absent, leading to the existence of a magnetic horizon for the propagation of cosmic rays. This effect has been previously studied by many authors, including Mollerach & Roulet [@mollerach2013], who developed a parametrization for it, under the assumption of Kolmogorov turbulence. In this work we generalize their result for the case of inhomogeneous extragalactic magnetic fields. Magnetic suppression ==================== The diffusive cosmic ray spectrum for an expanding universe can be written as [@berezinsky2006] $$j(E) = \frac{c}{4\pi} \int\limits_{0}^{z_{max}} dz \left\| \frac{dt}{dz} \right\| Q(E_g(E,z),z) \frac{dE_g}{dE} \left( \int\limits_{0}^\infty dB \frac{1}{N_s} \sum\limits_{i=0}^{N_s} \frac{\exp\left( -\frac{r_g^2}{\lambda^2} \right)}{(4\pi\lambda^2)^{3/2}} p(B) \right), \label{eq:spectot2}$$ where $p(B)$ is the probability distribution of the magnetic field strength $B$, $r_g$ is the comoving distance of the source and $\lambda$ is the so-called Syrovatskii variable, given by: $$\lambda^2(E,z,B)= \int\limits_0^z dz' \left\| \frac{dt}{dz'} \right\| \frac{1}{a^2(z')} \left[ \frac{cl_c(z)}{3} \left( a_L \left( \frac{E}{E_c(z,B)} \right)^{\frac{1}{3}} + a_H \left( \frac{E}{E_c(z,B)} \right)^2 \right) \right], \label{eq:syro2}$$ with $a = 1/1+z$ being the scale factor of the universe and $l_c(z) = l_{c,0} a(z)$ the coherence length of the field at redshift $z$. The parameters $a_L$ and $a_H$ are, respectively, 0.3 and 4. $E_c$ is the critical energy, defined as the energy for which a particle has a Larmor radius equal to the coherence length of the magnetic field. The probability distribution functions can be obtained from magnetohydrodynamical (MHD) simulations of the local universe. In this work we considered four different cosmological simulations, namely the ones performed by Miniati [@miniati2002], Dolag [et al.]{} [@dolag2005], Das [et al.]{} [@das2008], Donnert [et al.]{} [@donnert2009]. If the term in parentheses in equation \[eq:spectot2\] is equal to 1, then the magnetic field dependence will vanish and the shape of the spectrum will be independent of the modes of propagation. This result is known as the propagation theorem [@aloisio2004], and states that if the separation between the sources in a uniform distribution is much smaller than the characteristic propagation lengths, the UHECR spectrum will have a universal shape. This spectrum ($j_0$) will be henceforth called universal. We have not considered the actual time evolution of these cosmological simulations. Instead we assume a magnetic field distribution at $z=0$ and extrapolate it to higher redshifts: $B=B_0(1+z)^{2-m}$, with $m$ designating the evolution parameter. Moreover, we assume a Kolmogorov magnetic field with strengths taken from the simulations. The suppression factor $G$ can be written as: $$G = \frac{j(E)}{j_0(E)} \approx \exp \left[ - \frac{(aX_s)^\alpha}{x^\alpha + bx^\beta} \right],$$ with $x\equiv E/\langle E_c\rangle$, $\alpha$, $\beta$, $a$ and $b$ the best fit parameters obtained by fitting $j(E)/j_0(E)$ with the function in the right-hand side of the equation. The complete list of best fit parameters for these extragalactic magnetic field models can be found in ref. [@alvesbatista2014]. In this expression $X_s = d_s / \sqrt{R_H l_c}$, where $d_s$ is the source separation and $R_H$ the Hubble radius. Magnetic horizons ================= In this work the magnetic horizon is defined as the mean distance that a cosmic ray can propagate away from the source in a Hubble time. In figure \[fig:maghorizon\] $\lambda/\sqrt{R_H l_c}$ is displayed as a function of the redshift. In this case $\lambda$ can be understood as the average distance a particle can propagate away from the source in a time interval corresponding to a redshift $z$. ![Volume-averaged Syrovatskii variable for an $E/Z=$10$^{16}$ eV, $m=$1, $\gamma=$2 and $z_{max}=$4. Solid lines correspond to the extragalactic magnetic field distribution, dashed lines correspond to the values obtained using the mean magnetic field strengths obtained from these models, and dotted dashed lines are two limiting cases with high and low magnetic field strengths.[]{data-label="fig:maghorizon"}](AlvesBatista_Rafael_fig1.eps){width="0.58\columnwidth"} In this figure we notice that the magnetic horizons for the case of extragalactic magnetic field distributions from cosmological simulations are larger compared to the case of a Kolmogorov turbulent field with $B_{rms}$ equal to the mean magnetic field strength from the distributions. This happens due to the fact that the voids fill most of the volume, dominating the magnetic field distribution and hence the volume-averaged Syrovatskii variable. We can calculate the energy ($E_e$) for which the suppression factor is $G = 1/e \approx 0.37$ of its original value, as a function of the coherence length. The results are shown in figure \[fig:upperLimit\]. ![Upper limit on the energy for which the flux of cosmic rays is suppressed to $1/e$ ($\approx$37%) of its former value, as a function of the coherence length. Solid lines correspond to the indicated extragalactic magnetic field model, and dashed lines to constant magnetic field strengths. This particular case is for a source density of 6$\times$10$^{-6}$ Mpc$^{-3}$ and $Z$=26.[]{data-label="fig:upperLimit"}](AlvesBatista_Rafael_fig2.eps){width="0.685\columnwidth"} The magnetic suppression due to magnetic horizon effects starts to become relevant for $E\lesssim$10$^{17}$ eV, for the most optimistic choice of parameters (heavy composition, large coherence length and low source density). The curves in figure \[fig:upperLimit\] reflect the behavior of the diffusion coefficient, shown in equation \[eq:syro2\] within square brackets, which is proportional to $l_c^{-1}$ for small values of the coherence length, and to $l_c^{2/3}$ for large $l_c$. Discussion and outlook ====================== We have parametrized the suppression of the cosmic ray flux at energies $\lesssim$Z$\times$10$^{18}$ eV. The method to obtain this parametrization can be adapted to any magnetic field distribution from cosmological simulations (for details see ref. [@alvesbatista2014]). Moreover, we have also derived upper limits for this suppression to occur, as a function of the coherence length. The results here described suggest that the suppression sets in at energies below $\sim$10$^{17}$ eV. This has profound implications for the interpretation of current experimental data. For instance, recently there has been several attempts [@aloisio2013; @taylor2014] to perform a combined spectrum-composition fit to data from the Pierre Auger Observatory [@auger2010; @auger2014]. These results indicate that the spectral indexes of the sources are hard ($\gamma \sim$1.0-1.6), which contradicts the current acceleration paradigm, in which UHECRs are accelerated to the highest energies through Fermi-like mechanisms ($\gamma \sim$2.0-2.2). In ref. [@mollerach2013] it was shown that the existence of a magnetic horizon around 10$^{18}$ eV can affect the results of these combined fits, softening the spectral index to $\gamma \sim$2. We have shown that if one considers a more realistic extragalactic magnetic field model, the contribution of the voids is dominant and since the field strengths in these regions are low, the suppression will also be small compared to the case of a simple Kolmogorov turbulent magnetic field. In this case, the combined spectrum-composition fits would again favor scenarios in which the sources have hard spectral index. Acknowledgements {#acknowledgements .unnumbered} ================ RAB acknowledges the support from the Forschungs- und Wissenschaftsstiftung Hamburg. GS was supported by the State of Hamburg, through the Collaborative Research program “Connecting Particles with the Cosmos” and by BMBF under grant 05A11GU1. [99]{} A. Neronov and I. Vovk, Science [328]{} 73 (2010) S. Mollerach and E. Roulet, JCAP [10]{} 13 (2013) V. Berezinsky and A. Z. Gazizov, Astrophys. J. [643]{} 8 (2013) F. Miniati, MNRAS [337]{} 199 (2002) K. Dolag [et al.]{}, JCAP [1]{} 9 (2005) S. Das [et al.]{}, Astrophys. J. [682]{} 29 (2008) J. Donnert [et al.]{}, MNRAS [392]{} 1008 (2009) R. Aloisio and V. Berezinsky, Astrophys. J. [612]{} 900 (2004) R. Alves Batista and G. Sigl. arXiv:1407.6150 ([to appear in JCAP]{}) R. Aloisio, V. Berezinsky and P. Blasi. JCAP [10]{} 020 (2014) A. Taylor, Astropart. Phys. [54]{} 48 (2014) Pierre Auger Collaboration, Phys. Lett. B [685]{} 239 (2010) Pierre Auger Collaboration. arXiv:1409.5083 ([to appear in Phys. Rev. D]{})
--- abstract: 'We apply the Statefinder hierarchy and the growth rate of matter perturbations to discriminate modified Chaplygin gas (MCG), generalized Chaplygin gas (GCG), superfluid Chaplygin gas (SCG), purely kinetic k-essence (PKK), and $\Lambda$CDM model. We plot the evolutional trajectories of these models in the statefinder plane and in the composite diagnostic plane. We find that GCG, MCG, SCG, PKK, and $\Lambda$CDM can be distinguished well from each other at the present epoch by using the composite diagnostic $\{\epsilon(z), S^{(1)}_{5}\}$. Using other combinations, such as $\{S^{(1)}_{3}, S^{(1)}_4\}$, $\{S^{(1)}_{3}, S_{5}\}$, $\{\epsilon(z), S^{(1)}_{3}\}$, and $\{\epsilon(z), S_4 \}$, some of these five dark energy models cannot be distinguished.' author: - Jun Li - 'Rongjia Yang[^1]' - Bohai Chen title: Discriminating dark energy models by using the statefinder hierarchy and the growth rate of matter perturbations --- Introduction ============ Observations of supernovae, large scale structures, and the cosmic microwave background have confirmed that the Universe is experiencing accelerated expansion. An additional component in the matter sector, dubbed as dark energy, is usually introduced to explain this phenomenon in the framework of general relativity. The simplest and most theoretically appealing candidate of dark energy is the vacuum energy with a constant equation of state (EoS) parameter $w=-1$ ($\Lambda$CDM). This scenario is consistent with most of the current astronomical observations but suffers from the cosmological constant problem [@weinberg; @Carroll2001] and age problem (which is still an open problem) [@Yang2010]. It is possible that other unknown matters may be responsible for the accelerated expansion of the Universe. Over the past years, numerous dark energy models have been proposed, such as quintessence, phantom, k-essence, tachyon, (generalized) Chaplygin gas ((G)CG), etc. As more and more dark energy models have been proposed, it is important to discriminate various dark energy models. A geometrical diagnostic, called statefinder, is introduced in [@V; @U]. It has been used to distinguish a number of dark energy models, such as $\Lambda$CDM, quintessence [@V; @Sirichai; @Linder], GCG [@Gorini; @Writambhara; @Li], DGP [@Grigoris; @Myrzakulov], Galileon-modified gravity [@Sami; @Myrzakulov], purely kinetic k-essence model (PKK) [@Gao2010], holographic dark energy [@Granda; @Zhang], Ricci Dark Energy model [@Feng], Agegraphic Dark Energy Model [@Wei], quintom dark energy model [@Wu], and spatial Ricci scalar dark energy (SRDE) [@Yang2012]. In addition to statefinder, another method, called $Om(z)$ diagnostic [@V2], has been proposed to distinguish dark energy models. $Om(z)$ is constructed from the Hubble parameter and provides a null test of the $\Lambda$CDM model, namely, if the value of $Om(z)$ is identical at different redshift, then dark energy is $\Lambda$ precisely. $Om(z)$ has been used to compare $\Lambda$CDM with some dark energy models, such as quintessence[@V2], phantom[@V2], PKK [@Gao2010], holographic dark energy [@Granda], and SRDE [@Yang2012]. A simple extension of the $Om(z)$ diagnostic, called $Om3(z)$, combining standard ruler information from BAO with standard candle information from type Ia supernovae, yield a powerful novel null diagnostic of the cosmological constant hypothesis [@Shafieloo2012]. It has been shown, however, that one cannot distinguish PKK from the $\Lambda$CDM model at $68.3\%$ confidence level by using $Om(z)$ and the statefinder $\{r, s\}$ [@Gao2010]. So it is natural to look for new methods to distinguish PKK (or other dark energy models) from the $\Lambda$CDM model. In [@Maryam], it has been shown that the Statefinder hierarchy combined with the growth rate of matter perturbations defines a composite null diagnostic which can distinguish DGP from the $\Lambda$CDM model. So it is natural to ask whether it can distinguish PKK from the $\Lambda$CDM model? In this paper, we use the statefinder hierarchy and the growth rate of matter perturbations proposed in [@Maryam] to distinguish GCG, modified Chaplygin gas (MCG), superfluid Chaplygin gas (SCG), and PKK from $\Lambda$CDM model. We find that GCG, MCG, SCG, and PKK can be discriminated from the $\Lambda$CDM model by using the statefinder hierarchy and the growth rate of matter perturbations. The rest of the paper is organized as follows. In Sec. II, we will briefly review the Statefinder hierarchy and the growth rate of matter perturbations. In Sec. III, we use the Statefinder hierarchy and the growth rate of matter perturbations to distinguish GCG, MCG, SCG, and PKK from the $\Lambda$CDM model. In the last section some conclusions and discussions are presented. The Statefinder hierarchy and the growth rate of matter perturbations ===================================================================== In this section, we first introduce the usual formalism of Statefinder hierarchy, and then briefly describe the growth rate of matter perturbations. The Statefinder hierarchy ------------------------- Like the Statefinder $\{r, s\}$ which is related to the third derivative of the expansion factor and is defined as [@V; @U] $$\begin{aligned} r\equiv \frac{\dddot{a}}{aH^3},~~~~s\equiv\frac{r-1}{3(q-1/2)},\end{aligned}$$ where $H=\dot{a}/a$ is the Hubble parameter and $q\equiv -\ddot{a}/(aH^2)$ is the deceleration parameter, the “Statefinder hierarchy" includes higher derivatives of the scale factor $d^{n}a/dt^{n}$, $n\geq2$. It has been demonstrate that all members of the Statefinder hierarchy can be expressed in terms of elementary functions of the deceleration parameter $q$ (equivalently the matter energy density parameter $\Omega_{\rm m}(z)=8\pi G\rho_{\rm m}/(3H^2)$) [@Maryam]. The scale factor $a(t)/a_{0}=(1+z)^{-1}$ can be Taylor expanded around the present time $t_{0}$ as follows: $$\begin{aligned} \frac{a(t)}{a_{0}}=1+\sum^{\infty}_{n=1}\frac{A_{n}(t_{0})}{n!}[H_{0}(t-t_{0})]^{n},\end{aligned}$$ where $$\begin{aligned} A_{n}\equiv\frac{a^{(n)}}{aH^{n}},\end{aligned}$$ with $a^{(n)}=d^na/dt^n$ and $n$ is a positive integer. Historically $-A_2$ has been called deceleration parameter $q$, $A_3$ is the Statefinder $r$ [@V; @U] or the jerk $j$ [@Visser], $A_4$ is the snap [@Visser; @Capozziello; @Dabrowski; @Dunajski] and $A_5$ the lerk $l$ [@Visser; @Capozziello; @Dabrowski; @Dunajski]. For $\Lambda$CDM in a spatially flat, homogeneous, and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) universe, we can easily get: $$\begin{aligned} &A_{2}=1-\frac{3}{2}\Omega_{\rm m},\\ &A_{3}=1,\\ &A_{4}=1-\frac{3^2}{2}\Omega_{\rm m},\\ &A_{5}=1+3\Omega_{\rm m}+\frac{3^3}{2}\Omega_{\rm m}^{2},....\end{aligned}$$ For $\Lambda$CDM we have $\Omega_{m}=\frac{2}{3}(1+q)$. The Statefinder hierarchy, $S_{\rm n}$, is defined as [@Maryam]: $$\begin{aligned} &S_{2}=A_{2}+\frac{3}{2}\Omega_{\rm m},\\ &S_{3}=A_{3},\\ &S_{4}=A_{4}+\frac{3^2}{2}\Omega_{\rm m},\\ &S_{5}=A_{5}-3\Omega_{\rm m}-\frac{3^3}{2}\Omega_{\rm m}^{2},....\end{aligned}$$ The Statefinder hierarchy is unchanged for $\Lambda$CDM during the expansion of the universe since [@Maryam] $$\begin{aligned} \label{Sn} S_{n}\|_{\Lambda \rm{CDM}}=1,~~~~{\rm{for}}~~~~ n=2,3,4,....\end{aligned}$$ These equations define a null diagnostic for $\Lambda$CDM, since for evolving dark energy models some of these equalities in (\[Sn\]) may be violated. When $n\geq3$, more than one way can be adopted to define a null diagnostic, see for example [@Maryam] one series of Statefinders can be defined as: $$\begin{aligned} &S^{(1)}_{3}=S_{3},\\ &S^{(1)}_{4}=A_{4}+3(1+q)\\ &S^{(1)}_{5}=A_{5}-2(4+3q)(1+q), ....\end{aligned}$$ This series of Statefinders also stays at unity for $\Lambda$CDM during the expansion of the universe [@Maryam] $$\begin{aligned} S^{(1)}_{n}\|_{\Lambda \rm{CDM}}=1.\end{aligned}$$ In other words, $\{S_{n}, S^{(1)}_{n}\}\|_{\Lambda \rm{CDM}}=\{1,1\}$. For other dark energy models, however, the pair $\{S_{n}, S^{(1)}_{n}\}$ are expected to give different results. A second member of the Statefinder hierarchy could be constructed from the Statefinder $S^{(1)}_{n}$ as follows [@Maryam]: $$\begin{aligned} \label{sn} S^{(2)}_{n}=\frac{S^{(1)}_{n}-1}{3(q-1/2)}.\end{aligned}$$ For $\Lambda$CDM, $S^{(1)}_{n}=1$ and $S^{(2)}_{n}=0$, namely $\{S_{n}, S^{(2)}_{n}\}\|_{\Lambda \rm{CDM}}=\{1,0\}$. The pair $\{S_{n}, S^{(2)}_{n}\}$ are also expected to give different results for other dark energy models. Consequently the Statefinder hierarchy $\{S_{n}, S^{(1)}_{n}\}$ or $\{S_{n}, S^{(2)}_{n}\}$ can be used as an excellent means of distinguish dynamical dark energy models from $\Lambda$CDM. In [@Maryam], CG, DPG, $w$CDM (models with a constant EoS), and $\Lambda$CDM have been distinguished by using $\{S_3^{(1)}, S_4^{(1)}\}$. ![Evolution trajectories in the $\Omega_{\rm m}-w$ plane for five dark energy models. The long-dash, solid, dash, dot, dash-dot line represents the evolution of $\Lambda$CDM, MCG, PKK, SCG, and GCG, respectively. \[omw\]](omw.eps){width="9cm"} The growth rate of matter perturbations --------------------------------------- In this subsection, we will examine the linearized density perturbation. The equation of the linear matter density contrast, $\delta=\delta\rho_{\rm m}/\rho_{\rm m}$, is given by $$\begin{aligned} \label{pert} \ddot{\delta}+2H\dot{\delta}=4\pi G\rho_{\rm m}\delta,\end{aligned}$$ where the dot denotes a derivative with respect to time $t$. This equation holds only if dark energy arises because of modifications to the right hand side of the Einstein equation and in the absence of anisotropic stress. In this case Eq. (\[pert\]) holds on scales much smaller than the Jeans length for dark energy perturbations. The matter perturbation $\delta$ contains essentially the same information as the expansion rate $H(z)$ [@LP; @Alam2009]. The evolution equations of background in a flat universe are: $$\begin{aligned} \label{frei} H^2=\frac{8\pi G}{3}(\rho_{\rm m}+\rho_{\rm DE}),\end{aligned}$$ $$\begin{aligned} 2\frac{\ddot{a}}{a}+H^2=-8\pi G w\rho_{\rm DE},\end{aligned}$$ where $w$ is the EoS of dark energy: $w\equiv p_{\rm DE}/\rho_{\rm DE}$. Using the definition of matter energy density parameter, we can derive $$\begin{aligned} \label{acc} \frac{\ddot{a}}{a}=-\frac{1}{2}H-\frac{3}{2}w(1-\Omega_{\rm m})H.\end{aligned}$$ By combining Eqs. (\[pert\]) and (\[acc\]) we rewrite the linear perturbation equation as [@LP] $$\begin{aligned} \label{per1} \frac{d^2\ln\delta}{d\ln a^2}+\left(\frac{d\ln\delta}{d\ln a}\right)^2+\left[\frac{1}{2}-\frac{3}{2}w(1-\Omega_{\rm m})\right]\frac{d\ln\delta}{d\ln a}=\frac{3}{2}\Omega_{\rm m}.\end{aligned}$$ The fractional growth parameter $\epsilon(z)$ used with statefinder can be defined as follows [@VADS]: $$\begin{aligned} \label{12}\epsilon(z)=\frac{f(z)}{f_{\Lambda \rm{CDM}}(z)},\end{aligned}$$ where$f(z)=d \ln\delta/d\ln a$ describes the growth rate of linear density perturbations [@LP] and can be parameterized as $$\begin{aligned} \label{13}f(z)\simeq\Omega^{\gamma}_{\rm m}(z),\end{aligned}$$ where $\gamma(z)$ is the growth index parameter. The expression of $\gamma(z)$ can be obtained as follows. Using Eqs. (\[frei\]) and conservation of the stress energy, we have $$\begin{aligned} \label{om} \frac{d\Omega_{\rm m}}{d\ln a}=3w(1-\Omega_{\rm m})\Omega_{\rm m}.\end{aligned}$$ Combining Eqs. (\[per1\]) and (\[om\]), we obtain the equation for $f$ in terms of $\Omega_{\rm m}$ [@LP] $$\begin{aligned} \label{per2} 3w\Omega_{\rm m}(1-\Omega_{\rm m})\frac{df}{d\Omega_{\rm m}}+\left[\frac{1}{2}-\frac{3}{2}w(1-\Omega_{\rm m})\right]f+f^2=\frac{3}{2}\Omega_{\rm m}.\end{aligned}$$ Taking into account Eq. (\[13\]), Eq. (\[per2\]) becomes [@LP] $$\begin{aligned} \begin{split} 3w(1-\Omega_{\rm m})\Omega_{\rm m}\ln\Omega_{\rm m}\frac{d\gamma}{d\Omega_{\rm m}}-3w(\gamma-\frac{1}{2})\Omega_{\rm m}+\Omega_{\rm m}^{\gamma} -\frac{3}{2}\Omega_{\rm m}^{1-\gamma}+3w\gamma-\frac{3}{2}w +\frac{1}{2}=0. \end{split}\end{aligned}$$ For slowly varying EoS which satisfies $\|dw/d\Omega_{\rm m}\|\ll (1-\Omega_{\rm m})^{-1}$, we can have [@LP] $$\begin{aligned} \label{14} \gamma(z)\simeq\frac{3}{5-\frac{w}{1-w}}+\frac{3}{125}\frac{(1-w)(1-\frac{3}{2}w)}{(1-\frac{6}{5}w)^{3}}(1-\Omega_{\rm m}).\end{aligned}$$ The above approximation works reasonably well for physical dark energy models with a constant or a slowly variational EoS. For $\Lambda$CDM, it is obvious that $$\begin{aligned} \epsilon(z)\|_{\Lambda \rm{CDM}}=1.\end{aligned}$$ So we can combine the fractional growth parameter $\epsilon(z)$ and the Statefinders to define a composite null diagnostic: $\{\epsilon(z), S_{n}\}$ or $\{\epsilon(z), S^{(m)}_{n}\}$ [@Maryam]. Using $\{\epsilon(z), S_3^{(1)}\}$, DPG, $w$CDM, and $\Lambda$CDM have been distinguished in [@Maryam]. ![The evolutions of the linear matter density contrast $\delta$ for $\Lambda$CDM, MCG, PKK, SCG, and GCG models. \[per\]](per.eps){width="9cm"} Dark energy models and discriminations ====================================== In this section, we will use the Statefinder hierarchy and the growth rate of matter perturbations to distinguish some dark energy models. $\Lambda$CDM, $w$CDM, CG, and DGP models have been distinguished by using these two methods in Ref. [@Maryam]. Here our discussions focus on the following models: $\Lambda$CDM, generalized Chaplygin gas (GCG), modified Chaplygin gas (MCG), superfluid Chaplygin gas (SCG), and purely kinetic k-essence model (PKK). For simplicity we neglect the contributions of radiation and spatial curvature. Dark energy models ------------------ In Chaplygin gas unifying dark matter and dark energy, such as CG, GCG, and MCG, the negligible sound speed may produce unphysical oscillations and an exponential blowup in the dark matter power spectrum [@Sandvik]. This problem can be solved by decomposing the energy density into dark matter and dark energy [@Bento]. So here we treat GCG and MCG only as dark energy, not as models unifying dark matter and dark energy. \(1) GCG as dark energy is characterized with $p=-A/\rho^\alpha$ with $A$ a positive constant and $0<\alpha\leq 1$. For $\alpha=1$, GCG reduces to the CG model. The action for GCG can be written as a generalized Born-Infeld form [@Bento2002; @Gorini2004; @Chimento] whose parameters has been constrained with observational data in [@Yang2008]. The EoS of GCG and the expansion rate of a universe containing the GCG dark energy and pressureless matter are given by, respectively $$\begin{aligned} \label{5}w=-\frac{A_{\rm s}}{A_{\rm s}+(1-A_{\rm s})(1+z)^{3(1+\alpha)}},\end{aligned}$$ $$\begin{aligned} \label{4}E(z)=\frac{H(z)}{H_0}=\bigg[\Omega_{\rm m0}(1+z)^3+(1-\Omega_{\rm m0}) \bigg(A_{\rm s}+(1-A_{\rm s})(1+z)^{3(1+\alpha)}\bigg)^{\frac{1}{1+\alpha}}\bigg]^{\frac{1}{2}}.\end{aligned}$$ where $A_{\rm s}=A/\rho^{1+\alpha}_{\rm{gcg0}}$ with $\rho_{\rm{gcg0}}$ being the present value of the energy density of GCG. GCG as dark energy has been successfully confronted with observational tests [@GSA]. The values of parameters we take in the following content are constrained from large-scale structure observation: $A_{\rm s}=0.764$, $\alpha=-1.436$, and $\Omega_{\rm m0}=0.2895$ [@GSA]. \(2) MCG is considered as dark energy with $P=B\rho-A/\rho^\alpha$, where $A$, $B$, and $\alpha$ are positive constants with $0<\alpha\leq 1$. For $B=0$, MCG reduces to the GCG model. For $B=0$ and $\alpha=1$, MCG reduces to the CG model. For $A=0$, MCG reduces to dark energy models with a constant EoS $w=B$. The EoS of MCG and the normalized Hubble parameter take the form, respectively $$\begin{aligned} \label{7}w=B-\frac{A_{\rm s}(1+B)}{A_{\rm s}+(1-A_{\rm s})(1+z)^{3(1+B)(1+\alpha)}},\end{aligned}$$ $$\begin{aligned} \label{6}E(z)=\bigg[\Omega_{\rm m0}(1+z)^3+(1-\Omega_{\rm m0})\bigg(A_{\rm s}+(1-A_{\rm s})(1+z)^{3(1+B)(1+\alpha)}\bigg)^{\frac{1}{1+\alpha}}\bigg]^{\frac{1}{2}}.\end{aligned}$$ where $A_{\rm s}=A/(1+B)\rho^{1+\alpha}_{\rm{mcg0}}$ with $\rho_{\rm{mcg0}}$ being the present value of the energy density of MCG. Compared with GCG the proposed MCG is suitable to describe the evolution of the universe over a wide range of epoch [@UAS]. The best-fit values of parameters we take are: $A_{\rm s}=0.769$, $B=0.008$, $\alpha=0.002$, and $\Omega_{\rm m0}=0.262$ [@BP]. \(3) SCG is a model unifying dark matter and dark energy [@VAP]. It involves a Bose-Einstein condensate as dark energy possessing the EoS of Chaplygin gas and an excited state acts as dark matter. Though the component of dark energy possesses the EoS of the Chaplygin gas, but the evolution of the universe provided by SCG is different from that in the two-component model with the Chaplygin gas and cold dark matter as well as from that in the GCG unifying dark energy and dark matter [@VAP; @Yang2013]. The EoS of SCG and the the normalized Hubble parameter are given by, respectively $$\begin{aligned} \label{9}w=-\frac{\big[(1+z)^{-3}+k_{0}\big]^2}{k^{2}+\big[(1+z)^{-3}+k_{0}\big]^{2}},\end{aligned}$$ $$\begin{aligned} \label{8}E(z)=(1+z)^3\bigg[\frac{k^2+\big(k_0+(1+z)^{-3}\big)^2}{k^2+(1+k_0)^2}\bigg]^{\frac{1}{2}}.\end{aligned}$$ where $k$ and $k_0$ are positive constants. $k$ gives an initial normalized total particle number density. $k_0$ is associated to the ratio of normal and condensate density evaluated at present. The best-fit values of parameters we take are: $k=0.173$ and $k_{0}=0.297$ [@RAV]. \(4) PKK is a class of k-essence with Lagrangian: $p(X)=-V_0\sqrt{1-2X}$, where $X\equiv \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$ is the the kinetic energy and $V_0$ is a constant potential [@Chim04; @yang08]. The expansion rate of a universe containing the PKK dark energy and pressureless matter takes the form [@yang09] $$\begin{aligned} \label{10}E(z)=\bigg[\Omega_{\rm m0}(1+z)^3+(1-\Omega_{\rm m0})f(z)\bigg]^{\frac{1}{2}},\end{aligned}$$ where $$\begin{aligned} f(z)=\exp\bigg[3\int_{0}^{z}\frac{1+w(z')}{1+z'}dz'\bigg],\end{aligned}$$ with EoS $$\begin{aligned} \label{11} w=-\frac{1}{1+2k_{0}^{2}(1+z)^{6}},\end{aligned}$$ where $k_0$ is a constant. For $k_0=0$, the above EoS reduces to $-1$; meaning the $\Lambda$CDM model is contained in PKK model as one special case. The behavior of the EoS (\[11\]), runs closely to zero in the early Universe, while runs closely to $-1$ in the future. The best-fit values of parameters we take are: $\Omega_{\rm m0}=0.36$ and $k_{0}=0.067$ [@yang09]. ![The evolutions of the fractional growth parameter $\epsilon(z)$ for $\Lambda$CDM, MCG, PKK, SCG, and GCG models. \[ez\]](ez.eps){width="9cm"} Discriminations with the Statefinder hierarchy and the growth rate of matter perturbations ------------------------------------------------------------------------------------------ Now we will use Statefinder hierarchy and the growth rate of matter perturbations to distinguish dark energy models described above. The deceleration parameter $q$, $A_3$, $A_4$, and $A_5$ can be rewritten as $$\begin{aligned} \label{q} q &&=(1+z)\frac{1}{E}\frac{dE}{dz}-1,\\ \label{A3} A_3 &&=(1+z)\frac{1}{E^2}\frac{d[E^2(1+q)]}{dz}-3q-2,\\ \label{A3} A_4 &&=\frac{-(1+z)}{E^3}\frac{d[E^3(2+3q+A_3)]}{dz}+4A_3+3q(q+4)+6,\\ \label{A5} A_5 &&=\frac{-(1+z)}{E^4}\frac{d[E^4(A_4-4A_3-3q(q+4)-6)]}{dz}+5A_4-10A_3(q+2)-30q(q+2)-24.\end{aligned}$$ In order to distinguish dark energy models by using Statefinder hierarchy and the growth rate of matter perturbations, we need another important parameter, the dimensionless matter density parameter. The dimensionless matter density parameter for GCG, MCG, and PKK is $\Omega_{\rm m}=\Omega_{\rm m0}(1+z)^3/E^2$ and for SCG is $\Omega_{\rm m}=k_0/[k_0+(1+z)^{-3}]$. ![The Statefinder $\{S^{(1)}_{3}, S^{(1)}_{4}\}$ are shown for $\Lambda$CDM, MCG, PKK, SCG, and GCG models. The arrows show time evolution and the present epoch in different models is shown as a dot. \[nS41S31\]](S41S31.eps){width="9cm"} Before using the fractional growth parameter $\epsilon(z)$ to distinguish dark energy models, we investigate whether the EoSs of MCG, PKK, SCG, and GCG vary slowly so that Eq. (\[14\]) can be used. In Fig. \[omw\], we plot the evolutional trajectories in $\Omega_{\rm m}-w$ plane for five dark energy models considered here. The EoS of PKK varies very slowly at the present epoch but varies fast when $\Omega_{\rm m}\geq 0.7$. The EoS of SCG varies more quickly than that of PKK at the present epoch but varies more slowly than that of PKK when $\Omega_{\rm m}\geq 0.7$. The EoS of GCG varies quickly when $\Omega_{\rm m}\leq 0.2$ and then varies more slowly when $\Omega_{\rm m}$ increases. The EoS of MCG varies more quickly than those of PKK, SCG, and GCG when $\Omega_{\rm m}$ increases from $0.2$ to $0.4$. For $z=0$, we find $\|dw/d\Omega_{\rm m}\|=0.079\ll 1/(1-\Omega_{\rm m})\simeq 1.56$ for PKK, $\|dw/d\Omega_{\rm m}\|=0.152< 1/(1-\Omega_{\rm m})\simeq 1.297$ for SCG, $\|dw/d\Omega_{\rm m}\|=0.495< 1/(1-\Omega_{\rm m})\simeq 1.41$ for GCG, and $\|dw/d\Omega_{\rm m}\|=1.22< 1/(1-\Omega_{\rm m})\simeq 1.36$ for MCG. Obviously the EoSs of MCG and GCG donot vary so slowly that the condition $\|dw/d\Omega_{\rm m}\|\ll 1/(1-\Omega_{\rm m})$ is not satisfied well by them, though the best-fit values of parameters of MCG and GCG we take are constrained from observational growth data by using the Eq. (\[14\]) (under the condition $\|dw/d\Omega_{\rm m}\|\ll 1/(1-\Omega_{\rm m})$)[@GSA; @BP]. ![The composite diagnostic $\{\epsilon(z), S_{3}^{(1)}\}$ is plotted for $\Lambda$CDM, MCG, PKK, SCG, and GCG models. The arrows show time evolution and the present epoch in different models is shown as a dot. \[S31e\]](S31e.eps){width="9cm"} In Fig. \[per\], we solve Eq. (\[per1\]) numerically for a relevant choice of parameters. The deviations of the linear matter density contrast among $\Lambda$CDM, SCG, and GCG are very small, even smaller than $0.1$. But the deviations of the linear matter density contrast among $\Lambda$CDM, PKK, and MCG are large, especially the growth of the linear matter density contrast $\delta$ for MCG increases very slowly. In Fig. \[ez\], we plot the evolutions of the fractional growth parameter $\epsilon(z)$ for $\Lambda$CDM, MCG, PKK, SCG, and GCG models. Obviously the evolution of $\epsilon(z)$ for MCG is significantly different from that of $\Lambda$CDM, but those of SCG and GCG run closely to that of $\Lambda$CDM for $z>2$, meaning that SCG, GCG and $\Lambda$CDM cannot be distinguished from each other at hight redshift. At the present epoch ($z=0$), the deviations of $\epsilon(z)$ among $\Lambda$CDM, MCG, PKK, SCG, and GCG can be discriminated. These results for $\epsilon(z)$ match the results obtained by solving Eq. (\[per1\]) numerically. ![The Statefinder $\{S_4, S^{(1)}_{4}\}$ are shown for $\Lambda$CDM, MCG, PKK, SCG, and GCG models. \[kS41S4\]](S41S4.eps){width="9cm"} In [@Maryam], it has been shown that CG, DPG, $w$CDM, and $\Lambda$CDM can be distinguished by using $\{S_3^{(1)}, S_4^{(1)}\}$ and DPG, $w$CDM, and $\Lambda$CDM can be distinguished by using $\{\epsilon(z), S_3^{(1)}\}$. We also use the Statefinder $\{S_3^{(1)}, S_4^{(1)}\}$ to discriminate GCG, MCG, SCG, PKK, and $\Lambda$CDM. We plot the evolutional trajectories of these models in the $\{S_3^{(1)}, S_4^{(1)}\}$ plane. It is obvious that the trajectory of GCG is very different from those of MCG, SCG, PKK, and $\Lambda$CDM, and it can be distinguished well from them at the present epoch. SCG and PKK may be distinguished from $\Lambda$CDM at a certain epoch but they cannot be distinguished from each other at the present epoch. MCG and $\Lambda$CDM also cannot be distinguished from each other at the present epoch by using $\{S_3^{(1)}, S_4^{(1)}\}$, as shown in Fig. \[nS41S31\]. ![The Statefinder $\{S_{3}^{(1)}, S_5\}$ are shown for $\Lambda$CDM, MCG, PKK, SCG, and GCG models. \[S31S5\]](S31S5.eps){width="9cm"} Using the composite diagnostic $\{\epsilon(z), S_3^{(1)}\}$ to distinguish these dark energy models, we find that at the present epoch GCG, MCG, SCG can be distinguished well from $\Lambda$CDM or PKK, but $\Lambda$CDM and PKK cannot be distinguished from each other, as shown in Fig. \[S31e\]. In the Statefinder $\{S_4, S^{(1)}_{4}\}$ plane, it is obvious that the evolutional trajectories of these models are different from each other. We find that at the present epoch MCG and GCG can be distinguished from $\Lambda$CDM, PKK, or SCG; but PKK and SCG cannot be distinguished from each other, as shown in Fig. \[kS41S4\]. Using Statefinder $\{S_{3}^{(1)}, S_5\}$, MCG and GCG can be distinguished from $\Lambda$CDM, PKK, or SCG; PKK and SCG may can be distinguished from $\Lambda$CDM but cannot be distinguished from each other at present epoch, as shown in Fig. \[S31S5\]. ![The composite diagnostic $\{\epsilon(z), S_4\}$ is plotted for $\Lambda$CDM, MCG, PKK, SCG, and GCG models. \[S4e\]](S4e.eps){width="9cm"} In the composite diagnostic $\{\epsilon(z), S_4\}$ plane, we find that evolutional trajectories of these models are very different from each other; MCG, SCG, and GCG can be distinguished well from PKK or $\Lambda$CDM, but PKK and $\Lambda$CDM cannot be distinguished well from each other at the present epoch, as shown in Fig. \[S4e\]. In Fig. \[S51\], we find the trajectories of MCG and SCG are similar but other models’ trajectories are different in the composite diagnostic $\{\epsilon(z), S_5^{(1)}\}$ plane, MCG, SCG, GCG, PKK, and $\Lambda$CDM can be distinguished well from each other at the present epoch. ![The composite diagnostic $\{\epsilon(z), S_{5}^{(1)}\}$ is plotted for $\Lambda$CDM, MCG, PKK, SCG, and GCG models. \[S51\]](S51e.eps){width="9cm"} We also use other pairs, such as $\{S_3^{(1)}, S_5^{(1)}\}$, $\{S_4, S_5^{(1)}\}$, $\{S_5, S_4^{(1)}\}$, $\{\epsilon(z), S_4^{(1)}\}$, etc., to distinguish these models and find they cannot be distinguished well, compared with the results obtained by using pairs presented above. Conclusions and discussions =========================== We have used the Statefinder hierarchy and the growth rate of matter perturbations to discriminate MCG, GCG, SCG, PKK, and $\Lambda$CDM dark energy models. The evolutional trajectories of these dark energy models in the statefinder hierarchy, such as $\{S^{(1)}_{3}, S^{(1)}_4\}$, $\{S_4, S^{(1)}_{4}\}$, and $\{S^{(1)}_{3}, S_{5}\}$, and in the composite diagnostic, such as $\{\epsilon(z), S^{(1)}_{3}\}$, $\{\epsilon(z), S_4 \}$, and $\{\epsilon(z), S^{(1)}_{5}\}$, have been plotted. We have found that GCG, MCG, SCG, PKK, and $\Lambda$CDM can only be distinguished well from each other at the present epoch by using the composite diagnostic $\{\epsilon(z), S^{(1)}_{5}\}$. Using the Statefinder $\{S^{(1)}_{3}, S^{(1)}_4\}$, MCG cannot be distinguished from $\Lambda$CDM, and SCG cannot be distinguished from PKK at the present epoch. With $\{\epsilon(z), S^{(1)}_{3}\}$ or $\{\epsilon(z), S_4 \}$, we cannot distinguish PKK from $\Lambda$CDM at the present epoch but the remaining dark energy models are easily distinguished as shown in Figs. \[S31e\] and \[S4e\]. 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--- abstract: 'Kernel methods offer the flexibility to learn complex relationships in modern, large data sets while enjoying strong theoretical guarantees on quality. Unfortunately, these methods typically require cubic running time in the data set size, a prohibitive cost in the large-data setting. Random feature maps (RFMs) and the method both consider low-rank approximations to the kernel matrix as a potential solution. But, in order to achieve desirable theoretical guarantees, the former may require a prohibitively large number of features ${{J_{+}}}$, and the latter may be prohibitively expensive for high-dimensional problems. We propose to combine the simplicity and generality of RFMs with a data-dependent feature selection scheme to achieve desirable theoretical approximation properties of with just $O(\log {{J_{+}}})$ features. Our key insight is to begin with a large set of random features, then reduce them to a small number of weighted features in a data-dependent, computationally efficient way, while preserving the statistical guarantees of using the original large set of features. We demonstrate the efficacy of our method with theory and experiments—including on a data set with over 50 million observations. In particular, we show that our method achieves small kernel matrix approximation error and better test set accuracy with provably fewer random features than state-of-the-art methods.' bibliography: - 'references.bib' - 'jhh-references.bib' ---
--- abstract: \| Recent results on solar neutrinos provide hints that the LMA MSW solution could be correct. We perform accurate calculations for potential ‘smoking-gun’ effects of the LMA solution in the SuperKamiokande solar neutrino experiment, including: (1) an almost constant reduction of the standard recoil electron energy spectrum (with a weak, $< 10$%, relative increase below $6.5$ MeV); (2) an integrated difference in day-night rates ($2$% to $14$%); (3) an approximately constant zenith-angle dependence of the nighttime event rate; (4) a new test for the difference in the shape of the equally-normalized day-night energy spectra ($\sim 1$%); and (5) annual variations of the signal due to the regeneration effect ($\sim 6$ times smaller than the integrated day-night effect). We also establish a relation between the integrated day-night asymmetry and the seasonal asymmetry due to LMA regeneration. As a cautionary example, we simulate the effect of an absolute energy calibration error on the shape (distortion) of the recoil energy spectrum. We compare LMA predictions with the available SuperKamiokande data and discuss the possibilities for distinguishing experimentally between LMA and vacuum oscillations. If LMA is correct, global solutions combining data from different types of measurements made by SuperKamiokande or by different solar neutrino experiments could reveal in the next few years a many $\sigma$ indication of neutrino oscillations. address: - \| School of Natural Sciences, Institute for Advanced Study\ Princeton, NJ 08540\ - \| University of Wisconsin–Madison, Department of Physics,\ Madison WI 53706 - 'International Center for Theoretical Physics, 34100 Trieste, Italy' author: - 'J. N. Bahcall' - 'P. I. Krastev' - 'A. Yu. Smirnov' title: 'Is Large Mixing Angle MSW the Solution of the Solar Neutrino Problems? ' --- psfig Introduction {#sec:introduction} ============ On April 1, 1996, solar neutrino research shifted from the pioneering phase of the chlorine (Homestake) experiment [@chlorine] and the exploratory phase of the Kamiokande [@kamiokande], SAGE [@SAGE], and GALLEX [@GALLEX] experiments, to the era of precision measurements inaugurated by the SuperKamiokande [@superkamiokande300; @neutrino98; @superkamiokandeDN; @superkamiokande504; @superkamiokande708; @suzuki99; @minikata] experiment. In this paper, we concentrate on the predictions for ‘smoking-gun’ evidence of new neutrino physics that can be observed in electron-neutrino scattering experiments like SuperKamiokande. There are four generic oscillation solutions (LMA, SMA, and LOW MSW solutions and vacuum oscillations, cf. Ref. [@bks98] for a recent discussion) of the solar neutrino problems that involve two neutrinos. These solutions are all consistent with the predictions of the standard solar model [@BP98] and the observed average event rates in the chlorine, Kamiokande, SuperKamiokande, GALLEX, and SAGE experiments. We limit ourselves to such globally consistent oscillation solutions and adopt the methodology used in our recent paper [@bks98]. Recent experimental results from SuperKamiokande [@neutrino98; @superkamiokandeDN; @superkamiokande504; @superkamiokande708] provide some encouragement for considering the LMA solution of the MSW effect [@msw]. No statistically significant distortion of the recoil electron energy spectrum has been discovered. No obvious change has been revealed in the slope of the ratio \[Measurement/(Standard Spectrum)\] as the SuperKamiokande measurements have been extended to progressively lower energies. Moreover, the excess of events at the highest energies has diminished. A larger flux has been observed at night than during the day, although the difference observed by SuperKamiokande [@neutrino98; @superkamiokandeDN; @superkamiokande504; @superkamiokande708] (which is about the size of what is expected from the LMA solution) is only $\sim 1.6\sigma$ after $708$ days of data collection [@inoue99]. In addition, no large enhancement of the rate has been found when the neutrino trajectories cross the earth’s core. Moreover, the excess that is found appears already when the sun is just below the horizon. These results are, as we shall see in detail in this paper, expected on the basis of the LMA MSW solution but are not expected on the basis of the SMA solution. There are also some non-solar indications that large mixing angles may be plausible. First, and most important, SuperKamiokande and other experiments on atmospheric neutrinos have provided strong evidence that large lepton mixing is occurring at least in one channel involving muon neutrinos [@fukuda98]. Moreover, the mass required for the LMA solution ($m \sim \sqrt(\Delta m^2) \sim 10^{-2} {\rm eV} $) is only one order of magnitude smaller than the mass scale suggested by atmospheric neutrino oscillations. All of the other solar neutrino solutions imply a smaller or much smaller neutrino mass scale for solar neutrinos. It is however well known that a weak mass hierarchy allows a rather natural explanation of large mixing. The oscillation parameters of the LMA solar neutrino solution can give an observable effect for atmospheric neutrino experiments. In particular, the LMA solution could explain [@smirnov99] the excess of $e-$like events in the atmospheric data. In this paper, we provide accurate calculations for smoking-gun predictions of the LMA solution, including a simple new test (Sec. \[subsec:spectrumsolar\]). In particular, we calculate and compare with SuperKamiokande observations the following quantities: (1) the distortion of the electron recoil energy spectrum; (2) the zenith angle dependence of the observed counting rate; (3) the day-night spectrum test; and (4) the seasonal dependence (beyond the effect of the earth’s eccentric orbit) of the counting rate. All of these quantities are zero, independent of solar physics, if the standard model for electroweak interactions is correct. Therefore, measurements of the spectral energy distortion and the zenith angle and seasonal dependences constitute sensitive smoking-gun tests of new physics. As a cautionary example, we also simulate the effect of an energy calibration error on the apparent distortion of the electron recoil energy spectrum. [llcccc]{} Source&&Cl&Ga&Cl&Ga\ &&(SNU)&(SNU)&(SNU)&(SNU)\ &&[Standard]{}&[Standard]{}&[LMA]{}&[LMA]{}\ pp&$5.94 $&0.0&69.6&0.0 (0.0,0.0)&39.8 (47.1,37.5)\ pep&$1.39 \times 10^{-2}$&0.2&2.8&0.1 (0.1,0.1)&1.2 (1.8,1.4)\ hep&$2.10 \times 10^{-7}$&0.0&0.0&0.0 (0.0,0.0)&0.0 (0.0,0.0)\ ${\rm ^7Be}$&$4.80 \times 10^{-1}$&1.15&34.4&0.46 (0.65,0.60)&15.7 (21.5,17.4)\ ${\rm ^8B}$&$5.15 \times 10^{-4}$&5.9&12.4&1.8 (1.6,2.3)&3.7 (3.4,4.8)\ ${\rm ^{13}N}$&$6.05 \times 10^{-2}$&0.1&3.7&0.05 (0.06,0.05)&1.8 (2.4,1.9)\ ${\rm ^{15}O}$&$5.32 \times 10^{-2}$&0.4&6.0&0.16 (0.25,0.20)&2.5 (3.8,3.1)\ &&&&&\ Total&&$7.7^{+1.2}_{-1.0}$&$129^{+8}_{-6}$&2.6 (2.65,3.2)&64.7 (80,66)\ to Figure \[fig:zoom\] shows the allowed region of LMA solution in the plane of $\sin^2 2\theta$ and $\Delta m^2$. We have indicated the region allowed by the measured total rates in solar neutrino experiments by the continuous contours in Fig. \[fig:zoom\]; the contours are shown at $90$%, $95$%, and $99$% C.L. The constraints provided by the total rates in the Homestake, GALLEX, SAGE, and SuperKamiokande experiments are, at present, the most robust limitations on the allowed parameter space. The only slight change in the available data on total rates since the publication of Ref. [@bks98] is a small reduction after 708 days in the total error for the SuperKamiokande experiment [@superkamiokande708]; the allowed region based upon the total rates in solar neutrino experiments is essentially unchanged from our earlier study (cf. the similar results in Fig. 2 of Ref. [@bks98] for the case that only the total rates are considered). The dashed line in Fig. \[fig:zoom\] represents the $99$% allowed contour including both the measured SuperKamiokande day-night effect and the total rates. The SuperKamiokande collaboration has performed preliminary global analyses of the available $708$ days of data [@superkamiokande708; @suzuki99; @inoue99] . Table \[tab:LMArates\] contrasts the predictions of the standard model and the LMA for the chlorine and the gallium experiments. The LMA results that are not in parentheses are for $\Delta m^2 = 2\times 10^{-5} {\rm eV^2}$ and $\sin^2 2\theta = 0.8$. These neutrino parameters are close to the best-fit LMA values when only the total rates of the chlorine, Kamiokande, SuperKamiokande, GALLEX, and SAGE experiments are considered (see Ref. [@bks98]). In parentheses, we show the predictions for two rather extreme LMA solutions. We have evaluated the predictions for the chlorine and gallium experiments for a representative set of oscillation parameters and all of the values lie within the boundaries defined by the three solutions shown in Table \[tab:LMArates\]. Therefore, the predictions given in the table show the expected range of capture rates consistent with the LMA solution and the existing experiments. The range of production rates predicted by the set of LMA oscillation solutions considered in this paper is: $^{37}{\rm Cl} = 2.9 \pm 0.3$ SNU and $^{71}{\rm Ga} = 71 \pm 9$ SNU. The prediction of the LMA solution shown in Table \[tab:LMArates\] for the SuperKamiokande rate due to the $^8$B neutrino flux [@superkamiokande300; @neutrino98; @superkamiokande504; @superkamiokande708] is :${\rm Rate\ ({\rm measured}, ^8B) = 0.474\ Rate\ ({\rm BP98}, ^8B)}$. This paper is organized as follows. We discuss the distortion of the recoil electron energy spectrum in Sec. \[sec:spectrum\], the day-night differences of the event rate in Sec. \[sec:zenithdependence\], and the seasonal dependences in Sec. \[sec:seasonal\]. We describe in Sec. \[sec:vacuumvsLMAseasonal\] possibilities for distinguishing between LMA and vacuum oscillations. We summarize and discuss our main results in Sec. \[sec:discussion\]. Spectrum distortion {#sec:spectrum} =================== Figures \[fig:energyerror\] and \[fig:specconsttheta\] illustrate the main results of this section. The reader who wants to get quickly the essence of the physical situation can skip directly to the discussion of Fig. \[fig:energyerror\] in Sec. \[subsec:energyerror\] and Fig. \[fig:specconsttheta\] in Sec. \[subsec:predictedvsspectra\]. The observation of a distortion in the recoil electron energy spectrum produced by solar neutrinos scattering off electrons would be a definitive signature of physics beyond the Standard Electroweak Model. Neutrinos which correspond to recoil electrons with energies between $5$ MeV and $13$ MeV, which are most easily observed by the SuperKamiokande solar neutrino experiment [@superkamiokande300; @neutrino98; @superkamiokande504; @superkamiokande708], are predicted by the Standard Solar Model to be essentially all (more than 99%) from $^8$B beta decay [@BP98]. The shape of the $^8$B neutrino energy spectrum can be determined accurately from laboratory measurements [@balisi] and the influence of astrophysical factors is less than $1$ part in $10^5$ [@bahcall91]. For energies relevant to solar neutrino studies, the cross-sections for neutrino scattering have been calculated accurately in the standard electroweak model, including radiative corrections [@sirlin]. Therefore, if nothing happens to solar neutrinos on the way from the region of production in the solar interior to a detector on earth, the recoil electron energy spectrum from $^8$B neutrinos can be calculated precisely. Any measured deviation of the recoil electron energy spectrum from the standard shape would indicate new physics. The opposite, however, is not true: the absence of a measured distortion does not mean the absence of new physics. We first indicate in Sec. \[subsec:convolution\] how the neutrino energy spectrum is convolved with the electron recoil energy spectrum and with the measurement characteristics of the detector. We then summarize briefly in Sec. \[subsec:summaryofdata\] the characteristics of the measured SuperKamiokande energy spectrum. In Sec. \[subsec:energyerror\], we answer the question: How does an error in the energy calibration affect the shape of the recoil energy spectrum? We discuss in Sec. \[subsec:predictedvsspectra\] the comparison of the predicted and the measured electron energy spectra, which are presented in Fig. \[fig:specconsttheta\]. The convolution {#subsec:convolution} --------------- We concentrate in this paper on analyzing the ratio, $R(E_i)$, of the number of observed events, $N^{\rm Obs}_i$ in a given energy bin, $E_i$, to the number, $N^{\rm SSM}_i$, expected from the SSM [@BP98], where $$R(E_i) = \frac{N^{\rm Obs}_i}{N^{\rm SSM}_i} ~. \label{eq:Rdefinition}$$ For specific applications, we use the observed event numbers in the form provided by SuperKamiokande [@superkamiokande504; @superkamiokande708]; there are $17$ energy bins of $0.5$ MeV width from $5.5$ MeV to $14$ MeV and an $18$th bin that includes all events from $14$ MeV to $20$ MeV. When more data are available, it will be important to divide the events above $14$ MeV into several different bins [@bk98]. The number of events in a given energy bin, $i$, can be expressed as $$N_i = \int_{E_i}^{E_i + 0.5 MeV} dE_e \int^{\infty}_0 dE_e' f(E_e, E_e') \int^{\infty}_{E_e'} dE_{\nu} F(E_{\nu})\times$$ $$\left[ P(E_{\nu}) \frac{d\sigma_e (E_{\nu}, E_e') }{d E_e'} + (1 - P(E_{\nu})) \frac{d\sigma_{\mu} (E_{\nu}, E_e') }{d E_e'} \right], \label{numberi}$$ where $F(E_{\nu})$ is the flux of $^8$B neutrinos per unit energy at the detector, $f(E_e, E_e')$ is the energy resolution function which we take from [@superkamiokande300; @neutrino98]. $P(E_{\nu})$ is the survival probability $\nu_e \rightarrow \nu_e$, $d\sigma_e/d E_e'$ and $d\sigma_{\mu}/d E_e'$ are the differential cross-sections of the $\nu_e e - $ and $\nu_{\mu} e - $ scattering. There are practically no $^8$B neutrinos with energies greater than $15.5$ MeV, so that for practical purposes the upper limit of the integral over $E_\nu$ in Eq. (\[numberi\]) can be taken to be $16$ MeV. Including the effects of the finite energy resolution, the upper limit over the measured electron recoil energy, $E_e'$, can be taken to be $20$ MeV. To use Eq. (\[numberi\]) to calculate the combined prediction for the standard electroweak model and the standard solar model, set $P \equiv 1 $. The second term in the brackets of Eq. (\[numberi\]) then disappears. This term is also absent if the electron neutrinos are converted to sterile neutrinos. How does the convolution shown in Eq. (\[numberi\]) affect energy-dependent distortions (or conversions) of the incoming electron neutrinos? The convolution spreads out over a relatively wide energy range any energy-dependent features. In particular, the scattering process produces electrons with energies from zero to almost equal to the incoming neutrino energy (for neutrino energies much greater than the electron mass). If $\nu_e$ are converted to $\nu_{\mu}$ or/and $\nu_{\tau}$, then there will be neutral current scattering \[the second term in boxed brackets of Eq. (\[numberi\])\] which, while less probable, will also reduce the apparent effect of the conversion. Since the solar $^8$B neutrino spectrum decreases rapidly beyond about $10$ MeV, a distortion of the recoil electron spectrum at some energy $E_{e}$ ($E_{e} > 10$ MeV) is determined by the distortion of the neutrino spectrum at somewhat lower energy $E_{\nu} < E_{e}$. The energy resolution function has a crucial affect in smearing out distortion. The $2\sigma$ width for the energy resolution is about $3$ MeV at $E_e = 10$ MeV in SuperKamiokande [@superkamiokande300; @neutrino98; @superkamiokande504]. Smaller features will be smeared out. Summary of the data {#subsec:summaryofdata} ------------------- After $708$ days of data taking with SuperKamiokande, no unequivocal distortion of the electron recoil energy spectrum has been found. The data show essentially the spectrum shape expected for no distortion for $E < 13$ MeV with some excess events at $E > 13$ MeV. The excess electrons observed at higher energies may reflect: (1) a statistical fluctuation; (2) the contribution of $hep$ neutrinos [@bk98; @escribano; @fiorentini]; (3) a larger-than-expected error in the absolute energy normalization; or (4) new physics. The systematic effects due to $hep$ neutrinos or to an error in the energy normalization can hide a distortion due to neutrino conversion or cause an artificial distortion. For example, a suppression due to oscillations of the high energy part of the electron spectrum could be compensated for (hidden by) the effect of [hep]{} neutrinos. All of the four explanations for the high-energy excess have been described previously in the literature. However, the distortion due to an error in the absolute energy scale of the electrons has only been mentioned as a possibility [@bl96]. Therefore, we discuss in Sec. \[subsec:energyerror\] the effect on the energy spectrum of an error in the absolute energy scale. Another possible systematic error, a non-Gaussian tail to the energy resolution, could also produce an apparent distortion at the highest energies. However, this effect has been estimated by the SuperKamiokande collaboration and found to be small [@liu; @superkamiokande708]. To resolve the ambiguity at high recoil electron energies, one can proceed in at least two different ways. (1) One can exclude the spectral data above $13$ MeV, since the distortion due to both $hep$ neutrinos and the calibration of the energy scale normalization occur mainly at high energies. A comparison of the results of the analysis using all the data with the inferences reached using only data below $13$ MeV provides an estimate of the possible influence of systematic uncertainties. However, this procedure does throw away some important data. (2) One can treat the solar $hep$ neutrino flux as a free parameter in analyzing the energy spectrum [@bk98; @escribano]. In general, this is the preferred method of analysis and the one which we follow. How does an energy calibration error affect the recoil energy spectrum? {#subsec:energyerror} ----------------------------------------------------------------------- An error in the absolute normalization of the energy of the recoil electrons, $\delta = E_{\rm true} -E_{\rm measured}$, would lead to an apparent energy dependence in $R$ that is particularly important for the LMA solution, since the theoretical prediction is that $R$ is essentially constant in the higher energy region accessible to SuperKamiokande. The ratio $R(E)$ can be written as $$R^{\rm Obs}(E) ~= ~ \frac{N^{\rm Obs}(E + \delta)}{N^{\rm SM}(E)} ~\approx~ R_0^{\rm true}(E) \left( 1 + \frac{1}{N^{\rm SM}(E)} \frac{dN^{\rm SM}(E)}{dE} \delta \right)~, \label{eq:absolute}$$ where $N^{\rm SM}(E)$ is the standard model spectrum. Since $N^{\rm SM}$, $dN^{\rm SM/}dE$ and, in general, $\delta $ all depend on energy, the last term in Eq. (\[eq:absolute\]) gives rise to an apparent distortion of the observable spectrum. Moreover, since $N^{\rm SM}(E)$ is a decreasing function of energy, $\delta \times (dN^{\rm SM}/dE)/N^{\rm SM}$ increases with energy for negative $\delta $ . Thus, for negative $\delta $, $R(E)$ is enhanced at high energies. The SuperKamiokande collaboration has estimated the error in the absolute energy calibration as being $0.8\%$ at 10 MeV [@superkamiokande504], i.e., $\delta = 80~~ {\rm keV}$, $1\sigma$. We have simulated the effect of an error in the absolute energy calibration by convolving a standard model recoil energy spectrum with the energy resolution function of SuperKamiokande and introducing a constant offset error $\delta$. We calculated the ratio $R(E)$ directly from the equality in Eq. (\[eq:absolute\]), without making the approximation involved in the Taylor-series expansion. Figure \[fig:energyerror\] compares the computed $R(E)$ for three different values of $\delta$ with the observed SuperKamiokande data [@superkamiokande708] for $708$ days. For visual convenience, each curve is normalized somewhat differently, but all are normalized within the range allowed by the SuperKamiokande measurement of the total rate. We see from Fig. \[fig:energyerror\] that a $2\sigma$ ($160$ keV) energy calibration error can produce a significant pseudo-slope. More exotic effects can be produced if the one assumes that the energy error, $\delta$, is itself a function of energy. We conclude that future analyses of the distortion of the energy spectrum should include $\delta$ as one of the parameters that is allowed to vary in determining from the measurements the best-fit and the uncertainty in the energy distortion. Predicted versus measured spectra {#subsec:predictedvsspectra} --------------------------------- Figure \[fig:specconsttheta\] compares the predicted versus the measured ratio $R(E)$ \[see Eq. (\[eq:Rdefinition\])\] of the electron recoil energy spectra for a representative range of $\Delta m^2$ that spans the domain permitted by the global LMA solutions. All of the solutions shown have $\sin^2 2\theta = 0.8$ and are consistent with the average event rates in the chlorine, Kamiokande, SAGE, GALLEX, and SuperKamiokande experiments. Also, all of the histograms of neutrino solutions are normalized so as to yield the same value of the ratio $R(E)$ at $E = 10$ MeV. The measured values are taken from [@superkamiokande708]. The most conspicuous feature of Fig. \[fig:specconsttheta\] is the flatness of the ratio $R(E)$ that is predicted by the LMA solution with no enhancement of the $hep$ flux. The survival probability, $P$, without earth regeneration is $P(E) \approx \sin^2 \theta$ in the observable high energy part of the $^8$B neutrino spectrum. Regeneration in the earth can lead to a weak increase of the survival probability with energy. Excellent fits to the recoil energy spectra can be obtained by increasing the production cross section for the $hep$ flux by a factor of $10$ to $40$ over its nominal (see Ref. [@adelberger98]) best value. This result is illustrated in Fig. \[fig:specconsttheta\]. Results similar to Fig. \[fig:specconsttheta\] are obtained if $\sin^2 2\theta$ is allowed to vary over a representative range of the allowed global LMA solutions that are consistent with the average measured event rates. We find numerically that the dependence of the shape of the predicted spectrum upon $\sin^2 2\theta$ is very weak. Recent work by the SuperKamiokande collaboration has placed an experimental upper limit on the $hep$ neutrino flux at earth of at most $20$ times the nominal standard value [@neutrino98; @superkamiokande708; @suzuki99]. The fits shown in Fig. \[fig:specconsttheta\] demonstrate that we require a $hep$ flux at the sun of $10$ to $40$ times the standard value. However, LMA neutrino oscillations reduce the rate at the earth so that the observed rate is about a factor of two less than would correspond to a flux of purely $\nu_e$ $hep$ neutrinos. So the fits shown for variable $hep$ flux in Fig. \[fig:specconsttheta\] and are consistent with the SuperKamiokande upper limit on the measured $hep$ flux. What are the chances that SuperKamiokande can obtain smoking-gun evidence for a distortion of the energy spectrum if the LMA solution is correct? The lack of knowledge of the $hep$ production cross section prevents fundamental conclusions based upon the suggestive higher energy ($> 13$ MeV) events. The data reported by SuperKamiokande (at the time this paper is being written) above $14$ MeV are in a single bin ($14$ MeV to $20$ MeV). As emphasized in Ref. [@bk98], it is possible that measurements in smaller energy bins above $14$ MeV could reduce the uncertainty in the $hep$ flux and test the consistency of the LMA plus enhanced $hep$ predictions (for preliminary results see Ref. [@superkamiokande708]). For larger allowed values of $\Delta m^2$, the distortion curve turns up at low energies (cf. Fig. \[fig:specconsttheta\]). This upturn is due to the fact that for larger $\Delta m^2$ the low energy part of the $^8$B neutrino spectrum ($E \sim 5 - 6 $ MeV) is on the adiabatic edge of the suppression pit. For the smallest allowed $\Delta m^2$, LMA predicts a weak positive slope for $R(E)$ that is caused by the Earth regeneration effect. The predicted distortion at low energies, between, e.g., $5$ MeV and $6.5$ MeV, is small (typically $\sim$ a few percent up to as large as $10$ percent, cf. Fig. \[fig:specconsttheta\]) and would require approximately $10$ years of SuperKamiokande data in order to show up in a clear way. Earth regeneration effects {#sec:zenithdependence} ========================== If MSW conversions occur, the sun is predicted to be brighter at night in neutrinos than it is during the day [@earthreg] at those energies at which the survival probability of $\nu_e$ is less than one-half. This phenomenon, if observed, would be a dramatic smoking-gun indication of new physics independent of solar models[^1]. When the sun is below the horizon, neutrinos must traverse part of the earth in order to reach the detector. By interacting with terrestrial electrons, the more difficult to detect $\nu_\mu$ or $\nu_\tau$ can be reconverted to the more easily detectable $\nu_e$. The opposite process can also occur. The physics of this ‘day-night’ effect is well understood [@earthreg]. We adopt the methodology of Ref. [@brighter]. The SuperKamiokande collaboration has placed constraints on neutrino oscillation parameters from the measurement of the zenith angle dependence in $504$ days of data [@superkamiokande504]. We first discuss in Sec. \[subsec:integrated\] the day-night asymmetry averaged over all zenith angles and integrated over all energies and seasons of the year. We then analyze in Sec. \[subsec:zenith\] the dependence of the total rate upon the solar zenith angle. We then describe in Sec. \[subsec:spectrumsolar\] a new test, which we call the ‘day-night’ spectrum test. The Integrated Day-Night Effect {#subsec:integrated} ------------------------------- The average day-night asymmetry measured by the SuperKamiokande collaboration is [@inoue99] $$A_{\rm n-d} ~=~ 2\left[{\rm {\frac{night - day}{night + day} } } \right] ~=~ 0.060 \pm 0.036({\rm stat}) \pm 0.008 ({\rm syst}). \label{eq:daynightasym}$$ Here ‘night’ (‘day’) is the nighttime (daytime) signal averaged over energies above $6.5$ MeV and averaged over all zenith angles and seasons of the year.[^2] Table \[tab:daynight\] shows the calculated values for the day-night asymmetry for a range of different LMA solutions. For solutions within the allowed LMA domain, we find $A_{\rm n-d} = 0.02$ to $0.14$ . The dependence of the asymmetry on $\Delta m^2$ can be approximated with the LMA solution space by $$A_{\rm n-d} ~=~ 0.2 \left[\frac{10^{-5} {\rm eV}^2}{\Delta m^2} \right] \label{eq:daynightapprox}$$ for $\sin^2 2\theta = 0.8$ and for the range of $\Delta m^2$ shown in Table \[tab:daynight\]. This approximation depends only weakly on mixing angle. The $1 \sigma$ interval of $A_{\rm n-d}$ that follows from Eq. (\[eq:daynightasym\]) leads to an allowed range of $$\Delta m^2 = (2 - 8) \times 10^{-5} {\rm eV}^2, ~~~~~ 1 \sigma~. \label{eq:daynightmass}$$ Figure \[fig:zoom\] shows the approximately horizontal lines of equal Night-Day asymmetry in a $\Delta m^2$ - $\sin^2 2\theta$ plot together with region in parameter space (dashed contour) that is allowed by a combined fit of the total rates and the Night-Day asymmetry. The best-fit oscillation parameters for the combined fit are $$\label{bestall} \Delta m^2 = 2.7\times 10^{-5} {\rm eV}^2 ; ~\sin^22\theta = 0.76 .$$ The Night-Day asymmetry is $8$% for these best fit parameters. The Zenith Angle Dependence {#subsec:zenith} --------------------------- Figures \[fig:zenith\] and \[fig:zenithconstm\] compare the predicted and the observed (after $708$ days) dependence of the SuperKamiokande [@superkamiokande708] event rate upon the zenith angle of the sun, $\Theta$. The results are averaged over one year. In order to make the figures look most similar to figures published by the SuperKamiokande collaboration, we have constructed the plot using the nadir angle $\Theta_N = \pi - \Theta$. The predicted and the observed rates are averaged over all energies $> 6.5$ MeV. Three conclusions can be drawn from the results shown in Fig. \[fig:zenith\] and Fig. \[fig:zenithconstm\]. (1) The experimental error bars must be reduced by about a factor of two (total observation time of order eight years) before one can make a severe test of the average zenith-angle distribution predicted by the LMA solution. (2) Nevertheless, the available experimental results provide a hint of an effect: all five of the nighttime rates are larger than the average rate during daytime. (3) The zenith-angle dependence predicted by the LMA solution is relatively flat, i.e., the flux is approximately the same for all zenith-angles. The predicted LMA enhancement begins in the first nighttime bin and the enhancement is not significantly increased when the neutrinos pass through the earth’s core ($\cos(\Theta) > 0.8$) and can even decrease for some parameter choices. As $\Delta m^2$ decreases, the oscillation length increases and the effect of averaging over oscillation phases becomes less effective with the result that some structure appears in the calculated zenith angle dependence. For $\Delta m^2 < 10^{-5} \, {\rm eV^2}$, the departure from a flat zenith angle dependence becomes relatively large. Certain solutions of the SMA, those with $\sin^2 2\theta > 0.007$, predict a strong enhancement in the core that is not observed and therefore these SMA solutions are disfavored [@superkamiokande708]. Figures \[fig:zenith\] and \[fig:zenithconstm\] show that for $\Delta m^2 = (3-4)\times 10^{-5}$ eV$^2$ and a wide range of $\sin^2 2\theta$ the LMA solution provides an excellent fit to the zenith angle dependence. The theoretical uncertainties [@brighter] (due to the density distribution and composition of the earth and the predicted fluxes of the standard solar model) are at the level of $0.2$%, about an order of magnitude less than the effect that is hinted at by Fig. \[fig:zenith\] and Fig. \[fig:zenithconstm\]. ------------------------ ------------------------ ------------------- ------------------------ $\Delta m^2$ $A_{\rm n-d}$ $A_{\rm n-d}$ $A_{\rm n-d}$ $(10^{-5}~{\rm eV^2})$ $\sin^2 2\theta = 0.9$ $\sin^2 2\theta = $\sin^2 2\theta = 0.7$ 0.8$ 8 0.019 0.018 0.017 4 0.049 0.050 0.049 3 0.069 0.073 0.071 2 0.100 0.107 0.107 1.6 0.126 0.135 0.136 ------------------------ ------------------------ ------------------- ------------------------ : Day-Night Asymmetries\[tab:daynight\] The Day-Night Spectrum Test {#subsec:spectrumsolar} --------------------------- The spectral dependence of the day-night effect provides additional and independent information about neutrino properties that is not contained in the difference between the total day and the night event rates. The new physical information that is described by the difference between the day and the night energy spectra is the extent to which the different energies influence the total rates. Figure \[fig:daynightspect\] compares the calculated electron recoil energy spectra for the day and the night for a range of values of $\Delta m^2$ and $\sin^2 2\theta$. Again, we have plotted on the vertical axis of Fig. \[fig:daynightspect\] the ratio, $R(E)$ \[see Eq. (\[eq:Rdefinition\])\], of the measured recoil energy spectrum to the spectrum expected if there is no distortion [^3]. The night spectra lie above the day spectra for all relevant energies. Moreover, the difference of the nighttime and the daytime rates increases with energy, reflecting the fact that for the LMA solutions the regeneration effect increases with $\Delta m^2$. The systematic difference between the day and the night spectra is most easily isolated when the difference due to the average rates is removed from both the day and the night spectra. Figure \[fig:diffdaynightspect\] compares the daytime and the nighttime spectra when both spectra are normalized to the same total rate (the observed SuperKamiokande rate). This equal-normalization removes the difference that is normally referred to as the ‘day-night effect’. We refer to the equal-normalization procedure as the ‘day-night spectrum test’. Figure \[fig:diffdaynightspect\] shows that for energies below (above) $10$ MeV, the predicted differential daytime spectrum is higher (lower) than the predicted differential nighttime spectrum. The two normalized spectra become equal in rates at about $10$ MeV. LMA predicts that the nighttime spectrum contains relatively more high energy electrons than the daytime spectrum. Figure \[fig:diffdaynightspect\] shows that, when both spectra are normalized to have the same total rates, the average daytime spectrum between $5$ MeV and $10$ MeV is, for the parameters chosen, about $\sim 1.5 $% greater than the average nighttime spectrum. The oscillation parameters used to construct Fig. \[fig:diffdaynightspect\] are $\Delta m^2 = 2\times 10^{-5} {\rm eV^2}$ and $\sin^2 2\theta = 0.8$. We have plotted figures similar to Fig. \[fig:diffdaynightspect\] for a number of LMA solutions. For a fixed $\Delta m^2$, the equally normalized spectra are all very similar to each other. However, the amplitude of the difference between the night and the day spectra, about $3$% at $5$ MeV for the example shown in Fig. \[fig:diffdaynightspect\], decreases to about $2$% for $\Delta m^2 = 4\times 10^{-5} {\rm eV^2}$ and is only about $0.5$% for $\Delta m^2 = 8\times 10^{-5} {\rm eV^2}$ . Does the day-night spectrum test provide information independent of the information obtained from the integrated day-night rate effect or the more general zenith-angle effect? Yes, one can imagine that the day-night integrated effect (cf. Sec. \[subsec:integrated\]) has been measured to be of order $6$% in agreement with LMA predictions, but that when the day-night spectrum test is applied the appropriately normalized day rate at energies less than $10$ MeV lies below the similarly normalized nighttime rate. This later result would be inconsistent with the predictions of the LMA solution and therefore would provide information not available by just measuring the integrated difference of day and of night rates. The easiest way to apply the day-night spectrum test is to normalize the day and night spectra to the same total number of events and then compare the number of day events below $10$ MeV with the number of night events below $10$ MeV. In principle, one could divide the data into a number of different bins and test for the similarity to the predicted shape shown in Fig. \[fig:diffdaynightspect\], but the Poisson fluctuations would dominate if the data were divided very finely. The change in slope with energy, illustrated in Fig. \[fig:diffdaynightspect\], between equally normalized day and night recoil energy spectra may provide a new test of the LMA solution. It will also be useful to calculate the predicted change in slope for other solar neutrino solutions. Seasonal dependences {#sec:seasonal} ==================== If the LMA solution is correct, seasonal dependences occur as a result of the same physics that gives rise to the zenith-angle dependence of the event rates. At nighttime, oscillations in matter can reconvert $\nu_\mu$ or $\nu_\tau$ to $\nu_e$ or matter interactions can also cause the inverse process. The seasonal dependence arises primarily because the night is longer in the winter than in the summer. Early discussions of the seasonal effect with applications to radiochemical detectors is given in Refs. [@cribier]. A recent discussion of seasonal effects is presented in Ref. [@Valle][^4]. Predicted versus observed seasonal dependence {#subsec:predictedseasonal} --------------------------------------------- Figure \[fig:seasonalno\] shows the predicted LMA variation of the total event rates as a function of the season of the year. In constructing Fig. \[fig:seasonalno\], we corrected the counts for the effect of the eccentricity of the earth’s orbit. We show in Fig. \[fig:seasonalno\] the predictions for a threshold of $6.5$ MeV; Fig. \[fig:seasonal11pt5\] shows similar results for a threshold of $11.5$ MeV. In both cases, the seasonal effects are small, $\sim 1$% to $2$%, although the higher-energy events exhibit a somewhat larger ($30$% or $40$% larger) variation. The characteristic errors bars for SuperKamiokande measurements three years after beginning the operation are larger, typically $\sim 4$% (cf. Fig. \[fig:seasonal\]), than the predicted LMA seasonal variations. How do the SuperKamiokande data compare with the observations? For a threshold energy of $6.5$ MeV, the data corrected for the earth’s eccentricity are not yet available. Therefore, we compare in Fig. \[fig:seasonal\] the observed [@superkamiokande708] and the predicted seasonal plus eccentricity dependence. The eccentricity dependence is larger (for the allowed range of parameters) than the predicted LMA seasonal dependence; hence, the predicted variations in Fig. \[fig:seasonal\] are larger than in Fig. \[fig:seasonalno\]. The regeneration effect enhances the eccentricity effect in the northern hemisphere. In the southern hemisphere, the eccentricity and the regeneration-seasonal effects have opposite signs. This difference is in principle detectable; if the seasonal asymmetry is $7$% in the northern hemisphere then it would be about $3$% in the southern hemisphere. Unfortunately, the existing statistical error bars are too large to show evidence of either the eccentricity or the LMA seasonal dependence. If the LMA solution is the correct description of solar neutrino propagation, it appears likely from Fig. \[fig:seasonal\] that of order $10$ years of SuperKamiokande data will be required in order to see a highly significant seasonal effect due to LMA neutrino mixing. The relationship between the summer-winter and the day-night effect {#subsec:swvsdn} ------------------------------------------------------------------- Since the physics underlying the day-night and the seasonal dependences is the same, , non-resonant matter mediated neutrino oscillations, there must be a relationship between the two effects. We outline here a brief derivation of a formula that connects the size of the day-night asymmetry defined in Eq. (\[eq:daynightasym\]) and the winter-summer asymmetry, which we define as $$A_{\rm W ~-~ S} ~\equiv~ 2\left({ {\rm Winter - Summer} \over {\rm Winter + Summer} }\right) . \label{eq:defwintersummer}$$ Here ‘Winter’ (‘Summer’) is the signal averaged over the period from November 15 to February 15 (May 15 to August 15). Neutrinos reaching the earth from the sun will be in an incoherent mixture of mass eigenstates [@dighe99; @stodolsky98]. When the sun is below the horizon, the presence of electrons in the earth will cause some transitions to occur between different mass states. By solving the problem of matter induced neutrino transitions in a constant density medium, one can see that the characteristic oscillation length in the earth is always less than or equal to the vacuum oscillation length divided by $\sin 2\theta$. The transition probability is proportional to $\sin^2(\phi)$, where $$\phi ~\geq~ {{\pi R \sin 2\theta} \over {L_V} } = 17 {\sin 2\theta} \left({R \over R_{\rm earth}}\right) \left({{10 {\rm MeV} } \over {E} }\right) \left({ {\Delta m^2} \over {2\times 10^{-5} {\rm eV^2} } } \right), \label{eq:phidefn}$$ where $R$ is the distance traversed in the earth. For any of the nighttime bins discussed in Sec. \[sec:zenithdependence\], the phase of the oscillation is large so that even relatively small changes in distance and energy will lead to fast oscillatory behavior that causes the survival probability to average to a constant in all of the bins (cf. Eq. \[eq:phidefn\]). This is the reason why the zenith-angle enhancement shown in Fig. \[fig:zenith\] is approximately a constant, independent of zenith-angle. The event during the night and the event rate during the day are each approximately constant and may be represented as $R_N$ and $R_D$, respectively. Let the average duration of night be $t_S$ hours during the summer and $t_W$ hours during the winter. Then the summer signal is proportional to $\left[ R_N t_S + R_D(24 - t_S) \right]$. There is a similar expression for the winter signal: $\left[ R_N t_W + R_D(24 - t_W) \right]$. Then simple algebra shows that the seasonal asymmetry is $$A_{\rm W - S}~=~A_{\rm n -d} \left( {{t_W - t_S} \over 24 } \right). \label{eq:asymmetryidentity}$$ Thus the seasonal variations are proportional to the night-day asymmetry. The larger the day-night effect, the larger is the LMA predicted seasonal variations due to regeneration. For the location of SuperKamiokande, $t_W = 13.9$ hr and $t_S = 10.1$ hr, and the parenthetical expression in Eq. (\[eq:asymmetryidentity\]) is about $1/6$. Equation (\[eq:asymmetryidentity\]) gives the approximate relation between the seasonal and the day-night asymmetries and makes clear why the seasonal dependence is about $6$ times smaller than the already small day-night effect. For the central value of the measured range of day-night asymmetries [@inoue99] (cf. Eq. \[eq:daynightasym\]), we find from Eq (\[eq:asymmetryidentity\]) $A_{\rm W - S} \sim 1$ %, in agreement with our explicit calculations. Vacuum versus LMA oscillations {#sec:vacuumvsLMAseasonal} ============================== Table \[tab:confrontation\] summarizes the most striking differences in the predictions of the LMA and the vacuum oscillation solutions of the solar neutrino problems. We now discuss some aspects of these differences in more detail. --------------------------------- ------------------------ -------------------------------------- LMA Vacuum Day-night integrated effect small but non-zero zero Zenith-angle dependence of rate small but non-zero zero Day-night spectrum test small but non-zero zero Spectrum distortion flat relative spectrum can explain upturn at large energies Seasonal effect eccentricity dominates can be large and energy dependent --------------------------------- ------------------------ -------------------------------------- : LMA versus vacuum oscillations\[tab:confrontation\]. Section \[sec:zenithdependence\] contains a quantitative discussion of the day-night integrated effect, the zenith angle dependence of the rate, and the day-night spectrum test. Seasonal effects are discussed in Sec. \[sec:seasonal\] and spectrum distortion is discussed in Sec. \[sec:spectrum\]. The sharpest distinctions between vacuum and LMA predictions are expressed in the day-night differences. The LMA predicts non-zero day-night differences for all three of the tests discussed in Sec. \[sec:zenithdependence\] and listed as the first three phenomena in Table \[tab:confrontation\]. Vacuum oscillations predict zero for all these day-night phenomena. The spectrum distortion is predicted to be small for the LMA solution (cf. Fig. \[fig:specconsttheta\]). The ratio of the observed to the standard spectrum is essentially constant for energies above $6.5$ MeV, although a modest upturn can occur at lower energies. Vacuum oscillations allow a more varied set of spectral distortions and can accommodate, without enhanced $hep$ production, the possibly indicated upturn in the relative spectrum beyond $13$ MeV. One can use seasonal variations to distinguish between vacuum oscillations and LMA oscillations. For an early discussion of this possibility, see Ref. [@krastev95]. The LMA solution predicts that the seasonal effects are smaller than the geometrical effect arising from the eccentricity of the earth’s orbit. Moreover, there is only a weak enhancement of the LMA seasonal effect with increasing energy threshold. On the other hand, for vacuum oscillations the seasonal effects can be comparable with the geometrical effect and there can be a strong enhancement (or an almost complete suppression, even a reversal of the sign) of the seasonal effect as the threshold energy is increased (see first paper in Ref. [@krastev95]). For the LMA solution, regeneration occurs but only during the night. Therefore, the LMA solution predicts that there is no seasonal dependence of the daytime signal beyond that expected from the eccentricity of the earth’s orbit. The vacuum oscillation solutions predict that the day and the night seasonal dependences will be the same. Therefore, in principle one could distinguish between vacuum oscillations and LMA oscillations by comparing the seasonal dependence observed during daytime with the seasonal dependence observed at night. The LMA solution predicts for experiments measuring the low energy $^7$Be neutrino line (BOREXINO, KamLAND, LENS) that there will not be a significant contribution to the seasonal effect beyond that expected from the eccentricity of the earth’s orbit. This is because the day-night difference is very small at low energies in the LMA solution [@brighter; @bks98]. For vacuum oscillations, there can be significant seasonal dependences of the $^7$Be line in addition to the purely geometrical effect . SuperKamiokande data can be used to test for an enhancement of the seasonal variations as the energy threshold for selecting events is increased. Such an enhancement might be produced by vacuum oscillations. Vacuum oscillations with $\Delta m^2 \sim 4 \times 10^{-10}~{\rm eV^2}$, which give the best description of the recoil energy spectrum, provide a distinct pattern for the enhancement: the effect of eccentricity is larger than the effect of oscillations for a threshold energy of $6.5$ MeV, while the effect of oscillations is comparable with the eccentricity effect for a threshold of $11.5$ MeV [@krastev95]. Comparing Fig. \[fig:seasonal11pt5\] and Fig. \[fig:seasonal\], we see that the enhancement with energy threshold predicted by LMA is weaker than for vacuum oscillations. For LMA, the increase in the threshold only enhances the seasonal variations by of order $30$% to $40$% of an already small effect. This insensitivity to changes in energy threshold is in agreement with our calculations of the day and night recoil energy spectra (see Sec. \[subsec:swvsdn\]). Practically speaking, LMA predicts for SuperKamiokande that there will be no measurable change in the seasonal effect with increasing energy threshold. Summary and discussion {#sec:discussion} ====================== The current situation {#subsec:current} --------------------- The Large Mixing Angle solution is consistent with all the available data from solar neutrino experiments. Figure \[fig:zoom\] shows the allowed region of the LMA parameters in the approximation of two oscillating neutrinos. We have investigated in this paper the LMA predictions for SuperKamiokande of the distortion of the electron recoil energy spectrum, the integrated day-night effect, the zenith-angle dependence of the event rate, and the seasonal dependences. We have also evaluated and discussed an independent test, which we call the day-night spectrum test. Finally, we have analyzed the possibilities for distinguishing between LMA and vacuum oscillations (see Table \[tab:confrontation\] and the discussion in Sec. \[sec:vacuumvsLMAseasonal\]). We previously showed that the LMA solution is globally consistent with the measured rates in the chlorine, Kamiokande, SAGE, GALLEX, and SuperKamiokande experiments [@bks98]. The electron recoil energy spectrum predicted by the LMA solution is practically undistorted at energies above $7$ MeV, i.e., the $\nu_e$ survival probability is essentially independent of energy at high energies. However, the recoil energy spectrum that is measured by SuperKamiokande suggests an increased rate, relative to the standard recoil energy spectrum, for energies above $13$ MeV [@superkamiokande300; @neutrino98; @superkamiokande504; @superkamiokande708]. We have discussed in Sec. \[subsec:summaryofdata\] two experimental possibilities for explaining the upturn at large energies of the recoil energy spectrum: (1) a statistical fluctuation that will go away as the data base is increased; and (2) an error in the absolute energy calibration. Figure \[fig:energyerror\] shows that a best-fit to the upturn at large energies would require a several sigma error in the absolute energy calibration. Both of these seemingly unlikely possibilities, a large statistical fluctuation and a large error in the energy calibration, will be tested by future measurements with SuperKamiokande. There are at least two possibilities for explaining the spectral upturn that do not involve experimental errors or statistical fluctuations: (1) a $hep$ flux that is approximately $10$ to $30$ times larger than the conventional nuclear physics estimate; and (2) vacuum oscillations. We have concentrated in this paper on the explanation that invokes a larger-than-standard $hep$ flux. Our results are illustrated in Fig. \[fig:specconsttheta\] Even when the enhanced rate is used in the calculations, the $hep$ neutrinos are so rare that they do not effect anything measurable except the energy spectrum. At energies below $7$ MeV, the LMA solution predicts a slight upturn in the recoil energy spectrum relative to the standard model energy spectrum. This effect is shown in Fig. \[fig:specconsttheta\] and may be detectable in the future with much improved statistics. However, Fig. \[fig:energyerror\] shows that even a modest error in the absolute energy calibration could give rise to an apparently significant energy distortion. In future analyses, it will be important to include the absolute energy calibration error as one of the parameters that is allowed to vary in determining the best fit and the uncertainty in the spectrum shape. The zenith-angle dependence and the integrated day-night effect observed by SuperKamiokande are consistent with the predictions of the LMA and, in fact, show a hint of an effect ($\sim 1.6\sigma$) that is predicted by LMA oscillations (see Ref. [@inoue99] and Eq. \[eq:daynightasym\]). Figures \[fig:zenith\] and \[fig:zenithconstm\] show that the predicted and the observed nighttime enhancement are both relatively flat, approximately independent of the zenith angle. Moreover, all five of the measured nighttime rates are above the average daytime rate. However, the results are not very significant statistically. If the LMA description is correct, then another $5$ to $10$ years of SuperKamiokande data taking will be required in order to reveal a several standard deviation effect. The difference between the daytime and the nighttime recoil energy spectra may be detectable in the future. Figure \[fig:daynightspect\] shows separately the predicted day and the predicted night spectral energy distributions. The most useful way to test for differences between the shapes of the daytime and the nighttime energy spectra is to normalize both spectra to the same value. Figure \[fig:diffdaynightspect\] shows the predicted difference between the spectra observed during the day and the spectra observed at night. It will be extremely interesting to test the hint that a day-night effect has been observed with SuperKamiokande by comparing, as in Fig. \[fig:diffdaynightspect\], the equally-normalized day and night spectra. The simplest way of performing this test would be to compare the total area of the energy spectrum that is observed below $10$ MeV at night with the total area observed below $10$ MeV during the day. The predicted average difference in the day-night spectrum test is about $1.5$% for $\Delta m^2 = 2\times 10^{-5} {\rm eV^2}$ and decreases to $\sim 0.5$% for $\Delta m^2 = 8\times 10^{-5} {\rm eV^2}$. If the LMA solution is correct, then when the day and night spectra are normalized so that they have equal total areas, then the area under the daytime spectrum curve below $10$ MeV must be larger than the area of the nighttime spectrum below $10$ MeV. No significant difference between the daytime and the nighttime spectra is expected if vacuum oscillations are the correct solution of the solar neutrino problems. Seasonal differences are predicted to be small for the LMA solution. Equation (\[eq:asymmetryidentity\]) shows that seasonal differences are expected to be reduced relative to the day-night asymmetry by a factor of order of $6$ for the location of SuperKamiokande, i.e., the difference between the average length of the night in the winter and the summer divided by $24$ hours. The predicted and the observed seasonal dependences are shown in Fig. \[fig:seasonalno\] and Fig. \[fig:seasonal\]. If the LMA solution is correct, it will require many years of SuperKamiokande measurements in order to detect a statistically significant seasonal dependence. How will it all end? {#subsec:howend} -------------------- If there is new physics in the neutrino sector, then experimentalists need to provide two different levels of evidence in order to “solve” the solar neutrino problems. First, measurements must be made that are inconsistent with standard electroweak theory at more than the $3\sigma$ level of significance. Second, diagnostic measurements must be made that uniquely select a single non-standard neutrino solution. Where are we in this program? We are much of the way toward completing the first requirement in a global sense. A number of authors have shown [@bks98; @hata94; @parke95; @robertson] that an arbitrary linear combination of fluxes from different solar nuclear reactions, each with an undistorted neutrino energy spectrum, is inconsistent at about $3\sigma$ or more with a global description of all of the available solar neutrino data. The data sets used in these calculations have gradually been expanded to include the results of all five solar neutrino experiments: Homestake, Kamiokande, SAGE, GALLEX, and SuperKamiokande. The results have become stronger as more data were added. The precise agreement [@BP98] of the calculated sound speeds of the standard solar model with the measured helioseismological data has provided a further argument in support of the conclusion that some new physics is occurring. However, we do not yet have a measurement of a smoking-gun phenomenon, seen in a single experiment, that is by itself significant at a many sigma level of significance. This has been the goal of the current generation of solar neutrino experiments. What can we say about the possibility of achieving this goal with SuperKamiokande? There may of course be new physical phenomena that have not been anticipated theoretically and which will show up with a high level of significance in the SuperKamiokande experiment. We cannot say anything about this possibility. If the LMA description of solar neutrinos is correct, we can use the results shown in Fig. \[fig:energyerror\]–Fig. \[fig:seasonal11pt5\] to estimate how long SuperKamiokande must be operated in order to reduce the errors so that a single effect (spectrum distortion, zenith-angle dependence, or seasonal dependence) is significant at more than the $3\sigma$ level. Figures \[fig:energyerror\]–\[fig:seasonal11pt5\] show that the errors must be reduced for any one phenomenon by at least a factor of two in order to reach a multi-sigma level of significance. Since the available data comprise $708$ days of operation, over a calendar period of order three years, it seems likely that SuperKamiokande will require of order a decade of running in order to isolate a single direct proof of solar neutrino oscillations, provided the LMA description is correct. Fortunately, SuperKamiokande can make a global test of the standard electroweak hypothesis that nothing happens to neutrinos after they are created in the center of the sun. This global test could become significant at the several sigma level within a few years even if LMA is correct. For example, the combined effect of the zenith-angle dependence plus a possible spectral distortion at low energies might show up as a clear signal. Or, both the integrated day-night effect and the day-night spectrum test could be observed. The combined statistical significance of several difference tests could be very powerful. With the great precision and the high statistical significance of the SuperKamiokande data, we think that there is a good chance that a many sigma result may be obtainable in a relatively few years. [Note added in proof.]{} Recently, the SuperKamiokande collaboration has made available the data for $825$ days of observations (T. Kajita, talk given at the Second Int. Conf. “Beyond the Desert,” Castle Ringberg, Tegernsee, Germany, June 6 - 12, 1999). In this more complete data set, the significance of the excess of events at high energies has further decreased and the Day-Night asymmetry has increased to approximately the $2\sigma$ level. Both of these developments strengthen slightly the case for the LMA solution. We are grateful to the SuperKamiokande collaboration for continuing to make available their superb preliminary data and for many stimulating discussions. We are also grateful to David Saxe for skillfully performing some numerical calculations. 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Robertson, Phys Rev Lett [77]{}, 3720 (1996). =5.5in =5.5in =5.5in .5truein .5truein .5truein .5truein =5.5in =5.5in =5.5in [^1]: If neutrino oscillations occur, the predicted zenith-angle dependence changes slightly for different solar models since the flavor content of the calculated solar flux arriving at earth is influenced somewhat by the assumed standard model fluxes. See Ref. [@brighter], Sec. XB. [^2]: The definition of $A_{\rm n-d}$ given in Eq. (\[eq:daynightasym\]) is twice as large as in Ref. [@brighter]; we changed our definition to the one given in the present paper in order to conform to the usage by the SuperKamiokande collaboration. Similarly, we do not discuss the valuable constraints provide by moments of the day-night effect because the SuperKamiokande collaboration has not yet provided an analysis of their results in terms of moments. [^3]: The average energy spectra during the day and during the night have been calculated also in Ref. [@marispetcov97]. However, these authors did not include the SuperKamiokande energy resolution function, which leads to a significant smearing effect (see the discussions in Refs. [@bl96; @brighter; @guth99]). [^4]: The principal results given in this section, including Eq. (\[eq:asymmetryidentity\]), have been presented and discussed by A. Yu. Smirnov at the Moriond meeting for 1999 [@smirnovmoriond]. The seasonal variations found in Ref. [@Valle] are in good agreement with our results for the same values of the oscillation parameters.
--- abstract: 'This paper discusses an electron energy analyzer with a cylindrically symmetrical electrostatic field, designed for rapid Auger analysis. The device was designed and built. The best parameters of the analyzer were estimated and then experimentally verified.' author: - 'P. Čižmár' - 'I. Müllerová' - 'M. Jacka' - 'A. Pratt' title: 'New multi-channel electron energy analyzer with cylindrically symmetrical electrostatic field' --- [address=[Institute of Scientific Instruments, ASCR, Královopolská 147, Brno, CZ-61264, Czech Republic]{}, ,email=[[email protected]]{}]{} [address=[Institute of Scientific Instruments, ASCR, Královopolská 147, Brno, CZ-61264, Czech Republic]{}, ,email=[[email protected]]{}]{} [address=[formerly the University of York, Heslington, York, YO10 5DD, U.K.]{}, ,email=[[email protected]]{}]{} [address=[The University of York, Heslington, York, YO10 5DD, U.K.]{}, ,email=[[email protected]]{}]{} Introduction ============ One of the nearly nondestructive methods to examine surfaces of materials is the analysis of Auger electrons. These have energies from the range roughly from 50 to 2000 eV and are emitted from the top few nanometers giving unique valuable surface sensitivity. Most common sequential analyzers, such as CMA or CHA, are used. Although many tens of percents of emitted electrons can be collected by some analyzers, this still may not be sufficient for fast analysis, because energies are analyzed sequentially. Each time the particular detection energy is changed, there is a dead time needed by the system to get into the desired state. Such analyzers then need much more time to obtain a spectrum. This can be a serious issue for time dependent experiments, if the sample can be easily damaged by the electron beam, or if a spectrum is acquired for each pixel in an entire image. In general, in order to reduce the time needed to acquire a spectrum, either the solid angle intercepted by the analyzer can be increased, or parallel detection can be employed. It was shown in [@marcus1] that it is possible to acquire the entire energy spectrum of interest simultaneously. The basis of the analyzer used was the two dimensional hyperbolic field [@marcus2][@walker1]. The approach in this work is the development of an analyzer that keeps all the advantages of parallel acquisition and that also has a possibility to increase the solid angle by adding cylindrical symmetry with a new focusing property. The advantage of this solution is a further decrease of the time needed to acquire the spectrum. ![The hyperbolic field (left-hand side) and the cylindrically symmetrical field (right-hand side). The trajectories in the cylindrically symmetrical field are focused onto the detector, increasing the detected signal. Another focusing property is added.[]{data-label="fig_symetrie"}](symetrie.eps){width="7.5cm"} Cylindrically symmetrical electrostatic field ============================================= There are several conditions for the electrostatic field to be usable for electron energy analysis. First, Laplace’s equation has to be satisfied. The trajectories of the electrons analyzed by the field have to be focused in the detector plane. In this case, because the field has cylindrical symmetry, there must be the axis to axis focusing property of the field [@read1]. This means that the electrons starting from one point on the axis are focused back to the axis of symmetry, on which the detector is situated. The electrostatic field satisfying all above conditions may be defined by the potential: $$\varphi = V_0 z \log(r/R_0), \label{art_potdef}$$ where $V_0$ is a constant characterizing the strength of the field, $R_0$ is the internal radius, below which there is no field, $r$ and $z$ are cylindrical coordinates. Knowing the potential, the equations of motion in the cylindrical coordinate system can be written: $$\begin{aligned} m(\ddot r - r\dot \theta^2) &=& -qV_0z/r\nonumber\\ m(r \ddot\theta + 2\dot r\dot\theta) &=& 0\\ m\ddot z &=& -qV_0\log(r/R_0)\nonumber.\end{aligned}$$ When the axis to axis focusing mode is employed, it can be supposed that the particles are starting from the axis, and thus the angular component of the velocity is zero. Then the equations of motion can be simplified to: $$\begin{aligned} m\ddot r &=& -qV_0zR_0/r\nonumber\\ \ddot\theta &=& 0\\ m\ddot z &=& -qV_0\log(r/R_0)\nonumber.\end{aligned}$$ In contrast to the hyperbolic field case [@marcus2], this set of differential equations does not have an analytical solution; it has to be integrated numerically. The trajectories were calculated using the Runge-Kutta integration method [@rungekutta]. ![The cylindrically symmetrical field. a) Equipotentials and trajectories of electrons in the field for energies in the range of 900—2100 eV. The energy step is 100 eV. Trajectories start at \[0,0\]m, b) Dependence of the z-coordinate of the trajectory endpoint on the entry angle. (The presence of a maximum indicates first order focusing), c) The point spread function (PSF) for the energy of 2100 eV d) Dependence of the z-coordinate of the endpoints on energy (integral of the dispersion)[]{data-label="fig_field"}](field.eps){width="7.5cm"} It is also necessary to find the parameters of the field that produce the best focusing and thus also the best resolution for a specified solid angle of acceptance. One possible solution of this problem is using a minimization algorithm. In this case for various energies several trajectories were modeled. The sum of squares of deviations of the endpoint positions is a satisfactory function to minimize. From the trajectories of electrons in the field it is possible to calculate the dispersion and the best reachable resolution of the analyzer employing this kind of electrostatic field. The dispersion can be calculated just from the central trajectory for each energy. Instead of the traditional definition of the dispersion, $$D_r(E) = E \partial z / \partial E,$$ where the dispersion is relative to the energy, the absolute dispersion, $$D_a(E) = \partial z / \partial E,$$ is more suitable. For the CMA or other analyzers, where the detected energy is tuned, the relative definition is more applicable, because then the dispersion is nearly independent of energy. For the parallel analyzer the absolute definition is more suitable for the same reason. A similar problem is the definition of the resolution. In case of the CMA the relative definition is used. $$R_r(E) = \frac{E}{\Delta E(E)}$$ The reason is again the fact that this value is almost constant. For the same reason, in the case of the cylindrically symmetrical field analyzer or hyperbolic field analyzer (HFA) the absolute definition is more suitable. $$R_a(E) = \Delta E(E)$$ To be able to calculate the best possible resolution (the resolution affected only by the properties of the electrostatic field) for each energy a set of calculated endpoints of electron trajectories is needed. Because the analytical expressions of the trajectories are not known, whole trajectories have to be calculated instead of the endpoints only. The calculation then takes more time than in case of the HFA. Calculation of the trajectories and their endpoints showed that for different angles of entry different endpoints are obtained as expected. When the entry angle is increased, the endpoint is getting farther, until a turning-point is reached. Then the detected coordinate is decreasing. See Fig. \[fig\_field\]b. In fact, the existence of this turning-point enables focusing (first order focusing in this case). The dependence of the endpoint coordinate on the angle of entry can be very well approximated by a cubic polynomial for all energies. The coefficients of such polynomials are then dependent on energy and can be also very well approximated by quadratic polynomials. The detected coordinate of the endpoint can then be expressed as $$z(E,\psi) = \sum\limits_{i=0}^2\sum\limits_{j=0}^3 k_{ij}E^i\psi^j. \label{rov_kij}$$ The coefficients $k_{ij}$ can be calculated, and then from (\[rov\_kij\]) any number of endpoints can be interpolated. Then it is possible to calculate resolution from the histogram of positions, (Fig. \[fig\_field\]c) and dependence of the endpoint position on electron energy, which is in fact the integral of the dispersion. See Fig. \[fig\_field\]d. For the resolution, the $\Delta z$ is defined. In this case, the density of endpoint positions (PSF in this case) is divergent, because of the existence of turning-points. $\Delta z$ must be defined as the distance between 20%—80% of the distribution function. The $\Delta z$ varies with energy. The calculation showed that the absolute dispersion $D_a$ is very close to a constant. See Fig. \[fig\_field\]d. Thus $$\Delta E = \Delta z/D_a.$$ The modeling of the trajectories in the analytical field showed that the best focus is obtained when the energies of the analyzed electrons are between 1000 eV and 2000 eV instead of the desired range of 50 eV—1000 eV, considering that the position of the detector is given. It is possible to use this higher range of energies by placing an accelerator in front of the analyzer entrance. Simulations =========== To create a real analyzer with a field that has the same analytical properties as the field defined by Eq. \[art\_potdef\], electrodes of a particular shape have to be used. The outer shape of the analytical area of the analyzer is determined by the geometry of the chamber, electron column and detectors. The electrodes must be placed where the equipotentials cross the outer shape of the analyzer. These electrode shapes are then analytically calculated, and the shape functions are obtained. In some cases the problem leads to transcendental equations, and numerical methods must then be used. To examine the behavior of the analyzed electrons in the real analyzer, which is created by charged electrodes, simulation software can be used. The CPO [@cpo] software was suitable for this 3D pure electrostatic problem. This software uses the boundary integral method to calculate the value of the electrostatic potential at any particular point in the analyzer. It can also be used to calculate the trajectories of analyzed electrons. The boundary integral method is based on calculations of charge distributions on electrodes. Therefore the electrode shapes had to be divided into a set of smaller triangular or rectangular segments. When the charge distribution is known, electron motion within the field can be simulated and for each electron a trajectory endpoint can be obtained. For the endpoints obtained from CPO-3D simulation Eq. (\[rov\_kij\]) is also valid and the coefficients may be calculated. Out of them it is possible to figure out the resolution and the dispersion. ![The 3D simulation with the CPO-3D software. a) Subdivided electrodes and description of the electrodes, b) Side view at electrodes and simulated trajectories. The sample is positioned in the center of the hemispherical accelerator. c) Dependence of the analyzer resolution on kinetic energy of analyzed electrons at the sample.[]{data-label="fig_simulace"}](cpo3d.eps){width="7.5cm"} The device ========== Fig. \[fig\_spektrum\]a shows a photograph of the device. The analyzer must satisfy several conditions to be usable in an ultra-high-vacuum electron microscope system: - The analyzer must fit into the chamber and not collide with other parts of the microscope.\ - Only ultra-high-vacuum compatible materials must be used.\ - All parts of the analyzer have to be bakeable. They must stable at higher temperatures.\ - Magnetic materials must be avoided.\ The conditions above strongly limit the usable materials. The device had to be designed with respect to the system used. In this case the experiment took place in the electron microscopy laboratory at the Department of Physics of the University of York, UK. In the past a hyperbolic field analyzer was used in this system and the new cylindrically symmetrical field system was designed to work in the same position in the microscope. Therefore, the new analyzer had to have similar shape. The shapes of the electrodes were then calculated according to this requirement. PEEK and Kapton materials were used for the insulating parts of the device. Both materials are very stable at high temperatures and utra-high-vacuum compatible. The PEEK was used to make insulating spacers. The top cover was made of Kapton. The electrodes were accurately etched of stainless steal sheet. The side covers and lower cylindrical electrode were made of an aluminium alloy. A set of two concentric hemispheres was used as the accelerator, which produces the radial electrostatic field, accelerates the electrons emitted from the sample, and decelerates primary electrons before they land on the sample. The point where the primary beam hits the specimen has to be in the center of the hemispheres. As an inlet and outlet for electrons, two round holes need to be drilled in the hemispheres. These affect the electron trajectories because they form a lens, but the negative effect on the trajectories entering the analyzer is significant only at the lowest energies. At higher energies this effect is negligible. The analyzer was developed and built in the Institute of Scientific Instruments, Academy of Sciences of the Czech Republic. Experiment ========== ![a) Photograph of the analyzer. b) Photograph of the phosphor screen when acquiring an uncorrected electron energy spectrum emitted from the contaminated copper sample taken at primary beam energy of 2500 eV. c) A sum of three acquisitions of the same spectrum. d) Detail of the relaxation peak displayed as a bar graph.[]{data-label="fig_spektrum"}](spectra.eps){width="7.5cm"} The analyzer was used to verify the previous ideas and calculations and to demonstrate that the cylindrically symmetrical field can be successfully used in parallel energy analysis. The analyzer was inserted into an ultra high vacuum system equipped with an electrostatic column. The pressure in the sample chamber was on the order of $10^{-8}$ Pa. As a sample a piece of copper foil was used, although the sample was not cleaned. For a detector the electrons were multiplied by micro channel plate and then converted to light quanta with a phosphor screen. The image on the screen was then photographed through a vacuum window using Konica-Minolta Z3 digital camera. Conclusion ========== From the simulation, the absolute dispersion is nearly constant at $1.97\times 10^{-2} {\rm mm\cdot eV^{-1}}$ The resolution varies from 10 eV to 2 eV, for most energies keeps below 2 eV, which well corresponds to [@read2]. The result of the experiment is a 400 pixel long spectrum. (See Fig. \[fig\_spektrum\].) The relaxation peak is 1 pixel wide, which shows that the energy resolution is better than 3 eV (Fig. \[fig\_spektrum\]d) at 1500 eV, which satisfies the theoretical estimations. The experiment also showed the cylindrical focusing. In the photograph of the screen (Fig. \[fig\_spektrum\]b) different widths of illuminated area can be seen. These may be caused by a slight misalignment that occurred during bake-out. This fact also affects the signal levels of different channels. The peak between 300 eV and 500 eV is partially caused by carbon (the main matter covering the surface of the sample) because the sample was not cleaned. Acknowledgment ============== The work was supported by the ASCR grant agency project number IAA1065304. We also gratefully acknowledge help of Prof. B. Lencova and Mr. Pavel Klein. [99]{} M. Jacka, A. Kale, N. Traitler, Rev. Sci. Instrum. 74(2003)4298. M. Jacka, M. Kirk, M.M. ElGomati and M. Prutton, Rev. Sci. Instr. and Meth. A 519(2004)338. F.H. Read, Rev. of Sci. Instrum. 73(2002)1129. F.H. Read, D. Cubric, S. Kumashiro and A. Walker, Nucl. Instr. and Meth. A 519, 338 (2004). CPO Programs, Available on the web site http://www.electronoptics/.com. J.R. Dormand and P.J. Prince, J. Comp. Appl. Math., Vol. 6(1980)19. Ch.G.H. Walker, A. Walker, R. Badheka, S. Kumashiro, M. Jacka, M.M. El Gomati, M. Prutton and F.H. Read, in Proc. SPIE, Charged Particle Optics IV, ed by Eric Munro, Denver Colorado, 3777(1999)252.
--- abstract: 'We study how the order of $N$ independent random walks in one dimension evolves with time. Our focus is statistical properties of the inversion number $m$, defined as the number of pairs that are out of sort with respect to the initial configuration. In the steady-state, the distribution of the inversion number is Gaussian with the average $\langle m\rangle \simeq N^2/4$ and the standard deviation $\sigma\simeq N^{3/2}/6$. The survival probability, $S_m(t)$, which measures the likelihood that the inversion number remains below $m$ until time $t$, decays algebraically in the long-time limit, $S_m\sim t^{-\beta_m}$. Interestingly, there is a spectrum of $N(N-1)/2$ distinct exponents $\beta_m(N)$. We also find that the kinetics of first-passage in a circular cone provides a good approximation for these exponents. When $N$ is large, the first-passage exponents are a universal function of a single scaling variable, $\beta_m(N)\to \beta(z)$ with . In the cone approximation, the scaling function is a root of a transcendental equation involving the parabolic cylinder equation, $D_{2\beta}(-z)=0$, and surprisingly, numerical simulations show this prediction to be exact.' author: - 'E. Ben-Naim' title: On the Mixing of Diffusing Particles --- Introduction ============ Consider the permutation $3142$ of the four elements $\{1,2,3,4\}$. Three pairs: $(1,3)$, $(2,3)$, and $(2,4)$ are inverted in this permutation. The inversion number, defined as the total number of pairs that are out of sort, provides a natural measure for how “scrambled” a list of elements is. This basic combinatorial quantity [@pam; @wf; @gea; @mb] is helpful in many contexts. In computer science, the inversion number plays an important role in sorting and ranking algorithms [@dek]. Common on the web (“customers who like $\ldots$ may also like $\ldots$”), recommendations for books, songs, and movies use inversions to quantify how close the preferences of two customers are [@kt]. The number of inversions can also be used to measure how the order of a group of particles in one dimension changes with time. Figure \[fig-xt\] illustrates a space-time diagram of four diffusing particles. The number of inversions changes whenever two trajectories cross. Depending on the initial order of the two respective particles, a crossing may either introduce a new inversion or undo an existing one. Consequently, the inversion number either increases or decreases by one. Therefore, the inversion number equals the difference between the number of crossings of the first kind and the number of crossings of the second kind. Mixing dynamics has been extensively studied in the context of fluids [@jmo; @vhg] and granular materials [@ok], but much less attention has been given to mixing in the context of diffusion [@ya; @yal]. In this study, we consider an ensemble of $N$ diffusing particles in one-dimension, a system that is widely used to model the transport of colloidal and biological particles in narrow channels [@wbl; @cdl]. We use the inversion number to measure the degree to which particles mix. Clearly, a persistent small inversion number indicates a poorly mixed system, while a large inversion number implies that the opposite is true. We first study how the distribution of the inversion number evolves with time. We find that there is a transient regime in which the average inversion number as well as the standard deviation in this quantity both grow as the square-root of time. The distribution of the inversion number is stationary beyond this transient regime. When the number of particles is sufficiently large, the probability distribution function is always Gaussian, whether in the transient regime or in the steady-state. ![Space-time diagram of a four-particle system. The circled $+$s and $-$s indicate whether the inversion number increases or decreases when two trajectories cross. Four out of the five crossings increase the inversion number, and accordingly, the inversion number increases from $m=0$ to $m=3$.[]{data-label="fig-xt"}](fig1){width="30.00000%"} Our main focus is first-passage properties [@sr] of the inversion number. We ask: what is $S_m(t)$, the probability that the inversion number remains smaller than $m$ up to time $t$. For small values of $m$, the survival probability $S_m$ measures the likelihood that the particles remain poorly mixed throughout the evolution. Generally, the probabilities $S_m$ decay as a power law at large times, $S_m\sim t^{-\beta_m}$. In general, there is a broad spectrum of $N(N-1)/2$ distinct exponents, , that governs the asymptotic decay of the survival probabilities. We heavily use first-passage kinetics of a single particle that diffuses inside a circular cone [@rdd; @bs1; @bk] to understand the asymptotic behavior of $S_m$. We first utilize two-dimensional cones to obtain the first-passage exponents for a three-particle system exactly. Furthermore, we employ circular cones in $N-1$ dimensions and find good approximate values for the first-passage exponents. The cone approximation correctly predicts that when the number of particles is large, the exponents become a universal function, $\beta_m(N)\to \beta(z)$, of the scaling variable $z=(m-\langle m\rangle)/\sigma$. Here, $\langle m\rangle$ and $\sigma$ are the average and standard deviation of the distribution of inversion number, at the steady-state. Interestingly, our numerical simulations show that the cone approximation yields the exact scaling function $\beta(z)$ as a root of a transcendental equation involving the parabolic cylinder function. The rest of this paper is organized as follows. In section II, we introduce our basic system and define the inversion number. Stationary and transient properties of the distribution of the inversion number are discussed in sections III and IV, respectively. In section V, we use the cone approximation to understand first-passage properties of the inversion number. Scaling and extremal properties of the first-passage exponents are the focus of section VI. We conclude in section VII. The Inversion Number ==================== Our goal is to characterize how the order of an ensemble of diffusing particles changes with time. We conveniently use an ordinary random walk [@ghw; @hcb; @rg] to model the trajectory of a diffusing particle [@bdup]. Our system includes $N$ identical particles that move on an unbounded one-dimensional lattice. The particles are completely independent: at each step one particle is selected at random and it moves, with an equal probability, either to the left, $x\to x-1$, or to the right, $x\to x+1$. After each elementary step, time is augmented by the inverse number of particles, $t\to t+1/N$, so that each particle moves once per unit time. We index the particles according to their initial position with the leftmost particle labeled $n=1$ and the rightmost particle labeled $n=N$ (Figure \[fig-xt\]). Let $x_n(t)$ be the position of the $n$th particle at time $t$. By definition, $$\label{initial} x_1(0)<x_2(0)<\cdots <x_{N-1}(0)<x_N(0),$$ but the initial order unravels with time. Consider, for example, the four-particle system illustrated in Figure \[fig-xt\]. The particles reach a state where with three pairs, $(1,2)$, $(1,3)$, and $(2,3)$ being out of sort compared with time $t=0$. In general, a pair of particles for which $x_i(t)>x_j(t)$ and $i<j$ constitutes an inversion. Formally, the total number of inversions, $m$, is given by $$\label{inversion} m(t)=\sum_{i=1}^N\sum_{j=i+1}^N \Theta\big(x_i(t)-x_j(t)\big).$$ Here, $\Theta(x)$ is the Heaviside step function: $\Theta(x)=1$ for $x>0$ and $\Theta(x)=0$ for $x\leq 0$. The total number of pairs is $M=\binom{N}{2}$, and hence, the variable $m$ is within the bounds $0\leq m \leq M$ with $$\label{bounds} M=\frac{N(N-1)}{2}.$$ The inversion number is minimal, $m=0$, when the order is exactly the same as in the initial configuration, and it is maximal, $m=M$, when the order is the mirror image of the initial state. The inversion number changes whenever two trajectories cross (Figure \[fig-xt\]). A crossing either adds a new inversion or removes an existing one. Thus, we may assign a positive or a negative “charge” to each crossing as illustrated in Figure \[fig-xt\]. The inversion number, $m(t)$, is simply the sum of all of the charges up to time $t$. The Mahonian Distribution ========================= We first discuss basic statistical characteristics of the inversion number including the average, the variance, and more generally, the probability distribution function. At large time $t$, each random walk explores a region of size $\sqrt{t}$, and the probability of finding the particle at any lattice site inside this region is effectively uniform. This simple fact already implies that memory of the initial position fades with time. We thus expect that after sufficient time elapses, there is no memory of the initial order, and the order of the particles is completely random. To understand statistics of the inversion number for randomly ordered particles we consider the set of all $N!$ permutations of the $N$ elements $\{1,2,\ldots,N\}$. In the random state, each permutation of these elements occurs with probability $1/N!$. The probability $P_m(N)$ that the inversion number equals $m$ for a random permutation is well known as the Mahonian distribution in probability theory [@pam; @dek; @wf]. We highlight key features of this probability distribution as it plays a central role in our study. Let $Q_m(N)=N!P_m(N)$ be the number of permutations of $N$ elements with exactly $m$ inversions. For example, when $N=3$, one permutation ($123$) is free of inversions, there are two permutations with one inversion ($213$, $132$), two permutations with two inversions ($312$, $231$), and a single permutation with three inversions ($321$). Hence, $Q_0(3)=Q_3(3)=1$ while $Q_1(3)=Q_2(3)=2$. We list the distribution of the inversion number for $N\leq 4$, $$(P_0,P_1,\ldots,P_M)=\frac{1}{N!}\times \begin{cases} (1) & N=1,\\ (1,1) & N=2,\\ (1,2,2,1)& N=3,\\ (1,3,5,6,5,3,1) &N=4. \end{cases}$$ Of course, $P_m(N)$ is nonzero if and only if $0\leq m\leq M$. Since the mirror image of a configuration with $m$ inversions necessarily has $M-m$ inversions, the probability distribution satisfies $P_m=P_{M-m}$. Hence, the distribution is symmetric about $m=M/2$, and the average $\langle m\rangle\equiv \sum_m m P_m$ is simply $$\label{average} \langle m\rangle= \frac{N(N-1)}{4}.$$ Therefore, the average grows quadratically with the total number of particles when $N\gg 1$. The Mahonian distribution satisfies the simple recursion relation $$\label{recursion} P_m(N)=\frac{1}{N}\sum_{l=0}^{N-1}P_{m-l}(N-1),$$ with $P_m(1)=\delta_{m,0}$. This recursion reflects that every permutation of $N$ elements can be generated from a permutation of $N-1$ elements by inserting the $N$th element in any of the $N$ possible positions. Depending on where this last element is added, the number of inversions increases by an amount $\Delta m=0,1,\ldots,N-1$. Let us now introduce the generating function, $$\label{generating-def} {\cal P}(s,N)=\sum_{m=0}^M P_m(N)s^m.$$ For instance, ${\cal P}(s,1)=1$, ${\cal P}(s,2)=(1+s)/2!$ and ${\cal P}(s,3)=(1+s)(1+s+s^2)/3!$. In general, the generating function is given by the product [@dek] $$\label{generating} {\cal P}(s,N)=\frac{1}{N!}\prod_{n=1}^N (1+s+s^2+\cdots + s^{n-1}),$$ as also follows from the recursion . We can confirm the average by differentiating the generating function once, ${\cal P}'(s=1)=\langle m\rangle$, where the prime represents differentiation with respect $s$. By differentiating the generating function twice and using ${\cal P}''(s=1)=\langle m(m-1)\rangle$, we obtain the variance [@dek], $\sigma^2=\langle m^2\rangle -\langle m\rangle^2$, $$\label{variance} \sigma^2=\frac{N(N-1)(2N+5)}{72}.$$ This expression is obtained from $\sigma^2=\sum_{l=1}^N \frac{l^2-1}{12}$. Therefore, the standard deviation is rather large, $\sigma\simeq N^{3/2}/6$, when $N\gg 1$. The mean and the variance fully specify the probability distribution function for an asymptotically large number of particles. The Mahonian distribution becomes a function of a single variable, $P_m(N)\to \Phi(z)$, with the scaling variable $$\label{scaling-def} z=\frac{m-\langle m\rangle}{\sigma}.$$ The probability distribution function, $\Phi(z)$, is normal, that is, a Gaussian with zero mean and unit variance [@wf; @cjz], $$\label{normal} \Phi(z)\simeq\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{z^2}{2}\right).$$ To see that the central limit theorem applies, we convert the generating function into a Fourier transform, and then show that the Fourier transform is Gaussian in the large-$N$ limit [@krb]. The variable $z$ is a more transparent measure in the following sense. A value of $z$ of order one implies fairly random order. Indeed, according to the normal distribution , the inversion number falls within three standard deviations from the mean, $\|z\|<3$, with probability $0.997$. A large value, $\|z\|\gg 1$, indicates that the particle order strongly resembles the initial configuration (if $z>0$) or its mirror image (if $z<0$). Transient Behavior ================== By definition, the inversion number is zero initially, $m(0)=0$. At least partially, the initial order is preserved in the early stages of evolution, and the number of inversions must be substantially lower than . We consider the natural initial condition where the particles occupy $N$ consecutive lattice sites: $x_i(0)=i$, for all $i=1,2,\ldots,N$. Early on, particles “interact” only within their local neighborhood. The interaction length, $\ell$, grows diffusively with time, $\ell\sim \sqrt{t}$. On this length scale, particles are well-mixed, and according to , the number of inversions [per particle]{} is proportional to the number of interacting particles, $\ell$. Hence, the average number of inversions grows according to $\langle m(t)\rangle\sim N\ell \sim N\sqrt{t}$. As a consequence, $$\label{mt} \langle m(t)\rangle\simeq \begin{cases} {\rm const.}\times N\sqrt{t}\quad & 1\ll t\ll N^2,\\ N^2/4\quad & N^2\ll t, \end{cases}$$ when $N\gg 1$. The two expressions match at , a diffusive time scale that can be viewed as the mixing time. Therefore, there is a transient regime in which the inversion number grows as the square-root of time, followed by a steady-state, in which the average is given by . We obtain the variance using a similar heuristic argument. According to , the variance per particle is quadratic in the number of interacting particles, . Therefore, in the transient regime, $$\label{sigmat} \sigma(t) \simeq \begin{cases} {\rm const}\times \sqrt{N\,t}\quad & 1\ll t\ll N^2,\\ N^{3/2}/6\quad & N^2\ll t. \end{cases}$$ As expected, the transient behavior matches the steady-state behavior at the diffusive time scale $t\sim N^2$. Like the average, the standard deviation also grows as the square root of time. As shown in Figure \[fig-average\], results of numerical simulations confirm the scaling behavior . Moreover, the numerically measured average matches the steady-state value corresponding to the Mahonian distribution. We also verified that the stationary distribution is Gaussian with the variance . The simulations also show that the time-dependent distribution of inversion number, $p_m(N,t)$, is Gaussian throughout the transient regime (Figure \[fig-distribution\]): $$\label{normal1} p_m(N,t)\simeq \frac{1}{\sqrt{2\pi\sigma^2(t)}}\exp\left[-\frac{(m-\langle m(t)\rangle)^2}{2\sigma^2(t)}\right].$$ This behavior provides further support for our heuristic argument. Indeed, if the particles are well-mixed locally, then the distribution of the number of inversions per particle is Gaussian, and as the sum of $N$ Gaussian variables, the total inversion number must also have a Gaussian distribution. ![The normalized average $\langle m(t)\rangle/[N(N-1)/4]$ versus time $t$. The results correspond to an average over $10^2$ independent realizations of a system with $N=10^3$ random walks. Also shown for reference is a line with slope $1/2$.[]{data-label="fig-average"}](fig2){width="43.00000%"} ![The distribution of the inversion number in the intermediate time regime $1\ll t\ll N^2$. Shown is the distribution $p_m\equiv p_m(N,t)$ versus the variable $\big(m-\langle m(t)\rangle\big)/\sigma(t)$. The results are from $10^5$ independent realizations of a system with $N=10^3$ random walks. The distribution is shown at times $t=10^2$ (diamonds), $t=10^3$ (squares), and $t=10^4$ (circles). Also shown for reference is the normal distribution .[]{data-label="fig-distribution"}](fig3){width="44.00000%"} We used two different algorithms to simulate the diffusion process. In the naive algorithm, we randomly select a particle and move it to a randomly-chosen neighboring site. We increase time by $1/N$ after each jump. To calculate the inversion number, we use the formula , but since this enumeration requires ${\cal O}(N^2)$ operations, this simulation method is inefficient at large $N$. To overcome this difficulty, we introduced a variant where each lattice site may be occupied by at most one particle. At each step we pick one particle at random and attempt to move it by one site. This move is always accepted if the neighboring site is vacant, but otherwise, it is accepted with probability $1/2$. In the latter case, we merely exchange the identities of the respective particles, and as appropriate, update the inversion number by either $+1$ or $-1$. In our implementation, there are two arrays: the first lists the particle positions, [in order]{}, and the second lists the original position of each particle in the first list. This algorithm has a fixed computational cost per step, and it automatically keeps track of the inversion number. We rely on the fact that in one dimension, noninteracting random walks are equivalent to random walks that interact by exclusion [@teh; @dgl; @ra; @bs]. Still, we verified that the two algorithms yield essentially the same results. We utilized the naive algorithm to simulate small systems with $N<10$, but otherwise, we used the efficient algorithm. First-Passage Kinetics ====================== We have seen that the inversion number, which grows quadratically with the number of particles, can be quite large. Yet, if the mixing is poor and the particle trajectories rarely cross, the inversion number remains small. To quantify how common such a scenario is, we study first-passage kinetics [@sr; @snm]. In particular, we ask: what is the probability, $S_m(t)$, that the inversion number remains smaller than $m$ until time $t$. This “survival” probability is closely related to the first-passage probability as $[-dS_m/dt]\times dt$ is the probability that the inversion number reaches $m$ for the first time during the infinitesimal time interval $(t,t+dt)$. The quantity $S_1$ is the probability that the original order is perfectly maintained, or equivalently, the likelihood that none of the trajectories cross. This survival probability decays as a power law, with a rather large exponent, $$\label{first} S_1\sim t^{-N(N-1)/4},$$ in the long-time limit [@mef; @hf; @fg; @gz; @djg; @ck]. Our goal is to understand how this asymptotic behavior changes as the threshold $m$ increases. When $N=2$, the separation between the two random walks itself undergoes a random walk. Hence, $S_1$ is equivalent to the survival probability of a one-dimensional random walk in the vicinity of a trap, and $S_1\sim t^{-1/2}$ in agreement with . When $N=3$, we conveniently map the three random walks onto a single “compound” random walk in three dimensions with the coordinates $(x_1,x_2,x_3)$. To find $S_1$, we require that the compound walk remains inside the region $x_1<x_2<x_3$. We may view the boundary of this region as absorbing, and then, $S_1(t)$ equals the likelihood that that the compound walk survives at time $t$. The absorbing boundary forms a wedge because it is defined by the intersection of two planes, $x_1=x_2$ and . Generally, the survival probability of a particle that diffuses inside an absorbing wedge decays algebraically, $$S\sim t^{-1/(4V)},$$ where $V=\alpha/\pi$ is the normalized opening angle [@fs]. (The opening angle $0<\alpha\leq \pi$ is the angle between the wedge axis and the wedge boundary.) Alternatively, $0<V\leq 1$ is the fraction of the total solid angle enclosed by the wedge. The region $x_1<x_2<x_3$ occupies a fraction $V_1=\frac{1}{3!}=\frac{1}{6}$ of space and hence, $S_1\sim t^{-3/2}$, as also follows from . To find $S_2$ and $S_3$, we note that the regions in which the compound walk is allowed to move are always wedges (the three planes $x_1=x_2$, $x_1=x_3$, and $x_2=x_3$ divide space into six equal wedges [@bjmkr].) Moreover, the fraction of total solid angle enclosed by the absorbing boundaries is given by the cumulative distribution of inversion number: and . Hence, all three survival probabilities decay algebraically with time [@sr], $$\label{three} S_1\sim t^{-3/2},\qquad S_2\sim t^{-1/2},\qquad S_3\sim t^{-3/10},$$ and there are three distinct first-passage exponents. ![The survival probability $S_m(t)$ versus $t$ for a four-particle system. Shown are the quantities $S_2$ (bottom curve), $S_3$, $\ldots$, $S_6$ (top curve). The number of independent Monte Carlo runs varies from $10^6$ for the slowest decay to $10^{12}$ for the fastest decay.[]{data-label="fig-beta4"}](fig4){width="43.00000%"} The asymptotic behaviors suggest that all of the survival probabilities decay algebraically, $$\label{decay} S_m\sim t^{-\beta_m},$$ in the long-time limit. Moreover, there is a large family of exponents $$\label{family} \beta_1>\beta_2>\cdots>\beta_{N(N-1)/2},$$ that characterizes the power-law decay . We stress that the exponents depend on two variables, the threshold $m$ and the number of particles $N$, . We already know the exact values $\beta_1(3)=3/2$, $\beta_2(3)=1/2$, and $\beta_3(3)=3/10$ as well as $\beta_1(N)=N(N-1)/4$. Our numerical simulations confirm that indeed, there is a large spectrum of exponents. As shown in Figure \[fig-beta4\], there are six decay exponents when $N=4$. Table I lists the numerically measured values $\beta_m$, obtained from the local slope $d\ln S_m/d\ln t$. In general, the compound walk is confined to a certain “allowed” region of space. This region is bounded by multiple intersecting planes of the type $x_i=x_j$ with $i\neq j$, and generally, this unbounded domain has a complicated geometry. The boundary of this region encloses a fraction $V_m(N)$ of the total solid angle. On combinatorial grounds alone, we conveniently deduce that this fraction is given by the cumulative Mahonian distribution $$\label{cumulative-def} V_m(N)=\sum_{l=0}^{m-1} P_l(N).$$ Since the Mahonian distribution is symmetric, we have $V_m+V_{M+1-m}=1$. To evaluate $V_m(N)$, we expand the generating function , and for $m\leq 4$, we have [@dek] $$V_m(N)\!=\!\frac{1}{N!}\!\times\! \begin{cases} 1 & m=1,\\ N & m=2,\\ \frac{1}{2}(N-1)(N+2) & m=3,\\ \frac{1}{6}(N+1)(N^2+2N-6) & m=4.\\ \end{cases}$$ We have seen that the allowed region is a wedge when $N=3$. To obtain an approximation for the first-passage exponents, we follow an approach that proved useful in other first-passage problems involving multiple random walks and replace the boundary of the allowed region with a suitably chosen cone in $N-1$ dimensions [@bk1]. An unbounded cone with opening angle $\alpha$ occupies a fraction $V(\alpha)$ of the total solid angle, given by $$\label{Valpha} V(\alpha)=\frac{\int_0^\alpha d\theta\,(\sin\theta)^{N-3}} {\int_0^\pi d\theta\,(\sin\theta)^{N-3}}.$$ In $d$ dimensions, we have $d\Omega\propto (\sin\theta)^{d-2}d\theta$ where $\Omega$ is the solid angle and $\theta$ is the polar angle in spherical coordinates. In the cone approximation, we require $$\label{Valpha-equal} V(\alpha)=V_m$$ with $V_m$ given in . $m$ $1$ $2$ $3$ $4$ $5$ $6$ ---------------------- ---------------- --------------- --------------- --------------- --------------- ----------------- $V_m$ $\frac{1}{24}$ $\frac{1}{6}$ $\frac{3}{8}$ $\frac{5}{8}$ $\frac{5}{6}$ $\frac{23}{24}$ \[2pt\] $\alpha_m$ $0.41113$ $0.84106$ $1.31811$ $1.82347$ $2.30052$ $2.73045$ $\beta_m^{\rm cone}$ $2.67100$ $1.17208$ $0.64975$ $0.39047$ $0.24517$ $0.14988$ $\beta_m$ $3$ $1.39$ $0.839$ $0.455$ $0.275$ $0.160$ \[2pt\] : The six first-passage exponents for a four-particle system. The values $\beta_m$ are from the Monte Carlo simulation results shown in Figure \[fig-beta4\]. The values $\beta_m^{\rm cone}$ were obtained using the cone approximation, specified in Eqs. -. The cumulative Mahonian distribution, $V_m$, and the opening angle, $\alpha_m$, are listed as well. In a cone, the first-passage exponent $\beta\equiv \beta(\alpha)$ decreases as the opening angle $\alpha$ increases. In particular, $\beta=\pi/4\alpha$ in two dimensions, and $\beta=(\pi-\alpha)/2\alpha$ in four dimensions. Generally, however, $\beta$ is the smallest root of the following transcendental equation involving the associated Legendre functions [@NIST] of degree $2\beta+\gamma$ and order $\gamma=\frac{N-4}{2}$ [@bk] $$\begin{aligned} \label{cone} \begin{split} Q_{2\beta+\gamma}^\gamma(\cos\alpha) &= 0\qquad N\ {\rm odd},\\ P_{2\beta+\gamma}^\gamma(\cos\alpha) &= 0\qquad N\ {\rm even}. \end{split}\end{aligned}$$ Regardless of the dimension, the surface of a cone with $\alpha=\pi/2$ is a plane, and hence, $\beta(\pi/2)=1/2$. For example, to find $\beta_1(4)$, we first determine the fraction $V_1(4)=\frac{1}{4!}=\frac{1}{24}$ using . Then, we calculate the opening angle $\alpha=0.41113$ using equations - and finally determine the exponent $\beta_1(4)=2.67100$ as the appropriate root of equation . By construction, the cone approximation is exact for three particles. This approach gives a useful approximation to the six first-passage exponents when $N=4$ (Table I). Remarkably, the cone approximation continues to be a good approximation as the number of particles increases (Figure \[fig-beta5\]). ![The first-passage exponent $\beta_m$ versus $m$ for $N=4,5,6,7$. Shown are simulation results (circles) and the outcome of the cone approximation (squares).[]{data-label="fig-beta5"}](fig5){width="50.00000%"} The Scaling Function ==================== We are especially interested in the behavior when the number of particles is large. Let us first evaluate the cumulative Mahonian distribution in the large-$N$ limit. Since the Mahonian distribution is normal, the cumulative distribution is given by the error function, $$\label{Vm-scaling} V_m(N)\to \frac{1}{2}+\frac{1}{2}\,{\rm erf} \left(\frac{z}{\sqrt{2}}\right),$$ when $N\to\infty$. Here, $z$ is the scaling variable defined in and ${\rm erf}(\xi)=(2/\sqrt{\pi})\int_0^\xi \exp(-u^2)du $. To obtain Eq. , we substitute into and convert the sum into an integral. Equation is relevant in the limit $N\to\infty$, $m\to\infty$ with the scaling variable $z$ finite. Next, we evaluate the solid angle enclosed by an unbounded cone when the dimension is large. The dominant contribution to the integral in comes from a narrow region of order $1/\sqrt{N}$ centered on $\alpha=\pi/2$ where the integrand is Gaussian, $$(\sin\theta)^{N-2} \simeq e^{-N(\pi/2-\theta)^2/2}.$$ Using , we find that the fraction $V(\alpha)$ has the scaling form $$\label{Valpha-scaling} V(\alpha,N)\to \frac{1}{2}+\frac{1}{2}\,{\rm erf}\left(\frac{-y}{\sqrt{2}}\right),$$ with $y=(\cos\alpha)\sqrt{N}$. In writing this equation, we used the facts that and ${\rm erf}(\xi)=-{\rm erf}(-\xi)$. Equation holds in the limit $N\to\infty$, $\alpha\to \pi/2$, with the scaling variable $y$ finite. Asymptotic analysis of equation shows that the exponent $\beta(\alpha)$ adheres to the scaling form [@bk] $$\label{betaalpha-scaling} \beta(\alpha,N)\to \beta(y)\quad{\rm with}\quad y=(\cos\alpha)\sqrt{N},$$ in the limit $N\to\infty$, $\alpha\to\pi/2$ with the scaling variable $y$ finite. The scaling function, $\beta(y)$, is specified by the transcendental equation $D_{2\beta}(y)=0$, where $D_\nu$ is the parabolic cylinder function of order $\nu$ [@NIST]. The smallest root is the appropriate one [@bk]. ![The exponent $\beta$ versus the scaling variable $z$, shown using: (a) a linear-linear plot and (b) a linear-log plot. The simulation results are from Monte Carlo runs with $N=50$ (diamonds), $N=100$ (squares), and $N=200$ (circles) particles. The number of independent realizations varies from $10^4$ for slow first-passage processes to $10^8$ for fast one. The solid line shows the theoretical prediction .[]{data-label="fig-beta"}](fig6a "fig:"){width="42.50000%"} ![The exponent $\beta$ versus the scaling variable $z$, shown using: (a) a linear-linear plot and (b) a linear-log plot. The simulation results are from Monte Carlo runs with $N=50$ (diamonds), $N=100$ (squares), and $N=200$ (circles) particles. The number of independent realizations varies from $10^4$ for slow first-passage processes to $10^8$ for fast one. The solid line shows the theoretical prediction .[]{data-label="fig-beta"}](fig6b "fig:"){width="44.00000%"} By comparing equations and , we find our main result: the first-passage exponent depends on a single scaling variable, $$\label{scaling-form} \beta_m(N) \to \beta(z) \quad {\rm with}\quad z=\frac{m-\langle m\rangle}{\sigma},$$ in the large-$N$ limit. We reiterate that the average $\langle m\rangle$ and the standard-deviation $\sigma$ correspond to the steady-state values and , respectively. Using $y=-z$, the scaling function $\beta(z)$ is the smallest root of the transcendental equation $$\label{scaling-function} D_{2\beta}(-z)=0,$$ involving the parabolic cylinder function. When $\beta$ is a half-integer, the parabolic cylinder function is related to the Hermite polynomials [@NIST] and using this equivalence, we have $\beta(0)=1/2$, $\beta(-1)=1$, and $\beta(-\sqrt{3})=3/2$. Our numerical simulations (Figure \[fig-beta\]) confirm that the exponents $\beta_m(N)$ have the scaling form . Interestingly, the simulations strongly suggest that the scaling function predicted by the cone approximation is [exact]{}. We note that the convergence to the infinite-particle limit is very fast for positive $z$, but much slower for negative $z$ [@bk]. With the power-law decay , the mean first-passage time diverges whenever $\beta<1$, but it is finite otherwise. Since $\beta(z=-1)=1$, the time required for the inversion number to reach one standard deviation from the mean is infinite, on average. Regardless of the threshold $z$, there is a considerable chance that the random walks are poorly mixed because the survival probability decays algebraically. The scaling behavior is remarkable for a number of reasons. First, the form of the scaling variable, , is quite unusual. Second, there are roughly $N^2/2$ first-passage exponents and numerical evaluation of this large spectrum is daunting. Yet, the scaling form gives the range of parameters for which $\beta$ is of order one, and hence, numerically measurable. (It is difficult to measure a vanishing exponent, $\beta\to 0$, or a divergent exponent, $\beta\to \infty$.) Last, the emergence of scaling laws for a family of scaling exponents is also intriguing. Typically, in Statistical Physics, the opposite is true as one or two scaling exponents characterize a scaling law [@hes]. The extremal behaviors of the roots of the transcendental equation are derived in ref. [@bk], $$\label{scaling-limits} \beta(z)\sim \begin{cases} z^2/8 &z\to-\infty,\\ \sqrt{z^2/8\pi}\exp\left(-z^2/2\right)& z\to\infty. \end{cases}$$ The first-passage exponent is algebraically large if $z$ is large and negative, but it is exponentially small if $z$ is large and positive. $N$ $3$ $4$ $5$ $6$ $7$ $8$ ------------------------------ ---------------- ----------- ------------ ---------------- ---------------- ------------- $\beta_1$ $\frac{3}{2}$ $3$ $5$ $\frac{15}{2}$ $\frac{21}{2}$ $14$ \[1pt\] $\beta_1^{\rm cone}$ $\frac{3}{2}$ $2.67100$ $4.08529$ $5.73796$ $7.62336$ $9.73686$ \[1pt\] $\beta_M^{\rm cone}$ $\frac{3}{10}$ $0.14988$ $0.061195$ $0.019895$ $0.0050713$ $0.0010266$ \[2pt\] : The largest exponent, $\beta_1^{\rm cone}$, and the smallest exponent, $\beta_M^{\rm cone}$, obtained using the cone approximation for $N\leq 8$. Also listed for reference, is the exact value $\beta_1$. The exponential decay in implies that it is extremely unlikely that the initial order is perfectly reversed. The smallest exponent $\beta_M$ characterizes the probability $S_M$ that the order of the walkers does not turn into the mirror image of the initial state, that is, the probability that the compound walk remains in the [exterior]{} of the so-called “Weyl chamber” $x_1<x_2<\cdots<x_N$ [@mef; @hf; @fg; @gz; @djg; @ck]. This domain has $V_M=1-\frac{1}{N!}$, and Table II lists the outcome of the cone approximation for small $N$. To find the outcome of the cone approximation at large $N$, we first estimate the opening angle, $\pi-\alpha\simeq e/N$ by using Eq. and the Stirling formula $N!\simeq \sqrt{2\pi N}(N/e)^N$. From the asymptotic behavior for wide cones at large dimensions, [@bk], we conclude [@bk1] $$\label{smallest} \beta_M\simeq \frac{N^4}{2\,e^3\,N!}.$$ This value is extremely small, decaying roughly as the inverse of a factorial, and it is impossible to measure such a minuscule quantity using numerical simulations. The largest exponent describes the probability that the particles maintain the initial order or that the compound walk remains in the [interior]{} of the Weyl chamber with $V_1=\frac{1}{N!}$. Table II compares the outcome of the cone approximation with the exact value $\beta_1=N(N-1)/4$. The quality of the cone approximation worsens as $N$ grows. Nevertheless, the cone approximation is qualitatively correct. By substituting the opening angle $\alpha\simeq e/N$ into the thin-cone asymptotic behavior $\beta(\alpha)\simeq N\alpha^{-1}/4$ [@bk], we find $\beta_1\simeq N^2/(4e)$. This expression captures the quadratic growth of the exponent. Remarkably, the cone approximation is exact inside the scaling region, but it is only approximate outside this region. Conclusions =========== In summary, we used the number of pair inversions to measure the one-dimensional mixing of independent diffusing trajectories. A high inversion number typifies strong mixing whereas a persistent small inversion number indicates poor mixing. In the steady-state, the distribution of inversion number is given by the well-known Mahonian distribution, and consequently, it is Gaussian when the number of particles is large. Preceding the steady-state is a transient regime in which both the average inversion number and the standard deviation grow diffusively with time. We focused on first-passage statistics and showed that the probability that the inversion number does not exceed a certain threshold decays as power law with time. Moreover, we found that a large spectrum of decay exponents characterizes the asymptotic behavior. When the number of particles is large, the exponents obey a universal scaling function. The scaling variable equals the distance between the threshold inversion number and the average inversion number, measured in terms of the standard deviation. The cone approximation, which replaces the region in which the compound random walk is allowed to move with an unbounded circular cone, plays a central role in our analysis. This approach is exact for three particles, it produces very good estimates in higher dimensions, and remarkably, this framework yields the exact scaling function. The cone approximation gives lower bounds for the decay exponents because, among all unbounded domains with the same solid angle, the perfectly circular cone maximizes the survival probability [@bk1; @jwsr; @ch]. The cone approximation is useful in answering other first-passage questions such as the probability that the $n$th rightmost random walk does not cross the origin and the probability that the original rightmost particle always remains ahead of at least $n$ other particles [@bk1]. In both cases, there are as many exponents as there are particles, and curiously, the circular cone framework produces the scaling function governing the first-passage exponents approximately in the first case and exactly in the second case. Understanding when the cone approximation is exact and when it is approximate is an interesting challenge, with implications well beyond first-passage [@cj; @jdj; @ntv]. The first-passage exponent is directly related to the lowest eigenvalue of the Laplace operator, and therefore, we conclude that the lowest eigenvalue of the Laplacian similarly obeys scaling laws in high dimensions. The shape of the scaling function depends on the actual geometry [@ic]. 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--- bibliography: - 'BDPS.bib' --- [PSI-PR-19-14]{} [Full two-loop QCD corrections to the Higgs mass]{} 0.3cm [in the MSSM with heavy superpartners]{} [Emanuele Bagnaschi,$^{\!\!\!\,a}$ Giuseppe Degrassi,$^{\!\!\!\,b,\,c}$ Sebastian Pa[ß]{}ehr$^{\,d}$]{} [and Pietro Slavich$^{\,d}$]{} [${}^a$ Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland]{}\ ${}^b$ Dipartimento di Matematica e Fisica, Università di Roma Tre Via della Vasca Navale 84, I-00146 Rome, Italy \ [${}^c$ INFN, Sezione di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy]{}\ ${}^d$ Laboratoire de Physique Théorique et Hautes Energies (LPTHE), UMR 7589, Sorbonne Université et CNRS, 4 place Jussieu, 75252 Paris Cedex 05, France. Introduction {#sec:intro} ============ The Minimal Supersymmetric Standard Model (MSSM) is one of the best-motivated extensions of the Standard Model (SM), and probably the most studied. The Higgs sector of the MSSM consists of two $SU(2)$ doublets, but the model allows for a so-called “decoupling limit” in which a combination of the two doublets has SM-like couplings to matter fermions and gauge bosons – so that its neutral scalar component $h$ can be identified with the Higgs boson discovered at the LHC [@Aad:2012tfa; @Chatrchyan:2012xdj], which itself is broadly SM-like [@Khachatryan:2016vau] – while the orthogonal combination of doublets is much heavier. An important aspect of the MSSM is the existence of relations between the quartic Higgs couplings and the electroweak (EW) gauge couplings. In the decoupling limit, these relations induce a tree-level prediction $(\mh^2)^{\rm tree} \approx \MZ^2\cos^2 2\beta$ for the squared mass of the SM-like scalar, where $\MZ$ is the $Z$-boson mass, and the angle $\beta$ is related to the ratio of the vacuum expectation values (vevs) of the two Higgs doublets by $\tan\beta =$ $v_2/v_1$, and determines their admixture into $h$. Consequently, in the MSSM, the tree-level contribution can only make up for at most half of the squared mass of the observed Higgs boson, $(\mh^2)^{\rm obs} \approx (125~{\rm GeV})^2$ [@Aad:2015zhl]. The rest must arise from radiative corrections. It has been known since the early 1990s [@Okada:1990vk; @Ellis:1990nz; @Haber:1990aw; @Okada:1990gg; @Ellis:1991zd; @Brignole:1991pq] that the most relevant corrections to the Higgs mass are those controlled by the top Yukawa coupling, $g_t \sim {\cal O}(1)$, which involve the top quark and its superpartners, the stop squarks. These corrections are enhanced by logarithms of the ratio between stop and top masses, and also show a significant dependence on the value of the left–right stop mixing parameter $X_t$. In particular, for values of $\tan\beta$ large enough to saturate the tree-level prediction, a Higgs mass around $125$ GeV can be obtained with an average stop mass $\MS$ of about $1$–$2$ TeV when $X_t/\MS \approx 2$, whereas for vanishing $X_t$ the stops need to be heavier than $10$ TeV. Over the years, the crucial role of the radiative corrections stimulated a wide effort to compute them with the highest possible precision, in order to keep the theoretical uncertainty of the Higgs-mass prediction under control. By now, that computation is indeed quite advanced: full one-loop corrections [@Chankowski:1991md; @Brignole:1991wp; @Brignole:1992uf; @Chankowski:1992er; @Dabelstein:1994hb; @Pierce:1996zz] and two-loop corrections in the limit of vanishing external momentum [@Hempfling:1993qq; @Heinemeyer:1998jw; @Heinemeyer:1998kz; @Zhang:1998bm; @Heinemeyer:1998np; @Espinosa:1999zm; @Espinosa:2000df; @Degrassi:2001yf; @Brignole:2001jy; @Brignole:2002bz; @Martin:2002iu; @Martin:2002wn; @Dedes:2003km; @Heinemeyer:2004xw] are available, and the dominant momentum-dependent two-loop corrections [@Martin:2003qz; @Martin:2003it; @Martin:2004kr; @Borowka:2014wla; @Degrassi:2014pfa] as well as the dominant three-loop corrections [@Harlander:2008ju; @Kant:2010tf; @Harlander:2017kuc; @Stockinger:2018oxe; @R.:2019ply; @R.:2019irs] have also been obtained.[^1] However, when the SUSY scale $\MS$ is significantly larger than the EW scale (which we can identify, e.g., with the top mass $\mt$), any fixed-order computation of $\mh$ may become inadequate, because radiative corrections of order $n$ in the loop expansion contain terms enhanced by as much as $\ln^n(\MS/\mt)$. In the presence of a significant hierarchy between the scales, the computation of the Higgs mass needs to be reorganized in an effective field theory (EFT) approach: the heavy particles are integrated out at the scale $\MS$, where they only affect the matching conditions for the couplings of the EFT valid below $\MS$; the appropriate renormalization group equations (RGEs) are then used to evolve those couplings between the SUSY scale and the EW scale, where the running couplings are related to physical observables such as the Higgs-boson mass and the masses of fermions and gauge bosons. In this approach, the computation is free of large logarithmic terms both at the SUSY scale and at the EW scale, while the effect of those terms is accounted for to all orders in the loop expansion by the evolution of the couplings between the two scales. More precisely, large corrections can be resummed to the (next-to)$^n$-leading-logarithmic (N$^n$LL) order by means of $n$-loop calculations at the SUSY and EW scales combined with $(n\!+\!1)$-loop RGEs. The EFT approach to the computation of the MSSM Higgs mass dates back to the early 1990s [@Barbieri:1990ja; @Espinosa:1991fc; @Casas:1994us], and it has also been exploited in the past [@Haber:1993an; @Carena:1995bx; @Carena:1995wu; @Haber:1996fp; @Carena:2000dp; @Degrassi:2002fi; @Martin:2007pg] to determine analytically the coefficients of the logarithmic terms in the Higgs-mass corrections, by solving perturbatively the appropriate systems of boundary conditions and RGEs. In recent years, after the LHC results pushed the expectations for the SUSY scale into the TeV range, the realization that an accurate prediction for the Higgs mass in the MSSM cannot prescind from the resummation of the large logarithmic corrections brought the EFT computation under renewed focus [@Hahn:2013ria; @Draper:2013oza; @Bagnaschi:2014rsa; @Vega:2015fna; @Lee:2015uza; @Bagnaschi:2015pwa; @Bahl:2016brp; @Athron:2016fuq; @Staub:2017jnp; @Bagnaschi:2017xid; @Bahl:2017aev; @Athron:2017fvs; @Allanach:2018fif; @Bahl:2018jom; @Harlander:2018yhj; @Bahl:2018ykj]. In the simplest scenario in which all of the SUSY particles as well as the heavy Higgs doublet of the MSSM are clustered around a single scale $\MS$, so that the EFT valid below that scale is just the SM, the state of the art now includes: full one-loop and partial two-loop matching conditions for the quartic Higgs coupling at the SUSY scale, computed for arbitrary values of the relevant SUSY parameters [@Bagnaschi:2014rsa; @Bagnaschi:2017xid]; full three-loop RGEs for all of the parameters of the SM Lagrangian [@Mihaila:2012fm; @Chetyrkin:2012rz; @Mihaila:2012pz; @Bednyakov:2012en; @Chetyrkin:2013wya; @Bednyakov:2013eba]; full two-loop relations at the EW scale between the running SM parameters and a set of physical observables which include the pole Higgs mass [@Buttazzo:2013uya; @Kniehl:2015nwa; @Martin:2019lqd]. The combination of these results allows for a full NLL resummation of the large logarithmic corrections to the Higgs mass, whereas the NNLL resummation can only be considered partial, because in refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid] the two-loop matching conditions for the quartic Higgs coupling were computed in the “gaugeless limit” of vanishing EW gauge couplings.[^2] Beyond the pure EFT calculation of the Higgs mass, different “hybrid” approaches to combine the existing “diagrammatic” (i.e., fixed-order) calculations with a resummation of the logarithmic corrections have been proposed [@Hahn:2013ria; @Bahl:2016brp; @Athron:2016fuq; @Staub:2017jnp]. The aim is to include terms suppressed by powers of $v^2/\MS^2\,$ (where we denote by $v$ the vev of a SM-like Higgs scalar) up to the perturbative order accounted for by the diagrammatic calculation. In the EFT calculation, those terms can be mapped to the effect of non-renormalizable, higher-dimensional operators, and they are neglected when the theory valid below the matching scale is taken to be the plain SM in the unbroken phase of the EW symmetry. To avoid double counting, the hybrid approaches require a careful subtraction of the terms that are accounted for by both the diagrammatic and the EFT calculations, and indeed a few successive adjustments [@Bahl:2017aev; @Athron:2017fvs; @Bahl:2018ykj] were necessary to obtain predictions for $\mh$ that, in the limit of very heavy SUSY masses in which the ${\cal O}(v^2/\MS^2)$ terms are certainly negligible, show the expected agreement with the pure EFT calculation. The comparison between the predictions of the hybrid and pure EFT calculations, as well as a direct study [@Bagnaschi:2017xid] of the effects of non-renormalizable operators in the EFT, also show that the ${\cal O}(v^2/\MS^2)$ corrections are significantly suppressed for the values of $\MS$ that are large enough to allow for $\mh \approx 125$ GeV. Other recent developments of the EFT approach include: the study of MSSM scenarios in which both Higgs doublets are light, so that the effective theory valid below the SUSY scale is a two-Higgs-doublet model (THDM) [@Lee:2015uza; @Bagnaschi:2015pwa; @Bahl:2018jom]; the application of functional techniques to the full one-loop matching of the MSSM onto the SM [@Wells:2017vla]; the calculation of one-loop matching conditions between the couplings of two generic renormalizable theories [@Braathen:2018htl; @Gabelmann:2018axh], which can then be adapted to SUSY (or non-SUSY) models other than the MSSM. In this paper we focus again on the simplest EFT setup in which the theory valid below the SUSY scale is the SM, and we take a further step towards the full NNLL resummation of the large logarithmic corrections. In particular, we compute the two-loop threshold corrections to the quartic Higgs coupling that involve both the strong and the EW gauge couplings. Combined with the “gaugeless” results of refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid], this completes the calculation of the two-loop threshold corrections that involve the strong gauge coupling. We also discuss the necessary inclusion of contributions beyond the gaugeless limit in the relation between the pole Higgs mass and the $\msbar$-renormalized quartic Higgs coupling at the EW scale, and we compare the results of the full two-loop calculations of that relation given in refs. [@Buttazzo:2013uya] and [@Kniehl:2015nwa], respectively. Finally, considering a representative scenario for the MSSM with heavy superpartners, we find that the numerical impact of the new corrections on the prediction for the Higgs mass is modest, but comparable to the accuracy of the Higgs-mass measurement at the LHC. Two-loop matching of the quartic Higgs coupling {#sec:matching} =============================================== In this section we describe our calculation of the two-loop QCD contributions to the matching condition for the quartic Higgs coupling. We consider the setup in which all SUSY particles as well as a linear combination of the two Higgs doublets of the MSSM are integrated out at a common renormalization scale $Q\approx\MS$, so that the EFT valid below the matching scale is the SM. In our conventions the potential for the SM-like Higgs doublet $H$ contains the quartic interaction term $\frac\lambda 2 \,\|H\|^4$, and the tree-level squared mass of its neutral scalar component is $(\mh^2)^{\rm tree} = 2\lambda v^2$, with $v = \langle H^0 \rangle \approx 174$ GeV. Then the two-loop matching condition for the quartic coupling takes the form \[looplam\] (Q)= 14 \^2 (Q) + \^[1]{} + \^[2]{} , where $g$ and $\gp$ are the EW gauge couplings, $\beta$ can be interpreted as the angle[^3] that rotates the two original MSSM doublets into a light doublet $H$ and a massive doublet $A$, and $\Delta \lambda^{n\ell}$ is the $n$-loop threshold correction to the quartic coupling arising from integrating out the heavy particles at the scale $Q$. The complete result for the one-loop correction $\Delta \lambda^{1\ell}$, valid for arbitrary values of all the relevant SUSY parameters, can be found in refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid]. It is computed under the assumptions that $\lambda$, $g$ and $\gp$ in eq. (\[looplam\]) are $\msbar$-renormalized parameters of the SM, and that $\beta$ is defined beyond tree level as described in section 2.2 of ref. [@Bagnaschi:2014rsa], removing entirely the contributions of the off-diagonal wave-function renormalization (WFR) of the Higgs doublets. As to the two-loop correction $\Delta \lambda^{2\ell}$, ref. [@Bagnaschi:2014rsa] provided the contributions of ${\cal O}(g_t^4\,\gs^2)$, where $\gs$ is the strong gauge coupling, for arbitrary values of all the relevant SUSY parameters; ref. [@Bagnaschi:2017xid] provided in addition the two-loop contributions involving only the third-family Yukawa couplings, $g_t$, $g_b$ and $g_\tau$, again for arbitrary SUSY parameters, and also discussed some subtleties in the derivation of the ${\cal O}(g_b^4\,\gs^2)$ contributions from the known results for the ${\cal O}(g_t^4\,\gs^2)$ ones. Altogether, the results of refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid] amounted to a complete determination of $\Delta \lambda^{2\ell}$ in the limit of vanishing EW gauge (and first-two-generation Yukawa) couplings. In this paper we take a step beyond this “gaugeless limit” and compute the remaining corrections that involve the strong gauge coupling, namely those of ${\cal O}(g_{t,b}^2\,g^2 \gs^2)$, those of ${\cal O}(g_{t,b}^2 \, \gpq \gs^2)$ and those involving only gauge couplings, i.e. of ${\cal O}(g^4 \gs^2)$ and ${\cal O}(\gpqq \gs^2)$. We can decompose the “mixed” threshold correction to the quartic Higgs coupling into three terms: \[threeparts\] \^[2,[[QCDEW]{}]{}]{} = \^[2,[1[PI]{}]{}]{} + \^[2,[[WFR]{}]{}]{} + \^[2,[[RS]{}]{}]{} . The first term on the r.h.s. of the equation above denotes the contributions of two-loop, one-particle-irreducible (1PI) diagrams with four external Higgs fields involving both strong-interaction vertices (namely squark–gluon, four-squark or quark–squark–gluino vertices) and $D$-term-induced quartic Higgs–squark vertices proportional to $g^2$ or to $\gpq$, and possibly also vertices controlled by the Yukawa couplings. This term can be computed with a relatively straightforward adaptation of the effective-potential approach developed in refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid] for the corresponding calculation in the gaugeless limit. In contrast, the remaining terms on the r.h.s. of eq. (\[threeparts\]) require different approaches. The second term involves the two-loop, ${\cal O}(g_{t,b}^2 \, \gs^2)$ squark contributions to the WFR of the Higgs field, which multiply the tree-level quartic coupling, see eq. (\[looplam\]), giving rise to ${\cal O}(g_{t,b}^2\,g^2 \gs^2)$ and ${\cal O}(g_{t,b}^2 \, \gpq \gs^2)$ corrections. This term requires a computation of the external-momentum dependence of the two-loop self-energy of the Higgs boson, analogous to the one performed in refs. [@Martin:2003qz; @Martin:2003it; @Martin:2004kr; @Borowka:2014wla; @Degrassi:2014pfa; @Borowka:2018anu] for the “diagrammatic” case. Finally, the last term on the r.h.s. of eq. (\[threeparts\]) arises from the fact that, while our calculation of the matching condition for the quartic Higgs coupling $\lambda$ is performed in the $\drbar$ renormalization scheme assuming the field content of the MSSM, in the EFT valid below the SUSY scale – i.e., the SM – $\lambda$ is interpreted as an $\msbar$-renormalized quantity. Moreover, we find it convenient to use $\msbar$-renormalized parameters of the SM also for the EW gauge couplings entering the tree-level part of the matching condition, see eq. (\[looplam\]), and for the top Yukawa coupling entering the one-loop part (but not for the bottom Yukawa coupling, as discussed in ref. [@Bagnaschi:2017xid]). The corrections arising from the change of renormalization scheme for $\lambda$ can in turn be extracted from the two-loop self-energy diagrams for the Higgs boson, those arising from the change of scheme and model for the EW gauge couplings require a computation of the external-momentum dependence of the two-loop self-energy of the $Z$ boson, while those arising from the change of scheme and model for the top Yukawa coupling are easier to obtain, being just the product of one-loop terms. In the rest of this section we will describe in more detail our computation of each of the three terms on the r.h.s. of eq. (\[threeparts\]). 1PI contributions {#subsec:1PI} ----------------- The 1PI, two-loop contribution to the matching condition for the quartic Higgs coupling can be expressed as \[effpot\] \^[2,[1[PI]{}]{}]{} = 12. \|\_[H=0]{} , where $\Delta V^{2\ell,\,\tilde q}$ denotes the contribution to the MSSM scalar potential from two-loop diagrams involving the strong gauge interactions of the squarks, and the derivatives are computed at $H=0$ because we perform the matching between the MSSM and the SM in the unbroken phase of the EW symmetry. Since the strong interactions do not mix different types of squarks, we will now describe the derivation of the stop contribution to $\Delta \lambda^{2\ell,{\rm 1{\scriptscriptstyle PI}}}$, and later describe how to translate it into the sbottom contribution and into the contributions of the squarks of the first two generations. It is convenient to start from the well-known expression for the stop contribution of ${\cal O}(\gs^2)$ to the MSSM scalar potential in the broken phase of the EW symmetry [@Zhang:1998bm], \[v2as\] V\^[2,t]{} & = & \^2\^2C\_F N\_c { 2I(,,0) + 2L(,,) - 4m\_tI(,,)\ && +(1-2)J(,) + J(,) + }, where $\kappa = 1/(16\pi^2)$ is a loop factor, $C_F=4/3$ and $N_c=3$ are color factors, the loop integrals $I(x,y,z)$, $L(x,y,z)$ and $J(x,y)$ in eq. (\[v2as\]) are defined, e.g., in appendix D of ref. [@Degrassi:2009yq], $\mg$ stands for the gluino mass, $\tul$ and $\tdl$ are the two stop-mass eigenstates, and $ \theta_{t}$ denotes the stop mixing angle. The latter is related to the top and stop masses and to the left–right stop mixing parameter by \[s2t\] = . We recall that $X_t = A_t - \mu\cot\beta$, where $A_t$ is the trilinear soft SUSY-breaking Higgs–stop interaction term and $\mu$ is the Higgs-higgsino mass parameter in the superpotential (in fact, those two parameters enter our results only combined into $X_t$). To compute the fourth derivative of the effective potential entering eq. (\[effpot\]) we express the stop masses and mixing angle as functions of a field-dependent top mass $m_t = \hat g_t\,\|H\|$, where by $\hat g_t$ we denote [^4] a SM-like Yukawa coupling related to its MSSM counterpart $\hat y_t$ by $\hat g_t = \hat y_t\sin\beta$, and of a field-dependent $Z$-boson mass $\MZ = \hgz \,\|H\|$, where we define $\hgz^2 = (\hat g^2 + \hat g^{\prime\,2})/2$. We then obtain \[derivs\] . \|\_[H=0]{} &=& \_[m\_t,0]{} + , where the last term is obtained from the previous ones by swapping top and $Z$, and we used the shortcuts \[shortcut\] V\^[(k)]{}\_[p\_1… p\_k]{} = , (p\_i = t,Z) . The terms proportional to $g_t^4$ in the second line of eq. (\[derivs\]) give rise to the ${\cal O}(g_t^4\,\gs^2)$ contributions to $\Delta \lambda^{2\ell,{\rm 1{\scriptscriptstyle PI}}}$ already computed in ref. [@Bagnaschi:2014rsa], and we will not consider them further. We will focus instead on the mixed contributions to $\Delta \lambda^{2\ell,{\rm 1{\scriptscriptstyle PI}}}$ arising from the terms in eq. (\[derivs\]) that are proportional to $g_t^2\,\gz^2$ and to $\gz^4$. To obtain the derivatives with respect to $\mt$ and $\MZ$ of the stop masses and mixing entering $\Delta V^{2\ell,\,\tilde t}$ we exploit the relations \[derivst\] = 1 , = , \[derivsZ\] = , = -(d\_\^t - d\_\^t) , where \[dtLdtR\] d\_\^t = 12 - 23 \^2\_ , d\_\^t = 23 \^2\_ , $\theta_\smallW$ being the Weinberg angle. After taking the required derivatives of $\Delta V^{2\ell,\,\tilde t}$ with respect to $\mt^2$ and $\MZ^2$, we use eq. (\[s2t\]) to make the dependence of $\theta_t$ on $\mt$ explicit; we expand the function $\Phi(m^2_{\tilde t_{i}},\g,\mt^2)$ entering the loop integrals (see appendix D of ref. [@Degrassi:2009yq]) in powers of $\mt^2$; we take the limit $\|H\|\rightarrow$ $0$, leading to $m_t,\MZ\,\rightarrow0$; and we identify $\tul$ and $\tdl$ with the soft SUSY-breaking stop mass parameters $m_{Q_3}$ and $m_{U_3}$. We remark that, differently from the ${\cal O}(g_t^4\,\gs^2)$ contributions computed in ref. [@Bagnaschi:2014rsa] and the ${\cal O}(g_t^6)$ contributions computed in ref. [@Bagnaschi:2017xid], the mixed contributions to $\Delta \lambda^{2\ell,{\rm 1{\scriptscriptstyle PI}}}$ computed here do not contain infrared divergences that need to be canceled out in the matching of the quartic Higgs coupling between MSSM and SM. Also, the terms proportional to $ g_t^2\gz^2$ and to $\gz^4$ in eq. (\[derivs\]) that involve more than two derivatives of the two-loop effective potential vanish directly when we take the limit $\|H\|\rightarrow 0$, thus the mixed contributions to $\Delta \lambda^{2\ell,{\rm 1{\scriptscriptstyle PI}}}$ can be related as usual to the corresponding two-loop corrections to the Higgs mass. In order to obtain the contributions to $\Delta \lambda^{2\ell,{\rm 1{\scriptscriptstyle PI}}}$ from diagrams involving sbottoms, it is sufficient to perform the replacements $g_t\rightarrow g_b\,$, $X_t\rightarrow X_b\,$, $m_{U_3}\rightarrow m_{D_3}$ and $d_{\smallL,\smallR}^t\rightarrow d_{\smallL,\smallR}^b$ in the contributions from diagrams involving stops, with \[dbLdbR\] d\_\^b = -12 + 13 \^2\_ , d\_\^b = -13 \^2\_ . Finally, we recall that our calculation neglects the Yukawa couplings of the first two generations. The contributions to $\Delta \lambda^{2\ell,{\rm 1{\scriptscriptstyle PI}}}$ from diagrams involving up-type (or down-type) squarks of the first two generations can be obtained by setting $g_t =0$ (or $g_b =0$) in the contributions from diagrams involving stops (or sbottoms), and replacing the soft SUSY-breaking stop (or sbottom) mass parameters with those of the appropriate generation. The result for $\Delta \lambda^{2\ell,{\rm 1{\scriptscriptstyle PI}}}$ with full dependence on all of the input parameters is lengthy and not particularly illuminating, and we make it available upon request – together with all of the other corrections computed in this paper – in electronic form. We show here a simplified result valid in the limit in which all squark masses ($m_{Q_i},\, m_{U_i},\, m_{D_i},\,$ with $i=1,2,3$) as well as the gluino mass $\mg$ are set equal to a common SUSY scale $\MS$. In units of $\kappa^2\,\gs^2\,C_FN_c$, we find \^[2,[1[PI]{}]{}]{} &=& 34 (g\^4 + g\^[4]{})\^2\ &-&(g\^2 + g\^[2]{}) , where $(t\rightarrow b)$ denotes terms obtained from the previous ones within square brackets via the replacements $g_t\rightarrow g_b$ and $X_t \rightarrow X_b$. We remark in passing that there are no contributions of ${\cal O}(g^2\,g^{\prime\,2}\,\gs^2)$, because the corresponding diagrams involve the trace of the generators of the $SU(2)$ gauge group. WFR contributions {#subsec:WFR} ----------------- Differently from the case of the “gaugeless” corrections computed in refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid], where the quartic Higgs coupling can be considered vanishing at tree level, the mixed corrections include a contribution in which the tree-level coupling is combined with the two-loop WFR of the Higgs field. This reads \[dlamWFR\] \^[2,[[WFR]{}]{}]{} = -12(g\^2 + g\^[2]{})\^2 ( .\|\_[H=0]{} + ), where $\hat \Pi_{hh}^{2\ell,\,\tilde q}$ denotes the contribution to the renormalized Higgs-boson self-energy[^5] from two-loop diagrams involving the strong gauge interactions of the squarks, and the derivative is taken with respect to the external momentum $p^2$. In this case the notation $H=0$ means that, after having taken the derivative, we take the limit $v\rightarrow 0$. This implies $m_q,\MZ\rightarrow 0$ as well as $p^2\rightarrow 0$ (because $p^2$ is ultimately set to $2\lambda v^2$). Finally, the shift $\dWFR$ stems from the matching of the one-loop WFR, and will be discussed below. The relevant two-loop self-energy diagrams are generated with `FeynArts`[@Hahn:2000kx], using a modified version of the original MSSM model file [@Hahn:2001rv] that implements the QCD interactions in the background field gauge. The color factors are simplified with a private package and the Dirac algebra is handled by `TRACER` [@Jamin:1991dp]. In order to obtain a result valid in the limit $v\rightarrow 0$, we performed an asymptotic expansion of the self-energy in the heavy superparticle masses analogous to the one described in section 3 of ref. [@Degrassi:2010eu]. As a useful cross-check of our result, we verified that it agrees with the one that can be obtained by taking appropriate limits in the explicit analytic formulae for the Higgs-boson self-energy given in refs. [@Martin:2003qz; @Martin:2003it; @Martin:2004kr]. Our result for the derivative of the two-loop self-energy with full dependence on all of the input parameters is in turn made available upon request. In the simplified scenario of degenerate superparticle masses we find, in units of $\kappa^2\,\gs^2\,C_FN_c$, \[dPidp2\] .\|\_[H=0]{} &=& g\_t\^2 + (tb) . We remark that there are no contributions proportional to the EW gauge couplings. This is due to the fact that, in the limit of unbroken EW symmetry, the $D$-term-induced ${\cal O}(g^2)$ and ${\cal O}(g^{\prime\,2})$ couplings of the Higgs boson to squarks enter only self-energy diagrams that do not depend on the external momentum. We also remark that the derivative of the two-loop self-energy has logarithmic infra-red (IR) divergences in the limit $p^2\rightarrow 0$, see the last term within square brackets in the second line of eq. (\[dPidp2\]). These divergences cancel out in the matching of the Higgs-boson WFR between the MSSM and the SM. Indeed, in the limit $v\rightarrow0$ the one-loop contribution of top and bottom quarks to the derivative of the self-energy, which is present both above and below the matching scale, reads \[dPidp2oneloop\] .\|\_[H=0]{} = N\_c (g\_t\^2 + g\_b\^2) (1-) . However, the top and bottom Yukawa couplings must be interpreted as the ones of the MSSM above the matching scale, and the ones of the SM below the matching scale. Thus, in the matching between the MSSM and the SM the derivative of the two-loop Higgs self-energy receives the shift \[shiftWFR\] = 2N\_c (g\_t\^2 g\_t\^[t,\^2]{} + g\_b\^2 g\_b\^[b,\^2]{}) (1-) , where $\Delta g_t^{\tilde t,\,\gs^2}$ denotes the one-loop, ${\cal O}(\gs^2)$ contribution from diagrams involving stops to the difference between the MSSM coupling $\hat g_t$ and the SM coupling $g_t$, \[deltagtsusy\] g\_t\^[t,\^2]{} = - \^2C\_F , and the analogous shift in the bottom Yuakwa coupling, $\Delta g_b^{\tilde b,\,\gs^2}$, can be obtained from eq. (\[deltagtsusy\]) with the replacements $m_{U_3}\rightarrow m_{D_3}$ and $X_t\rightarrow X_b$. The loop functions $\wt F_6(x)$ and $\wt F_9(x,y)$ are defined in the appendix A of ref. [@Bagnaschi:2014rsa]. Using the limits $\wt F_6(1)=0$ and $\wt F_9(1,1)=1$ it is easy to see that, for degenerate superparticle masses, the shifts in eq. (\[shiftWFR\]) cancel out entirely the terms within square brackets in the second line of eq. (\[dPidp2\]). We have of course checked that the cancellation of the IR divergences in the matching holds even when we retain the full dependence on all of the relevant superparticle masses. Contributions arising from the definitions of the couplings {#subsec:RSC} ----------------------------------------------------------- The third contribution to the mixed correction to the quartic Higgs coupling in eq. (\[threeparts\]), which we denoted as $\Delta \lambda^{2\ell,\,{\rm {\scriptscriptstyle RS}}}$, collects in fact three separate contributions arising from differences in the renormalization scheme used for the couplings of the MSSM and for those of the EFT valid below the matching scale (i.e., the SM): \[dlambdascheme\] \^[2,[[RS]{}]{}]{} = \_\^[2,[[RS]{}]{}]{} + \_[g]{}\^[2,[[RS]{}]{}]{} + \_[g\_t]{}\^[2,[[RS]{}]{}]{} . The first contribution in the equation above stems from the fact that supersymmetry provides a prediction for the $\drbar$-renormalized quartic Higgs coupling, whereas we interpret the parameter $\lambda$ in the EFT as renormalized in the $\msbar$ scheme. The difference between $\lambda^{\smalldrbar}$ and $\lambda^{\smallmsbar}$ contains terms of ${\cal O}(\lambda\,g_t^2\,\gs^2)$ and ${\cal O}(\lambda\,g_b^2\,\gs^2)$, arising from the dependence on the regularization method of the two-loop quark–gluon contributions to the Higgs WFR, which translate to mixed terms when $\lambda$ is replaced by its tree-level MSSM prediction. In the limit $v\rightarrow 0$ we find \[WFRreg\] .\|\_[H=0]{}\^ - .\|\_[H=0]{}\^ = 2\^2\^2(g\_t\^2+g\_b\^2)C\_FN\_c (1-) - 12 \^2\^2(g\_t\^2+g\_b\^2)C\_FN\_c , where DRED and DREG stand for dimensional reduction and dimensional regularization, respectively. This is again in agreement with the result that can be obtained by taking the appropriate limits in the analytic formulae of refs. [@Martin:2003qz; @Martin:2003it; @Martin:2004kr]. The first term on the r.h.s. of eq. (\[WFRreg\]) contains an IR divergence for $p^2\rightarrow 0$, but that term cancels out in the matching between MSSM and SM when the top and bottom Yukawa couplings entering the one-loop quark contribution to the WFR, see eq. (\[dPidp2oneloop\]), are translated from the $\drbar$ scheme to the $\msbar$ scheme according to \[dgtreg\] g\_q\^ = g\_q\^ (1+g\_q\^[,\^2]{}), [with]{} g\_q\^[,\^2]{} = -\^2C\_F . The surviving term on the r.h.s. of eq. (\[WFRreg\]) then leads to the following correction to the quartic Higgs coupling: \_\^[2,[[RS]{}]{}]{} = 14 \^2\^2C\_FN\_c(g\_t\^2+g\_b\^2)(g\^2+g\^[2]{})\^2 . Supersymmetry connects the tree-level quartic Higgs coupling to the $\drbar$-renormalized EW gauge couplings of the MSSM. Since we choose instead to express the tree-level part of the matching condition for $\lambda$, see eq. (\[looplam\]), in terms of $\msbar$-renormalized couplings of the SM, the threshold correction $\Delta\lambda^{2\ell}$ receives an additional shift, i.e. the second contribution in eq. (\[dlambdascheme\]). The relation between the two sets of EW gauge couplings reads g\^2 + g \^[2]{} = (g\^2 + g \^[2]{}) ( 1 + … + .\|\_[H=0]{}\^- .\|\_[H=0]{}\^+ .\|\_[H=0]{} ) , where the ellipsis denotes one- and two-loop terms that are not of ${\cal O}(g^4\,\gs^2)$ or ${\cal O}(g^{\prime\,4}\,\gs^2)$. We denote by $\hat\Pi_{\smallZ\smallZ}^{2\ell,\,q}$ the two-loop quark–gluon contribution to the transverse part of the renormalized $Z$-boson self-energy, computed either in dimensional reduction or in dimensional regularization, and by $\hat\Pi_{\smallZ\smallZ}^{2\ell,\,\tilde q}$ the contribution from two-loop diagrams involving the strong gauge interactions of the squarks.[^6] The notation $H=0$ means again the limit $v\rightarrow 0$, which in this case can be obtained by Taylor expansion in $m_q^2$ and $p^2$, since both the squark contribution and the DRED–DREG difference of the quark contribution are free of IR divergences. In units of $\kappa^2\,\gs^2\,C_FN_c$, we find \[shiftEW\] \_g\^[2,[[RS]{}]{}]{} &=& {- - .\ && - . \_[i=1]{}\^3 },\ where the two terms within curly brackets in the first line account for the $\drbar$–$\msbar$ conversion of $g^2$ and $g^{\prime\,2}$, respectively, whereas the two terms in the second line (where the sum runs over three squark generations) account for their MSSM–SM threshold correction. The function $F(\msqq,\mg^2)$ is defined as \[funcF\] F(,\^2) = (7+) + +(1-) . We checked that the explicit renormalization-scale dependence of the ${\cal O}(g^4\,\gs^2)$ and ${\cal O}(g^{\prime\,4}\,\gs^2)$ threshold corrections to $g^2$ and $g^{\prime\,2}$ is consistent with what can be inferred from the difference between their $\beta$-functions in the SM [@Machacek:1983tz] and in the MSSM [@Jones:1974pg; @Jones:1983vk; @Martin:1993zk]. Finally, the third contribution in eq. (\[dlambdascheme\]) arises from the fact that we choose to express the ${\cal O}(g^2\,g_t^2)$ and ${\cal O}(g^{\prime\,2}\,g_t^2)$ terms in the one-loop threshold correction to the quartic Higgs coupling in terms of the $\msbar$-renormalized top Yukawa coupling of the SM. The resulting shift in the two-loop correction reads \[shiftgt\] \_[g\_t]{}\^[2,[[RS]{}]{}]{} = 2(g\_t\^[t,\^2]{} +g\_q\^[,\^2]{}) \^[1, t\_]{} , where $\Delta g_t^{\tilde t,\,\gs^2}$ and $\Delta g_q^{{\scriptscriptstyle {\rm reg}},\,\gs^2}$ are the one-loop, ${\cal O}(g_s^2)$ shifts given in eqs. (\[deltagtsusy\]) and (\[dgtreg\]), respectively, and \^[1, t\_]{} &=& N\_cg\_t\^2{ 12 (g\^2-) +.\ && . +} ,\ where $\xqu=m_{Q_3}/m_{U_3}$ and the loop functions $\wt F_3(x)$, $\wt F_4(x)$ and $\wt F_5(x)$ are defined in the appendix A of ref. [@Bagnaschi:2014rsa]. Note that all three functions are equal to 1 for $x=1$. Combining all contributions --------------------------- We now provide a result that combines all of the contributions discussed in the previous sections, valid in the limit of degenerate superparticle masses $m_{Q_i}=m_{U_i}=m_{D_i}=\mg=\MS$. In units of $\kappa^2\,\gs^2\,C_FN_c$, we find \[dlambdatotal\] \^[2,[[QCDEW]{}]{}]{} &=& \^2[@[3]{}]{}{-(g\^4+g\^[4]{}) (1+4)\ && +(g\^2 + g\^[2]{}){ (g\_t\^2+ g\_b\^2)(--32 -12\^2) .\ && +g\_t\^2\ && . +g\_b\^2}[@[3]{}]{}}\ &+&(g\^2 + g\^[2]{}){ g\_t\^2\ && .+g\_b\^2 } . As a non-trivial check of our final result, we verified that by taking the derivative of the r.h.s. of eq. (\[looplam\]) with respect to $\ln Q^2$ we can recover the ${\cal O}(\lambda\,g_t^2\,\gs^2)$ and ${\cal O}(\lambda\,g_b^2\,\gs^2)$ terms of the $\beta$-function for the quartic Higgs coupling of the SM [@Machacek:1984zw]: \[rgelam\] 40\^2\^2(g\_t\^2+g\_b\^2) . To this effect, we must combine the explicit scale dependence of our result for $\Delta \lambda^{2\ell,\,{\rm {\scriptscriptstyle QCD\mbox{-}EW}}}$ with the implicit scale dependence of the parameters that enter the tree-level and one-loop parts of the matching condition (for the squark contributions to the latter, see the appendix of ref. [@Bagnaschi:2017xid]). In particular, we need the terms that involve the strong gauge coupling in the two-loop $\beta$-functions of the EW gauge couplings [@Machacek:1983tz], \[rgeEW\] 12\^2g\^4\^2 , \^2g\^[4]{}\^2 , in the two-loop $\beta$-function of $\cdbe$ [@Sperling:2013xqa], (3+ 16\^2\^2) , and in the one-loop $\beta$-functions of the parameters that enter $\Delta \lambda^{1\ell}$, -\^2\^2 , \^2 , \[rgeYuk\] -8g\_t\^2\^2 , -g\_b\^2\^2 . Note that in eqs. (\[rgeEW\])–(\[rgeYuk\]) we distinguish the (hatted) $\drbar$-renormalized couplings of the MSSM from the (unhatted) $\msbar$-renormalized couplings of the SM (however, the distinction is irrelevant for the strong gauge coupling, which enters only the two-loop part of the calculation). Finally, to recover eq. (\[rgelam\]) we need to convert the remaining MSSM Yukawa couplings in the one-loop part of the derivative of $\lambda$ into their SM counterparts, and exploit in the two-loop part the tree-level MSSM relation to replace $(g^2+g^{\prime\,2})\,\cdbe^2$ with $4\lambda$. The EW-scale determination of the Higgs mass {#sec:weakscale} ============================================ A consistent determination of the Higgs mass in the EFT approach requires that the relation between the pole Higgs mass and the $\msbar$-renormalized quartic Higgs coupling at the EW scale be computed at the same perturbative order in the various SM couplings as the SUSY-scale threshold correction $\Delta\lambda$. In particular, the “gaugeless” calculation of refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid] requires a full determination of the one-loop corrections to the Higgs mass at the EW scale, combined with the two-loop corrections obtained in the limit $g=g^{\prime}=\lambda=0$. Denoting the scale at which we perform the calculation of the Higgs mass as $\qew$, this approximation implies \[2loopSMgl\] m\_h\^2 &=&\ &+& 8\^2C\_FN\_cg\_t\^2\^2m\_t\^2(3\_t\^2 + \_t) - 2\^2N\_cg\_t\^4m\_t\^2(9\_t\^2 - 3\_t + 2 + ) , where $G_F$ is the Fermi constant, $\delta^{1\ell}$ is the one-loop correction first computed in ref. [@Sirlin:1985ux], and $\ell_t =$ $\ln(\mt^2/\qew)$. The two-loop terms in the second line of eq. (\[2loopSMgl\]) are taken from ref. [@Degrassi:2012ry]. Note that the form of the ${\cal O}(g_t^4\,\mt^2)$ terms implies that the top-quark contribution to $\delta^{1\ell}$ is expressed in terms of $\msbar$-renormalized top and Higgs masses. Also, note that eq. (\[2loopSMgl\]) omits for conciseness all corrections involving the bottom and tau Yukawa couplings, which in the SM are greatly suppressed with respect to their top-only counterparts. In the SUSY-scale calculation of $\Delta\lambda^{2\ell}$ described in section \[sec:matching\] we go beyond the gaugeless limit, and we include contributions that involve both the EW gauge couplings and the strong gauge coupling. Strictly speaking, to match the accuracy of that calculation at the EW scale we only need to replace the ${\cal O}(g_t^2\,\gs^2\,\mt^2)$ terms in eq. (\[2loopSMgl\]) with the complete contributions arising from two-loop diagrams involving quarks and gluons, retaining the dependence on the external momentum in the Higgs self-energy. Explicit formulae for those contributions are provided in ref. [@Bezrukov:2012sa]. However, the full two-loop contributions to the relation between the pole Higgs mass, $\lambda(\qew)$ and $G_F$ in the SM are also available [@Buttazzo:2013uya; @Kniehl:2015nwa; @Martin:2019lqd]. These contributions can in principle be implemented in our EFT calculation, to prepare the ground for the eventual completion of the NNLL resummation of the large logarithmic corrections. We stress that the inclusion at the EW scale of two-loop corrections whose counterparts are still missing at the SUSY scale cannot be claimed to improve the overall accuracy of the calculation, but it does not degrade it either. Indeed, in the EFT approach the EW-scale and SUSY-scale sides of the calculation are separately free of log-enhanced terms, and the inclusion of additional pieces in only one side does not entail the risk of spoiling crucial cancellations between large corrections. The results of the full two-loop calculation of ref. [@Buttazzo:2013uya] were made public in the form of interpolating formulae. In particular, the relation between the Higgs quartic coupling and the pole Higgs and top masses reads [^7] \[interp\] (=) = 0.25208 + 0.00412( - 125.15) - 0.00008( - 173.34) 0.00060\_[th]{} , where the remaining input parameters – namely, the gauge-boson masses, $G_F$ and $\gs(\MZ)$ – are set to the central values listed in ref. [@Buttazzo:2013uya]. Eq. (\[interp\]) can be exploited to treat the measured value of the Higgs mass as an input parameter: it is then possible to evolve $\lambda$ to the SUSY scale using the RGEs of the SM, and use the threshold condition in eq. (\[looplam\]) to determine one of the MSSM parameters (e.g., a common mass scale for the stops, or the stop mixing parameter $X_t\,$, or $\tan\beta$) as a function of the others. In alternative, eq. (\[interp\]) can be inverted to predict the Higgs mass starting from a full set of MSSM parameters, using the value of $\lambda(\mt)$ obtained by evolving the $\lambda(\MS)$ computed in eq. (\[looplam\]) down to the EW scale. In the latter approach, a phenomenological analysis of the MSSM may well encounter points of the parameter space in which the prediction for the Higgs mass is several GeV away from the measured value. It is then legitimate to wonder about the range of validity of the linear interpolation involved in eq. (\[interp\]), which was obtained in a pure-SM context with the (small) uncertainty of the Higgs mass measurement in mind. ![Difference (in MeV) between the Higgs mass given as input to the code [mr]{} and the Higgs mass obtained by inserting in eq. (\[interp\]) the value of $\lambda(\mt)$ computed by [mr]{}, as a function of the input Higgs mass.[]{data-label="fig:buttvsmr"}](plots/figure1){width="13cm"} To test eq. (\[interp\]), we compare its predictions against those of the independent two-loop calculation presented in ref. [@Kniehl:2015nwa]. The latter is made available in the public code [mr]{} [@Kniehl:2016enc], which computes the $\msbar$-renormalized parameters of the SM Lagrangian from a set of physical observables that includes $G_F$ and the pole masses of the Higgs and gauge bosons and of the top and bottom quarks. We start from an input value for the pole Higgs mass ranging between $120$ GeV and $130$ GeV, feed it into [mr]{}, then insert the value of $\lambda(\mt)$ computed by [mr]{} into eq. (\[interp\]) to obtain a new prediction for $\mh$ according to the calculation of ref. [@Buttazzo:2013uya]. The remaining input parameters are fixed to the central values considered in ref. [@Buttazzo:2013uya]. In figure \[fig:buttvsmr\] we plot the difference between the initial and final values of $\mh$, which can be taken as a measure of the discrepancy between the two calculations, as a function of the initial value. In the vicinity of $\mh = 125$ GeV, where the interpolation involved in eq. (\[interp\]) can be expected to be accurate, the two values of $\mh$ differ by about $20$ MeV, i.e. by less than $0.02\%$. This is well within the theoretical uncertainty estimated in the last term of eq. (\[interp\]), which implies a shift in $\mh$ of about $150$ MeV. Such a good numerical agreement is particularly remarkable in view of the fact that the calculations of refs. [@Buttazzo:2013uya] and [@Kniehl:2015nwa] differ substantially in what concerns the renormalization of the Higgs vev and the corresponding treatment of the tadpole contributions (they also differ in the treatment of higher-order QCD corrections to the top Yukawa coupling). When we move away from the observed value of the Higgs mass, the discrepancy between the two calculations varies, reaching up to about $120$ MeV for the lowest considered value $\mh=120$ GeV. While such discrepancy remains within the theoretical uncertainty of eq. (\[interp\]), the behavior of the blue line in figure \[fig:buttvsmr\] suggests that, for values of $\mh$ a few GeV away from the observed one, the linear interpolation loses accuracy and the full dependence on the value of the Higgs mass should be taken into account. Finally, we performed an analogous test on the $\mt$ dependence of eq. (\[interp\]), keeping the value of the pole Higgs mass that we feed into [mr]{} fixed to $125.15$ GeV, and varying the pole top mass by $\pm 2$ GeV around its central value of $173.34$ GeV. We find that the difference between the initial value of $\mh$ and the one obtained by inserting in eq. (\[interp\]) the value of $\lambda(\mt)$ computed by [mr]{} varies only by about $10$ MeV in the considered range of $\mt$. This suggests that the linear interpolation of the dependence on the pole top mass in eq. (\[interp\]) is not problematic. Impact of the mixed corrections {#sec:tlnumbers} ================================= In this section we investigate the numerical impact of the mixed threshold corrections to the quartic Higgs coupling on the prediction for the Higgs mass in the MSSM with heavy superpartners. We use the code [mr]{} [@Kniehl:2016enc] to extract – at full two-loop accuracy – the $\msbar$-renormalized parameters of the SM Lagrangian from a set of physical observables, and to evolve them up to the SUSY scale using the three-loop RGEs of the SM. As mentioned in the previous section, in our EFT approach the fact that we combine a full two-loop calculation at the EW scale with an incomplete two-loop calculation at the SUSY scale does not entail the risk of spoiling crucial cancellations between large corrections. Throughout the section we use the world average $\mt = 173.34$ GeV [@ATLAS:2014wva] for the pole top mass, and fix the remaining physical inputs (other than the Higgs mass) to their current PDG values [@Tanabashi:2018oca], namely $G_F= 1.1663787 \times 10^{-5}$ GeV$^{-2}$, $\MZ = 91.1876$ GeV, $\MW = 80.385$ GeV, $\mb = 4.78$ GeV and $\alpha_s(\MZ)=0.1181$. In order to obtain a prediction for the Higgs mass from a full set of MSSM parameters, we vary the value of the pole mass $\mh$ that we give as input to [mr]{} until the value of the $\msbar$-renormalized SM parameter $\lambda(Q)$ returned by the code at the SUSY scale $Q=\MS$ coincides with the MSSM prediction of eq. (\[looplam\]). In addition to the mixed corrections to the quartic Higgs coupling computed in this paper, we use the results of refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid] for the full one-loop correction $\Delta \lambda^{1\ell}$ and for the “gaugeless” part of the two-loop correction $\Delta \lambda^{2\ell}$. We recall that, as discussed in ref. [@Bagnaschi:2017xid], we must also convert the $\msbar$-renormalized bottom Yukawa coupling $g_b(\MS)$ returned by [mr]{} into its $\drbar$-renormalized MSSM couterpart, $\hat g_b(\MS)$, to avoid the occurrence of potentially large $\tb$-enhanced terms in $\Delta \lambda^{2\ell}$. To this effect, we make use of eqs. (8), (9), (12) and (13) of ref. [@Bagnaschi:2017xid]. In figure \[fig:dmh\] we show the difference (in MeV) between the EFT predictions for the Higgs mass obtained with and without the inclusion of the mixed corrections to the quartic Higgs coupling. We consider a simplified MSSM scenario in which the masses of all superparticles (sfermions, gauginos and higgsinos) as well as the mass of the heavy Higgs doublet are set equal to the common SUSY scale $\MS$, which we vary between $1$ TeV and $20$ TeV. The left–right stop mixing parameter is fixed either as $X_t=\sqrt 6 \,\MS$ (lower, blue lines) or $X_t=2\,\MS$ (upper, red lines). In each set of lines the solid one is obtained with $\tan\beta=20$ and the dashed one with $\tan\beta=5$. The remaining input parameters are the trilinear Higgs–sfermion interaction terms for sbottoms and staus, which we fix as $A_b=A_\tau=A_t$. We remark that all of the MSSM parameters are interpreted as $\drbar$-renormalized quantities expressed at the renormalization scale $Q=\MS$. The star on each line marks the value of $\MS$ for which the improved calculation of $\Delta \lambda^{2\ell}$ (i.e., including the mixed corrections) leads to the observed value of the Higgs mass, $\mh = 125.09$ GeV [@Aad:2015zhl]. Figure \[fig:dmh\] shows that, in the considered scenario, the mixed corrections computed in this paper are fairly small, shifting the MSSM prediction for $\mh$ downwards by ${\cal O}(100)$ MeV. The fact that the corrections are reduced (in absolute value) for larger values of $\MS$ is partially due to the scale dependence of the relevant couplings: with the exception of $\gp$, they all decrease with increasing $Q=\MS$. The comparison between the solid and dashed lines in each set shows that the dependence of the mixed corrections on $\tb$ is rather mild (in contrast, the overall prediction for $\mh$ depends strongly on $\tb$, as shown by the relative position of the stars on the solid and dashed lines). Finally, the comparison between the lower (blue) and upper (red) sets of lines shows that the mixed corrections depend rather strongly on the ratio $\|X_t/\MS\|$, with smaller ratios leading to smaller corrections. Indeed, we checked that for $\|X_t/\MS\| < 1$ the effect of the corrections can be at most of ${\cal O}(10)$ MeV in the considered scenario. ![Difference (in MeV) between the predictions for the Higgs mass obtained with and without the inclusion of the mixed corrections to the quartic Higgs coupling, as a function of a common SUSY scale $\MS$, for $X_t=\sqrt{6}\,\MS$ (lower, blue lines) or $X_t=2\,\MS$ (upper, red lines), and $A_b=A_\tau=A_t$. In each set of lines the solid one is obtained with $\tan\beta=20$ and the dashed one with $\tan\beta=5$. The star on each line marks the value of $\MS$ for which the improved calculation of $\Delta \lambda^{2\ell}$ leads to $\mh = 125.09$ GeV.[]{data-label="fig:dmh"}](plots/figure2){width="13cm"} An alternative way to assess the effect of the newly-computed corrections consists in taking the measured value of the Higgs mass as an input parameter, and using the matching condition on the quartic Higgs coupling at the SUSY scale, eq. (\[looplam\]), to constrain the MSSM parameters. In figure \[fig:MSXt\] we show the values of $\MS$ and $X_t$ that lead to $\mh = 125.09$ GeV, in the simplified scenario with degenerate superparticle and heavy-Higgs masses, for $\tan\beta=20$ and $A_b=A_\tau=A_t$. We focus on values of the ratio $X_t/\MS$ between $2$ and $2.5$, which allow for SUSY masses around 2 TeV (i.e., roughly at the limit of the HL-LHC reach [@CidVidal:2018eel]). Once again, all of the MSSM parameters are interpreted as $\drbar$-renormalized quantities expressed at the renormalization scale $Q=\MS$. The (black) dotted line in figure \[fig:MSXt\] is obtained including only the one-loop threshold corrections to the Higgs quartic coupling, the (blue) dashed line includes the two-loop corrections in the gaugeless limit, and the (red) solid line includes also the effect of the mixed corrections. Unsurprisingly, the comparison between the three lines shows that the mixed corrections are sub-dominant with respect to the two-loop corrections computed in the gaugeless limit. Nevertheless, they can shift the value of $\MS$ that leads to the observed Higgs mass by ${\cal O}(100)$ GeV in the direction of heavier superparticles. ![Values of the common SUSY scale $\MS$ and of the stop mixing term $X_t$ that lead to $\mh = 125.09$ GeV, for $\tan\beta=20$ and $A_b=A_\tau=A_t$. The dotted line is obtained including only the one-loop threshold corrections to the Higgs quartic coupling, the dashed line includes the two-loop corrections in the gaugeless limit, and the solid line includes also the effect of the mixed corrections.[]{data-label="fig:MSXt"}](plots/figure3){width="12.5cm"} Conclusions {#sec:conclusions} =========== If the MSSM is realized in nature, both the measured value of the Higgs mass and the negative results of the searches for superparticles at the LHC suggest some degree of separation between the SUSY scale and the EW scale. In this scenario, the MSSM prediction for the Higgs mass is subject to potentially large logarithmic corrections, making a fixed-order calculation of $\mh$ inadequate and calling for an all-orders resummation in the EFT approach. In this paper we improved the EFT calculation of the Higgs mass in the MSSM, by computing the class of two-loop threshold corrections to the quartic Higgs coupling that involve both the strong and the EW gauge couplings. Combined with the ${\cal O}(g_t^4\,g_s^2)$ and ${\cal O}(g_b^4\,g_s^2)$ corrections previously provided in refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid], this completes the calculation of the two-loop threshold corrections that involve the strong gauge coupling. Our calculation involves novel complications with respect to the case of the “gaugeless” two-loop corrections of refs. [@Bagnaschi:2014rsa; @Bagnaschi:2017xid]. While in the latter all two-loop diagrams could be computed in the effective potential approach (i.e., for vanishing external momenta), the mixed corrections include contributions from the ${\cal O}(p^2)$ parts of the two-loop self-energies of the Higgs and gauge bosons. We obtained results for the threshold corrections to the quartic Higgs coupling valid for generic values of all the relevant SUSY parameters, which we make available on request in electronic form. For the sake of illustration, in section \[sec:matching\] we provided explicit formulae in the simplified limit of degenerate superparticle masses. We remark that our calculation can be trivially adapted also to the split-SUSY scenario in which the gluino is much lighter than the squarks, by taking the limit of vanishing gluino mass in our full results. On the other hand, in scenarios in which the gluino is heavier than the squarks the two-loop corrections to the quartic Higgs coupling contain potentially large terms enhanced by powers of the ratios between the gluino mass and the squark masses. This is a well-known aspect of the $\drbar$ renormalization of the squark masses and trilinear couplings [@Degrassi:2001yf; @Vega:2015fna; @Bagnaschi:2017xid; @Braathen:2016mmb], which could be addressed either by devising an “on-shell” scheme adapted to the heavy-SUSY setup, or by building a tower of EFTs in which the gluino is independently decoupled at a higher scale than the squarks. In the EFT approach, the inclusion of new threshold corrections to the quartic Higgs coupling at the SUSY scale mandates that corrections of the same perturbative order in the relevant couplings be included in the calculation of the pole Higgs mass at the EW scale. In the simplest heavy-SUSY setup in which the effective theory valid below the SUSY scale is just the SM, we can exploit the full two-loop calculations of the relation between $\mh$, $\lambda(\qew)$ and $G_F$ presented in refs. [@Buttazzo:2013uya; @Kniehl:2015nwa; @Martin:2019lqd]. In section \[sec:weakscale\] we compared the results of two of those calculations, refs. [@Buttazzo:2013uya] and [@Kniehl:2015nwa], discussing the range of validity of an interpolating formula provided in ref. [@Buttazzo:2013uya]. In section \[sec:tlnumbers\] we investigated the numerical impact of the mixed corrections to the quartic Higgs coupling. We considered a simplified MSSM scenario with degenerate masses for all superparticles and for the heavy Higgs doublet, focusing on the region of the parameter space in which the prediction for the Higgs mass is close to the observed value and the stop squarks are in principle still accessible at the HL-LHC. We used the code [ mr]{} [@Kniehl:2016enc], based on the calculation of ref. [@Kniehl:2015nwa], to extract all of the SM couplings from a set of physical observables and to evolve them up to the SUSY scale, where we compare the value of the quartic Higgs coupling with its MSSM prediction. We found that the impact of the newly-computed two-loop corrections on the prediction for the Higgs mass tends to be small, and it is certainly sub-dominant with respect to the impact of the “gaugeless” two-loop corrections. In the considered scenario, the mixed corrections can shift the prediction for the Higgs mass by ${\cal O}(100)$ MeV, and they can shift the values of the stop masses required to obtain the observed value of $\mh$ by ${\cal O}(100)$ GeV. We stress that the smallness of these effects is in fact a desirable feature of the EFT approach to the calculation of the Higgs mass. While the logarithmically enhanced corrections are accounted for by the evolution of the parameters between the matching scale and the EW scale, and high-precision calculations at the EW scale can be borrowed from the SM, the small impact of new two-loop corrections computed at the SUSY scale suggests that the uncertainty associated to uncomputed higher-order terms should be well under control in the considered scenario. On the other hand, we recall that the accuracy of the measurement of the Higgs mass at the LHC has already reached the level of $100\!-\!200$ MeV [@Tanabashi:2018oca] – i.e., it is comparable to the effects of the corrections discussed in this paper – and will improve further when more data are analyzed. If SUSY eventually shows up at the TeV scale, the mass and couplings of the SM-like Higgs boson will serve as precision observables to constrain MSSM parameters that might not be directly accessible by experiment. To this purpose, the accuracy of the theoretical predictions will have to match the experimental one, making a full inclusion of two-loop effects in the Higgs-mass calculation unavoidable. Our results should be viewed as a necessary step in that direction. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Mark Goodsell for useful discussions. The work of S. P. and P. S. is supported in part by French state funds managed by the Agence Nationale de la Recherche (ANR), in the context of the LABEX ILP (ANR-11-IDEX-0004-02, ANR-10-LABX-63) and of the grant “HiggsAutomator” (ANR-15-CE31-0002). G. D. acknowledges warm hospitality at LPTHE and support from “HiggsAutomator” during the completion of this work. [^1]: We focus here on the MSSM with real parameters. Significant efforts have also been devoted to the Higgs-mass calculation in the presence of CP-violating phases [@Pilaftsis:1999qt; @Choi:2000wz; @Carena:2000yi; @Frank:2006yh; @Heinemeyer:2007aq; @Hollik:2014wea; @Hollik:2014bua; @Hollik:2015ema; @Goodsell:2016udb; @Passehr:2017ufr; @Borowka:2018anu], as well as in non-minimal SUSY extensions of the SM. [^2]: A partial N$^3$LL resummation of the corrections involving only the highest powers of the strong gauge coupling is also available, combining the three-loop matching condition of ref. [@Harlander:2018yhj] with SM results from refs. [@Bednyakov:2013eba; @Martin:2013gka; @Martin:2014cxa; @Martin:2015eia]. [^3]: Here and thereafter, we use the shortcuts $c_\phi\equiv\cos\phi$ and $s_\phi\equiv\sin\phi$ for a generic angle $\phi$. [^4]: We denote with a hat $\drbar$-renormalized couplings of the MSSM, and without a hat $\msbar$-renormalized couplings of the SM. However, in the two-loop part of the corrections the distinction between hatted and un-hatted couplings amounts to a higher-order effect, thus we will drop the hats there to reduce clutter. [^5]: For the self-energies of both the Higgs boson and the $Z$ boson, we adopt in this paper the sign convention according to which $m^2_{\rm pole}= m^2_{\rm run} - \Pi(m^2)$. The WFR for the Higgs boson is then $Z_h = 1 - d\Pi_{hh}(p^2)/dp^2$. Note that this is the opposite of the sign convention adopted in refs. [@Martin:2003qz; @Martin:2003it; @Martin:2004kr] and [@Braathen:2018htl]. [^6]: The fact that these mixed corrections should depend only on the quark and squark contributions to the gauge-boson self-energy can be easily inferred by considering the renormalization of the gauge couplings of the leptons. [^7]: Note that our normalization for $\lambda$ differs from the one in refs. [@Buttazzo:2013uya; @Kniehl:2015nwa; @Martin:2019lqd] by a factor 2.
--- bibliography: - 'biblio.bib' title: 'Measurement of -boson production at large rapidities in collisions at ${\mathbf {\snn=5.02}}$ TeV' --- Acknowledgements {#acknowledgements .unnumbered} ================ The ALICE Collaboration would like to thank H. Paukkunen for providing the pQCD calculations with EPS09 and EPPS16, and F. Lyonnet and A. Kusina for providing the pQCD calculations with nCTEQ15. The ALICE Collaboration {#app:collab} =======================
--- abstract: 'Torsion is a geometrical object, required by quantum mechanics in curved spacetime, which may naturally solve fundamental problems of general theory of relativity and cosmology. The black-hole cosmology, resulting from torsion, could be a scenario uniting the ideas of the big bounce and inflation, which were the subject of a recent debate of renowned cosmologists.' author: - '[Nikodem Pop[ł]{}awski]{}' title: THE SIMPLEST ORIGIN OF THE BIG BOUNCE AND INFLATION --- [International Journal of Modern Physics D]{}\ Vol. [27]{}, No. 14 (2018) 1847020\ ©World Scientific Publishing Company I have recently read an interesting article, [Pop Goes the Universe]{}, in the January 2017 edition of Scientific American, written by Anna Ijjas, Paul Steinhardt, and Abraham Loeb (ISL) [@ISL1]. This article follows an article by the same authors, [Inflationary paradigm in trouble after Planck 2013]{} [@ISL2]. They state that cosmologists should reassess their favored inflation paradigm because it has become nonempirical science, and consider new ideas about how the Universe began, namely, the big bounce. Their statements caused a group of 33 renowned physicists, including 4 Nobel Prize in Physics laureates, to write a reply, [A Cosmic Controversy]{}, categorically disagreeing with ISL about the testability of inflation and defending the success of inflationary models [@reply]. I agree with the ISL critique of the inflation paradigm, but I also agree with these 33 physicists that some models of inflation are testable. As a solution to this dispute, I propose a scenario, in which every black hole creates a new universe on the other side of its event horizon. Accordingly, our Universe may have originated from a black hole existing in another universe. This scenario considers a geometrical property of spacetime called torsion, which can generate both the big bounce and inflationary dynamics, and I published it as [Universe in a black hole in Einstein–Cartan gravity]{} in [@ApJ]. The current theory of the origin of our Universe, which is based on Einstein’s general theory of relativity (GR), assumes that our Universe has started more than 13 billion years ago from an extremely hot and dense state called the big bang. The big-bang cosmology successfully describes primordial nucleosynthesis (production of the lightest elements in the early Universe) and predicts the cosmic microwave background (CMB) radiation, which was emitted about 370,000 years after the big bang and which we observe coming from all directions in the sky. In order to explain why the Universe that we observe today appears at the largest scales flat (not curved) and nearly uniform all over space and in every direction, the theory of cosmic inflation has been proposed, according to which the early Universe went through an extremely accelerated (exponential) expansion by an enormous factor in volume [@infl]. Inflation also can predict the form of primordial density fluctuations observed in the CMB, which seed the structure formation in the Universe: stars, galaxies, and galaxy clusters. Inflation requires that the Universe be filled with an exotic form of high-density energy that gravitationally self-repels, enhancing and speeding up the expansion of the Universe. This inflationary energy is hypothetical and we have no evidence that it exists. There have been hundreds of models regarding the origin of inflation in the last 36 years, generating different rates of inflation. The most common models attribute inflation to a hypothetical scalar field called inflaton. Even with the inflaton, inflation is not a precise theory but rather a highly flexible framework that admits many possibilities. Moreover, inflation does not tell us why the big bang happened or what created the initial volume of space that evolved into the Universe observed today. According to the scientists analyzing the results from a Planck satellite of the European Space Agency, the new map of the CMB confirms inflation [@Planck2013]. ISL do not agree with this interpretation of the Planck 2013 results. These results eliminate a wide range of more complex inflationary models and favor models with a single scalar field, as reported by the Planck Collaboration. Among single-scalar-field models, Planck 2013 disfavors the simplest (power-law) inflaton models relative to models with plateau-like potentials. However, as ISL point out, plateau-like models have serious problems: they require an initially smooth Universe (the initial conditions problem), are in the class of eternally inflating models (which leads to unpredictable creation of new universes in the multiverse), and are unlikely compared to power-law inflation (have much smaller scalar-field range and amount of expansion) [@ISL1]. In addition, scalar-field plateau-like models require at least three parameters. The big bang itself is also unphysical: the big-bang Universe started from a point of infinite density, called singularity. ISL advocate for another scenario in which the Universe began with a big bounce, a transition from a contracting cosmological phase to the current expanding phase. They write [@ISL1]: “Although most cosmologists assume a bang, there is currently no evidence — zero — to say whether the event that occurred 13.7 billion years ago was a bang or a bounce. Yet a bounce, as opposed to a bang, does not require a subsequent period of inflation to create a universe like the one we find, so bounce theories represent a dramatic shift away from the inflation paradigm.” In bounce theories, contraction before a bounce can smooth and flatten the Universe, which is what inflation was supposed to do when it was proposed. In addition, bounce theories do not produce multiple universes which are predicted by inflationary scenarios. The most natural theoretical-physics mechanism which generates a bounce comes from an old (the 1920s) extension of general relativity, called the Einstein–Cartan (EC) or Einstein–Cartan–Sciama–Kibble theory of gravity [@SK; @EC]. This theory extends GR by removing its artificial symmetry constraint on the spacetime affine connection (the connection is a geometrical quantity which tells us how to do calculus in a curved space). Instead, a part of the affine connection called the torsion tensor can be different from zero and turns out to be related to the quantum-mechanical, intrinsic angular momentum of elementary particles called spin, as shown by Dennis Sciama and Tom Kibble in the 1960s [@SK]. Even though the spin is a quantum phenomenon, it originates, like the mass, from the Casimir invariants of the Poincaré algebra. The conservation law for the total angular momentum (orbital plus spin) of a particle in curved spacetime that admits the exchange between its orbital and intrinsic components (spin–orbit interaction) requires torsion. The field equations give a linear differential relation between the curvature and energy–momentum of matter, as in GR, and a linear algebraic relation between the torsion and spin of matter. These two relations introduce effective corrections to the energy–momentum tensor of matter, which are significant only at extremely high densities, much larger than the density of nuclear matter, existing in black holes and near the big bang. In a May 2012 article in Inside Science, [Every Black Hole Contains a New Universe]{}, I wrote [@Inside]: “In this picture, spins in particles interact with spacetime and endow it with a property called torsion. To understand torsion, imagine spacetime not as a two-dimensional canvas, but as a flexible, one-dimensional rod. Bending the rod corresponds to curving spacetime, and twisting the rod corresponds to spacetime torsion. If a rod is thin, you can bend it, but it is hard to see if it is twisted or not.” At such high densities, torsion manifests itself as a force that counters gravity, which was discovered by Andrzej Trautman and Friedrich Hehl and their collaborators in the 1970s [@avert]. As in GR, very massive stars end up as black holes: regions of space from which nothing, not even light, can escape. Gravitational attraction due to curvature initially overcomes repulsion due to torsion and matter in a black hole collapses, but eventually the coupling between torsion and spin (acting like gravitational repulsion) becomes very strong and prevents the matter from compressing indefinitely to a singularity. The matter instead reaches a state of finite, extremely large density, stops collapsing, undergoes a bounce like a compressed spring, and starts rapidly expanding. Extremely strong gravitational forces near this state cause an intense, quantum particle production, increasing the mass inside a black hole by many orders of magnitude and strengthening gravitational repulsion that powers the bounce. The rapid recoil after such a big bounce could be what has led to our expanding Universe. It also explains why the present large-scale Universe appears at flat and nearly uniform all over space and in every direction, without needing scalar-field inflation, which I showed in: [Cosmology with torsion: An alternative to cosmic inflation]{} [@cosmology]. The energy of matter at the big bounce is an order of magnitude higher than the Planck energy. Recent observations of high-energy photons from gamma-ray bursts, however, indicate that spacetime may behave classically even at scales above the Planck energy. The classical spin-torsion mechanism of the bounce may thus be justified. Furthermore, the EC theory passes all tests of GR because both theories give significantly different predictions only at extremely high densities that exist in black holes or in the very early Universe. Torsion in the EC gravity therefore provides a theoretical explanation of a scenario (suggested also by Igor Novikov, Lee Smolin, and Stephen Hawking [@BH]), according to which every black hole produces a new, baby universe inside and becomes an Einstein–Rosen bridge (wormhole) that connects this universe to the parent universe in which the black hole exists: [Radial motion into an Einstein–Rosen bridge]{} [@bridge]. In the new universe, the parent universe appears as the other side of the only white hole, a region of space that cannot be entered from the outside and which can be thought of as the time reverse of a black hole. Accordingly, our own Universe could have originated from the interior of a black hole existing in another universe (our Universe as the interior of a black hole was proposed by Raj Pathria [@Pat]). The motion of matter through the black hole’s boundary called an event horizon can only happen in one direction, providing a past–future asymmetry at the horizon and thus everywhere in the baby universe. The arrow of time in such a universe would therefore be inherited, through torsion, from the parent universe [@cosmology]. The new universe in a black hole is closed: finite and without boundaries (with the exception of the white hole, which connects it to the parent universe). It can be thought of as a three-dimensional analogue of the two-dimensional surface of a sphere: if you continue to go in any direction, you eventually would come back from the opposite direction. The formation and evolution of such a universe is not visible for external observers in the parent universe, for whom the event horizon’s formation and all subsequent processes would occur after an infinite amount of time had elapsed (because of the time dilation by gravity). A baby universe is thus a separate branch of spacetime with its own timeline. ISL write [@ISL1]: “bouncing theories have an important advantage compared with inflation: they do not produce a multimess.” The black-hole cosmology presented here produces other universes but only beyond the event horizons of black holes, thus the resulting multiverse is organized. In a recent article, [Non-parametric reconstruction of an inflaton potential from Einstein–Cartan–Sciama–Kibble gravity with particle production]{}, written with Shantanu Desai [@SD], we analyzed numerically the dynamics of the early Universe based on the EC theory of gravity with quantum particle production and proposed in my ApJ 2016 article [@ApJ]. The scale factor of the Universe $a$ and its temperature $T$ satisfy the Friedmann equations: $$\begin{aligned} & & \frac{{\dot{a}}^2}{c^2}+1=\frac{1}{3}\kappa\tilde{\epsilon}a^2=\frac{1}{3}\kappa(h_\star T^4-\alpha h_{n\textrm{f}}^2 T^6)a^2, \label{dynamics1} \\ & & \frac{\dot{a}}{a}+\frac{\dot{T}}{T}=\frac{cK}{3h_{n1}T^3}, \label{dynamics2}\end{aligned}$$ where dot denotes the time derivative, $\tilde{\epsilon}$ is the effective energy density (the positive term proportional to $T^4$ comes from relativistic matter and the negative term proportional to $T^6$ comes from the spin-torsion gravitational repulsion), $\kappa=8\pi G/c^4$, $h$-constants depend on the number of the thermal degrees of freedom (the number of elementary particle species), and $K=\beta(\kappa\tilde{\epsilon})^2$ is the particle production rate. This dynamics contains only one unknown parameter, the particle production coefficient $\beta$. We found that for different values of this coefficient, the universe can have different numbers of cycles, each of which begins with a nonsingular bounce followed by expansion to a crunch, followed by contraction to the next bounce. Each new cycle, compared to the previous cycle, lasts longer and represents the universe with a larger amount of matter. The last bounce: the big bounce, is what we refer to as the big bang. From the obtained time dependence of the scale factor (size) of the universe, we reconstructed a scalar field potential which would give the same dynamics of the early universe. We did it because the CMB quantities measured by the Planck satellite can be directly calculated for scalar-field inflation models. For a particular range of the particle production coefficient, we obtained one bounce (the big bounce) followed by a nearly exponential expansion of the Universe, which lasted for an extremely short amount of time and which smoothed out and flattened the Universe to the observed values, as shown in Fig. \[scale\] [@SD]. We found that the reconstructed potential is of a plateau-like shape, as shown in Fig. \[potential\] [@SD], which is supported by the Planck 2013 data [@Planck2013]. From this potential, we calculated the CMB quantities measured by the Planck satellite, and found that they do not significantly depend on the scale factor at the big bounce. Our predictions for these quantities are consistent with the Planck 2015 observations [@Planck2015]. This scenario can thus produce big bounce and plateau-like inflation without a scalar field. ![The ratio of the scale factor $a(t)$ to its initial value (at the big bounce) as a function of time for a particlular value of the particle production coefficient $\beta$. The dashed magenta line represents the transition from an accelerating (torsion-dominated) phase to a decelerating (radiation-dominated) phase. We obtain about 60 $e$-folds.[]{data-label="scale"}](a.png){width="50.00000%"} ![The scalar-field potential $V(\phi)$ giving the same dynamics as Eqs. (\[dynamics1\]) and (\[dynamics2\]). The vertical line indicates the end of inflation when the Universe transitions from acceleration to deceleration. The shape of the potential is plateau-like.[]{data-label="potential"}](Vofphi.png){width="50.00000%"} Unlike plateau-like models based on a scalar field, our scenario escapes the problems noted by ISL: it does not rely on a hypothetical scalar field, does not depend on the initial conditions of the Universe, avoids eternal inflation, and is simpler — has only one parameter (which should be predicted by a future quantum theory of gravity). And most importantly, it eliminates the initial singularity problem, which is not addressed by scalar-field inflation [@ApJ]. This scenario is motivated by including quantum-mechanical spin in a theory of gravity, and the nonsingular big bounce and inflation are derived, not imposed. Inflation models assume that the inflaton starts inflation and then somehow decays into ordinary matter. The EC theory gives a simpler dynamics: spin-generated torsion produces the bounce, quantum particle production starts inflation (which was also suggested in [@Sing]), weakening of torsion ends inflation (without reheating), and weakening of curvature ends particle production, after which the Universe expands through standard radiation-dominated and matter-dominated eras. In addition, torsion modifies the Dirac equation that describes the quantum-mechanical behavior of fermions (quarks and leptons) that form ordinary matter. These particles must be spatially extended, which may solve the ultraviolet-divergence problems in quantum field theory arising from treating them as points. The spatial extension of fermions arising from torsion is at the level of the Cartan length, which for an electron is about 100 million Planck lengths and can be tested in the future: [Nonsingular Dirac particles in spacetime with torsion]{} [@nonsingular]. This modification may also be responsible for the observed imbalance of matter and antimatter in the Universe and could relate the apparently missing antimatter to dark matter, a mysterious form of matter that does not interact electromagnetically and accounts for a majority of the matter in the Universe: [Matter–antimatter asymmetry and dark matter from torsion]{} [@matter]. Finally, torsion may be the source of dark energy, a mysterious form of energy that permeates all of space and increases the present rate of expansion of the Universe, allowing it to grow infinitely large and to last infinitely long: [Affine theory of gravitation]{} [@affine]. ISL wrote [@ISL1]: “The fact that our leading ideas have not worked out is a historic opportunity for a theoretical breakthrough. Instead of closing the book on the early universe, we should recognize that cosmology is wide open”. The 33 physicists wrote in their reply [@reply]: “Like any scientific theory, inflation need not address all conceivable questions. Inflationary models, like all scientific theories, rest on a set of assumptions, and to understand those assumptions we might need to appeal to some deeper theory. This, however, does not undermine the success of inflationary models.” Here is my response: The EC theory of gravity may be that deeper theory which has been waiting for its breakthrough. Spacetime torsion is a geometrical phenomenon, required by quantum mechanics, which may naturally solve fundamental problems of GR and cosmology. It provides the simplest and most natural mechanism for a bounce that started the expansion of our Universe. With quantum particle production, it also provides the simplest model of inflation that does not need hypothetical fields, lasts for a finite time, and is consistent with the Planck data. 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--- abstract: 'In this paper we exploit the structural properties of standard and non-standard models of set theory to produce models of set theory admitting automorphisms that are well-behaved along an initial segment of their ordinals. $\mathrm{NFU}$ is Ronald Jensen’s modification of Quine’s ‘New Foundations’ set theory that allows non-sets into the domain of discourse. The axioms $\mathrm{AxCount}$, $\mathrm{AxCount}_\leq$ and $\mathrm{AxCount}_\geq$ each extend $\mathrm{NFU}$ by placing restrictions on the cardinality of a finite set of singletons relative to the cardinality of its union. Using the results about automorphisms of models of set theory we separate the consistency strengths of these three extensions of $\mathrm{NFU}$. We show that $\mathrm{NFU}+\mathrm{AxCount}$ proves the consistency of $\mathrm{NFU}+\mathrm{AxCount}_\leq$, and $\mathrm{NFU}+\mathrm{AxCount}_\leq$ proves the consistency of $\mathrm{NFU}+\mathrm{AxCount}_\geq$.' author: - Zachiri McKenzie bibliography: - 'automorphismsandnonstandardmodels24.bib' title: 'Automorphisms of models of set theory and extensions of $\mathrm{NFU}$' --- Introduction ============ In [@jen69] Ronald Jensen introduces a weakening of Quine’s ‘New Foundations’ ($\mathrm{NF}$), which he calls $\mathrm{NFU}$, by allowing urelements (non-sets) into the domain of discourse. Despite the innocuous appearance of this weakening Jensen, in the same paper, shows that $\mathrm{NFU}$ is equiconsistent with a weak subsystem of $\mathrm{ZFC}$ and unlike $\mathrm{NF}$ is consistent with both the Axiom of Choice and the negation of the Axiom of Infinity. In the early nineties Randall Holmes and Robert Solovay embarked upon the project of determining the relative consistency strengths of natural extensions of $\mathrm{NFU}$ and the strengths of these extensions relative to subsystems and extensions of $\mathrm{ZFC}$. The fruits of this work can be seen in [@hol01], [@solXX], [@solXY] and [@ena04] which pinpoint the exact strength of a variety of natural extensions of $\mathrm{NFU}$ relative to subsystems and extensions of $\mathrm{ZFC}$.\ \ Throughout this paper we will use $\mathrm{NFU}$ to denote the theory described by Jensen in [@jen69] supplemented with both the Axiom of Choice and the Axiom of Infinity. We will study three extensions of $\mathrm{NFU}$ that are obtained by adding the Axiom of Counting ($\mathrm{AxCount}$), $\mathrm{AxCount}_\leq$ and $\mathrm{AxCount}_\geq$ respectively. The first of these axioms was proposed in [@ros53] to facilitate induction in $\mathrm{NF}$. Both $\mathrm{AxCount}_\leq$ and $\mathrm{AxCount}_\geq$ are natural weakenings of $\mathrm{AxCount}$ that were introduced by Thomas Forster in his Ph.D. thesis [@for77]. The combined work of Rolland Hinnion [@hin75] and Jensen [@jen69] shows that $\mathrm{NFU}+\mathrm{AxCount}_\leq$ proves the consistency of $\mathrm{NFU}$. We improve this result by showing that $\mathrm{NFU}+\mathrm{AxCount}_\leq$ proves the consistency of $\mathrm{NFU}+\mathrm{AxCount}_\geq$. We also show that $\mathrm{NFU}+\mathrm{AxCount}$ proves the consistency of $\mathrm{NFU}+\mathrm{AxCount}_\leq$.\ \ The techniques developed in [@jen69] coupled with the observations in [@bof88] establish a strong link between models of $\mathrm{NFU}$ and models of subsystems of $\mathrm{ZFC}$ that admit non-trivial automorphism. In light of this connection and motivated by questions related to the strength of the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$ Randall Holmes asked the following: \[Q:HolmesQuestion\] Is there model $\mathcal{M} \models \mathrm{ZFC}$ that admits an automorphism $j: \mathcal{M} \longrightarrow \mathcal{M}$ such that - $j(n) \geq n$ for all $\mathcal{M} \models n \in \omega$, - $j(\alpha) < \alpha$ for some $\mathcal{M} \models \alpha \in \omega_1$? A model equipped with such an automorphism would yield a model of $\mathrm{NFU}+\mathrm{AxCount}_\leq$ in which the set of infinite cardinal numbers is countable. In Sections \[Sec:AutomorphismsFromStandardModels\] and \[Sec:AutomorphismsFromNonStandardModels\] of this paper we construct models of subsystems of $\mathrm{ZFC}$ equipped with automorphisms that are well-behaved along initial segment of their ordinals. In Section \[Sec:AutomorphismsFromStandardModels\] we will show that models of set theory admitting automorphisms that move no points down along an initial segment of their ordinals can be built from standard models of set theory. This result allows us to show that every complete consistent extension of $\mathrm{ZFC}$ has a model which does not move any ordinal down. We then show in Section \[Sec:AutomorphismsFromNonStandardModels\] that models of set theory admitting automorphism which move no natural number down but do move a recursive ordinal down can be built from non-standard $\omega$-models of set theory. This allows us to give a positive answer to Question \[Q:HolmesQuestion\] even when $\omega_1$ is replaced by $\omega_1^{\mathrm{ck}}$.\ \ In section \[Sec:NFU\] we describe the set theory $\mathrm{NFU}$. We will survey work in [@jen69] and [@bof88] which shows that models $\mathrm{NFU}$ can be built from models of subsystems of $\mathrm{ZFC}$ that admit non-trivial automorphism. We also survey Holmes’s adaption to $\mathrm{NFU}$ of techniques developed in [@hin75]. These techniques show that subsystems of $\mathrm{ZFC}$ can be interpreted in extensions of $\mathrm{NFU}$. In Sections \[Sec:NFUPlusAxCountGEQ\] and \[Sec:NFUPlusAxCountLEQ\] we apply the model theoretic results proved in Sections \[Sec:AutomorphismsFromStandardModels\] and \[Sec:AutomorphismsFromNonStandardModels\] to shed light of the strength of the theories $\mathrm{NFU}+\mathrm{AxCount}_\leq$ and $\mathrm{NFU}+\mathrm{AxCount}_\geq$. It is in these sections that we separate the consistency strengths of $\mathrm{NFU}+\mathrm{AxCount}$, $\mathrm{NFU}+\mathrm{AxCount}_\leq$ and $\mathrm{NFU}+\mathrm{AxCount}_\geq$.\ \ The study of models admitting automorphisms has also recently yielded model theoretic characterisations of set theories and other foundational theories. In [@ena04] Ali Enayat provides an elegant characterisation of a large cardinal extension of $\mathrm{ZFC}$ in terms of the existence of a model of a weak subsystem of $\mathrm{ZFC}$ that admits a well-behaved automorphism. Enayat’s work [@ena06] proves similar characterisations for a variety of subsystems of second order arithmetic. We have intentionally organised the background and results relating to automorphisms of models of subsystems of $\mathrm{ZFC}$ into Sections \[Sec:Background\], \[Sec:AutomorphismsFromStandardModels\] and \[Sec:AutomorphismsFromNonStandardModels\] so readers who are only interested in these results can skip Sections \[Sec:NFU\], \[Sec:NFUPlusAxCountGEQ\] and \[Sec:NFUPlusAxCountLEQ\]. Background {#Sec:Background} ========== Throughout this article we will use $\mathcal{L}$ to denote the language of set theory. If $A, B, C, \ldots$ are new relation, function or constant symbols then we will use $\mathcal{L}_{A, B, C, \ldots}$ to denote the language obtained by adding $A, B, C, \ldots$ to $\mathcal{L}$.\ \ Given an extension of the language of set theory $\mathcal{L}^\prime$, we will often have cause to consider the Lévy hierarchy of $\mathcal{L}^\prime$ formulae which we will denote $\Delta_0(\mathcal{L}^\prime)$, $\Sigma_1(\mathcal{L}^\prime), \Pi_1(\mathcal{L}^\prime), \ldots$. If $T$ is an $\mathcal{L}^\prime$-theory then we say that an $\mathcal{L}^\prime$-formula $\phi$ is $\Delta_n^T$ if and only if $\phi$ is provably equivalent in $T$ to both a $\Sigma_n(\mathcal{L}^\prime)$ and a $\Pi_n(\mathcal{L}^\prime)$ formula. In addition to the Lévy classes we will also have cause to consider a hierarchy of subclasses of $\mathcal{L}$ introduced by Moto-o Takahashi in [@tak72]. Following [@fk91] we define the class of $\Delta_0^\mathcal{P}$ formulae to be the class of well-formed formulae built up inductively from atomic formulae in the form $x \in y$ and bounded quantification of the form $\forall x \in y$, $\exists x \in y$, $\forall x \subseteq y$ and $\exists x \subseteq y$ using the connectives $\land$, $\lor$, $\neg$ and $\Rightarrow$. The class of all $\Sigma_{n+1}^\mathcal{P}$ formulae is defined inductively to be the class of all formulae in the form $\exists \vec{x} \psi$ where $\psi$ is $\Pi_n^\mathcal{P}$. And, the class of all $\Pi_{n+1}^\mathcal{P}$ formulae is similarly defined to be the class of all formulae in the form $\forall \vec{x} \psi$ where $\psi$ is $\Sigma_n^\mathcal{P}$.\ \ Let $\mathcal{L}^\prime$ be a language. We will denote $\mathcal{L}^\prime$-structures, which we will also call models, using the calligraphic roman letters ($\mathcal{M}$, $\mathcal{N}$, etc.). If $\mathcal{M}$ is an $\mathcal{L}^\prime$-structure and $A$ is a relation, function or constant symbol in $\mathcal{L}^\prime$ then we will use $M^\mathcal{M}$ to denote the underlying set of $\mathcal{M}$ and $A^\mathcal{M}$ to denote the interpretation of $A$ in $\mathcal{M}$; we will write $\mathcal{M}= \langle M^\mathcal{M}, \ldots, A^\mathcal{M}, \ldots \rangle$. If $\mathcal{M}$ is an $\mathcal{L}^\prime$-structure and $\phi$ is an $\mathcal{L}^\prime$ formula or term then we write $\phi^\mathcal{M}$ for the relativisation $\phi$ to $\mathcal{M}$. If $\mathcal{M}$ is an $\mathcal{L}^\prime$-structure and $a \in M^\mathcal{M}$ then we define $$a^= \{ x \in M^\mathcal{M} \mid \mathcal{M} \models x \in a \}.$$ Given $\mathcal{L}^{\prime\prime} \subseteq \mathcal{L}^\prime$ and an $\mathcal{L}^\prime$-structure $\mathcal{M}$, we will use $\mathcal{M} \mid_{\mathcal{L}^{\prime\prime}}$ to denote the $\mathcal{L}^{\prime\prime}$-reduct of $\mathcal{M}$ and $\mathbf{Th}_{\mathcal{L}^{\prime\prime}}(\mathcal{M})$ to denote the $\mathcal{L}^{\prime\prime}$-theory of $\mathcal{M}$.\ \ If $\mathcal{L}_{A, B, \ldots}$ is an extension of the language of set theory then we will use $\mathcal{L}_{A, B, \ldots}^$ to denote the Skolemisation of $\mathcal{L}_{A, B, \ldots}$. That is, for every $\mathcal{L}_{A, B, \ldots}^$-formula $\phi(x_1, \ldots, x_{n+1})$ there is an $n$-placed function symbol $\mathbf{t}_\phi$ in $\mathcal{L}_{A, B, \ldots}^$. We will use $\mathbf{Sk}(\mathcal{L}_{A, B, \ldots})$ to denote the $\mathcal{L}_{A, B, \ldots}^$-theory that includes the axioms $$\forall \vec{y}(\exists x \phi(x, \vec{y}) \Rightarrow \phi(\mathbf{t}_\phi(\vec{y}), \vec{y})) \textrm{ for every } \mathcal{L}_{A, B, \ldots}^ \textrm{-formula } \phi(x, \vec{y}).$$ It is well known that any $\mathcal{L}_{A, B, \ldots}$-structure $\mathcal{M}$ can be expanded to an $\mathcal{L}_{A, B, \ldots}^$-structure $\mathcal{M}^\prime$ such that $\mathcal{M}^\prime \mid_{\mathcal{L}_{A, B, \ldots}} = \mathcal{M}$ and $\mathcal{M}^\prime \models \mathbf{Sk}(\mathcal{L}_{A, B, \ldots})$. If $\mathcal{M}$ is an $\mathcal{L}_{A, B, \ldots}^$-structure and $X \subseteq M^\mathcal{M}$ then we can build a structure $\mathcal{H}_{\mathcal{L}_{A, B, \ldots}}^{\mathcal{M}}(X)$, by inductively closing under the function symbols in $\mathcal{L}_{A, B, \ldots}^$, with the property that $\|\mathcal{H}_{\mathcal{L}_{A, B, \ldots}}^{\mathcal{M}}(X)\|= \|X\| \cdot \aleph_0$. We call $\mathcal{H}_{\mathcal{L}_{A, B, \ldots}}^{\mathcal{M}}(X)$ the Skolem hull (of $\mathcal{M}$) generated by $X$. If $\mathcal{M} \models \mathbf{Sk}(\mathcal{L}_{A, B, \ldots})$ then $\mathcal{H}_{\mathcal{L}_{A, B, \ldots}}^{\mathcal{M}}(X) \prec \mathcal{M}$. Let $\mathcal{L}^\prime$ be a language and let $\mathcal{M}$ be an $\mathcal{L}^\prime$-structure. Let $\langle I, < \rangle$ be a linear order. We say that $\{ c_i \mid i \in I \} \subseteq M^\mathcal{M}$ is a class of order indiscernibles if and only if for all $\mathcal{L}^\prime$-formulae, $\phi(x_1, \ldots, x_n)$, and for all $i_1 < \ldots < i_n$ and $j_1 < \ldots < j_n$ in $I$, $$\mathcal{M} \models \phi(c_{i_1}, \ldots, c_{i_n}) \iff \phi(c_{j_1}, \ldots, c_{j_n}).$$ In [@ehr56] Ehrenfeucht and Mostowski use Ramsey’s Theorem [@ram30] to show that every theory with an infinite model has a model that with an infinite class of order indiscernibles indexed by $\mathbb{Z}$. We will make use of the infinite version of Ramsey’s Theorem in the constructions appearing in the next two sections. (Ramsey [@ram30]) Let $n, k \in \omega$. For all $f: [\omega]^n \longrightarrow k$, there exists an $H \subseteq \omega$ and $i \in k$ such that $\|H\|= \aleph_0$ and for all $x \in [H]^n$, $f(x)= i$. Equipped with a model that is endowed with a $\mathbb{Z}$-index class of order indiscernibles Ehrenfeucht and Mostowski observe that the Skolem hull generated by the class of indiscernibles admits a non-trivial automorphism. (Ehrenfeucht-Mostowski [@ehr56]) Let $\mathcal{L}^\prime$ be a language and let $\mathcal{M}$ be an $\mathcal{L}^\prime$-structure. Let $\langle I, < \rangle$ be a linear order and let $\sigma: I \longrightarrow I$ be an order automorphism. If $X= \{ c_i \mid i \in I \} \subseteq M^\mathcal{M}$ is a class of order indiscernibles then there is an automorphism $j:\mathcal{H}_{\mathcal{L}^\prime}^\mathcal{M}(X) \longrightarrow \mathcal{H}_{\mathcal{L}^\prime}^\mathcal{M}(X)$ such that for all $i \in I$, $j(c_i)= c_{\sigma(i)}$. This yields the following result: \[Th:EhrenfeuchtMostowskiTheorem\] (Ehrenfeucht-Mostowski [@ehr56]) Let $\mathcal{L}^\prime$ be a language. If $T$ is an $\mathcal{L}^\prime$-theory with an infinite model that there exists an $\mathcal{L}^\prime$-structure $\mathcal{M} \models T$ that admits a non-trivial automorphism $j: \mathcal{M} \longrightarrow \mathcal{M}$. When speaking about subsystems of $\mathrm{ZFC}$ we will refer to subschemes and extension of the separation and collection axiom schemes. If $\Gamma$ is a set of formulae then we use $\Gamma$-separation (-collection) to denote the scheme axioms that asserts separation (collection) for all formulae in $\Gamma$. We use $\mathrm{TCo}$ to abbreviate the axiom of transitive containment which says that every set is contained in a transitive set. In $\mathrm{ZFC}$ the axiom of foundation implies that every $\mathcal{L}$-definable class has an $\in$-minimal element; we call this consequence class foundation. Without transitive containment and full separation this implication breaks down. In light of this, if $\Gamma$ is a set of formulae then we use $\Gamma$-foundation to denote the scheme of axioms that asserts foundation for every class definable by a formula in $\Gamma$. Throughout this paper we will make reference to the following important subsystem of $\mathrm{ZFC}$: - $\mathrm{ZFC}^-$ is $\mathrm{ZFC}$, axiomatised with collection rather than replacement, minus the powerset axiom. - Mac Lane set theory ($\mathrm{Mac}$) is $\mathcal{L}$-theory axiomatised by the axioms of extensionality, emptyset, pairing, union, powerset, infinity, transitive containment, $\Delta_0(\mathcal{L})$-separation and foundation. - Kripke-Platek set theory ($\mathrm{KP}$) is the $\mathcal{L}$-theory obtained by deleting infinity and powerset from $\mathrm{Mac}$ and adding $\Delta_0(\mathcal{L})$-collection and $\Pi_1(\mathcal{L})$-foundation. - Zermelo set theory ($\mathrm{Z}$) is the $\mathcal{L}$-theory obtained by deleting transitive containment from $\mathrm{Mac}$ and adding full separation. - $\mathrm{KP}^{\mathcal{P}}$ is obtained by adding infinity, powerset, $\Delta_0^{\mathcal{P}}$-collection and $\Pi_1^{\mathcal{P}}$-foundation to $\mathrm{KP}$. The theories $\mathrm{Mac}$, $\mathrm{KP}$, $\mathrm{Z}$ and $\mathrm{KP}^\mathcal{P}$ are studied extensively in [@mat01] which compares the strength of these systems to a variety of other subsystems of $\mathrm{ZFC}$.\ \ Just as $\Delta_0(\mathcal{L})$-formulae are absolute between transitive classes, $\Delta_0^\mathcal{P}$-formulae are absolute between transitive classes that have the same notion of powerset. The following lemma is a straightforward modification of a result proved in [@fk91]: Let $\phi(x_1, \ldots, x_n)$ be an $\Delta_0^\mathcal{P}$-formula. If $M$ and $N$ are transitive classes such that - $M \subseteq N$, - $\langle M, \in \rangle \models \mathrm{Mac}$, - for all $x \in M$, $$\langle N, \in \rangle \models \mathcal{P}(x) \textrm{ exists},$$ - for all $x \in M$, $\mathcal{P}^M(x)=\mathcal{P}^N(x)$, then for all $a_1, \ldots, a_n \in M$, $$\langle M, \in \rangle \models \phi(a_1, \ldots, a_n) \textrm{ if and only if } \langle N, \in \rangle \models \phi(a_1, \ldots, a_n).$$ Let $\mathcal{L}^\prime$ be a countable extension of the language of set theory. We will write $\mathcal{L}_{\omega_1 \omega}^\prime$ for the infinitary language extending $\mathcal{L}^\prime$ that includes formulae built using conjunctions and disjunctions of countable length. We can fix a coding of the language $\mathcal{L}_{\omega_1 \omega}^\prime$ in $H_{\aleph_1}$ that is definable by a $\Delta_1^{\mathrm{KP}}$ formula over $H_{\aleph_1}$. If $\phi$ is an $\mathcal{L}_{\omega_1 \omega}^\prime$-formula then we will write $\ulcorner \phi \urcorner$ for the element of $H_{\aleph_1}$ that codes $\phi$. We say that a set $A$ is admissible if and only if it is a transitive model of $\mathrm{KP}$. If $A$ is a countable admissible set then we define $$(\mathcal{L}_{\omega_1 \omega}^\prime)_A= \{ \phi \in \mathcal{L}_{\omega_1 \omega}^\prime \mid \ulcorner \phi \urcorner \in A \}.$$ The deduction rules of first-order logic can be extended to fragments of the infinitary logic $\mathcal{L}_{\omega_1, \omega}^\prime$ — we will write $\Gamma \vdash_{(\mathcal{L}_{\omega_1, \omega}^\prime)_A} \sigma$ if the $\sigma$ is provable from $\Gamma$ in $(\mathcal{L}_{\omega_1, \omega}^\prime)_A$. As usual we will use $\omega_1^{\mathrm{ck}}$ to denote the Church-Kleene ordinal. If $\alpha$ is an ordinal the we will use $L_\alpha$ to denote the $\alpha^{\mathrm{th}}$ level of Gödel’s constructible hierarchy. It is well known that $L_{\omega_1^{\mathrm{ck}}}$ is an admissible set. Moreover, $\omega_1^{\mathrm{ck}}$ is the least $\alpha > \omega$ such that $L_\alpha$ is admissible. In [@bar67] Jon Barwise proves analogues of the compactness and completeness fragments of $\mathcal{L}_{\omega_1, \omega}^\prime$ coded in admissible sets. \[Th:BarwiseCompleteness\] (Barwise Completeness Theorem) Let $\mathcal{L}^\prime$ be a countable extension of the language of set theory. If $A$ is a countable admissible set then $$\{ \ulcorner \phi \urcorner \mid (\phi \in (\mathcal{L}^\prime_{\omega_1\omega})_A) \land (\vdash_{(\mathcal{L}^\prime_{\omega_1\omega})_A} \phi) \}$$ is definable by a $\Sigma_1(\mathcal{L})$ formula over $A$. \[Th:BarwiseCompactness\] (Barwise Compactness Theorem) Let $\mathcal{L}^\prime$ be a countable extension of the language of set theory. Let $A$ be a countable admissible set and let $T \subseteq (\mathcal{L}^\prime_{\omega_1\omega})_A$ be definable by a $\Sigma_1(\mathcal{L})$ formula over $A$. If for every $T^\prime \subseteq T$ with $T^\prime \in A$, $T^\prime$ has a model, then $T$ has a model. We will use the Barwise Compactness Theorem in sections \[Sec:AutomorphismsFromNonStandardModels\] and \[Sec:NFUPlusAxCountLEQ\] to build non-standard $\omega$-models of subsystems of $\mathrm{ZFC}$. Automorphisms from standard models of set theory {#Sec:AutomorphismsFromStandardModels} ================================================ In this section we will exploit the structural properties of models of set theory that are standard along an initial segment of the ordinals to produce models of set theory admitting automorphisms that are well-behaved along these initial segments. We begin by showing that the existence of an $\omega$-model of Mac Lane set theory implies the existence of an elementarily equivalent model admitting an automorphism that does not move any natural number down. Our models admitting automorphism will be built using a modification of the Ehrenfeucht-Mostowski method [@ehr56] using ideas from [@kor94]. The aim will be to build a fully Skolemised model with a $\mathbb{Z}$-indexed class of order indiscernible natural numbers. Special care will be taken in the construction of this model to ensure the following behaviour: - the class of order indiscernibles will be cofinal in the natural numbers of the Skolem hull that includes all of the order indiscernibles, - if a natural number sitting below all of the order indiscernibles corresponds to Skolem term $\Phi$ mentioning finitely many order indiscernibles then the same natural number also corresponds to the Skolem term $\Phi^\prime$ obtained by replacing the order indiscernibles in $\Phi$ with an order equivalent set of order indiscernibles. The Skolem hull of the resulting model which includes the $\mathbb{Z}$-indexed class of order indiscernibles admits an automorphism which moves every natural number sitting above one of the order indiscernibles up, and fixes every natural number that sits below all of the order indiscernibles. Therefore the cofinality of the class of order indiscernibles guarantees that no natural number is moved down.\ \ We begin our exposition of the results of this section by introducing an extension of the language of set theory. We define $\mathcal{L}^{\mathbf{ind}} \supseteq \mathcal{L}^$ by adding - a unary relation $\mathcal{C}$, - constant symbols $c_i$ for each $i \in \mathbb{Z}$. This language allows us to define the theory that will yield models of $\mathrm{Mac}$ admitting the desired automorphism. \[Df:TheoryW1\] We define the $\mathcal{L}^{\mathbf{ind}}$-theory $\mathbf{W}_1 \supseteq \mathrm{Mac} \cup \mathbf{Sk}(\mathcal{L})$ by adding the axioms - $\forall x (\mathcal{C}(x) \Rightarrow x \in \omega)$ - $(\forall x \in \omega)(\exists y \in \omega)(y > x \land \mathcal{C}(y))$, - $\mathcal{C}(c_i)$ for all $i \in \mathbb{Z}$, - $c_i < c_j$ for all $i, j \in \mathbb{Z}$ with $i < j$, - for all $\mathcal{L}^$-formulae $\phi(x_1, \ldots, x_n)$, $$\forall x_1 \ldots \forall x_n \forall y_1 \ldots \forall y_n \left( \begin{array}{c} x_1 < \ldots < x_n \land y_1 < \ldots < y_n \land \bigwedge_{1 \leq i \leq n} (\mathcal{C}(x_i) \land \mathcal{C}(y_i))\\ \Rightarrow (\phi(x_1, \ldots, x_n) \iff \phi(y_1, \ldots, y_n)) \end{array} \right),$$ - for all $\mathcal{L}^$ Skolem functions $\mathbf{t}(x_1, \ldots, x_n)$ and for all $i_1 < \ldots < i_n$ and $j_1 < \ldots < j_n$ in $\mathbb{Z}$, $$\forall x (\mathcal{C}(x) \Rightarrow \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < x) \Rightarrow (\mathbf{t}(c_{i_1}, \ldots, c_{i_n})= \mathbf{t}(c_{j_1}, \ldots, c_{j_n})).$$ It should be noted that (i) and (ii) of Definition \[Df:TheoryW1\] are single axioms and (iii), (iv), (v) and (vi) of Definition \[Df:TheoryW1\] are axiom schemes. Axioms (i), (ii) and scheme (v) in Definition \[Df:TheoryW1\] ensure that $\mathcal{C}$ is a class of indiscernibles that is cofinal in the natural numbers. We will see that the cofinality of $\mathcal{C}$ will ensure that the $c_i$s (a subclass of $\mathcal{C}$ by Definition \[Df:TheoryW1\](iii)) will be a cofinal subclass of the natural numbers in the Skolem hull that includes the $c_i$s. This idea of using a cofinal super class of indiscernibles to ensure that the $\mathbb{Z}$-indexed class of indiscernibles is cofinal in the Skolem hull first appears in [@kor94]. Axiom scheme (vi) in Definition \[Df:TheoryW1\] ensures that the Skolem terms sent below $\mathcal{C}$ are fixed when the sequence of $c_i$s appearing in the Skolem term are replaced with an order equivalent sequence. This combined with the cofinality of the $c_i$s will ensure that the automorphism generated by the $c_i$s moves all natural numbers in one direction.\ \ We are now able to show that the existence of an $\omega$-model of $\mathrm{Mac}$ implies the existence of an elementarily equivalent model of $\mathbf{W}_1$. \[Th:ConsistencyOfW1\] If $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M} \rangle$ is an $\omega$-model of $\mathrm{Mac}$ then $\mathbf{W}_1 \cup \mathbf{Th}_{\mathcal{L}}(\mathcal{M})$ is consistent. Let $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M} \rangle$ be an $\omega$-model of $\mathrm{Mac}$. Let $g: \omega \longrightarrow (\omega^\mathcal{M})^$ be an isomorphism guaranteed by the fact that $\mathcal{M}$ is an $\omega$-model. Using the Axiom of Choice in the ambient theory we can find interpretations for the Skolem functions and expand $\mathcal{M}$ to an $\mathcal{L}^$-structure $\mathcal{M}^\prime \models \mathbf{Sk}(\mathcal{L})$ with $\mathcal{M}^\prime \mid_{\mathcal{L}}= \mathcal{M}$. We will use the Compactness Theorem to show that $\mathbf{W}_1 \cup \mathbf{Th}_{\mathcal{L}}(\mathcal{M})$ is consistent. Let $\Delta \subseteq \mathbf{W}_1$ be finite. Our first task is to choose a suitable interpretation for the predicate $\mathcal{C}$. Suppose that $\Delta$ mentions instances of the scheme (v) of Definition \[Df:TheoryW1\] for the $\mathcal{L}^$-formulae $\phi_0, \ldots, \phi_{m-1}$. Without loss of generality we can assume that for all $0 \leq i < m$, $\phi_i$ has arity $k$. Define $H_1: [\omega]^k \longrightarrow \mathcal{P}(m)$ by $$H_1(\{x_1, \ldots, x_k \})= \{ i \in m \mid \mathcal{M}^\prime \models \phi_i(g(x_1), \ldots, g(x_k))\} \textrm{ where } x_1 < \ldots < x_k.$$ Now, by Ramsey’s Theorem there is an infinite $C^\prime \subseteq \omega$ such that $H_1``[C^\prime]^k= \{A\}$. Let $C= g``C^\prime$. Now, $C \subseteq (\omega^\mathcal{M})^$ is a set of elements that are order indiscernible with respect to the formulae $\phi_0, \ldots, \phi_{m-1}$. And $C$ is cofinal in $(\omega^\mathcal{M})^$, since $\langle (\omega^\mathcal{M})^, \in^\mathcal{M} \rangle$ is isomorphic to $\langle \omega, \in \rangle$. Therefore by interpreting $\mathcal{C}$ using $C$ we can expand $\mathcal{M}^\prime$ to a structure $\mathcal{M}^{\prime\prime}$ that satisfies (i) and (ii) of Definition \[Df:TheoryW1\] and all instances of Definition \[Df:TheoryW1\](v) that are mentioned in $\Delta$.\ Our task now turns to finding an interpretation for the $c_i$s that are mentioned in $\Delta$. Without loss of generality we can assume that $\Delta$ mentions exactly the constants $c_{-n}, \ldots, c_{n}$. Let $z$ be the least element of $C^\prime$. Now, suppose that $\Delta$ mentions instances of the scheme Definition \[Df:TheoryW1\](vi) for the Skolem terms $\mathbf{t}_0, \ldots, \mathbf{t}_{l-1}$. Without loss of generality we can assume that for all $0 \leq i < l$, $\mathbf{t}_i$ is function symbol with arity $v$. Define $H_2: [C^\prime]^v \longrightarrow (z+1)^l$ by $$H_2(\{ x_1, \ldots, x_v \})= \langle y_0, \ldots, y_{l-1} \rangle \textrm{ where } x_1 < \ldots < x_v \textrm{ and}$$ $$y_i= \left\{ \begin{array}{ll} \mathbf{t}_i(g(x_1), \ldots, g(x_v)) & \textrm{if } \mathcal{M}^{\prime\prime} \models \left(\begin{array}{c} (\mathbf{t}_i(g(x_1), \ldots, g(x_v)) \in \omega) \land \\ (\mathbf{t}_i(g(x_1), \ldots, g(x_v)) < g(z)) \end{array}\right)\\ z & \textrm{otherwise} \end{array} \right).$$ Using Ramsey’s Theorem we can find an infinite $D^\prime \subseteq C^\prime$ such that $H_2``[D^\prime]^v = \{\eta\}$. Let $D= g``D^\prime$. By interpreting $c_{(i-1)-n}$ by the $i^{\mathrm{th}}$ element of $D$ for $1 \leq i \leq 2n+1$ we expand $\mathcal{M}^{\prime\prime}$ to a structure $\mathcal{M}^{\prime\prime\prime}$ such that $$\mathcal{M}^{\prime\prime\prime} \models \Delta \cup \mathbf{Th}_{\mathcal{L}}(\mathcal{M}) \cup \mathbf{Sk}(\mathcal{L}).$$ Therefore by compactness $\mathbf{W}_1 \cup \mathbf{Th}_{\mathcal{L}}(\mathcal{M})$ is consistent. We now show that a model of $\mathbf{W}_1$ yields a model of $\mathrm{Mac}$ admitting an automorphism that does not move any natural number down. Lemma \[Th:IndiscerniblesCofinalInSkolemHull\] is due to Friederike Körner in [@kor94]. \[Th:IndiscerniblesCofinalInSkolemHull\] (Körner [@kor94]) Let $\mathcal{M} \models \mathbf{W}_1$. For all $\mathcal{L}^$ Skolem functions $\mathbf{t}(x_1, \ldots, x_n)$ and for all $i_1 < \ldots < i_n$ in $\mathbb{Z}$, if $\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) \in \omega$ then there exists a $k \in \mathbb{Z}$ such that $$\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < c_k.$$ Let $\mathcal{M} \models \mathbf{W}_1$. Let $\mathbf{t}(x_1, \ldots, x_n)$ be an $\mathcal{L}^$ Skolem function. Let $i_1 < \ldots < i_n$ in $\mathbb{Z}$ and assume that $\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) \in \omega$. By Definition \[Df:TheoryW1\](ii) there is an $a \in (\omega^\mathcal{M})^$ such that $$\mathcal{M} \models \left( \bigwedge_{1 \leq j \leq n} (c_{i_j} < a) \right) \land (\mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < a) \land \mathcal{C}(a).$$ Let $k > i_n$. By indiscernibility (Definition \[Df:TheoryW1\](vii)): $$\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < c_k.$$ \[Th:SkolemFunctionsFixedBelow\] Let $\mathcal{M} \models \mathbf{W}_1$. For all $\mathcal{L}^$ Skolem functions $\mathbf{t}(x_1, \ldots, x_n)$ and for all $i_1 < \ldots < i_n$ and $j_1 < \ldots < j_n$ in $\mathbb{Z}$, if for all $k \in \mathbb{Z}$, $\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < c_k$ then $$\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n})= \mathbf{t}(c_{j_1}, \ldots, c_{j_n}).$$ Let $\mathcal{M} \models \mathbf{W}_1$. Let $\mathbf{t}(x_1, \ldots, x_n)$ be an $\mathcal{L}^$ Skolem function and, let $i_1 < \ldots < i_n$ and $j_1 < \ldots < j_n$ be in $\mathbb{Z}$. Assume that for all $k \in \mathbb{Z}$, $\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < c_k$. Let $k \in \mathbb{Z}$ be such that $k < i_1$. By indiscernibility (Definition \[Df:TheoryW1\](vii)): $$\mathcal{M} \models \forall x(\mathcal{C}(x) \Rightarrow \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < x).$$ Therefore by Definition \[Df:TheoryW1\](viii): $$\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n})= \mathbf{t}(c_{j_1}, \ldots, c_{j_n}).$$ \[Th:AutomorphismMovingNoNaturalDown\] If $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M} \rangle$ is an $\omega$-model of $\mathrm{Mac}$ then there exists an $\mathcal{L}$-structure $\mathcal{N} \equiv \mathcal{M}$ admitting an automorphism $j: \mathcal{N} \longrightarrow \mathcal{N}$ such that - $\mathcal{N} \models j(n) \geq n$ for all $n \in (\omega^\mathcal{N})^$, - there exists an $n \in (\omega^\mathcal{N})^$ with $\mathcal{N} \models j(n) > n$. Let $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M} \rangle$ be an $\omega$-model of $\mathrm{Mac}$. By Lemma \[Th:ConsistencyOfW1\] there exists an $\mathcal{L}^{\mathbf{ind}}$-structure $\mathcal{Q} \models \mathbf{W}_1$ such that $\mathcal{Q} \mid_{\mathcal{L}} \equiv \mathcal{M}$. Let $C= \{ c_i^\mathcal{Q} \mid i \in \mathbb{Z} \}$. Let $\mathcal{N}= \mathcal{H}_{\mathcal{L}}^{\mathcal{Q}}(C)$. Therefore $\mathcal{N} \equiv \mathcal{M}$. Define $j: \mathcal{N} \longrightarrow \mathcal{N}$ by $$j(c_i^\mathcal{Q})= c_{i+1}^\mathcal{Q} \textrm{ for all } i \in \mathbb{Z},$$ $$\textrm{for all } \mathcal{L}^ \textrm{ Skolem functions } \mathbf{t}(x_1, \ldots, x_n) \textrm{ and for all } i_1 < \ldots < i_n \textrm{ in } \mathbb{Z},$$ $$j(\mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}))= \mathbf{t}(j(c_{i_1}^\mathcal{Q}), \ldots, j(c_{i_n}^\mathcal{Q})).$$ Now, $j$ is an automorphism of the $\mathcal{L}$-structure $\mathcal{N}$. If $\mathbf{t}(x_1, \ldots x_n)$ is an $\mathcal{L}^$ Skolem function and $i_1 < \ldots < i_n$ is in $\mathbb{Z}$ such that $\mathcal{Q} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) \in \omega$ then by Lemma \[Th:IndiscerniblesCofinalInSkolemHull\] either - there is $k \in \mathbb{Z}$ such that $$\mathcal{Q} \models c_k < \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) \leq c_{k+1}$$ (Note that $k$ might equal $i_j$ for some $1 \leq j \leq n$), - or, for all $k \in \mathbb{Z}$, $$\mathcal{Q} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < c_k.$$ If (ii) holds then Lemma \[Th:SkolemFunctionsFixedBelow\] implies that $$j(\mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}))= \mathbf{t}(c_{i_1+1}^\mathcal{Q}, \ldots, c_{i_n+1}^\mathcal{Q})= \mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}).$$ If (i) holds then indiscernibility (Definition \[Df:TheoryW1\](vii)) implies that $$\mathcal{Q} \models c_k < \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) \leq c_{k+1} \textrm{ if and only if } \mathcal{Q} \models c_{k+1} < \mathbf{t}(c_{i_1+1}, \ldots, c_{i_n+1}) \leq c_{k+2}.$$ $$\textrm{Therefore } j(\mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}))= \mathbf{t}(c_{i_1+1}^\mathcal{Q}, \ldots, c_{i_n+1}^\mathcal{Q}) >^\mathcal{Q} \mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}).$$ This shows that $\mathcal{N} \models j(n) \geq n$ for all $n \in (\omega^\mathcal{N})^$. And, $\mathcal{N} \models j(c_0^\mathcal{Q}) > c_0^\mathcal{Q}$. Theorem \[Th:AutomorphismMovingNoNaturalDown\] can be extended to show that the existence of a transitive model of Mac Lane set theory implies the existence of an elementarily equivalent model admitting a non-trivial automorphism that does not move any ordinal down. We begin by defining an extension of the $\mathcal{L}^{\mathbf{ind}}$-theory $\mathbf{W}_1$. \[Df:TheoryW2\] We define the $\mathcal{L}^{\mathbf{ind}}$-theory $\mathbf{W}_2 \supseteq \mathrm{Mac} \cup \mathbf{Sk}(\mathcal{L})$ by adding the axioms - $\forall x (\mathcal{C}(x) \Rightarrow x \in \omega)$ - $(\forall x \in \omega)(\exists y \in \omega)(y > x \land \mathcal{C}(y))$, - $\mathcal{C}(c_i)$ for all $i \in \mathbb{Z}$, - $c_i < c_j$ for all $i, j \in \mathbb{Z}$ with $i < j$, - for all $\mathcal{L}^$-formulae $\phi(x_1, \ldots, x_n)$, $$\forall x_1 \ldots \forall x_n \forall y_1 \ldots \forall y_n \left( \begin{array}{c} x_1 < \ldots < x_n \land y_1 < \ldots < y_n \land \bigwedge_{1 \leq i \leq n} (\mathcal{C}(x_i) \land \mathcal{C}(y_i))\\ \Rightarrow (\phi(x_1, \ldots, x_n) \iff \phi(y_1, \ldots, y_n)) \end{array} \right),$$ - for all $\mathcal{L}^$ Skolem functions $\mathbf{t}(x_1, \ldots, x_n)$ and for all $i_1 < \ldots < i_n$ and $j_1 < \ldots < j_n$ in $\mathbb{Z}$, $$\forall x (\mathcal{C}(x) \Rightarrow \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < x) \Rightarrow (\mathbf{t}(c_{i_1}, \ldots, c_{i_n})= \mathbf{t}(c_{j_1}, \ldots, c_{j_n})),$$ - for all $\mathcal{L}^$ Skolem functions $\mathbf{t}(x_1, \ldots, x_n)$ and for all $i_1 < \ldots < i_n$ in $\mathbb{Z}$, $$(\mathbf{t}(c_{i_1}, \ldots, c_{i_n}) \textrm{ is an ordinal}) \Rightarrow (\mathbf{t}(c_{i_1}, \ldots, c_{i_n}) \leq \mathbf{t}(c_{i_n+1}, \ldots, c_{i_n+1})).$$ It should be noted that (vii) of Definition \[Df:TheoryW2\] is an axiom scheme. This scheme will ensure that the ordinals in the Skolem hull of a model of $\mathbf{W}_2$ that includes the order indiscernible $c_i$s will not be moved down by the automorphism of this Skolem hull generated by sending $c_i$ to $c_{i+1}$ for all $i \in \mathbb{Z}$.\ \ We are now in a position to prove an analogue of Lemma \[Th:ConsistencyOfW1\]. \[Th:ConsistencyOfW2\] If $\langle M, \in \rangle$ is a transitive model of $\mathrm{Mac}$ then $\mathbf{W}_2 \cup \mathbf{Th}_\mathcal{L}(\langle M, \in \rangle)$ is consistent. Let $\mathcal{M}= \langle M, \in \rangle$ be a transitive model of $\mathrm{Mac}$. Therefore $\langle M, \in \rangle$ is an $\omega$-model and the $\omega$ of $\langle M, \in \rangle$ coincides with the $\omega$ of the ambient model. Again, we will use the Compactness Theorem to show that $\mathbf{W}_2 \cup \mathbf{Th}_\mathcal{L}(\langle M, \in \rangle)$ is consistent. Let $\Delta \subseteq \mathbf{W}_2$ be finite. Using the same arguments used in the proof of Lemma \[Th:ConsistencyOfW1\] we can find $C \subseteq \omega$ and expand $\mathcal{M}$ to a structure $\mathcal{M}^\prime$ in which $\mathcal{C}$ is interpreted using $C$ and - $\mathcal{M}^\prime \models \mathbf{Sk}(\mathcal{L})$, - $\mathcal{M}^\prime \mid_{\mathcal{L}}= \mathcal{M}$, - $\mathcal{M}^\prime$ satisfies (i) and (ii) of Definition \[Df:TheoryW2\], - $\mathcal{M}^\prime$ satisfies all instances of Definition \[Df:TheoryW2\](v) that appear in $\Delta$. We can also use the arguments utilised in the proof of Lemma \[Th:ConsistencyOfW1\] to find an infinite $D \subseteq C$ such that if $\mathbf{t}(x_1, \ldots, x_k)$ is a Skolem function mentioned in $\Delta$, and $a_1 < \ldots < a_k$ and $b_1 < \dots < b_k$ are in $D$ then $$\textrm{if } \mathbf{t}^{\mathcal{M}^\prime}(a_1, \ldots, a_k) \in \omega \textrm{ and } \mathbf{t}^{\mathcal{M}^\prime}(a_1, \ldots, a_k) < \min C$$ $$\textrm{then } \mathbf{t}^{\mathcal{M}^\prime}(a_1, \ldots, a_k)= \mathbf{t}^{\mathcal{M}^\prime}(b_1, \ldots, b_k).$$ Now, suppose that $\Delta$ only mentions the constant symbols $c_{-n}, \ldots, c_n$. Let $\mathbf{s}_0, \ldots, \mathbf{s}_{m-1}$ and $\eta_0, \ldots, \eta_{m-1}$ be such that - $\eta_i \in \mathbb{Z}^{<\omega}$ for all $i \in m$, - $\eta_i(p) < \eta_i(q)$ for all $i \in m$ and $p < q < \|\eta_i\|$, - $\mathbf{s}_i$ is an $\mathcal{L}^$ Skolem function of arity $\|\eta_i\|$ for all $i \in m$, - if the axiom $$(\mathbf{t}(c_{i_1}, \ldots, c_{i_l}) \textrm{ is an ordinal}) \Rightarrow (\mathbf{t}(c_{i_1}, \ldots, c_{i_l}) \leq \mathbf{t}(c_{i_1+1}, \ldots, c_{i_l+1})) \textrm{ appears in } \Delta$$ then there is an $j \in m$ such that $\mathbf{t}= \mathbf{s}_j$ and $\eta_j= \langle i_1, \ldots, i_l \rangle$. Note that if $i_1 < \ldots < i_l$ are in $\mathbb{Z}$ then $i_1 < i_1+1 \leq i_2 < i_2+1 \leq \ldots i_{l-1}+1 \leq i_l$. For each $0 \leq j < m$, let $\psi_j$ be the $\mathcal{L}^$-formula obtained by replacing $c_{i_1}, c_{i_1+1}, \ldots, c_{i_l}, c_{i_l+1}$ by free variables $x_1, \ldots, x_{u_j}$ in the $\mathcal{L}^{\mathbf{ind}}$-sentence $$\mathbf{s}_j(c_{i_1}, \ldots, c_{i_l}) \leq \mathbf{s}_j(c_{i_1+1}, \ldots, c_{i_l+1}) \textrm{ where } \eta_j= \langle i_1, \ldots, i_l \rangle.$$ Note that for all $0 \leq j < m$, the $\mathcal{L}^$-formula $\psi_j$ has arity $u_j$ for $\|\eta_j\|+ 1 \leq u_j \leq 2\cdot\|\eta_j\|$. For all $j \in m$ and $k \in \|\eta_j\|$ we define $\zeta_k^j$ by $$\zeta_0^j= 1 \textrm{ for all } j \in m,$$ $$\zeta_{k+1}^j= \left\{ \begin{array}{ll} \zeta_k^j+1 & \textrm{if } \eta_j(k+1) = \eta_j(k) + 1\\ \zeta_k^j+2 & \textrm{otherwise} \end{array}\right)$$ This ensures that if $j \in m$ and $x_1, \ldots, x_{u_j}$ are the free variables appearing in $\psi_j$ then $$\psi_j(x_1, \ldots, x_{u_j}) \textrm{ is } \mathbf{s}_j(x_{\zeta_0^j}, \ldots, x_{\zeta_{\|\eta_j\|-1}^j}) \leq \mathbf{s}_j(x_{\zeta_0^j+1}, \ldots, x_{\zeta_{\|\eta_j\|-1}^j+1}).$$ Now, consider the function $K_0: [D]^{u_0} \longrightarrow 3$ defined by $$K_0(\{x_1, \ldots, x_{u_0}\})= \left\{ \begin{array}{ll} 0 & \textrm{if } \mathcal{M}^{\prime} \models \left( \begin{array}{c} (\mathbf{s}_0(x_{\zeta_0^0}, \ldots, x_{\zeta_{\|\eta_0\|-1}^0}) \textrm{ is an ordinal})\\ \land (\neg \psi_0(x_1, \ldots, x_{u_0})) \end{array} \right) \\ 1 & \textrm{if } \mathcal{M}^{\prime} \models \left( \begin{array}{c} (\mathbf{s}_0(x_{\zeta_0^0}, \ldots, x_{\zeta_{\|\eta_0\|-1}^0}) \textrm{ is an ordinal})\\ \land(\psi_0(x_1, \ldots, x_{u_0})) \end{array}\right)\\ 2 & \textrm{otherwise} \end{array}\right)$$ $$\textrm{where } x_1 < \ldots < x_{u_0}.$$ By Ramsey’s Theorem there is an infinite $A_0 \subseteq D$ such that $K_0``[A_0]^{u_0}= \{p\}$ for some $p \in 3$. Now, for each $0 \leq j < m-1$ define $K_{j+1}: [A_j]^{u_{j+1}} \longrightarrow 3$ by $$K_{j+1}(\{x_1, \ldots, x_{u_{j+1}}\})= \left\{ \begin{array}{ll} 0 & \textrm{if } \mathcal{M}^{\prime} \models \left( \begin{array}{c} (\mathbf{s}_{j+1}(x_{\zeta_0^{j+1}}, \ldots, x_{\zeta_{\|\eta_{j+1}\|-1}^{j+1}}) \textrm{ is an ordinal})\\ \land(\neg \psi_{j+1}(x_1, \ldots, x_{u_{j+1}})) \end{array} \right)\\ 1 & \textrm{if } \mathcal{M}^{\prime} \models \left( \begin{array}{c} (\mathbf{s}_{j+1}(x_{\zeta_0^{j+1}}, \ldots, x_{\zeta_{\|\eta_{j+1}\|-1}^{j+1}}) \textrm{ is an ordinal})\\ \land(\psi_{j+1}(x_1, \ldots, x_{u_{j+1}})) \end{array} \right) \\ 2 & \textrm{otherwise} \end{array}\right)$$ $$\textrm{where } x_1 < \ldots < x_{u_{j+1}}.$$ And let $A_{j+1} \subseteq A_j$ be the infinite set guaranteed by Ramsey’s Theorem such that $K_{j+1}``[A_{j+1}]^{u_{j+1}}= \{p\}$ for some $p \in 3$.\ Claim: For all $0 \leq j < m$, $K_j``[A_{m-1}]^{u_j} \neq \{ 0 \}$.\ Suppose that $j \in m$ with $K_j``[A_{m-1}]^{u_j} = \{ 0 \}$. Let $h: \omega \longrightarrow A_{m-1}$ enumerate the elements of $A_{m-1}$ in order. $$\textrm{Let } B= \{ \mathbf{s}_j^{\mathcal{M}^{\prime}}(h(\zeta_0^j+k), \ldots, h(\zeta_{\|\eta_j\|-1}^j+k)) \mid k \in \omega \}.$$ Now, $B$ is a set of ordinals. Let $\alpha$ be the least element of $B$. Therefore there is a $k \in \omega$ such that $\alpha= \mathbf{s}_j^{\mathcal{M}^{\prime}}(h(\zeta_0^j+k), \ldots, h(\zeta_{\|\eta_j\|-1}^j+k))$. But now, $$\mathcal{M}^{\prime} \models \mathbf{s}_j(h(\zeta_0^j+k), \ldots, h(\zeta_{\|\eta_j\|-1}^j+k)) > \mathbf{s}_j(h(\zeta_0^j+(k+1)), \ldots, h(\zeta_{\|\eta_j\|-1}^j+(k+1)))$$ $$\textrm{and } \mathbf{s}_j^{\mathcal{M}^{\prime}}(h(\zeta_0^j+(k+1)), \ldots, h(\zeta_{\|\eta_j\|-1}^j+(k+1))) \in B$$ which contradicts the fact that $\mathbf{s}_j^{\mathcal{M}^{\prime\prime}}(h(\zeta_0^j+k), \ldots, h(\zeta_{\|\eta_j\|-1}^j+k))$ is least. This proves the claim.\ Using the first $2\cdot n +1$ elements of $A_{m-1}$ to interpret the constants $c_{-n}, \ldots, c_n$ we can expand $\mathcal{M}^{\prime}$ to a structure $\mathcal{M}^{\prime\prime}$ such that $$\mathcal{M}^{\prime\prime} \models \Delta \cup \mathbf{Th}_{\mathcal{L}}(\langle M, \in \rangle) \cup \mathbf{Sk}(\mathcal{L}).$$ Therefore, by compactness, $\mathbf{W}_2 \cup \mathbf{Th}_{\mathcal{L}}(\langle M, \in \rangle)$ is consistent. Since the theory $\mathbf{W}_2$ extends $\mathbf{W}_1$, Lemmas \[Th:IndiscerniblesCofinalInSkolemHull\] and \[Th:SkolemFunctionsFixedBelow\] apply to models of $\mathbf{W}_2$. This allows us to prove an analogue of Theorem \[Th:AutomorphismMovingNoNaturalDown\] showing that if there exists a transitive model of $\mathrm{Mac}$ then there exists an elementarily equivalent model admitting a non-trivial automorphism that does not move any ordinal down. \[Th:TransitiveModelsYieldAutomorphismMovingNoOrdinalDown\] If $\langle M, \in \rangle$ is a transitive model of $\mathrm{Mac}$ then there is an $\mathcal{L}$-structure $\mathcal{N} \equiv \langle M, \in \rangle$ admitting an automorphism $j:\mathcal{N} \longrightarrow \mathcal{N}$ such that - $\mathcal{N} \models j(\alpha) \geq \alpha$ for all $\alpha \in \mathrm{Ord}^\mathcal{N}$, - there exists an $n \in (\omega^\mathcal{N})^$ such that $\mathcal{N} \models j(n) > n$. Let $\langle M, \in \rangle$ be a transitive model of $\mathrm{Mac}$. By Lemma \[Th:ConsistencyOfW2\] there exists an $\mathcal{L}^{\mathbf{ind}}$-structure $\mathcal{Q} \models \mathbf{W}_2$ such that $\mathcal{Q} \mid_{\mathcal{L}} \equiv \langle M, \in \rangle$. Let $C= \{ c_i^\mathcal{Q} \mid i \in \mathbb{Z} \}$. Let $\mathcal{N}= \mathcal{H}_{\mathcal{L}}^{\mathcal{Q}}(C)$. Therefore $\mathcal{N} \equiv \langle M, \in \rangle$. Define $j: \mathcal{N} \longrightarrow \mathcal{N}$ by $$j(c_i^\mathcal{Q})= c_{i+1}^\mathcal{Q} \textrm{ for all } i \in \mathbb{Z},$$ $$\textrm{for all } \mathcal{L}^ \textrm{ Skolem functions } \mathbf{t}(x_1, \ldots, x_n) \textrm{ and for all } i_1 < \ldots < i_n \textrm{ in } \mathbb{Z},$$ $$j(\mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}))= \mathbf{t}(j(c_{i_1}^\mathcal{Q}), \ldots, j(c_{i_n}^\mathcal{Q})).$$ Now, $j$ is an automorphism of the $\mathcal{L}$-structure $\mathcal{N}$. The same arguments used in the proof of Theorem \[Th:AutomorphismMovingNoNaturalDown\] reveal that if $\mathbf{t}(x_1, \ldots, x_n)$ is an $\mathcal{L}^$ Skolem function and $i_1 < \ldots < i_n$ are in $\mathbb{Z}$ such that $\mathcal{N} \models \mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}) \in \omega$ then $$\mathcal{N} \models j(\mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}))= \mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}) \textrm{ or } \mathcal{N} \models j(\mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q})) > \mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}).$$ In particular, $$\mathcal{N} \models j(c_0^\mathcal{Q}) > c_0^\mathcal{Q}.$$ If $\mathbf{t}(x_1, \ldots, x_n)$ is an $\mathcal{L}^$ Skolem function and $i_1 < \ldots < i_n$ are in $\mathbb{Z}$ and $$\mathcal{Q} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) \textrm{ is an ordinal}$$ then the scheme (vii) of Definition \[Df:TheoryW2\] implies that $$j(\mathbf{t}^\mathcal{Q}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}))= \mathbf{t}^\mathcal{Q}(c_{i_1+1}^\mathcal{Q}, \ldots, c_{i_n+1}^\mathcal{Q}) \geq \mathbf{t}^\mathcal{Q}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}).$$ Therefore for all $\alpha \in \mathrm{Ord}^\mathcal{N}$, $$\mathcal{N} \models j(\alpha) \geq \alpha.$$ This theorem allows us to prove that every complete consistent extension of $\mathrm{ZFC}$ has a model admitting a non-trivial automorphism that does not move any ordinal down. Using the Reflection Principle and the Compactness Theorem we show that every complete consistent extension of $\mathrm{ZFC}$ has a model $\mathcal{M}$ with a point $\lambda \in M^\mathcal{M}$, which $\mathcal{M}$ believes is a limit ordinal, such that externally the structure $\langle V_\lambda^\mathcal{M}, \in^\mathcal{M} \rangle$ is elementarily equivalent to $\mathcal{M}$. Since $\mathcal{M}$ believes that $V_\lambda$ is a transitive model of $\mathrm{Mac}$, we can apply Lemma \[Th:ConsistencyOfW2\] inside $\mathcal{M}$ to obtain a model admitting the desired automorphism. \[Th:ZFCwithAutThatDoesNotMoveAnyOrdinalDown\] If $T \supseteq \mathrm{ZFC}$ is complete and consistent $\mathcal{L}$-theory then there is an $\mathcal{L}$-structure $\mathcal{N} \models T$ admitting an automorphism $j: \mathcal{N} \longrightarrow \mathcal{N}$ such that - $\mathcal{N} \models j(\alpha) \geq \alpha$ for all $\alpha \in \mathrm{Ord}^\mathcal{N}$, - there exists an ordinal $\alpha \in \mathrm{Ord}^\mathcal{N}$ such that $\mathcal{N} \models j(\alpha) > \alpha$. Let $T \supseteq \mathrm{ZFC}$ be complete and consistent $\mathcal{L}$-theory. Let $\bar{\lambda}$ be a new constant symbol. Define $T^\star \supseteq T$ to be the $\mathcal{L}_{\bar{\lambda}}$-theory that includes the axioms: - $\bar{\lambda}$ is a limit ordinal, - for all $\mathcal{L}$-sentences, $\sigma$, $$(\langle V_{\bar{\lambda}}, \in \rangle \models \sigma) \iff \sigma.$$ Let $\mathcal{M}$ be an $\mathcal{L}$-structure such that $\mathcal{M} \models T$. Let $\Delta \subseteq T^\star$ be finite. Using the Reflection Principle we can expand $\mathcal{M}$ to an $\mathcal{L}_{\bar{\lambda}}$-structure $\mathcal{M}^\prime$ such that $\mathcal{M}^\prime \models \Delta$. Therefore, by the Compactness Theorem, there is an $\mathcal{L}_{\bar{\lambda}}$-structure $\mathcal{Q}$ such that $$\mathcal{Q} \models T^\star.$$ We work inside $\mathcal{Q}$. Since $\bar{\lambda}$ is a limit ordinal, $V_{\bar{\lambda}}$ is a transitive model of $\mathrm{Mac}$. Therefore Theorem \[Th:TransitiveModelsYieldAutomorphismMovingNoOrdinalDown\] implies that there exists an $\mathcal{L}$-structure $\mathcal{N} \equiv \langle V_{\bar{\lambda}}, \in \rangle$ admitting an automorphism $j: \mathcal{N} \longrightarrow \mathcal{N}$ such that - $\mathcal{N} \models j(\alpha) \geq \alpha$ for all $\alpha \in \mathrm{Ord}^\mathcal{N}$, - there exists an ordinal $\alpha \in \mathrm{Ord}^\mathcal{N}$ such that $\mathcal{N} \models j(\alpha) > \alpha$. Axiom scheme (II) above ensures that $\mathcal{N} \models T$. Automorphisms from non-standard models {#Sec:AutomorphismsFromNonStandardModels} ====================================== In Theorem \[Th:AutomorphismMovingNoNaturalDown\] and Theorem \[Th:TransitiveModelsYieldAutomorphismMovingNoOrdinalDown\] we built models that admit automorphisms which are well-behaved in the sense that they do not move any point down along an initial segment of the ordinals. In this section we will use non-standard models of set theory with standard parts to construct models of set theory equipped with automorphisms which do not move any points down along an initial segment of the ordinals but do move points down above this initial segment. This allows us to answer Question \[Q:HolmesQuestion\].\ \ We begin by showing that models of set theory admitting an automorphism which does not move any natural number down and does move an ordinal above the natural numbers down can be built from a non-standard $\omega$-model of set theory. Throughout this section $\hat{f}$ will be a new unary function symbol and $\bar{\beta}$ will be a new constant symbol. This allows us to define the language $\mathcal{L}^{\mathbf{ind}}_{\hat{f}, \bar{\beta}}$ that extends $\mathcal{L}^_{\hat{f}, \bar{\beta}}$ in the same way that $\mathcal{L}^{\mathbf{ind}}$ extends $\mathcal{L}^$. We define $\mathcal{L}^{\mathbf{ind}}_{\hat{f}, \bar{\beta}} \supseteq \mathcal{L}^_{\hat{f}, \bar{\beta}}$ by adding - a unary predicate $\mathcal{C}$, - constant symbols $c_i$ for each $i \in \mathbb{Z}$. Equipped with this language we are able to define a theory extending $\mathbf{W}_1$ that yields models admitting the desired automorphism. \[Df:TheoryW3\] We define the $\mathcal{L}^{\mathbf{ind}}_{\hat{f}, \bar{\beta}}$-theory $\mathbf{W}_3 \supseteq \mathrm{Mac} \cup \mathbf{Sk}(\mathcal{L}_{\hat{f}, \bar{\beta}})$ by adding the axioms - $\forall x (\mathcal{C}(x) \Rightarrow x \in \omega)$ - $(\forall x \in \omega)(\exists y \in \omega)(y > x \land \mathcal{C}(y))$, - $\mathcal{C}(c_i)$ for all $i \in \mathbb{Z}$, - $c_i < c_j$ for all $i, j \in \mathbb{Z}$ with $i < j$, - $\bar{\beta}$ is an ordinal, - $(\forall x \in \omega)(\hat{f}(x) \in \bar{\beta})$, - $(\forall x \in \omega)(\forall y \in \omega)(x < y \Rightarrow \hat{f}(y) < \hat{f}(x))$, - for all $\mathcal{L}^_{\hat{f}, \bar{\beta}}$-formulae $\phi(x_1, \ldots, x_n)$, $$\forall x_1 \ldots \forall x_n \forall y_1 \ldots \forall y_n \left( \begin{array}{c} x_1 < \ldots < x_n \land y_1 < \ldots < y_n \land \bigwedge_{1 \leq i \leq n} (\mathcal{C}(x_i) \land \mathcal{C}(y_i))\\ \Rightarrow (\phi(x_1, \ldots, x_n) \iff \phi(y_1, \ldots, y_n)) \end{array} \right),$$ - for all $\mathcal{L}^_{\hat{f}, \bar{\beta}}$ Skolem functions $\mathbf{t}(x_1, \ldots, x_n)$ and for all $i_1 < \ldots < i_n$ and in $\mathbb{Z}$, $$\forall x (\mathcal{C}(x) \Rightarrow \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < x) \Rightarrow (\mathbf{t}(c_{i_1}, \ldots, c_{i_n})= \mathbf{t}(c_{j_1}, \ldots, c_{j_n})).$$ As we saw with the theory $\mathbf{W}_1$, axioms (i), (ii) and (iii) and axiom scheme (vii) of Definition \[Df:TheoryW3\] ensure that $\mathcal{C}$ defines a class of order indiscernibles that is cofinal in the natural numbers and the $c_i$s form a subclass of $\mathcal{C}$. Axiom scheme (ix) ensures that every Skolem term which is interpreted by a natural number less than every element in $\mathcal{C}$ is fixed when the sequence of $c_i$s appearing in this term is replaced by an order equivalent sequence of $c_i$s. Axioms (v), (vi) and (vii) ensure that $\hat{f}$ is an order reversing function from the natural numbers into the ordinal $\bar{\beta}$. As with $\mathbf{W}_1$, the Skolem hull of a model of $\mathbf{W}_3$ that contains the $c_i$s will admit an automorphism that does not move any natural number down. The behaviour of the unary function symbol $\hat{f}$ will ensure that any automorphism of a substructure of a model of $\mathbf{W}_3$ moves the point interpreting $\hat{f}(c_0)$ in the opposite direction to the point interpreting $c_0$.\ \ We are now in a position to show that the existence of an $\omega$-model of $\mathrm{Mac}$ that is non-standard below an ordinal $\bar{\beta}$ implies the existence of an elementarily equivalent model of $\mathbf{W}_3$. \[Th:ConsistencyOfW3\] If $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M}, \bar{\beta}^\mathcal{M} \rangle$ is an $\omega$-model of $\mathrm{Mac}$ such that\ $\mathcal{M} \models \bar{\beta} \textrm{ is an ordinal}$, and $\mathcal{M}$ is non-standard below $\bar{\beta}^\mathcal{M}$ then $\mathbf{W}_3 \cup \mathbf{Th}_{\mathcal{L}_{\bar{\beta}}}(\mathcal{M})$ is consistent. Let $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M}, \bar{\beta}^\mathcal{M} \rangle$ be an $\omega$-model of $\mathrm{Mac}$ such that $$\mathcal{M} \models \bar{\beta} \textrm{ is an ordinal}$$ and $\mathcal{M}$ is non-standard below $\bar{\beta}^\mathcal{M}$. Let $g: \omega \longrightarrow (\omega^\mathcal{M})^$ be the isomorphism guaranteed by the fact that $\mathcal{M}$ is an $\omega$-model. Let $f: \omega \longrightarrow (\bar{\beta}^\mathcal{M})^$ be such for all $n, m \in \omega$, $$n < m \textrm{ if and only if } f(m) <^\mathcal{M} f(n).$$ The existence of such a function is guaranteed by the fact that $\mathcal{M}$ is non-standard below $\bar{\beta}^\mathcal{M}$. As with the proof of Lemma \[Th:ConsistencyOfW1\] and Lemma \[Th:ConsistencyOfW2\] we will use compactness to show that $\mathbf{W}_3 \cup \mathbf{Th}_{\mathcal{L}_{\bar{\beta}}}$ is consistent. Let $\Delta \subseteq \mathbf{W}_3$ be finite. Using $f$ to interpret $\hat{f}$ we can expand $\mathcal{M}$ to an $\mathcal{L}_{\hat{f}, \bar{\beta}}$-structure $\mathcal{M}^\prime$ that satisfies (v), (vi) and (vii) of Definition \[Df:TheoryW3\]. Using the same arguments we used in the proof of Lemma \[Th:ConsistencyOfW1\] we can find $C \subseteq (\omega^\mathcal{M})^$ and expand $\mathcal{M}^\prime$ to a structure $\mathcal{M}^{\prime\prime}$ in which $\mathcal{C}$ is interpreted using $C$ and - $\mathcal{M}^{\prime\prime} \models \mathbf{Sk}(\mathcal{L}_{\hat{f}, \bar{\beta}})$, - $\mathcal{M}^{\prime\prime} \mid_{\mathcal{L}_{\hat{f}, \bar{\beta}}}= \mathcal{M}^\prime$, - $\mathcal{M}^{\prime\prime}$ satisfies (i) and (ii) of Definition \[Df:TheoryW3\], - $\mathcal{M}^{\prime\prime}$ satisfies all instances of \[Df:TheoryW3\](viii) appearing in $\Delta$. Without loss of generality we can assume that $\Delta$ only mentions the constant symbols $c_{-n}, \ldots, c_0, \ldots c_n$ and $\bar{\beta}$. Again, using the same arguments we used in the proof Lemma \[Th:ConsistencyOfW1\] we can find an infinite $D \subseteq C$ such that by interpreting $c_{-n}, \ldots, c_0, \ldots, c_n$ using the first $2\cdot n +1$ of $D$ we can expand $\mathcal{M}^{\prime\prime}$ to a structure $\mathcal{M}^{\prime\prime\prime}$ that satisfies all instance of (iii) and (ix) of Definition \[Df:TheoryW3\] which appear in $\Delta$. Therefore $$\mathcal{M}^{\prime\prime\prime} \models \Delta \cup \mathbf{Th}_{\mathcal{L}_{\bar{\beta}}}(\mathcal{M}) \cup \mathbf{Sk}(\mathcal{L}_{\hat{f}, \bar{\beta}}).$$ Therefore, by compactness, $\mathbf{W}_3 \cup \mathbf{Th}_{\mathcal{L}_{\bar{\beta}}}(\mathcal{M})$ is consistent. Axiom schemes (viii) and (ix) of Definition \[Df:TheoryW3\] allow us to prove analogues of Lemmas \[Th:IndiscerniblesCofinalInSkolemHull\] and \[Th:SkolemFunctionsFixedBelow\]. \[Th:IndiscerniblesCofinalInSkolemHull2\] (Körner [@kor94]) Let $\mathcal{M} \models \mathbf{W}_3$. For all $\mathcal{L}^_{\hat{f}, \bar{\beta}}$ Skolem functions $\mathbf{t}(x_1, \ldots, x_n)$ and for all $i_1 < \ldots < i_n$ in $\mathbb{Z}$, if $\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) \in \omega$ then there exists a $k \in \mathbb{Z}$ such that $$\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < c_k.$$ Using the same arguments as the proof of Lemma \[Th:IndiscerniblesCofinalInSkolemHull\]. \[Th:SkolemFunctionsFixedBelow2\] Let $\mathcal{M} \models \mathbf{W}_3$. For all $\mathcal{L}^_{\hat{f}, \bar{\beta}}$ Skolem functions $\mathbf{t}(x_1, \ldots, x_n)$ and for all $i_1 < \ldots < i_n$ and $j_1 < \ldots < j_n$ in $\mathbb{Z}$, if for all $k \in \mathbb{Z}$, $\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n}) < c_k$ then $$\mathcal{M} \models \mathbf{t}(c_{i_1}, \ldots, c_{i_n})= \mathbf{t}(c_{j_1}, \ldots, c_{j_n}).$$ Using the same arguments as the proof of Lemma \[Th:SkolemFunctionsFixedBelow\]. This allows us to show that $\mathcal{L}_{\bar{\beta}}$-structures which are $\omega$-models of $\mathrm{Mac}$ and non-standard below $\bar{\beta}$ yield elementarily equivalent models admitting an automorphism that does not move any natural number down but moves an ordinal less than $\bar{\beta}$ down. \[Th:AutomorphismsFromNonStandardOmegaModels\] If $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M}, \bar{\beta}^\mathcal{M} \rangle$ is an $\omega$-model of $\mathrm{Mac}$ such that\ $\mathcal{M} \models \bar{\beta} \textrm{ is an ordinal}$ and $\mathcal{M}$ is non-standard below $\bar{\beta}^\mathcal{M}$ then there exists an $\mathcal{L}_{\bar{\beta}}$-structure $\mathcal{N} \equiv \mathcal{M}$ admitting an automorphism $j: \mathcal{N} \longrightarrow \mathcal{N}$ such that - $\mathcal{N} \models j(n) \geq n$ for all $n \in (\omega^\mathcal{N})^$, - there exists $\alpha \in (\bar{\beta}^\mathcal{N})^$ such that $\mathcal{N} \models j(\alpha) < \alpha$. Let $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M}, \bar{\beta}^\mathcal{M} \rangle$ be an $\omega$-model of $\mathrm{Mac}$ such that $$\mathcal{M} \models \bar{\beta} \textrm{ is an ordinal}$$ and $\mathcal{M}$ is non-standard below $\bar{\beta}^\mathcal{M}$. By Lemma \[Th:ConsistencyOfW3\] there exists an $\mathcal{L}^{\mathbf{ind}}_{\hat{f}, \bar{\beta}}$-structure $\mathcal{Q} \models \mathbf{W}_3$ such that $\mathcal{Q} \mid_{\mathcal{L}_{\bar{\beta}}} \equiv \mathcal{M}$. Let $C= \{ c_i^\mathcal{Q} \mid i \in \mathbb{Z} \}$. Let $\mathcal{N}= \mathcal{H}_{\mathcal{L}_{\hat{f}, \bar{\beta}}}^{\mathcal{Q}}(C)$. Therefore $\mathcal{N} \mid_{\mathcal{L}_{\bar{\beta}}} \equiv \mathcal{M}$. Define $j: \mathcal{N} \longrightarrow \mathcal{N}$ by $$j(c_i^\mathcal{Q})= c_{i+1}^\mathcal{Q} \textrm{ for all } i \in \mathbb{Z},$$ $$\textrm{for all } \mathcal{L}^_{\hat{f}, \bar{\beta}} \textrm{ Skolem functions } \mathbf{t}(x_1, \ldots, x_n) \textrm{ and for all } i_1 < \ldots < i_n \textrm{ in } \mathbb{Z},$$ $$j(\mathbf{t}^\mathcal{Q}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}))= \mathbf{t}^\mathcal{Q}(j(c_{i_1}^\mathcal{Q}), \ldots, j(c_{i_n}^\mathcal{Q})).$$ Now, $j$ is an automorphism of the $\mathcal{L}_{\hat{f}, \bar{\beta}}$-structure $\mathcal{N}$. The same arguments used in the proof of Theorem \[Th:AutomorphismMovingNoNaturalDown\] reveal that if $\mathbf{t}(x_1, \ldots, x_n)$ is an $\mathcal{L}^_{\hat{f}, \bar{\beta}}$ Skolem function and $i_1 < \ldots < i_n$ are in $\mathbb{Z}$ such that $\mathcal{N} \models \mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}) \in \omega$ then $$\mathcal{N} \models j(\mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q})) \geq \mathbf{t}(c_{i_1}^\mathcal{Q}, \ldots, c_{i_n}^\mathcal{Q}).$$ Now, $j(\hat{f}^{\mathcal{Q}}(c_0^\mathcal{Q}))= \hat{f}^\mathcal{Q}(c_1^\mathcal{Q}) \in (\bar{\beta}^\mathcal{Q})^$, and $$\mathcal{Q} \models \hat{f}(c_1) < \hat{f}(c_0) < \bar{\beta}.$$ Therefore, $$\mathcal{N} \models j(\hat{f}(c_0^\mathcal{Q})) < \hat{f}(c_0^\mathcal{Q}) \textrm{ and } \hat{f}^\mathcal{N}(c_0^\mathcal{Q}) \in (\bar{\beta}^\mathcal{N})^.$$ Therefore $j$ is the desired automorphism of the model $\mathcal{N} \mid_{\mathcal{L}_{\bar{\beta}}}$. Theorem \[Th:AutomorphismsFromNonStandardOmegaModels\] allows us to give a positive answer to Question \[Q:HolmesQuestion\] by showing that every complete consistent extension of $\mathrm{ZFC}$ has a model admitting an automorphism that does not move any natural number down but does move an ordinal less than the first non-recursive ordinal down. Using the same argument that we used in the proof of Theorem \[Th:ZFCwithAutThatDoesNotMoveAnyOrdinalDown\] we will consider a model $\mathcal{M}$ of a complete consistent extension of $\mathrm{ZFC}$ with a limit ordinal $\lambda \in M^\mathcal{M}$ such that externally $\langle V_\lambda^\mathcal{M}, \in^\mathcal{M} \rangle$ is elementarily equivalent with $\mathcal{M}$. Using an argument developed by Harvey Friedman in [@fri73] we will apply the Barwise Compactness Theorem (Theorem \[Th:BarwiseCompactness\]) to $V_\lambda$ inside $\mathcal{M}$ to obtain an $\omega$-model of an extension of $\mathrm{Mac}$ that is non-standard below $\omega_1^{\mathrm{ck}}$. Applying Theorem \[Th:AutomorphismsFromNonStandardOmegaModels\] to this non-standard $\omega$-model will yield the model with the desired automorphism. \[Th:HolmesAutomorphism\] If $T \supseteq \mathrm{ZFC}$ is a complete and consistent $\mathcal{L}$-theory then there is an $\mathcal{L}$-structure $\mathcal{N} \models T$ admitting an automorphism $j: \mathcal{N} \longrightarrow \mathcal{N}$ such that - $\mathcal{N} \models j(n) \geq n$ for all $n \in (\omega^\mathcal{N})^$, - there exists an $\alpha \in ((\omega_1^{\mathrm{ck}})^\mathcal{N})^$ such that $\mathcal{N} \models j(\alpha) < \alpha$. Let $T \supseteq \mathrm{ZFC}$ be a complete and consistent $\mathcal{L}$-theory. Let $\bar{\lambda}$ be a new constant symbol. Define $T^\star \supseteq T$ to be the $\mathcal{L}_{\bar{\lambda}}$-theory that includes the axioms: - $\bar{\lambda} > \omega_1^{\mathrm{ck}}$ is a limit ordinal, - for all $\mathcal{L}$-sentences, $\sigma$, $$(\langle V_{\bar{\lambda}}, \in \rangle \models \sigma) \iff \sigma.$$ Let $\Delta \subseteq T^\star$ be finite. Let $\mathcal{M}^\prime$ be an $\mathcal{L}$-structure such that $\mathcal{M}^\prime \models T$. Using the Reflection Principle we can expand $\mathcal{M}^\prime$ to an $\mathcal{L}_{\bar{\lambda}}$-structure $\mathcal{M}^{\prime\prime}$ such that $$\mathcal{M}^{\prime\prime} \models \Delta.$$ Using the Compactness Theorem we can find an $\mathcal{L}_{\bar{\lambda}}$-structure $\mathcal{M} \models T^\star$ such that $\omega^\mathcal{M}$ is non-standard. Let $a \in (\omega^\mathcal{M})^$ be non-standard.\ We work inside $\mathcal{M}$. Let $A= L_{\omega_1^{\mathrm{ck}}}$. Let $\mathcal{L}^\prime$ be the extension of $\mathcal{L}$ obtained by adding - the constant symbol $\bar{\beta}$, - constant symbols $\hat{\alpha}$ for every $\alpha \in \omega_1^{\mathrm{ck}}$, - constant symbol $\mathbf{c}$. Let $\Gamma= \{ \phi \in \mathbf{Th}_{\mathcal{L}_{\bar{\beta}}}(\langle V_{\bar{\lambda}}, \in, \omega_1^{\mathrm{ck}} \rangle) \mid \ulcorner \phi \urcorner \in V_a \}$.\ Define the $(\mathcal{L}^\prime_{\omega_1 \omega})_A$-theory $Q$ with axioms - $\Gamma$, - $\hat{\zeta} \in \hat{\nu}$ for all $\zeta \in \nu \in \omega_1^{\mathrm{ck}}$, - for all $\nu \in \omega_1^{\mathrm{ck}}$, $$\forall x \left( x \in \hat{\nu} \Rightarrow \bigvee_{\zeta \in \nu} (x= \hat{\zeta})\right),$$ - $\mathbf{c}$ is an ordinal, - $\hat{\nu} \in \mathbf{c}$ for all $\nu \in \omega_1^{\mathrm{ck}}$, - $\forall x (x \in \mathbf{c} \cup \{\mathbf{c}\} \Rightarrow (x \textrm{ is not admissible}))$. Since $\Gamma$ is finite, $Q$ is $\Sigma_1(\mathcal{L})$ over $A$. Let $Q^\prime \subseteq Q$ be such that $Q^\prime \in A$. Therefore $$\{ \alpha \in \omega_1^{\mathrm{ck}} \mid \hat{\alpha} \textrm{ is mentioned in } Q^\prime \} \textrm{ is bounded in } \omega_1^{\mathrm{ck}}.$$ Therefore we can find a $\xi \in \omega_1^{\mathrm{ck}}$ which is greater than every $\alpha \in \omega_1^{\mathrm{ck}}$ such that $\hat{\alpha}$ is mentioned in $Q^\prime$. We expand $\langle V_{\bar{\lambda}}, \in, \omega_1^{\mathrm{ck}} \rangle$ by - interpreting $\mathbf{c}$ using $\xi$, - if $\gamma \leq \alpha$ and $\hat{\alpha}$ is mentioned in $Q^\prime$ then we interpret $\hat{\gamma}$ using $\gamma$, to obtain a structure $\mathcal{Q} \models Q^\prime$. Therefore, by the Barwise Compactness Theorem, the $(\mathcal{L}^\prime_{\omega_1 \omega})_A$-theory $Q$ is consistent. Let $\mathcal{N}^{\prime\prime} \models Q$ and let $\mathcal{N}^\prime= \langle (N^\prime)^{\mathcal{N}^\prime}, \in^{\mathcal{N}^\prime}, \bar{\beta}^{\mathcal{N}^\prime} \rangle$ be the $\mathcal{L}_{\bar{\beta}}$-reduct of $\mathcal{N}^{\prime\prime}$. The finite axiom scheme (i) ensures that $\mathcal{N}^\prime \models \mathrm{Mac}$. Axiom schemes (ii) and (iii) of $Q$ ensure that $\mathcal{N}^\prime$ is an $\omega$-model. Now, suppose that $\mathcal{N}^\prime$ is well-founded below $\bar{\beta}^{\mathcal{N}^\prime}$. Let $\eta$ be the least element of $$\{ \xi \in (N^{\prime})^{\mathcal{N}^\prime} \mid (\forall \nu \in \omega_1^{\mathrm{ck}}) \mathcal{N}^{\prime\prime} \models \hat{\nu} \in \xi \} .$$ Now, $\eta= \omega_1^{\mathrm{ck}}$ and $L_{\omega_1^{\mathrm{ck}}}= L_\eta^{\mathcal{N}^\prime}$. Therefore $$\mathcal{N}^\prime \models \eta \textrm{ is admissible}.$$ But this contradicts axiom (vi) of $Q$ since $$\mathcal{N}^{\prime\prime} \models \eta \leq \mathbf{c}.$$ Therefore $\mathcal{N}^\prime$ is an $\omega$-model of $\mathrm{Mac}$ that is non-standard below $\bar{\beta}^{\mathcal{N}^\prime}$. And by (i) of $Q$, $$\mathcal{N}^\prime \models \Gamma.$$ Therefore by Theorem \[Th:AutomorphismsFromNonStandardOmegaModels\] there exists an $\mathcal{L}_{\bar{\beta}}$-structure $\mathcal{N} \equiv \mathcal{N}^\prime$ admitting an automorphism $j: \mathcal{N} \longrightarrow \mathcal{N}$ such that - $\mathcal{N} \models j(n) \geq n$ for all $n \in (\omega^\mathcal{N})^$, - there exists $\alpha \in (\bar{\beta}^\mathcal{N})^$ such that $\mathcal{N} \models j(\alpha) < \alpha$. Now, $\mathcal{N} \models \bar{\beta}= \omega_1^{\mathrm{ck}}$ and $\mathcal{N} \models \Gamma$. Therefore the ambient arithmetic believes that $\mathcal{N} \models T$. And $j: \mathcal{N} \mid_{\mathcal{L}} \longrightarrow \mathcal{N} \mid_{\mathcal{L}}$ is an automorphism such that - $\mathcal{N} \mid_{\mathcal{L}} \models j(n) \geq n$ for all $n \in (\omega^{\mathcal{N} \mid_{\mathcal{L}}})^$, - there exists an $\alpha \in ((\omega_1^{\mathrm{ck}})^{\mathcal{N} \mid_{\mathcal{L}}})^$ such that $\mathcal{N} \mid_{\mathcal{L}} \models j(\alpha) < \alpha$. NFU {#Sec:NFU} === In this section we give a brief description of the modification of Quine’s ‘New Foundations’ set theory $\mathrm{NFU}$ that was first introduced by Ronald Jensen in [@jen69]. A detailed introduction to $\mathrm{NFU}$ which includes the specifics of how mathematics can be formalised in this theory can be found in the introductory textbook [@hol98]. We would also like to direct readers to [@solXX] which contains a more condensed introduction to $\mathrm{NFU}$.\ \ By weakening the extensionality axiom in Quine’s $\mathrm{NF}$ to allow objects that are not sets into the domain of discourse, Jensen [@jen69] defines the subsystem $\mathrm{NFU}$ of $\mathrm{NF}$ which he is able to show is equiconsistent with a weak subsystem of $\mathrm{ZFC}$. In contrast to $\mathrm{NF}$ (see [@spe53]), Jensen’s modified system is consistent with both the Axiom of Choice and the negation of the Axiom of Infinity. As we have already mentioned, in this paper we will use $\mathrm{NFU}$ to denote Jensen’s weakening of Quine’s $\mathrm{NF}$ fortified with both the Axiom of Infinity and the Axiom of Choice. Following [@hol98] and [@solXX] the Axiom of Infinity will be obtained in $\mathrm{NFU}$ by introducing a type-level ordered pair as a primitive notion. The underlying language of $\mathrm{NFU}$ is the language of set theory ($\mathcal{L}$) endowed with a unary predicate $\mathcal{S}$ and a ternary predicate $P$. The predicate $\mathcal{S}$ will be used to distinguish sets from non-sets. The ternary predicate $P$ will define a pairing function. Before presenting the axioms of $\mathrm{NFU}$ we first need to define the notion of a stratified formula. \[Df:StratifiedFormulae\] An $\mathcal{L}_{\mathcal{S}, P}$-formula $\phi$ is stratified if and only if there is a function $\sigma$ from the variables appearing in $\phi$ to $\mathbb{N}$ such that - if $\textrm{`}x=y\textrm{'}$ is a sub-formula of $\phi$ then $\sigma(\textrm{`}x\textrm{'})= \sigma(\textrm{`}y\textrm{'})$, - if $\textrm{`}x \in y\textrm{'}$ is a sub-formula of $\phi$ then $\sigma(\textrm{`}y\textrm{'})= \sigma(\textrm{`}x\textrm{'}) + 1$, - if $\textrm{`}P(x, y, z)\textrm{'}$ is a sub-formula of $\phi$ then $\sigma(\textrm{`}z\textrm{'})= \sigma(\textrm{`}x\textrm{'})= \sigma(\textrm{`}y\textrm{'})$. Equipped with this definition we are able to give the axioms of $\mathrm{NFU}$. \[Df:NFU\] We define $\mathrm{NFU}$ to be the $\mathcal{L}_{\mathcal{S}, P}$-theory with axioms - (Extensionality for Sets) $$\forall x \forall y(\mathcal{S}(x) \land \mathcal{S}(y) \Rightarrow (x=y \iff \forall z(z \in x \iff z \in y))),$$ - (Stratified Comprehension) for all stratified $\mathcal{L}_{\mathcal{S}, P}$-formulae $\phi(x, \vec{z})$, $$\forall \vec{z} \exists y (\mathcal{S}(y) \land \forall x(x \in y \iff \phi(x, \vec{z}))),$$ - (Axioms of Pairing) $$\forall x \forall y \exists z (P(x, y, z) \land \forall w (P(x, y, w) \Rightarrow w= z)),$$ $$\forall x \forall y \forall u \forall v \forall z (P(x, y, z) \land P(u, v, z) \Rightarrow u = x \land v = y),$$ - (Axiom of Choice) every set is well ordered. Stratified comprehension proves the existence of a universal set which we will denote $V$. As is suggested by the above axiomatisation, ordered pairs in $\mathrm{NFU}$ are coded using the predicate $P$. That is to say, we define $$\langle x, y \rangle= z \textrm{ if and only if } P(x, y, z).$$ The map $x \mapsto \langle x, x \rangle$ is a set by Stratified Comprehension, and the Axiom of Pairing implies that this map is an injection that is not a surjection. This shows that, as we mentioned above, the Axioms of Pairing imply the Axiom of Infinity.\ \ The existence of ‘large’ sets allows $\mathrm{NFU}$ to represent cardinal numbers as equivalence classes of equipollent sets. If $X$ is a set then we write $\|X\|$ for the cardinal to which $X$ belongs. In a similar fashion ordinals can be represented by equivalence classes of isomorphic well-orderings. If $R$ is a well-order then we will write $[R]$ for ordinal to which $R$ belongs. These representations allow the properties of being a cardinal number and being an ordinal number to be expressed using stratified formulae. This means that stratified comprehension proves the existence of a set of all cardinals, which we will denote $\mathrm{CN}$, and the existence of a set of all ordinals, which we will denote $\mathrm{ON}$. If $\alpha$ and $\beta$ are ordinals we say that $\alpha < \beta$ if and only if for every $R \in \alpha$ and $S \in \beta$, there exists an order isomorphism from $R$ into a proper initial segment of $S$. The order $<$ well-orders $\mathrm{ON}$ and we will use $\Omega$ to denote the ordinal to which $<$ belongs. The fact that cardinals and ordinals are coded using disjoint classes of objects endows importance to the following definition which mirrors the definition of initial ordinal in $\mathrm{ZFC}$. If $R$ is a binary relation then we use $\mathrm{Dom}(R)$ to denote the set $\{ x \mid \exists y (\langle x, y \rangle \in R \lor \langle y, x \rangle \in R)\}$. If $\kappa$ is a cardinal then we define $\mathrm{init}(\kappa)$ to be the least ordinal $\alpha$ such that there exists an $R \in \alpha$ and an $X \in \kappa$ such that $\mathrm{Dom}(R)= X$. We will use $\mathbb{N}$ to denote the set of all finite cardinal numbers and $\mathrm{CNI}$ to denote the set of all infinite cardinal numbers. If $\kappa$ and $\lambda$ are cardinal numbers then we say that $\kappa < \lambda$ if and only if for all $X \in \kappa$ and $Y \in \lambda$ there is an injection from $X$ into $Y$ but no bijection between $X$ and $Y$. The axiom of choice ensures that this ordering is a well-ordering. We will use $\aleph_0$ to denote the least infinite cardinal number and $\aleph_1$ to denote the least uncountable cardinal. If $X \subseteq \mathrm{CN}$ then we use $\sup X$ to denote the $<$-least $\kappa$ such that for all $\lambda \in X$, $\lambda \leq \kappa$. It should be noted that the restriction of comprehension to stratified formulae prevents $\mathrm{NFU}$ from proving the existence of von Neumann ordinals corresponding to well-orderings (see [@hen69] and [@fh09]).\ \ Another unorthodox feature of $\mathrm{NFU}$ is the fact that it fails to prove that an arbitrary set is the same size as its own set of singletons. We will use $\iota$ to denote the singleton operation. We say that a set $x$ is Cantorian if and only if $x$ is the same size as $\iota``x$. We say that $x$ is strongly Cantorian if and only if $\iota \upharpoonright x$ is a set. For example, if $x$ is a set that has size a concrete natural number then $\mathrm{NFU}$ proves that $x$ is strongly Cantorian. On the other hand, $\mathrm{NFU}$ proves that $V$ is not Cantorian. This behaviour motivates the definition of the $T$ operation on cardinals, ordinals and equivalence classes of well-founded relations. We define the $T$ operation on cardinals and equivalence classes of well-founded relations as follows: - if $\kappa$ is a cardinal then $$T(\kappa)= \|\iota``A\| \textrm{ where } A \in \kappa,$$ - if $[W]$ is the equivalence class of isomorphic well-founded relations with $W \in [W]$ then $$T([W])= \textrm{ the class of relations isomorphic to } \{ \langle \iota x, \iota y \rangle \mid \langle x, y \rangle \in W\}.$$ It should be noted that any stratification of the formula $x= T(y)$ assigns a type to $x$ that is one higher than the type of $y$. In fact, if $\mathrm{NFU}$ is consistent then the collection $\{ \langle x, y \rangle \mid x= T(y) \}$ is a proper class. Another definable set operation that is inhomogeneous in the sense that any stratification of the formula defining this operation assigns different types to the arguments and the result is the powerset operation ($\mathcal{P}$). This inhomogeneity prevents $\mathrm{NFU}$ from proving Cantor’s Theorem, and indeed the cardinal $\|V\|$ is an explicit counterexample to this result since $\mathcal{P}(V) \subseteq V$. Despite this failure, $\mathrm{NFU}$ is able to prove that for all sets $X$, $T(\|X\|) < \|\mathcal{P}(X)\|$. The following definition facilitates a type-homogeneous notion of cardinal exponentiation. If $\kappa$ is a cardinal then we define $$2^{\kappa}= \left\{\begin{array}{ll} \lambda & \textrm{if there exists an } X \in \kappa \textrm{ and a cardinal } \lambda \textrm{ s.t. } T(\lambda)= \|\mathcal{P}(X)\| \\ \emptyset & \textrm{otherwise} \end{array} \right)$$ The operation that sends $\kappa \mapsto 2^{\kappa}$ is definable by an $\mathcal{L}_{\mathcal{S}, P}$-formula that admits a stratification which assigns the same type to $\kappa$ and $2^{\kappa}$. The following result shows that this exponentiation operation possesses the strictly inflationary property that we intuitively associate with cardinal exponentiation. \[Th:ExpInflationary\] Let $\kappa$ be a cardinal. If $2^{\kappa} \neq \emptyset$ then $\kappa < 2^{\kappa}$. This allows us to define an analogue of the Beth operation. We define [$$\bar{\bar{\beth}}= \bigcap \left\{ X \subseteq \mathrm{CNI} \mid (\aleph_0 \in X) \land (\forall \kappa \in \mathrm{CNI})(\kappa \in X \Rightarrow 2^\kappa \in X) \land \forall Y ( Y \subseteq X \Rightarrow \sup Y \in X)\right\}.$$]{} Stratified Comprehension ensures that $\bar{\bar{\beth}}$ is a set and Lemma \[Th:ExpInflationary\] implies that $\bar{\bar{\beth}}$ is well-ordered by the natural ordering on cardinal numbers. If $\alpha$ is ordinal such that $\alpha < [< \cap(\bar{\bar{\beth}} \times \bar{\bar{\beth}})]$ then we define $\beth_\alpha^{TT}$ to be the $\alpha^{\textrm{th}}$ member of $\bar{\bar{\beth}}$. The $T$ operation allows us to make explicit the relationship between an ordinal $\alpha$ and the initial segment of $\mathrm{ON}$ defined by $\alpha$. \[Th:OrderTypeOfOrdinalsLessThan\] If $\alpha$ is an ordinal and $X= \{ \beta \in \mathrm{ON} \mid \beta < \alpha \}$ then $[< \cap X \times X]= T^2(\alpha)$. Asserting that the $T$ operation is well-behaved is a natural way of extending the axioms of $\mathrm{NFU}$. One family of examples of this type of extension is Rosser’s Axiom of Counting [@ros53] and two weakenings introduced in [@for77]. - ($\mathrm{AxCount}$) $(\forall n \in \mathbb{N}) T(n)=n$, - ($\mathrm{AxCount}_\leq$) $(\forall n \in \mathbb{N}) n \leq T(n)$, - ($\mathrm{AxCount}_\geq$) $(\forall n \in \mathbb{N}) n \geq T(n)$. In [@ros53] it is shown that $\mathrm{AxCount}$ facilitates induction for unstratified properties. This added strength allows $\mathrm{NFU}+\mathrm{AxCount}$ to prove the consistency of Zermelo set theory [@hin75] and implies that $\mathrm{NFU}+\mathrm{AxCount}$ is consistent with the existence of classical sets such as the von Neumann $\omega$ and $V_\omega$ [@for06]. In [@for77], [@hin75] and [@for06] it is observed that some of these consequences of $\mathrm{AxCount}$ also follow from the weakening $\mathrm{AxCount}_\leq$.\ \ In [@hol98] Holmes adapts the techniques developed by Rolland Hinnion in [@hin75] to show that fragments of Zermelo-Fraenkel set theory can be interpreted in classes of topped well-founded extensional relations in $\mathrm{NFU}$. Let $R$ be a well-founded relation. For all $x \in \mathrm{Dom}(R)$, define $\mathrm{seg}_R(x)$ by induction $$S_0= \{ \langle z, x \rangle \mid \langle z, x \rangle \in R \},$$ $$S_{n+1}= \{ \langle w, z \rangle \mid z \in \mathrm{Dom}(S_n) \land \langle w, z \rangle \in R \},$$ $$\mathrm{seg}_R(x)= \bigcup_{n \in \mathbb{N}} S_n.$$ \[Df:BFEXT\] We say that a relation $R$ is a topped well-founded extensional relation (BFEXT) if and only if $R$ is well-founded extensional and there exists an $x \in \mathrm{Dom}(R)$ such that $R= \mathrm{seg}_R(x)$. Well-foundedness implies that if $R$ is a BFEXT then the $x \in \mathrm{Dom}(R)$ such that $R= \mathrm{seg}_R(x)$ is unique— we will use $\mathbf{t}_R$ to denote this element. \[Df:EqualityAndMembershipForBFEXTs\] Let $R$ and $S$ be BFEXTs. We say that $R \cong S$ if and only if there is a bijection $f: \mathrm{Dom}(R) \longrightarrow \mathrm{Dom}(S)$ such that $$\langle x, y \rangle \in R \textrm{ if and only if } \langle f(x), f(y) \rangle \in S.$$ We say that $R \epsilon S$ if and only if there exists an $x \in \mathrm{Dom}(S)$ such that $\langle x, \mathbf{t}_S \rangle \in S$ and $R \cong \mathrm{seg}_S(x)$. If $R$ is a BFEXT then stratified comprehension allows $\mathrm{NFU}$ to prove that the following collections are sets: $$\label{Df:BFEXTEquivRelationInNFU} [R]= \{ S \mid (S \textrm{ is a BFEXT }) \land (R \cong S) \}$$ $$\label{Df:BFinNFU} \mathrm{BF}= \{ [R] \mid R \textrm{ is a BFEXT } \}$$ We abuse notation and define the relation $\epsilon$ on $\mathrm{BF}$. For all $X, Y \in \mathrm{BF}$ we define $$X \epsilon Y \textrm{ if and only if there exists } R \in X \textrm{ and } S \in Y \textrm{ such that } R \epsilon S.$$ In [@solXX] Robert Solovay observes, without proof, that the structure $\langle \mathrm{BF}, \epsilon \rangle$ looks like $\langle H_{\kappa^+}, \in \rangle$ in the sense that there is a cardinal $\kappa$ such that - $\langle \mathrm{BF}, \epsilon \rangle$ is well-founded and extensional, - for every BFEXT $R$, $$\|\{ X \in \mathrm{BF} \mid X \epsilon [R] \}\| \leq \kappa,$$ - if $X \subseteq \mathrm{BF}$ with $\|X\| \leq \kappa$ then there exists a BFEXT $R$ such that $$X = \{ Y \mid Y \epsilon [R] \}.$$ Property (I) above is proved by Hinnion in [@hin75]: (Hinnion [@hin75]) The theory $\mathrm{NFU}$ proves that the structure $\langle \mathrm{BF}, \epsilon \rangle$ is well-founded and extensional. For the sake of completeness we will prove that if $\kappa= T^2(\|V\|)$ then properties (II) and (III) above hold. \[Th:ExtensionsOfBFEXTsAreLessThanT2V\] The theory $\mathrm{NFU}$ proves that if $R$ is a BFEXT then $$\|\{ X \in \mathrm{BF} \mid X \epsilon [R] \}\| \leq T^2(\|V\|).$$ We work in $\mathrm{NFU}$. Let $R$ be a BFEXT. Let $\lhd$ be a well-ordering of $V$. Let $Y= \{ X \in \mathrm{BF} \mid X \epsilon [R] \}$. Define $f: Y \longrightarrow \iota^2``V$ by $$\label{Df:EmbeddingOfBFEXTensionIntoIotaV} \begin{array}{lcl} f(X)= \{\{x\}\} & \textrm{ if and only if } & S^{\prime\prime} \textrm{ is the } \lhd \textrm{-least element of } [R] \textrm{ and}\\ & & \textrm{there exists } S^\prime \in X \textrm{ and } f: S^\prime \longrightarrow S^{\prime\prime} \textrm{ witnessing}\\ & & \textrm{the fact that } S^\prime \cong \mathrm{seg}_{S^{\prime \prime}}(x) \textrm{ and } \langle x, \mathbf{t}_{S^{\prime\prime}} \rangle \in S^{\prime \prime}. \end{array}$$ It is clear from (\[Df:EmbeddingOfBFEXTensionIntoIotaV\]) that $f$ is defined by a stratified formula. Therefore $\mathrm{NFU}$ proves that $f$ is a set. The fact that $[R]$ is an equivalence class of isomorphic BFEXTs implies that $f$ is an injective function. Therefore $\|Y\| \leq T^2(\|V\|)$. An important tool for building BFEXTs is the following Lemma, proved in [@hol98], which shows that in $\mathrm{NFU}$ well-founded relations can be collapsed onto BFEXTs. \[Df:CollapsingLemmaForBFEXTs\] (Collapsing Lemma [@hol98]) The theory $\mathrm{NFU}$ proves that if $R$ is a well-founded relation then there exists an extensional well-founded relation $S$ and a map $g: \mathrm{Dom}(R) \longrightarrow \mathrm{Dom}(S)$ that is onto $\mathrm{Dom}(S)$ such that for all $x \in \mathrm{Dom}(R)$ and $z \in \mathrm{Dom}(S)$, $\langle z, g(x) \rangle \in S$ if and only if there exists a $y \in \mathrm{Dom}(R)$ such that $z= g(y)$ and $\langle y, x \rangle \in R$. Moreover, the relation $S$ is uniquely determined up to isomorphism by $R$. This allows us to prove that if $\kappa= T^2(\|V\|)$ then property (III) above holds. \[Th:SetOfBFEXTsLessThanT2VAreExtensions\] The theory $\mathrm{NFU}$ proves that if $X \subseteq \mathrm{BF}$ is such that $\|X\| \leq T^2(\|V\|)$ then there exists a BFEXT $W$ such that $$X= \{ Y \in \mathrm{BF} \mid Y \epsilon [W] \}.$$ We work in $\mathrm{NFU}$. Let $\lhd$ be a well-ordering of $V$. Let $X \subseteq \mathrm{BF}$ be such that $\|X\| \leq T^2(\|V\|)$. Let $f: X \longrightarrow \iota^2``V$ be an injection. Let $$D= \{ \mathrm{Dom}(R) \times \{a \} \mid (R \textrm{ is the } \lhd \textrm{-least element of }[R])\land ([R] \in X) \land(f([R])= \{\{a\}\})\}.$$ Let $\xi$ be such that $\xi \notin \bigcup D$. Define $S \subseteq \bigcup D \times \bigcup D$ by [$$\begin{array}{lcl} \langle x, y \rangle \in S & \textrm{ if and only if } & x= \langle w, a \rangle \textrm{ and } y= \langle z, a \rangle \textrm{ and } (R \textrm{ is the } \lhd \textrm{-least element of } [R])\\ & & \textrm{and }([R] \in X) \textrm{ and } (f([R])= \{\{a\}\}) \textrm{ and } (\langle w, z \rangle \in R) \textrm{ or},\\ & & x= \langle w, a \rangle \textrm{ and } y= \xi \textrm{ and } (R \textrm{ is the } \lhd \textrm{-least element of } [R])\\ & & \textrm{and } ([R] \in X) \textrm{ and } (f([R])= \{\{a\}\}) \textrm{ and } w= \mathbf{t}_S. \end{array}$$]{} Now, $S$ is a well-founded relation with a unique top $\xi$. Moreover, for all BFEXTs $R$, if $[R] \in X$ then there exists an $x \in \mathrm{Dom}(S)$ such that $\langle x, \xi \rangle \in S$ and $R \cong \mathrm{seg}_S(x)$. Let $W$ be the extensional collapse of $S$ obtained using Lemma \[Df:CollapsingLemmaForBFEXTs\]. Now, $W$ is a BFEXT and $$X= \{ Y \in \mathrm{BF} \mid Y \epsilon [W] \}.$$ This characterisation implies that the structure $\langle \mathrm{BF}, \epsilon \rangle$ is a model of $\mathrm{ZFC}^-$. \[Th:BFEXTsModelZFCminus\] (Hinnion [@hin75], Holmes [@hol98]) The theory $\mathrm{NFU}$ proves that $$\langle \mathrm{BF}, \epsilon \rangle \models \mathrm{ZFC}^-.$$ The operation $T$ restricted to the set $\mathrm{BF}$ is an embedding of the structure $\langle \mathrm{BF}, \epsilon \rangle$ into itself. That is to say, for all $[R], [S] \in \mathrm{BF}$, $$\textrm{both } T([R]) \textrm{ and } T([S]) \textrm{ are BFEXTs and}$$ $$[R] \epsilon [S] \textrm{ if and only if } T([R]) \epsilon T([S]).$$ Hinnion [@hin75] noted that the set $\mathrm{ON}$ can be bijectively mapped onto the von Neumann ordinals of the Zermelian structure $\langle \mathrm{BF}, \epsilon \rangle$. Define $\mathbf{k}: \mathrm{ON} \longrightarrow \mathrm{BF}$ by $$\label{Df:EmbeddingK} \mathbf{k}([R])= [R \cup \{ \langle x, x^ \rangle \mid x \in \mathrm{Dom}(R) \}] \textrm{ where } R \textrm{ is well-order such that } x^* \notin \mathrm{Dom}(R).$$ The definition of $\mathbf{k}$ admits a stratification that assigns the same type to both the result and the argument. Therefore stratified comprehension can be used to show that the map $\mathbf{k}: \mathrm{ON} \longrightarrow \mathrm{Ord}^{\langle \mathrm{BF}, \epsilon \rangle}$ is a set. It is also clear that the map $\mathbf{k}$ is order preserving and for all equivalence classes of well-orderings $[R]$, $$\mathbf{k}(T([R]))= T(\mathbf{k}([R])).$$ We can extend the notions of Cantorian and strongly Cantorian to ordinals and equivalence classes of BFEXTs. We say that an ordinal $\alpha$ is Cantorian if and only if $T(\alpha)= \alpha$. We say that $\alpha$ is strongly Cantorian if and only if for all $\beta \leq \alpha$, $T(\beta)= \beta$. \[Df:CandSCBFEXTs\] Let $R$ be a BFEXT. We say that $[R]$ is Cantorian if and only if $T([R])= [R]$. We say that $[R]$ is strongly Cantorian if and only if for all BFEXTs $S$ with $$\langle \mathrm{BF}, \epsilon \rangle \models [S] \epsilon \mathrm{TC}(\{[R]\}),$$ $T([S])= [S]$. In [@jen69] Jensen proves an analogue of Specker’s Theorem [@spe53] showing that models of $\mathrm{NFU}$ can be produced from models of a modification of Hao Wang’s theory of types (see [@wan52]), obtained by weakening the extensionality axiom scheme, that admit a type-shifting automorphism. The language $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$ is the $\mathbb{Z}$-sorted theory with variables\ $x_i^1, y_i^1, z_i^1, x_i^2, y_i^2, z_i^2, \ldots$ for each $i \in \mathbb{Z}$ and - binary predicates $\in_i$ for each $i \in \mathbb{Z}$, - binary predicates $=_i$ for each $i \in \mathbb{Z}$, - unary predicates $\mathcal{S}_i$ for each $i \in \mathbb{Z}$, - ternary predicates $P_i$ for each $i \in \mathbb{Z}$. Well-formed formulae of $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$ are built up inductively from the atomic formulae\ $x_i^n \in_i y_{i+1}^m$, $x_i^n =_i y_i^m$, $\mathcal{S}_i(x_i^n)$ and $P_i(x_i^m, y_i^k, z_i^n)$ for all $i \in \mathbb{Z}$ and for all $n, m, k \in \mathbb{N}$. An $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-structure $\mathcal{M}$ consists of domains $M_i^\mathcal{M}$ and relations $\in_i^\mathcal{M}$, $\mathcal{S}_i^{\mathcal{M}}$ and $P_i^{\mathcal{M}}$ for each $i \in \mathbb{Z}$. For each $i \in \mathbb{Z}$, the predicates $=_i^\mathcal{M}$ will be interpreted as true identity in the domain $M_i^\mathcal{M}$. Quantifiers appearing in well-formed $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-formulae range over objects of the same type as variable directly to the right of the quantifier. For example $\forall x_i$ reads ‘for all objects of type $i$’. We are now in a position to describe the theory of types, which we will call $\mathrm{T}\mathbb{Z}\mathrm{TU}$, related to $\mathrm{NFU}$ introduced in [@jen69]. We differ from [@jen69] by including both the axiom of infinity (at each type $i \in \mathbb{Z}$) and the axiom of choice (at each type $i \in \mathbb{Z}$) in our axiomatisation of $\mathrm{T}\mathbb{Z}\mathrm{TU}$. As with our axiomatisation of $\mathrm{NFU}$ we ensure that the axiom of infinity holds at each type $i \in \mathbb{Z}$ by specifying that for all $i \in \mathbb{Z}$, the predicate $P_i$ defines a type-level pairing function. We define $\mathrm{T}\mathbb{Z}\mathrm{TU}$ to be the $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-theory with axioms - (Extensionality for Sets) for all $i \in \mathbb{Z}$, $$\forall x_{i+1} \forall y_{i+1} \left( \begin{array}{c} \mathcal{S}_{i+1}(x_{i+1})\land \mathcal{S}_{i+1}(y_{i+1}) \Rightarrow\\ (x_{i+1} =_{i+1} y_{i+1} \iff \forall z_i(z_i \in_i x_{i+1} \iff z_i \in_i y_{i+1})) \end{array}\right),$$ - (Comprehension) for all $i \in \mathbb{Z}$ and for all $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-formula $\phi(x_i, \vec{z})$, $$\forall \vec{z} \exists y_{i+1}( \mathcal{S}_{i+1}(y_{i+1}) \land \forall x_i(x_i \in_i y_{i+1} \iff \phi(x_i, \vec{z}))),$$ - (Pairing) for all $i \in \mathbb{Z}$, $$\forall x_i \forall y_i \exists z_i (P_i(x_i, y_i, z_i) \land \forall w_i (P_i(x_i, y_i, w_i) \Rightarrow w_i =_i z_i)),$$ $$\forall x_i \forall y_i \forall u_i \forall v_i \forall z_i (P_i(x_i, y_i, z_i) \land P_i(u_i, v_i, z_i) \Rightarrow u_i =_i x_i \land v_i =_i y_i).$$ - (Axiom of Choice) for all $i \in \mathbb{Z}$, the set of all objects of type $i$ have a well ordering \[at type $i+1$\]. The theory $\mathrm{T}\mathbb{Z}\mathrm{TU}$ is Wang’s theory [@wan52] fortified with the axiom of infinity and axiom of choice (at each type $i \in \mathbb{Z}$), and with extensionality weakened (at each type $i \in \mathbb{Z}$) to allow urelements into the domain of discourse. If $\phi$ is an $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-formula then we define $\phi^\dagger$ to be the $\mathcal{L}_{\mathcal{S}, P}$-formula obtained by deleting the subscripts from the predicate and function symbols appearing in $\phi$. Note that if $\phi$ is a well-formed $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-formula then $\phi^\dagger$ is stratified. Let $\mathcal{M}= \langle (M_i^{\mathcal{M}})_{i \in \mathbb{Z}}, (\in_i^{\mathcal{M}})_{i \in \mathbb{Z}}, (\mathcal{S}_i^{\mathcal{M}})_{i \in \mathbb{Z}}, (P_i^{\mathcal{M}})_{i \in \mathbb{Z}} \rangle$ be an $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-structure. A type-shifting automorphism of $\mathcal{M}$ is a bijection $$\sigma: \bigcup_{i \in \mathbb{Z}} M_i^{\mathcal{M}} \longrightarrow \bigcup_{i \in \mathbb{Z}} M_i^{\mathcal{M}} \textrm{ such that for all } i \in \mathbb{Z},$$ - $\sigma \upharpoonright M_i^{\mathcal{M}}$ is a bijection onto $M_{i+1}^{\mathcal{M}}$, - for all $x \in M_i^\mathcal{M}$ and for all $y \in M_{i+1}^{\mathcal{M}}$, $$x \in_i^\mathcal{M} y \textrm{ if and only if } \sigma(x) \in_{i+1}^\mathcal{M} \sigma(y),$$ - for all $x \in M_i^\mathcal{M}$, $$\mathcal{S}_i(x) \textrm{ if and only if } \mathcal{S}_{i+1}(\sigma(x)),$$ - for all $x, y, z \in M_i^\mathcal{M}$, $$P_i(x, y, z) \textrm{ if and only if } P_{i+1}(\sigma(x), \sigma(y), \sigma(z)).$$ Equipped with this definition we are now in a position to recall Jensen’s analogue of Specker’s Theorem that links models of $\mathrm{NFU}$ to models of $\mathrm{T}\mathbb{Z}\mathrm{TU}$ admitting a type-shifting automorphism. Let $\mathcal{M}= \langle (M_i^\mathcal{M})_{i \in \mathbb{Z}}, (\in_i^\mathcal{M})_{i \in \mathbb{Z}}, (\mathcal{S}_i^\mathcal{M})_{i \in \mathbb{Z}}, (P_i^\mathcal{M})_{i \in \mathbb{Z}} \rangle$ be an $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-structure such that $\mathcal{M}\models \mathrm{T}\mathbb{Z}\mathrm{TU}$ with a type-shifting automorphism $\sigma$. Define for all $x, y \in M_0^{\mathcal{M}}$, $$\label{Df:NFUMembership} x \in_\sigma y \textrm{ if and only if } x \in_0^\mathcal{M} \sigma(y).$$ The membership relation defined on $M_0^{\mathcal{M}}$ by (\[Df:NFUMembership\]) yields an $\mathcal{L}_{\mathcal{S}, P}$-structure\ $\langle M_0^{\mathcal{M}}, \in_\sigma, \mathcal{S}_0^\mathcal{M}, P_0^\mathcal{M} \rangle$ that models $\mathrm{NFU}$. \[Th:NFUfromTZTU\] (Jensen) Let $\mathcal{M}= \langle (M_i^\mathcal{M})_{i \in \mathbb{Z}}, (\in_i^\mathcal{M})_{i \in \mathbb{Z}}, (\mathcal{S}_i^\mathcal{M})_{i \in \mathbb{Z}}, (P_i^\mathcal{M})_{i \in \mathbb{Z}} \rangle$ be an $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-structure. If $\mathcal{M}\models \mathrm{T}\mathbb{Z}\mathrm{TU}$ and $\sigma$ is a type-shifting automorphism of $\mathcal{M}$ then is a model of $\mathrm{NFU}$. Moreover, for all $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-formulae $\phi(x_{t_1}^1, \ldots, x_{t_n}^n)$ and for all $a_1, \ldots, a_n \in M_0^\mathcal{M}$, $$\mathcal{M} \models \phi(\sigma^{t_1}(a_1), \ldots, \sigma^{t_n}(a_n)) \textrm{ if and only if } \langle M_0^{\mathcal{M}}, \in_\sigma, \mathcal{S}_0^\mathcal{M}, P_0^{\mathcal{M}} \rangle \models \phi^\dagger(a_1, \ldots, a_n).$$ Maurice Boffa [@bof88] observed that models of $\mathrm{NFU}$ arise naturally from models of set theory that admit automorphism. Let $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M} \rangle$ be an $\mathcal{L}$-structure such that $\mathcal{M} \models \mathrm{Mac}$. Let $j: \mathcal{M} \longrightarrow \mathcal{M}$ be an automorphism and let $c \in M^\mathcal{M}$ be such that - $\mathcal{M} \models c \textrm{ is infinite}$, - $\mathcal{M} \models c \cup \mathcal{P}(c) \subseteq j(c)$. There is a natural model of $\mathrm{T}\mathbb{Z}\mathrm{TU}$ that arises from $\mathcal{M}$, $j$ and $c$. For all $i \in \mathbb{Z}$, define $$\label{Df:TZTUFromMacWithAut1} M_i= j^i(c) \textrm{ and } E_i= \in^\mathcal{M} \cap (j^i(c) \times \mathcal{P}^\mathcal{M}(j^i(c))) \textrm{ and } S_i= \mathcal{P}^\mathcal{M}(j^{i-1}(c)).$$ For all $i \in \mathbb{Z}$, we can use the Axiom of Choice in $\mathrm{Mac}$ to define a ternary relation $\bar{P}_i$ on $M_i$ such that $$\label{Df:TZTUFromMacWithAut2} \mathcal{M}_j^c= \langle (M_i)_{i \in \mathbb{Z}}, (E_i)_{i \in \mathbb{Z}}, (S_i)_{i \in \mathbb{Z}}, (\bar{P}_i)_{i \in \mathbb{Z}} \rangle \models \mathrm{T}\mathbb{Z}\mathrm{TU}.$$ Now, it is clear from this construction that $j$ is a type-shifting automorphism of the $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-structure $\mathcal{M}_j^c$. This yields the following result: \[Th:NFUFromMacWithAut\] Let $\mathcal{M}$ be an $\mathcal{L}$-structure such that $\mathcal{M} \models \mathrm{Mac}$. Let $j: \mathcal{M} \longrightarrow \mathcal{M}$ be an automorphism and let $c \in M^\mathcal{M}$ be such that - $\mathcal{M} \models c \textrm{ is infinite}$, - $\mathcal{M} \models c \cup \mathcal{P}(c) \subseteq j(c)$. If $\mathcal{M}_j^c$ is defined by (\[Df:TZTUFromMacWithAut1\]) and (\[Df:TZTUFromMacWithAut2\]) and $\in_j$ is defined by (\[Df:NFUMembership\]) from the type-shifting automorphism $j$ acting on $\mathcal{M}_j^c$ then $\langle M_0, \in_j, S_0, \bar{P}_0 \rangle \models \mathrm{NFU}$. Moreover, for all $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-formulae $\phi(x_{t_1}^1, \ldots, x_{t_n}^n)$ and for all $a_1, \ldots, a_n \in M_0$, $$\mathcal{M}_j^c \models \phi(j^{t_1}(a_1), \ldots, j^{t_n}(a_n)) \textrm{ if and only if } \langle M_0, \in_j, S_0, \bar{P}_0 \rangle \models \phi^\dagger(a_1, \ldots, a_n).$$ The results of [@bof88] reveal that condition (ii) of Theorem \[Th:NFUFromMacWithAut\] can be weakened to $$\mathcal{M} \models \|c \cup \mathcal{P}(c) \| \leq \|j(c)\|.$$ In [@jen69], Jensen identifies a weak subsystem of $\mathrm{ZFC}$ that is equiconsistent with $\mathrm{NFU}$. It should be noted that the arguments appearing in [@jen69] which claim to establish this equiconsistency are incomplete. However, using results proved by Thomas Forster and Richard Kaye that are reported in [@mat01], Jensen’s model constructions in [@jen69] yield the following equiconsistency result: (Jensen [@jen69]) $\mathrm{NFU}$ is equiconsistent with $\mathrm{Mac}$. The strength of $\mathrm{NFU}+\mathrm{AxCount}_\geq$ {#Sec:NFUPlusAxCountGEQ} ==================================================== In this section we will use Theorem \[Th:AutomorphismMovingNoNaturalDown\] to prove two results concerning the strength of the theory $\mathrm{NFU}+\mathrm{AxCount}_\geq$. We will first show that $\mathrm{NFU}+\mathrm{AxCount}_\geq$ is unable to prove that $\mathrm{CNI}$ is infinite. We will then use the fact that the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$ is able to build $\omega$-models of Zermelo set theory in the structure $\langle \mathrm{BF}, \epsilon \rangle$ to show that $\mathrm{NFU}+ \mathrm{AxCount}_\leq$ proves the consistency of $\mathrm{NFU}+\mathrm{AxCount}_\geq$. \[Th:ModelOfNFUAxCountGEQBethOmegaDoesNotExist\] There exists an $\mathcal{L}_{\mathcal{S}, P}$-structure $\mathcal{M}$ such that $\mathcal{M} \models \mathrm{NFU} + \mathrm{AxCount}_\geq$ and $$\mathcal{M} \models \mathrm{CNI} \textrm{ is finite}.$$ We work in the theory $\mathrm{ZFC}+\mathrm{GCH}$. The $\mathcal{L}$-structure $\langle V_{\omega+\omega}, \in \rangle$ is an $\omega$-model of Zermelo set theory plus $\mathrm{TCo}$. By Theorem \[Th:AutomorphismMovingNoNaturalDown\] there is an $\mathcal{L}$-structure $\mathcal{N} \equiv \langle V_{\omega+\omega}, \in \rangle$ admitting an automorphism $j: \mathcal{N} \longrightarrow \mathcal{N}$ such that - $\mathcal{N} \models j(n) \geq n$ for all $n \in (\omega^\mathcal{N})^$, - there exists an $n \in (\omega^\mathcal{N})^$ with $\mathcal{N} \models j(n) > n$. Since $\mathcal{N} \equiv \langle V_{\omega+\omega}, \in \rangle$, the rank function $\rho$ is well-defined and $$\mathcal{N} \models \forall \alpha ((\alpha \textrm{ is an ordinal}) \Rightarrow (V_\alpha \textrm{ exists})).$$ Let $n \in (\omega^\mathcal{N})^$ be such that $\mathcal{N} \models j(n) > n$. Recalling (\[Df:TZTUFromMacWithAut1\]) and (\[Df:TZTUFromMacWithAut2\]) we can use $j$ and $V_{\omega+n}$ to define an $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-structure $\mathcal{M}^{V_{\omega+n}^{\mathcal{N}}}_j \models \mathrm{T}\mathbb{Z}\mathrm{TU}$ such that $j$ is a type-shifting automorphism of $\mathcal{M}^{V_{\omega+n}^{\mathcal{N}}}_j$. Using (\[Df:NFUMembership\]) and Theorem \[Th:NFUFromMacWithAut\] we can define $\in_j$, $S$ and $\bar{P}$ on $V_{\omega+n}^\mathcal{N}$ so as $$\mathcal{M}= \langle V_{\omega+n}^\mathcal{N}, \in_j, S, \bar{P} \rangle \models \mathrm{NFU}.$$ By the properties of $j$, for all $n \in (\omega^\mathcal{N})^$, $$\mathcal{M}^{V_{\omega+n}^{\mathcal{N}}}_j \models \|\iota``n\| \leq j(n).$$ Therefore, by Theorem \[Th:NFUFromMacWithAut\], for all $n \in (\omega^\mathcal{N})^$, $$\mathcal{M} \models T(n) \leq n.$$ Therefore $\mathcal{M} \models \mathrm{AxCount}_\geq$. Now, let $\sigma_i$ be the $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-sentence that says: ‘the set of infinite cardinals \[at level $i$\] is finite’. It follows from the Generalised Continuum Hypothesis that for all $i \in \mathbb{Z}$, $$\mathcal{M}^{V_{\omega+n}^{\mathcal{N}}}_j \models \sigma_i.$$ Therefore $\mathcal{M} \models \mathrm{CNI} \textrm{ is finite}$. By observing that $\bar{\bar{\beth}} \subseteq \mathrm{CNI}$ we get the following result: There exists an $\mathcal{L}_{\mathcal{S}, P}$-structure $\mathcal{M}$ such that $\mathcal{M} \models \mathrm{NFU} + \mathrm{AxCount}_\geq$ and $$\mathcal{M} \models \beth_\omega^{TT} \textrm{ does not exist}.$$ We now turn to showing that the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$ is strictly stronger than $\mathrm{NFU}+\mathrm{AxCount}_\geq$. In [@hin75] Roland Hinnion shows that the ordinal strength endowed upon $\mathrm{NF}$ by $\mathrm{AxCount}_\leq$ means that there exists a point in the structure $\langle \mathrm{BF}, \epsilon \rangle$ that $\langle \mathrm{BF}, \epsilon \rangle$ believes is a transitive model of Zermelo set theory. Using the characterisation of the structure $\langle \mathrm{BF}, \epsilon \rangle$ proved in section \[Sec:NFU\] we will reprove Hinnion’s result in the context of $\mathrm{NFU}$. We make the following definitions in $\mathrm{NFU}$: Let $X \in \mathrm{BF}$. We define $$\mathrm{Pow}(X)= \{ Y \in \mathrm{BF} \mid \langle \mathrm{BF}, \epsilon \rangle \models Y \subseteq [R] \}.$$ Let $X \in \mathrm{BF}$. We define $$X^{\mathrm{ext}}= \{ Y \in \mathrm{BF} \mid \langle \mathrm{BF}, \epsilon \rangle \models Y \in [R] \}.$$ Corollary \[Th:BFEXTsModelZFCminus\] implies that $\mathrm{NFU}$ proves that $$\langle \mathrm{BF}, \epsilon \rangle \models V_\omega \textrm{ exists}.$$ Adding $\mathrm{AxCount}_\leq$ to $\mathrm{NFU}$ allows us to show that the structure $\langle \mathrm{BF}, \epsilon \rangle$ believes that $V_{\omega+\omega}$ exists. \[Th:SizeOfPow\] The theory $\mathrm{NFU}$ proves that if $X \in \mathrm{BF}$ then $$\|\mathrm{Pow}(X)\|= 2^{\|X^{\mathrm{ext}}\|}.$$ We work in $\mathrm{NFU}$. Let $X \in \mathrm{BF}$. Define $f: \iota``\mathrm{Pow}(X) \longrightarrow \mathcal{P}(X^{\mathrm{ext}})$ by $$\begin{array}{lcl} f(\{Y\})= U & \textrm{ if and only if } & \forall V(V \in U \iff \langle \mathrm{BF}, \epsilon \rangle \models V \in Y). \end{array}$$ Stratified comprehension ensures that $f$ is a set. It is clear that $f$ is a bijective function. Therefore $T(\|\mathrm{Pow}(X)\|)= \|\mathcal{P}(X^{\mathrm{ext}})\|$, which proves the lemma. This allows us to show that the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$ proves that for all internal natural numbers $n$, the structure $\langle \mathrm{BF}, \epsilon \rangle$ believes that $V_{\omega+n}$ exists. \[Th:BethNLessThanT2V\] The theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$ proves that for all $n \in \mathbb{N}$, $\beth_n^{TT} \leq T^2(\|V\|)$. We work in the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$. Let $n$ be least such that $T^2(\|V\|) < \beth_n^{TT} \leq \|V\|$. Therefore $T^4(\|V\|) < T^2(\beth_n^{TT}) \leq T^2(\|V\|)$. And so, $\beth_{T^2(n)}^{TT}= T^2(\beth_n^{TT}) < \beth_n^{TT}$. But this contradicts the fact that $T^2(n) \geq n$. \[Th:AxCountLEQImpliesVOmegaPlusNExists\] The theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$ proves that for all $n \in \mathbb{N}$, - $\langle \mathrm{BF}, \epsilon \rangle \models V_{\omega+\mathbf{k}(n)} \textrm{ exists}$, - $\|(V_{\omega+\mathbf{k}(n)}^{\langle \mathrm{BF}, \epsilon \rangle})^{\mathrm{ext}}\|= \beth_n^{TT}$. We work in the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$. We prove the theorem by induction. Corollary \[Th:BFEXTsModelZFCminus\] implies that - $\langle \mathrm{BF}, \epsilon \rangle \models V_\omega \textrm{ exists}$, - $\|(V_{\omega}^{\langle \mathrm{BF}, \epsilon \rangle})^{\mathrm{ext}}\|= \aleph_0= \beth_0^{TT}$. Suppose that the theorem holds for $n \in \mathbb{N}$. Now, by Lemma \[Th:SizeOfPow\], $$\|\mathrm{Pow}(V_{\omega+\mathbf{k}(n)}^{\langle \mathrm{BF}, \epsilon \rangle})\| = 2^{\|(V_{\omega+\mathbf{k}(n)}^{\langle \mathrm{BF}, \epsilon \rangle})^{\mathrm{ext}}\|}= 2^{\beth_n^{TT}}= \beth_{n+1}^{TT}.$$ Therefore, by Lemma \[Th:BethNLessThanT2V\], $\|\mathrm{Pow}(V_{\omega+\mathbf{k}(n)}^{\langle \mathrm{BF}, \epsilon \rangle})\| \leq T^2(\|V\|)$. And so, by Theorem \[Th:SetOfBFEXTsLessThanT2VAreExtensions\], there exists an $X \in \mathrm{BF}$ such that for all $Y \in \mathrm{BF}$, $$Y \in \mathrm{Pow}(V_{\omega+\mathbf{k}(n)}^{\langle \mathrm{BF}, \epsilon \rangle}) \textrm{ if and only if } \langle \mathbf{BF}, \epsilon \rangle \models Y \in X.$$ Now, $\langle \mathbf{BF}, \epsilon \rangle \models X= V_{\omega+\mathbf{k}(n+1)}$. Our next task is to confirm that, in $\mathrm{NFU}$, an ordinal $\alpha$ is finite if and only if $\mathbf{k}(\alpha)$ is finite in the structure $\langle \mathrm{BF}, \epsilon \rangle$. \[Th:FiniteOrdinalsInNFUAreFiniteOrdinalsInBF\] The theory $\mathrm{NFU}$ proves that for all ordinals $\alpha$, $$(\alpha \textrm{ is finite}) \textrm{ if and only if } \langle \mathrm{BF}, \epsilon \rangle \models (\mathbf{k}(\alpha) \textrm{ is finite}).$$ We work in $\mathrm{NFU}$. We prove the contra-positive of both directions. Let $\alpha$ be an ordinal. Let $X= \{ \beta \in \mathrm{ON} \mid \beta < \alpha \}$.\ Suppose that $\alpha$ is infinite. Therefore $T^2(\alpha)$ is infinite. Let $f: X \longrightarrow X$ be an injection that is not a surjection. Define $G \subseteq \mathrm{BF}$ such that for all $x \in \mathrm{BF}$, $$\begin{array}{lcl} x \in G & \textrm{ if and only if } & \langle \mathrm{BF}, \epsilon \rangle \models (x= \langle \mathbf{k}(\beta), \mathbf{k}(\gamma) \rangle) \textrm{ and } f(\beta)= \gamma. \end{array}$$ Theorem \[Th:SetOfBFEXTsLessThanT2VAreExtensions\] implies that there exists $Y \in \mathrm{BF}$ such that for all $x \in \mathrm{BF}$, $$x \in G \textrm{ if and only if } \langle \mathrm{BF}, \epsilon \rangle \models x \in Y.$$ Now, $$\langle \mathrm{BF}, \epsilon \rangle \models (Y \textrm{ is an injection from } \mathbf{k}(\alpha) \textrm{ into } \mathbf{k}(\alpha) \textrm{ that is not surjective}).$$ Conversely, suppose that $\alpha$ is such that $$\langle \mathrm{BF}, \epsilon \rangle \models (\mathbf{k}(\alpha) \textrm{ is infinite}).$$ Let $G \in \mathrm{BF}$ be such that $$\langle \mathrm{BF}, \epsilon \rangle \models (G \textrm{ is an injection from } \mathbf{k}(\alpha) \textrm{ into } \mathbf{k}(\alpha) \textrm{ that is not surjective}).$$ Define $f: X \longrightarrow X$ such that for all $\beta, \gamma < \alpha$, $$f(\beta)= \gamma \textrm{ if and only if } \langle \mathrm{BF}, \epsilon \rangle \models (G(\mathbf{k}(\beta))= \mathbf{k}(\gamma)).$$ Now, $f$ witnesses the fact that $T^2(\alpha)$ is infinite. Therefore $\alpha$ is infinite. Theorem \[Th:AxCountLEQImpliesVOmegaPlusNExists\] and Lemma \[Th:FiniteOrdinalsInNFUAreFiniteOrdinalsInBF\] imply Hinnion’s result about the strength of $\mathrm{AxCount}_\leq$ in the context of $\mathrm{NFU}$. \[Th:AxCountLEQProveVOmegaExists\] (Hinnion [@hin75]) $$\mathrm{NFU}+ \mathrm{AxCount}_\leq \vdash (\langle \mathrm{BF}, \epsilon \rangle \models V_{\omega+\omega} \textrm{ exists}).$$ We work in $\mathrm{NFU}+\mathrm{AxCount}_\leq$. Theorem \[Th:AxCountLEQImpliesVOmegaPlusNExists\] combined with Lemma \[Th:FiniteOrdinalsInNFUAreFiniteOrdinalsInBF\] imply that $$\langle \mathrm{BF}, \epsilon \rangle \models (\forall n \in \mathbb{N})(V_{\omega+n} \textrm{ exists}).$$ Therefore, by collection in $\langle \mathrm{BF}, \epsilon \rangle$, $$\langle \mathrm{BF}, \epsilon \rangle \models V_{\omega+\omega} \textrm{ exists}.$$ \[Th:NFUPlusAxCountLEQProvesConZ\] (Hinnion [@hin75]) $$\mathrm{NFU}+\mathrm{AxCount}_\leq \vdash \mathrm{Con}(\mathrm{Z}).$$ It follows from Theorem \[Th:AutomorphismMovingNoNaturalDown\] that the structure $\langle \mathrm{BF}, \epsilon\rangle$ in the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$ can also see a model of $\mathrm{NFU}+ \mathrm{AxCount}_\geq$. $$\mathrm{NFU}+\mathrm{AxCount}_\leq \vdash \mathrm{Con}(\mathrm{NFU}+\mathrm{AxCount}_\geq).$$ We work inside $\langle \mathrm{BF}, \epsilon \rangle$. Theorem \[Th:AxCountLEQProveVOmegaExists\] shows that $V_{\omega+\omega}$ exists. Using the same arguments used in proof of Theorem \[Th:ModelOfNFUAxCountGEQBethOmegaDoesNotExist\] we can see that this implies that there exists an $\mathcal{L}_{\mathcal{S}, P}$-structure $\mathcal{M}$ such that $$\mathcal{M} \models \mathrm{NFU}+ \mathrm{AxCount}_\geq.$$ These results still leave the following questions unanswered: What is the exact strength of the theory $\mathrm{NFU}+\mathrm{AxCount}_\geq$ relative to a subsystem of $\mathrm{ZFC}$? Does $\mathrm{NFU}+\mathrm{AxCount}_\geq$ prove the consistency of $\mathrm{NFU}$? The strength of $\mathrm{NFU}+\mathrm{AxCount}_\leq$ {#Sec:NFUPlusAxCountLEQ} ==================================================== The aim of this section is to use the results of section \[Sec:AutomorphismsFromNonStandardModels\] to shed light on the strength of the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$. We begin by showing that there is a model $\mathrm{NFU}+\mathrm{AxCount}_\leq$ in which the order-type of $\mathrm{CNI}$ equipped with the natural ordering is recursive. We will then show that $\mathrm{NFU}+\mathrm{AxCount}$ proves the consistency of the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$.\ \ We begin by introducing an extension of Zermelo set theory that includes function symbols whose intended interpretations are the rank function and the function $\alpha \mapsto V_\alpha$. This theory will have the property that the standard part of a non-standard $\omega$-model of this theory will be a model $\mathrm{KP}^{\mathcal{P}}$. Let $\hat{\rho}$ and $\bar{V}$ be new unary function symbols. We define Zermelo with ranks set theory ($\mathrm{ZR}$) to be the $\mathcal{L}_{\hat{\rho}, \bar{V}}$-theory with axioms - all of the axioms of $\mathrm{Z}$, - $\forall x(\hat{\rho}(x) \textrm{ is the least ordinal s.t. } (\forall y \in x)(\hat{\rho}(y) < \hat{\rho}(x)))$, - $\forall x((x \textrm{ is an ordinal}) \Rightarrow \forall y(y \in \bar{V}(x) \iff \hat{\rho}(y) < x))$, - separation for all $\mathcal{L}_{\hat{\rho}, \bar{V}}$-formulae. By considering the set $V_{\omega+\omega}$ and interpreting $\hat{\rho}$ using $\rho$ and $\bar{V}$ using the function $\alpha \mapsto V_\alpha$ we see that $\mathrm{KP}^\mathcal{P}$ proves the consistency of the theory $\mathrm{ZR}$.\ \ We are now in a position to define the standard part of a non-standard model of $\mathrm{ZR}$. \[Df:StandardOrdinal\] Let $\mathcal{M}$ be an $\mathcal{L}_{\hat{\rho}, \bar{V}}$-structure such that $\mathcal{M} \models \mathrm{ZR}$. We define $$\varpi_o(\mathcal{M})= \{\alpha \mid (\exists x \in M^\mathcal{M})((\mathcal{M} \models x \textrm{ is an ordinal}) \land (\langle x, \in^\mathcal{M} \rangle \cong \langle \alpha, \in \rangle)) \}.$$ \[Df:StandardPartZR\] Let $\mathcal{M}$ be an $\mathcal{L}_{\hat{\rho}, \bar{V}}$-structure such that $\mathcal{M} \models \mathrm{ZR}$. We define $$\mathbf{std}^{\mathrm{ZR}}(\mathcal{M})= \{ x \in M^\mathcal{M} \mid (\exists \alpha \in \varpi_o(\mathcal{M}))(\langle \hat{\rho}^\mathcal{M}(x), \in^\mathcal{M} \rangle \cong \langle \alpha, \in \rangle)\}.$$ We are now able to prove that the standard part of a non-standard $\omega$-model of $\mathrm{ZR}$ is a model of $\mathrm{KP}^{\mathcal{P}}$. The idea of building models of $\mathrm{KP}^{\mathcal{P}}$ from the standard part of a non-standard model first appears in [@fri73]. \[Th:StandardPartOfModelOfZR\] If $\mathcal{M} \models \mathrm{ZR}$ is a non-standard $\omega$-model then $$\langle \mathbf{std}^{\mathrm{ZR}}(\mathcal{M}), \in^\mathcal{M} \rangle \models \mathrm{KP}^\mathcal{P}.$$ Let $\mathcal{M}= \langle M^\mathcal{M}, \in^\mathcal{M}, \hat{\rho}^\mathcal{M}, \bar{V}^\mathcal{M} \rangle$ be a non-standard $\omega$-model such that $\mathcal{M} \models \mathrm{ZR}$. Let $\mathcal{N}= \langle \mathbf{std}^{\mathrm{ZR}}(\mathcal{M}), \in^\mathcal{M} \rangle$. From the point of view of $\mathcal{M}$, $\mathbf{std}^{\mathrm{ZR}}(\mathcal{M})$ is a transitive subclass of the universe such that for all $\alpha \in \mathbf{std}^{\mathrm{ZR}}(\mathcal{M})$ with $\mathcal{N} \models \alpha \textrm{ is an ordinal}$, $$\bar{V}^\mathcal{M}(\alpha) \in \mathbf{std}^{\mathrm{ZR}}(\mathcal{M}).$$ Therefore $\mathcal{N} \models \mathrm{Mac}$ and if $\phi(\vec{x})$ is a $\Delta_0^\mathcal{P}$-formula then for all $\vec{a} \in \mathbf{std}^{\mathrm{ZR}}(\mathcal{M})$, $$\mathcal{N} \models \phi(\vec{a}) \textrm{ if and only if } \mathcal{M} \models \phi(\vec{a}).$$ And so, $\mathcal{N}$ also satisfies $\Delta_0^\mathcal{P}$-Separation. We need to verify that $\Delta_0^\mathcal{P}$-Collection holds in $\mathcal{N}$. Let $\phi(x, y, \vec{z})$ be a $\Delta_0^\mathcal{P}$-formula. Let $\vec{a}, b \in \mathbf{std}^{\mathrm{ZR}}(\mathcal{M})$ be such that $$\label{Df:CollectionInStandardModelOfZR} \mathcal{N} \models (\forall x \in b) \exists y \phi(x, y, \vec{a}).$$ Let $\alpha \in M^\mathcal{M}$ be non-standard such that $\mathcal{M} \models \alpha \textrm{ is an ordinal}$. Working inside $\mathcal{M}$, let $$A= \{ \beta \in \bar{V}(\alpha) \mid (\exists x \in b)(\exists y(\phi(x, y, \vec{a}) \land \hat{\rho}(y)= \beta) \land \forall y(\phi(x, y, \vec{a}) \Rightarrow \hat{\rho}(y) \geq \beta)) \}$$ $$\textrm{and } B= \{ \beta \in \bar{V}(\alpha) \mid (\beta \textrm{ is an ordinal}) \land (\beta \notin A) \}.$$ The fact that $\mathcal{M} \models \mathrm{ZR}$ ensures that $A$ and $B$ are sets. The fact that $\phi$ is absolute between $\mathcal{M}$ and $\mathcal{N}$ and (\[Df:CollectionInStandardModelOfZR\]) ensures that $A^ \subseteq \mathbf{std}^{\mathrm{ZR}}(\mathcal{M})$. Let $\gamma$ be the $\in^\mathcal{M}$-least element of $B$. Therefore $\gamma \in \mathbf{std}^{\mathrm{ZR}}(\mathcal{M})$. Therefore, working inside $\mathcal{N}$, let $$C= \{ y \in \bar{V}(\gamma) \mid (\exists x \in b) \phi(x, y, \vec{a}) \}.$$ $\Delta_0^\mathcal{P}$-separation in $\mathcal{N}$ ensures that $C$ is a set in $\mathcal{N}$. Moreover, $$\mathcal{N} \models (\forall x \in b)(\exists y \in C) \phi(x, y, \vec{a}).$$ Therefore $\Delta_0^{\mathcal{P}}$-collection holds in $\mathcal{N}$.\ The fact that the relation $\in^\mathcal{N}$ is well-founded on $\mathbf{std}^{\mathrm{ZR}}(\mathcal{M})$ guarantees that $\mathcal{N}$ satisfies full class foundation. Using the Barwise Compactness Theorem we can prove (in $\mathrm{ZFC}^-$) that if $V_{\omega_1^{\mathrm{ck}}}$ exists then there exists a non-standard $\omega$-model of $\mathrm{ZR}$. \[Th:NonStandardOmegaModelInZFCMinus\] The theory $\mathrm{ZFC}^-+ V_{\omega_1^{\mathrm{ck}}} \textrm{ exists}$ proves that there exists an $\mathcal{L}_{\hat{\rho}, \bar{V}}$-structure $\mathcal{N}$ such that $\mathcal{N} \models \mathrm{ZR}$ is a non-standard $\omega$-model. We work in the theory $\mathrm{ZFC}^-+ V_{\omega_1^{\mathrm{ck}}} \textrm{ exists}$. Let $\rho$ be the rank function. Define $V_: \omega_1^{\mathrm{ck}} \longrightarrow V_{\omega_1^{\mathrm{ck}}}$ by $\alpha \mapsto V_\alpha$. The $\mathcal{L}_{\hat{\rho}, \bar{V}}$-structure $\mathcal{M}= \langle V_{\omega_1^{\mathrm{ck}}}, \in, \rho \cap (V_{\omega_1^{\mathrm{ck}}} \times V_{\omega_1^{\mathrm{ck}}}), V_ \rangle$ is such that $\mathcal{M} \models \mathrm{ZR}$.\ Let $A= L_{\omega_1^{\mathrm{ck}}}$. Let $\mathcal{L}^\prime$ be the extension of $\mathcal{L}_{\hat{\rho}, \bar{V}}$ obtained by adding - constant symbols $\hat{\alpha}$ for every $\alpha \in \omega_1^{\mathrm{ck}}$, - constant symbol $\mathbf{c}$. Define the $(\mathcal{L}^\prime_{\omega_1 \omega})_A$-theory $Q$ with axioms - all of the axioms of $\mathrm{ZR}$, - $\hat{\zeta} \in \hat{\nu}$ for all $\zeta \in \nu \in \omega_1^{\mathrm{ck}}$, - for all $\nu \in \omega_1^{\mathrm{ck}}$, $$\forall x \left( x \in \hat{\nu} \Rightarrow \bigvee_{\zeta \in \nu} (x= \hat{\zeta})\right),$$ - $\mathbf{c}$ is an ordinal, - $\hat{\nu} \in \mathbf{c}$ for all $\nu \in \omega_1^{\mathrm{ck}}$, - every ordinal is recursive. The theory $Q$ is $\Sigma_1(\mathcal{L})$ over $A$. Let $Q^\prime \subseteq Q$ be such that $Q^\prime \in A$. Therefore the set $$B= \{ \alpha \in \omega_1^{\mathrm{ck}} \mid \hat{\alpha} \textrm{ is mentioned in } Q^\prime \}$$ is bounded in $\omega_1^{\mathrm{ck}}$. Therefore we can expand $\mathcal{M}$ to a structure $\mathcal{M}^\prime$ that satisfies $Q^\prime$. Therefore by the Barwise Compactness Theorem (Theorem \[Th:BarwiseCompactness\]) there exists a structure $\mathcal{N}^\prime$ that satisfies $Q$. Let $\mathcal{N}= \langle N^\mathcal{N}, \in^\mathcal{N}, \hat{\rho}^\mathcal{N}, \bar{V}^\mathcal{N} \rangle$ be the $\mathcal{L}_{\hat{\rho}, \bar{V}}$ reduct of $\mathcal{N}^\prime$. It follows from axiom scheme (i) of $Q$ that $\mathcal{N} \models \mathrm{ZR}$. Axiom schemes (ii) and (iii) of $Q$ ensure that $\mathcal{N}$ is an $\omega$-model. Let $$C= \{ \beta \in N^\mathcal{N} \mid (\mathcal{N}^\prime \models \beta \textrm{ is an ordinal})\land (\forall \nu \in \omega_1^{\mathrm{ck}}) (\mathcal{N}^\prime \models \hat{\nu} \in \beta) \}.$$ By (iv) and scheme (v) of $Q$, $C$ is non-empty. Suppose that $\mathcal{N}$ is well-founded. Let $\alpha \in N^\mathcal{N}$ be the least element of $C$. But, $\langle \alpha, \in^\mathcal{N} \rangle \cong \langle \omega_1^{\mathrm{ck}}, \in \rangle$, which contradicts (vi) of $Q$. Therefore $\mathcal{N}$ is a non-standard $\omega$-model. Combining Lemma \[Th:StandardPartOfModelOfZR\] and Lemma \[Th:NonStandardOmegaModelInZFCMinus\] immediately yields: $\mathrm{ZFC}^-+ V_{\omega_1^{\mathrm{ck}}} \textrm{ exists} \vdash \mathrm{Con}(\mathrm{KP}^{\mathcal{P}})$. The non-standard $\omega$-model built in Lemma \[Th:NonStandardOmegaModelInZFCMinus\] also has the property that every ordinal is recursive. By assuming that we are working in a model of the generalised continuum hypothesis we can also ensure that every set of infinite sets with distinct cardinalities is isomorphic to a recursive ordinal when ordered by the natural ordering on cardinals. \[Th:NonStandardOmegaModelFromZFCPlusGCH\] The theory $\mathrm{ZFC}+\mathrm{GCH}$ proves that there exists an $\mathcal{L}_{\hat{\rho}, \bar{V}}$-structure $\mathcal{N}$ such that $$\mathcal{N} \models \mathrm{ZR} +\textrm{every ordinal is recursive},$$ $$\label{Df:CNIAreRecursive} \textrm{and } \mathcal{N} \models \forall X \left( \begin{array}{c} (\forall x, y \in X)(\|x\| \geq \omega \land \|y\| \geq \omega \land (\|x\| < \|y\| \lor \|y\| < \|x\| \lor x=y)) \Rightarrow\\ \exists f \exists \alpha\left( \begin{array}{c} (\alpha \textrm{ is a recursive ordinal}) \land (f: X \longrightarrow \alpha)\\ \land (\forall x, y \in X)(\|x\| < \|y\| \Rightarrow f(x) < f(y)) \end{array}\right) \end{array}\right),$$ and $\mathcal{N}$ is a non-standard $\omega$-model. We work in the theory $\mathrm{ZFC}+\mathrm{GCH}$. Let $\rho$ be the rank function. Define $V_: \omega_1^{\mathrm{ck}} \longrightarrow V_{\omega_1^{\mathrm{ck}}}$ by $\alpha \mapsto V_\alpha$. The $\mathcal{L}_{\hat{\rho}, \bar{V}}$-structure $\mathcal{M}= \langle V_{\omega_1^{\mathrm{ck}}}, \in, \rho \cap (V_{\omega_1^{\mathrm{ck}}} \times V_{\omega_1^{\mathrm{ck}}}), V_ \rangle$ is such that $\mathcal{M} \models \mathrm{ZR}$ and $\mathcal{M} \models \textrm{every ordinal is recursive}$. The Generalised Continuum Hypothesis ensures that $\mathcal{M}$ satisfies (\[Df:CNIAreRecursive\]). Using the same arguments used in the proof of Lemma \[Th:NonStandardOmegaModelInZFCMinus\] we can find an $\mathcal{L}_{\hat{\rho}, \bar{V}}$-structure $\mathcal{N}$ such that $$\mathcal{N} \models \mathrm{ZR} +\textrm{every ordinal is recursive},$$ $$\textrm{and } \mathcal{N} \models \forall X \left( \begin{array}{c} (\forall x, y \in X)(\|x\| \geq \omega \land \|y\| \geq \omega \land (\|x\| < \|y\| \lor \|y\| < \|x\| \lor x=y)) \Rightarrow\\ \exists f \exists \alpha\left( \begin{array}{c} (\alpha \textrm{ is a recursive ordinal}) \land (f: X \longrightarrow \alpha)\\ \land (\forall x, y \in X)(\|x\| < \|y\| \Rightarrow f(x) < f(y)) \end{array}\right) \end{array}\right),$$ and $\mathcal{N}$ is a non-standard $\omega$-model. By applying Theorem \[Th:AutomorphismsFromNonStandardOmegaModels\] to the model $\mathcal{N}$ built in Lemma \[Th:NonStandardOmegaModelFromZFCPlusGCH\] we obtain a model of $\mathrm{NFU}+\mathrm{AxCount}_\leq$ in which $\langle \mathrm{CNI}, < \rangle$ is isomorphic to a recursive ordinal. \[Th:WeakModelOfNFUPlusAxCountLEQ\] There exists an $\mathcal{L}_{\mathcal{S}, P}$-structure $\mathcal{M}$ such that $\mathcal{M} \models \mathrm{NFU}+\mathrm{AxCount}_\leq$ and $$\mathcal{M} \models \langle \mathrm{CNI}, < \rangle \textrm{ is isomorphic to a recursive ordinal}.$$ We work in the theory $\mathrm{ZFC}+\mathrm{GCH}$. By Lemma \[Th:NonStandardOmegaModelFromZFCPlusGCH\] there exists a non-standard $\omega$-model $\mathcal{Q} \models \mathrm{ZR} + \textrm{every ordinal is recursive}$ such that $$\label{Df:CNIRecursiveII} \mathcal{Q} \models \forall X \left( \begin{array}{c} (\forall x, y \in X)(\|x\| \geq \omega \land \|y\| \geq \omega \land (\|x\| < \|y\| \lor \|y\| < \|x\| \lor x=y)) \Rightarrow\\ \exists f \exists \alpha\left( \begin{array}{c} (\alpha \textrm{ is a recursive ordinal}) \land (f: X \longrightarrow \alpha)\\ \land (\forall x, y \in X)(\|x\| < \|y\| \Rightarrow f(x) < f(y)) \end{array}\right) \end{array}\right),$$ Let $\mathcal{Q}^\prime$ be the $\mathcal{L}$ reduct of $\mathcal{Q}$. Let $\mathcal{Q}^{\prime\prime}$ be the expansion of $\mathcal{Q}^\prime$ to an $\mathcal{L}_{\bar{\beta}}$-structure satisfying the assumptions of Theorem \[Th:AutomorphismsFromNonStandardOmegaModels\]. Therefore by Theorem \[Th:AutomorphismsFromNonStandardOmegaModels\] there exists an $\mathcal{L}_{\bar{\beta}}$-structure $\mathcal{N} \equiv \mathcal{Q}^{\prime\prime}$ admitting an automorphism $j: \mathcal{N} \longrightarrow \mathcal{N}$ such that - $\mathcal{N} \models j(n) \geq n$ for all $n \in (\omega^\mathcal{N})^$, - there exists $\alpha \in (\bar{\beta}^\mathcal{N})^$ such that $\mathcal{N} \models j(\alpha) < \alpha$. Since $\mathcal{N} \equiv \mathcal{Q}^{\prime\prime}$, the rank function $\rho$ is well-defined and $$\mathcal{N} \models \forall \alpha ((\alpha \textrm{ is an ordinal}) \Rightarrow (V_\alpha \textrm{ exists})).$$ Let $\alpha \in (\bar{\beta}^\mathcal{N})^$ be such that $\mathcal{N} \models j(\alpha) < \alpha$. Recalling (\[Df:TZTUFromMacWithAut1\]) and (\[Df:TZTUFromMacWithAut2\]) we can use $j^{-1}$ and $V_{\alpha}^\mathcal{N}$ to define an $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-structure $\mathcal{M}_{j^{-1}}^{V_\alpha^\mathcal{N}} \models \mathrm{T}\mathbb{Z}\mathrm{TU}$ such that $j^{-1}$ is a type-shifting automorphism of $\mathcal{M}_{j^{-1}}^{V_\alpha^\mathcal{N}}$. Using (\[Df:NFUMembership\]) and Theorem \[Th:NFUFromMacWithAut\] we can define $\in_{j^{-1}}$, $S$ and $\bar{P}$ on $V_{\alpha}^\mathcal{N}$ so as $$\mathcal{M}= \langle V_{\alpha}^\mathcal{N}, \in_{j^{-1}}, S, \bar{P} \rangle \models \mathrm{NFU}.$$ By the properties of $j^{-1}$, for all $n \in (\omega^\mathcal{N})^$, $$\mathcal{M}^{V_{\alpha}^{\mathcal{N}}}_{j^{-1}} \models j^{-1}(n) \leq \|\iota``n\|.$$ Therefore, by Theorem \[Th:NFUFromMacWithAut\], for all $n \in (\omega^\mathcal{N})^$, $$\mathcal{M} \models T(n) \geq n.$$ Therefore $\mathcal{M} \models \mathrm{AxCount}_\leq$. Now, let $\sigma_i$ by the $\mathcal{L}_{\mathrm{T}\mathbb{Z}\mathrm{TU}}$-sentence that says: ‘the set of infinite cardinals \[at level $i$\] equipped with the natural ordering is isomorphic to a recursive ordinal’. It follows from (\[Df:CNIRecursiveII\]) that for all $i \in \mathbb{Z}$, $$\mathcal{M}^{V_{\alpha}^{\mathcal{N}}}_{j^{-1}} \models \sigma_i.$$ Therefore $$\mathcal{M} \models \textrm{the order-type of } \langle \mathrm{CNI}, < \rangle \textrm{ is recursive}.$$ Again, since $\bar{\bar{\beth}} \subseteq \mathrm{CNI}$, we get: There exists an $\mathcal{L}_{\mathcal{S}, P}$-structure $\mathcal{M}$ such that $\mathcal{M} \models \mathrm{NFU}+\mathrm{AxCount}_\leq$ and $$\mathcal{M} \models \beth_{\omega_1^{\mathrm{ck}}}^{TT} \textrm{ does not exist}.$$ Our aim now turns to proving that the theory $\mathrm{NFU}+\mathrm{AxCount}$ is strictly stronger than $\mathrm{NFU}+\mathrm{AxCount}_\leq$. Our tactic for proving this result will be to apply Lemma \[Th:NonStandardOmegaModelInZFCMinus\] and Theorem \[Th:AutomorphismsFromNonStandardOmegaModels\] inside the structure $\langle \mathrm{BF}, \epsilon \rangle$ in the theory $\mathrm{NFU}+\mathrm{AxCount}$. In order to apply Lemma \[Th:NonStandardOmegaModelInZFCMinus\] inside $\langle \mathrm{BF}, \epsilon \rangle$ we first need to show that $\mathrm{NFU}+\mathrm{AxCount}$ proves that $$\langle \mathrm{BF}, \epsilon \rangle \models V_{\omega_1^{\mathrm{ck}}} \textrm{ exists}.$$ We will prove this result using the same tactics employed in section \[Sec:NFUPlusAxCountGEQ\]. \[Th:CountableOrdinalsAreStronglyCantorian\] The theory $\mathrm{NFU}+\mathrm{AxCount}$ proves that if $\alpha$ is a countable then\ $T(\alpha)= \alpha$. We work in the theory $\mathrm{NFU}+\mathrm{AxCount}$. Let $\alpha$ be a countable ordinal. Let $R \in \alpha$ be such that $\mathrm{Dom}(R) \subseteq \mathbb{N}$. Let $$S= \{ \langle \{x\}, \{y\} \rangle \mid \langle x, y \rangle \in R \}.$$ Define $f: \mathrm{Dom}(R) \longrightarrow \mathrm{Dom}(S)$ by $$\begin{array}{lcl} f(x)= \{y\} & \textrm{ if and only if } & T(y)=x \end{array}.$$ Stratified Comprehension ensures that $f$ exists. Since $T$ is the identity on $\mathrm{Dom}(R)$, $f$ witnesses the fact that $R \cong S$. Therefore $T(\alpha)= \alpha$. This allows us to prove an extension of Lemma \[Th:BethNLessThanT2V\] in the presence of $\mathrm{AxCount}$. \[Th:BethAlphaLEQT2V\] The theory $\mathrm{NFU}+\mathrm{AxCount}$ proves that for all countable ordinals $\alpha$, $\beth_\alpha^{TT} \leq T^2(\|V\|)$. We work in the theory $\mathrm{NFU}+\mathrm{AxCount}$. Let $\alpha$ be the least countable ordinal such that $T^2(\|V\|) < \beth_\alpha^{TT} \leq \|V\|$. But this immediately leads to contradiction because, $$\beth_\alpha^{TT}= \beth_{T^2(\alpha)}^{TT}= T^2(\beth_\alpha^{TT}) \leq T^2(\|V\|).$$ This extension facilitates the proof of an analogue of Theorem \[Th:AxCountLEQImpliesVOmegaPlusNExists\]. \[Th:AxCountProvesCountableRanksExist\] The theory $\mathrm{NFU}+\mathrm{AxCount}$ proves that for all countable ordinals $\alpha$, - $\langle \mathrm{BF}, \epsilon \rangle \models V_{\mathbf{k}(\alpha)} \textrm{ exists}$, - $\|(V_{\mathbf{k}(\alpha)}^{\langle \mathrm{BF}, \epsilon \rangle})^{\mathrm{ext}}\|= \beth_\alpha^{TT}$. We work in the theory $\mathrm{NFU}+\mathrm{AxCount}$. We prove the theorem by induction. Corollary \[Th:BFEXTsModelZFCminus\] implies that - $\langle \mathrm{BF}, \epsilon \rangle \models V_\omega \textrm{ exists}$, - $\|(V_{\omega}^{\langle \mathrm{BF}, \epsilon \rangle})^{\mathrm{ext}}\|= \aleph_0= \beth_0^{TT}$. Let $\alpha$ be a countable ordinal and suppose that the Theorem holds for all $\beta < \alpha$. If $\alpha$ is a limit ordinal then collection in $\langle \mathrm{BF}, \epsilon \rangle$ implies that $$\langle \mathrm{BF}, \epsilon \rangle \models V_{\mathbf{k}(\alpha)} \textrm{ exists}.$$ Moreover, $$\|(V_{\mathbf{k}(\alpha)}^{\langle \mathrm{BF}, \epsilon \rangle})^{\mathrm{ext}}\|= \sup \{ \|(V_{\mathbf{k}(\beta)}^{\langle \mathrm{BF}, \epsilon \rangle})^{\mathrm{ext}}\| \mid \beta < \alpha \}= \beth_\alpha^{TT}.$$ Now, suppose that $\alpha= \gamma + 1$. By Lemma \[Th:SizeOfPow\], $$\|\mathrm{Pow}(V_{\mathbf{k}(\gamma)}^{\langle \mathrm{BF}, \epsilon \rangle})\| = 2^{\|(V_{\mathbf{k}(\gamma)}^{\langle \mathrm{BF}, \epsilon \rangle})^{\mathrm{ext}}\|}= 2^{\beth_\gamma^{TT}}= \beth_{\alpha}^{TT}.$$ Therefore, by Lemma \[Th:BethAlphaLEQT2V\], $\|\mathrm{Pow}(V_{\mathbf{k}(\gamma)}^{\langle \mathrm{BF}, \epsilon \rangle})\| \leq T^2(\|V\|)$. And so, by Theorem \[Th:SetOfBFEXTsLessThanT2VAreExtensions\], there exists an $X \in \mathrm{BF}$ such that for all $Y \in \mathrm{BF}$, $$Y \in \mathrm{Pow}(V_{\mathbf{k}(\gamma)}^{\langle \mathrm{BF}, \epsilon \rangle}) \textrm{ if and only if } \langle \mathbf{BF}, \epsilon \rangle \models Y \in X.$$ Now, $\langle \mathbf{BF}, \epsilon \rangle \models X= V_{\mathbf{k}(\alpha)}$. We now turn to extending Lemma \[Th:FiniteOrdinalsInNFUAreFiniteOrdinalsInBF\] to show that, in $\mathrm{NFU}$, an ordinal $\alpha$ is countable if and only if $\mathbf{k}(\alpha)$ is countable in $\langle \mathrm{BF}, \epsilon \rangle$. \[Th:CountableOrdinalsInNFUAreCountableInBF\] The theory $\mathrm{NFU}$ proves that for all ordinals $\alpha$, $$(\alpha \textrm{ is countable}) \textrm{ if and only if } \langle \mathrm{BF}, \epsilon \rangle \models (\mathbf{k}(\alpha) \textrm{ is countable}).$$ We work in $\mathrm{NFU}$. Let $\alpha$ be an ordinal. Let $X= \{ \beta \in \mathrm{ON} \mid \beta < \alpha \}$.\ Suppose that $\alpha$ is countable. Therefore $T^2(\alpha)$ is countable. Let $f: X \longrightarrow \mathbb{N}$ be an injection. Define $G \subseteq \mathrm{BF}$ such that for all $x \in \mathrm{BF}$, $$\begin{array}{lcl} x \in G & \textrm{ if and only if } & \langle \mathrm{BF}, \epsilon \rangle \models (x= \langle \mathbf{k}(\beta), \mathbf{k}(n) \rangle) \textrm{ and } f(\beta)= n. \end{array}$$ Theorem \[Th:SetOfBFEXTsLessThanT2VAreExtensions\] implies that there exists $Y \in \mathrm{BF}$ such that for all $x \in \mathrm{BF}$, $$x \in G \textrm{ if and only if } \langle \mathrm{BF}, \epsilon \rangle \models x \in Y.$$ Now, $$\langle \mathrm{BF}, \epsilon \rangle \models (Y \textrm{ is an injection from } \mathbf{k}(\alpha) \textrm{ into } \omega).$$ Conversely, suppose that $\alpha$ is such that $$\langle \mathrm{BF}, \epsilon \rangle \models (\mathbf{k}(\alpha) \textrm{ is countable}).$$ Let $G \in \mathrm{BF}$ be such that $$\langle \mathrm{BF}, \epsilon \rangle \models (G \textrm{ is an injection from } \mathbf{k}(\alpha) \textrm{ into } \omega).$$ Define $f: X \longrightarrow \mathbb{N}$ such that for all $\beta < \alpha$ and for all $n \in \mathbb{N}$, $$f(\beta)= n \textrm{ if and only if } \langle \mathrm{BF}, \epsilon \rangle \models (G(\mathbf{k}(\beta))= \mathbf{k}(n)).$$ Now, Lemma \[Th:FiniteOrdinalsInNFUAreFiniteOrdinalsInBF\] implies that $f$ is a function witnessing the fact that $T^2(\alpha)$ is countable. Therefore $\alpha$ is countable. Theorem \[Th:AxCountProvesCountableRanksExist\] combined with Lemma \[Th:CountableOrdinalsInNFUAreCountableInBF\] show that, in the theory $\mathrm{NFU}+\mathrm{AxCount}$, the structure $\langle \mathrm{BF}, \epsilon \rangle$ believes that $V_{\omega_1^{\mathrm{ck}}}$ exists. \[Th:NFUPlusAXCountProvesVOmega1CKExists\] $$\mathrm{NFU}+\mathrm{AxCount} \vdash (\langle \mathrm{BF}, \epsilon \rangle \models V_{\omega_1^{\mathrm{ck}}} \textrm{ exists}).$$ Applying Lemma \[Th:StandardPartOfModelOfZR\] inside the structure $\langle \mathrm{BF}, \epsilon \rangle$ immediately shows that adding $\mathrm{AxCount}$ to $\mathrm{NFU}$ allows us to strengthen Corollary \[Th:NFUPlusAxCountLEQProvesConZ\]. $$\mathrm{NFU}+\mathrm{AxCount} \vdash \mathrm{Con}(\mathrm{KP}^\mathcal{P}).$$ Corollary \[Th:NFUPlusAXCountProvesVOmega1CKExists\] also allows us to show that the theory $\mathrm{NFU}+\mathrm{AxCount}$ is strictly stronger than $\mathrm{NFU}+\mathrm{AxCount}_\leq$. $$\mathrm{NFU}+\mathrm{AxCount} \vdash \mathrm{Con}(\mathrm{NFU}+\mathrm{AxCount}_\leq).$$ We work in the theory $\mathrm{NFU}+\mathrm{AxCount}$. By Corollary \[Th:NFUPlusAXCountProvesVOmega1CKExists\]: $$\langle \mathrm{BF}, \epsilon \rangle \models V_{\omega_1^{\mathrm{ck}}} \textrm{ exists}.$$ By applying Lemma \[Th:NonStandardOmegaModelInZFCMinus\] and Theorem \[Th:AutomorphismsFromNonStandardOmegaModels\] and using the same arguments used in the proof of Theorem \[Th:WeakModelOfNFUPlusAxCountLEQ\] we can build a model of $\mathrm{NFU}+\mathrm{AxCount}_\leq$ inside the structure $\langle \mathrm{BF}, \epsilon \rangle$. This section still leaves the following question unanswered: \[Df:StrengthOfAxCountLEQQuestion\] What is the exact strength of the theory $\mathrm{NFU}+\mathrm{AxCount}_\leq$ relative to a subsystem of $\mathrm{ZFC}$? Acknowledgements.* This research was completed while I was a Ph.D. student in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge. I would like to thank my supervisor Thomas Forster for all of his help and support. I would also like to thank Ali Enayat, Randall Holmes and Andrey Bovykin for many helpful discussions. My Ph.D. studies were supported the Cambridge Commonwealth Trusts.
--- abstract: 'Total Internal Reflection Microscopy (TIRM) is a sensitive non-invasive technique to measure the interaction potentials between a colloidal particle and a wall with femtonewton resolution. The equilibrium distribution of the particle-wall separation distance $z$ is sampled monitoring the intensity $I$ scattered by the Brownian particle under evanescent illumination. Central to the data analysis is the knowledge of the relation between $I$ and the corresponding $z$, which typically must be known [a priori]{}. This poses considerable constraints to the experimental conditions where TIRM can be applied (short penetration depth of the evanescent wave, transparent surfaces). Here, we introduce a method to experimentally determine $I(z)$ by relying only on the distance-dependent particle-wall hydrodynamic interactions. We demonstrate that this method largely extends the range of conditions accessible with TIRM, and even allows measurements on highly reflecting gold surfaces where multiple reflections lead to a complex $I(z)$.' address: - '$^{1}$Max-Planck-Institut für Metallforschung, Heisenbergstra[ß]{}e 3, 70569 Stuttgart, Germany' - '$^{2}$2. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany' author: - 'Giovanni Volpe,$^{1,2,}$ Thomas Brettschneider,$^{2}$ Laurent Helden,$^{2}$ and Clemens Bechinger$^{1,2}$' title: Novel perspectives for the application of total internal reflection microscopy --- [10]{} url \#1[`#1`]{}urlprefix\[2\]\[\][[\#2](#2)]{} J. Walz, “Measuring particle interactions with total internal reflection microscopy,” Curr. Opin. Colloid. Interface Sci. 1997, 600–606 (2). D. C. Prieve, “Measurement of colloidal forces with TIRM,” Adv. Colloid. Interface Sci. 82, 93–125 (1999). G. Binnig, C. F. Quate, and C. 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Walz, “A new approach for analyzing particle motion near an interface using total internal reflection microscopy,” J. Colloid. Interface Sci. 284, 323–331 (2005). M. D. Carbajal-Tinoco, R. Lopez-Fernandez, and J. L. Arauz-Lara, “Asymmetry in colloidal diffusion near a rigid wall,” Phys. Rev. Lett. 99, 138,303 (2007). G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084,701 (2007). Introduction ============ Total Internal Reflection Microscopy (TIRM) [@Walz1997; @Prieve1999] is a fairly new technique to optically measure the interactions between a single colloidal particle and a surface using evanescent light scattering. The distribution of the separation distances sampled by the particle’s Brownian motion is used to obtain the potential energy profile $U(z)$ of the particle-surface interactions with sub-$k_BT$ resolution, where $k_BT$ is the thermal energy. Amongst various techniques available to probe the mechanical properties of microsystems, the strength of TIRM lies in its sensitivity to very weak interactions. Atomic Force Microscopy (AFM) [@Binnig1986; @duc91] requires a macroscopic cantilever as a probe and is typically limited to forces down to several piconewton ($10^{-12}\, N$); the sensitivity of Photonic Force Microscopy (PFM) [@Ghislain1993; @Berg-sorensen2004; @Volpe2007A] can even reach a few femtonewtons ($10^{-15}\, N$), but this method is usually applied to bulk measurements far from any surface. TIRM, instead, can measure forces with femtonewton resolution acting on a particle near a surface. Over the last years, TIRM has been successfully applied to study electrostatic [@bik90; @Grunberg2001], van der Waals [@bev99], depletion [@Pie02b; @Bevan2002; @Helden2003; @Kleshchanok2006], magnetic [@Blickle2005], and, rather recently, critical Casimir [@Hertlein2008CC] forces. ![Total Internal Reflection Microscopy (TIRM). (a) Schematic of a typical TIRM setup: a Brownian particle moves in the evanescent electromagnetic field generated by total internal reflection of a laser beam; its scattering is collected by an objective lens; and the scattering intensity is recorded using a photomultiplier (PMT). (b) Typical experimental scattering intensity time-series (polystyrene particle in water, $R = 1.45\, \mu m$). (c) Exponential intensity-distance relation ($\beta = 120\, nm$). (d) Particle position distribution (acquisition time $1200\,s$, sampling rate $500\, Hz$). (e) Experimental (dots) and theoretical (line) potential obtained from the position distribution using the Boltzmann factor.[]{data-label="Fig1"}](fig_1.eps){width="8cm"} A schematic sketch of a typical TIRM setup is presented in Fig. \[Fig1\](a). To track the Brownian trajectory of a spherical colloidal particle diffusing near a wall, an evanescent field is created at the substrate-liquid interface. The scattered light is collected with a microscope objective and recorded with a photomultiplier connected to a data acquisition system. Fig. \[Fig1\](b) shows a typical example of an experimentally measured intensity time-series $I_t$ of a polystyrene particle with radius $R=1.45\, \mu m$ in water. Due to the evanescent illumination, the intensity of the light scattered by the particle is quite sensitive to the particle-wall distance. If the corresponding intensity-distance relation $I(z)$ is known (and monotonic), the vertical component of the particles trajectory $z_t$ can be deduced from $I_t$. To obtain $I(z)$, it is in principle required to solve a rather complex Mie scattering problem, i.e. the scattering of a micron-sized colloidal particle under evanescent illumination close to a surface [@Chew1979; @liu95], where multiple reflections between the particle and the substrate and Mie resonances must be accounted for [@wan99; @liu00; @Helden2006; @rie07]. When such effects can be neglected, the scattering intensity is proportional to the evanescent field intensity and, since the latter decays exponentially, TIRM data are typically analyzed using a purely exponential $I(z) = I_0 e^{-z/\beta}$ [@Walz1997; @Prieve1999; @Chew1979; @liu95; @pri93] (Fig. \[Fig1\](c)), where $\beta = \lambda/4\pi\sqrt{n_s^2\sin^2\theta - n_m^2}$ is the evanescent field penetration depth, $\lambda$ the incident light wavelength, $n_s$ the substrate refractive index, $n_m$ the liquid medium refractive index, and $\theta$ the incidence angle, which must be larger than the critical angle $\theta_c = \arcsin(n_m/n_s)$. $I_0$ is the scattering intensity at the wall, which can be determined e.g. using a hydrodynamic method proposed [@Bevan2000]. From the obtained $z_t$ the particle-wall interaction potential, i.e. $U(z)$, is easily derived by applying the Boltzmann factor $U(z) = -k_B T \ln{p(z)}$ to the calculated position distribution $p(z)$ (Fig. \[Fig1\](d,e)). For an electrically charged dielectric particle suspended in a solvent, the interaction potential typically corresponds to $U(z) = B\exp{(-\kappa z)}+[\frac{4}{3}\pi R^3 (\rho_{p}-\rho_{m})g - F_s] z$. The first term is due to double layer forces with $\kappa^{-1}$ the Debye length and $B$ a prefactor depending on the surface charge densities of the particle and the wall [@Walz1997; @Prieve1999; @Grunberg2001]. The second term describes the effective gravitational contributions with $\rho_{p}$ and $\rho_{m}$ the particle and solvent density and $g$ the gravitational acceleration constant; $F_s$ takes into account additional optical forces, which may result from a vertically incident laser beam often employed as a two-dimensional optical trap to reduce the lateral motion of the particle [@wal92]. Depending on the experimental conditions, additional interactions, such as depletion or van der Waals forces, may arise. Despite the broad range of phenomena that have successfully been addressed with TIRM, most studies have been carried out with small penetration depths (at most $\beta \approx 100\, nm$), and have therefore been limited to rather small particle-substrate distances $z$. In addition, no TIRM studies on highly reflecting walls, e.g. gold surfaces, have been reported, although such surfaces are interesting since they can support surface plasmons enhancing the evanescent field [@Marti1993] and the optical near-field radiation forces [@Volpe2006A; @Righini2007; @Righini2008]. Furthermore, gold coatings can be easily functionalized [@ulm96], which would allow to apply TIRM to e.g. biological systems. The reason for these limitations is the aforementioned problem to obtain a reliable $I(z)$ relationship under these conditions. For example, it has been demonstrated that large penetration depths (e.g. above $\approx 200\, nm$ in Ref. [@Hertlein2008TIRM]) increase the multiple optical reflections between the particle and the wall, which in turn leads to a non-exponential $I(z)$ [@Helden2006; @Hertlein2008TIRM]. Experiments combining TIRM and AFM found deviations from simple exponential behaviour very close to the wall even for shorter penetration depths [@McKee2005]. In principle, such effects can be included into elaborate scattering models, however, this requires precise knowledge of the system properties and, in particular, of the refractive indices of particle, wall, and liquid medium [@Hertlein2008TIRM]. Since the latter are prone to significant uncertainties (in particular for the colloidal particles), the application of TIRM under such conditions remains inaccurate. Here, we introduce a method to experimentally determine $I(z)$ by making solely use of the experimentally acquired $I_t$ and of the distance-dependent hydrodynamic interactions between the particle and the wall. In particular, no knowledge about the shape of the potential $U(z)$ is required. We demonstrate the capability of this method by experiments and simulations, and we also apply it to experimental conditions with long penetration depths ($\beta= 720\,nm$) and even with highly reflective gold surfaces. Theory ====== Diffusion coefficient and skewness of Brownian motion near a wall ----------------------------------------------------------------- ![Vertical diffusion coefficient $D_{\bot}(z)$ near a wall (Eq. (\[Brennerformula\])).[]{data-label="Fig2"}](fig_2.eps){width="7.5cm"} Colloidal particles immersed in a solvent undergo Brownian motion due to collisions with solvent molecules. This erratic motion leads to particle diffusion with the Stokes-Einstein diffusion coefficient $D_{SE} = k_BT/6\pi\eta R$, where $\eta$ is the shear viscosity of the liquid. It is well known that this bulk diffusion coefficient decreases close to a wall due to hydrodynamic interactions. From the solution of the creeping flow equations for a spherical particle in motion near a wall assuming nonslip boundary conditions and negligible inertial effects, one obtains for the diffusion coefficient in the vertical direction [@Brenner1961], $$\label{Brennerformula} D_{\bot}(z) = \frac{D_{SE}}{l(z)} \mbox{,}$$ where $l(z) = \frac{4}{3}\sinh{\left( \alpha(z) \right)} \sum_{n=1}^{\infty} \frac{n(n+1)}{(2n-1)(2n+3)} \left[ \frac{ 2\sinh{\left( (2n+1)\alpha(z) \right)} + (2n+1)\sinh{\left( 2\alpha(z) \right)} }{ 4\sinh^2{\left( (n+0.5)\alpha(z) \right)} - (2n+1)^2\sinh^2{\left( \alpha(z) \right)}} -1 \right]$ and $\alpha(z) = \cosh^{-1}\left( 1+\frac{z}{R}\right)$. As shown in Fig. \[Fig2\], $D_{\bot}$ first increases with $z$ approaching the corresponding bulk value at a distance of several particle radii away from the wall. Experimentally, the diffusion coefficient can be obtained from the mean square displacement (MSD) calculated from a particle trajectory. For the $z$-component this reads ${\langle (z_{t+\Delta t} - z_t)^2\rangle} = 2 D_{SE} \Delta t$, where $\langle ... \rangle$ indicates average over time $t$. To account for a $z$-dependent diffusion coefficient close to a wall, one has to calculate the conditional MSD given that the particle is at time $t$ at position $z$, i.e. $\langle (z_{t+\Delta t} - z_t)^2 \mid z_t=z \rangle = 2 D_{\bot}(z) \Delta t$ where the equality is valid for $\Delta t \rightarrow 0$; in such limit, this expression is only determined by the particle diffusion even if the particle is exposed to an external potential $U(z)$. From this follows that $D_{\bot}(z)$ can be directly obtained from the particle’s trajectory $$\label{Z_Brenner} D_{\bot}(z) = \lim_{\Delta t \rightarrow 0} \frac{1}{2\Delta t} \left\langle (z_{t+\Delta t} - z_t)^2 \mid z_t=z \right\rangle \mbox{.}$$ Eq. (\[Z\_Brenner\]) was employed already by several groups [@Oetama2005; @Carbajal-Tinoco2007] to validate Eq. (\[Brennerformula\]). The distribution of particle displacements $h(z; z_0,\Delta t)$ around a given distance $z_0$ converges to a gaussian for $\Delta t \rightarrow 0$ and therefore its [skewness]{} – i.e. the normalized third central moment – converges to zero. Accordingly, $$\label{Z_Skewness} S (z) \equiv \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t^2} \left\langle \left( \frac{z_{t+\Delta t} - z_t - M (z, \Delta t)}{ \sqrt{2D_{\bot}(z)} } \right)^3 \mid z_t=z \right\rangle = 0 \mbox{,}$$ where $M (z, \Delta t) = \langle z_{t+\Delta t} - z_t \mid z_t=z \rangle = \underset{\hat{z}}{\arg \max} \; h(\hat{z}; z, \Delta t)$, where $\underset{\hat{z}}{\arg \max}$ indicates the argument that maximize the given function. Mean square displacement and skewness of the scattering intensity ----------------------------------------------------------------- In a TIRM experiment, $h(z; z_0, \Delta t)$ is translated into a corresponding intensity distribution $h(I; I_0, \Delta t)$ around intensity $I_0 = I(z_0)$, whose shape strongly depends on $I(z)$. In Fig. \[Fig3\] we demonstrate how a particle displacement distribution $h(z; z_0, \Delta t)$, which is gaussian for small $\Delta t$, translates into the corresponding scattered intensity distribution $h(I; I_0, \Delta t)$ for an arbitrary non-exponential $I(z)$ dependence. In the linear regions of $I(z)$, the corresponding $h(I; I_0, \Delta t)$ are also gaussian with the half-width determined by the slope of the $I(z)$ curve. In the non-linear part of $I(z)$, however, a non-gaussian intensity histogram with finite skewness is obtained. ![ Relation between position distributions and intensity distributions. Brownian diffusion of a particle around a point is symmetric, leading for small $\Delta t$ to a gaussian distribution $h(z;z_0,\Delta t)$ (bottom). According to $I(z)$ this leads to the scattered intensity histograms $h(I;I_0,\Delta t)$ (left): in the linear region of $I(z)$, $h(I;I_0,\Delta t)$ is also gaussian with the width depending on $I'$ (Eq. (\[I\_MSD\])); in the nonlinear region of $I(z)$ (central histogram), $h(I;I_0,\Delta t)$ deviates from a gaussian and has a finite skewness depending on $I''$ (Eq. (\[I\_Skewness\])).[]{data-label="Fig3"}](fig_3.eps){width="7.5cm"} In the following we calculate the MSD and the skewness of $h(I; I_0, \Delta t)$ for an arbitrary $I(z)$, which we assume to be a continuous function with well defined first and second derivates $I'$ and $I''$. In the vicinity of $z_0$, $I(z)$ can be therefore expanded in a Taylor series $I(z) = I(z_0) + I'(z_0)(z-z_0) + \frac{1}{2} I''(z_0)(z-z_0)^2$ for $z \rightarrow z_0$, where $I' = \frac{dI}{dz}$ and $I'' = \frac{d^2I}{dz^2}$. The MSD of $h(I; I_0, \Delta t)$ is $$\label{I_MSD} \mathrm{MSD}(I) \equiv \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \left\langle (I_{t+\Delta t}-I_t)^2 \mid I_t = I \right\rangle = I'^2 (z) \cdot 2 D_{\bot} (z) \mbox{,}$$ where $\left\langle (I_{t+\Delta t}-I_t)^2 \mid I_t = I \right\rangle = I'^2 \left\langle (z_{t+\Delta t}-z_t)^2 \mid I_t = I \right\rangle$ for $\Delta t \rightarrow 0$ and Eq. (\[Z\_Brenner\]) has been used. The skewness of $h(I; I_0, \Delta t)$ is $$\label{I_Skewness} \mathrm{S}(I) \equiv \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t^2} \left\langle \left( \frac{I_{t+\Delta t} - I_t - M(I,\Delta t)}{ \sqrt{\mathrm{MSD}(I)} } \right)^3 \mid I_t = I \right\rangle = \frac{9}{2} \frac{I''(z)}{\|I'(z)\|} \cdot \sqrt{2 D_{\bot}(z)} \mbox{,}$$ with $M(I,\Delta t) = \underset{\hat{I}}{\arg \max} \; h(\hat{I}; I, \Delta t)$ and $\left\langle (I_{t+\Delta t} - I_t - M(I,\Delta t))^3 \mid I_t = I \right\rangle = I'^3 \left\langle (z_{t+\Delta t} - z_t - M(z,\Delta t))^3 \mid z_t = z \right\rangle + \frac{3}{2} I'^2 I'' \left\langle (z_{t+\Delta t} - z_t - M(z,\Delta t) )^4 \mid z_t = z \right\rangle$ for $\Delta t \rightarrow 0$ where the first term is null because of Eq. (\[Z\_Skewness\]), and the second term is calculated using the properties of the momenta of a gaussian distribution $\left\langle (...)^4 \right\rangle = 3 \left\langle (...)^2 \right\rangle^2$. In Fig. \[Fig4\] we applied Eqs. (\[I\_MSD\]) and (\[I\_Skewness\]) to the intensity time-series corresponding to a particle trajectory simulated using a Langevin difference equation assuming various $I(z)$ (Fig. \[Fig4\](a,c,e)). The solid lines in Fig. \[Fig4\](b,d,f) show the theoretical $\mathrm{MSD}(I)$ (black) and $\mathrm{S}(I)$ (red) and the dots the ones obtained from the simulations. When $I(z)$ is linear (Fig. \[Fig4\](a)), $\mathrm{MSD}(I)$ is proportional to Eq. (\[Brennerformula\]) and $\mathrm{S}(I)$ vanishes (Fig. \[Fig4\](b)). When $I(z)$ is exponential (Fig. \[Fig4\](c)) or a sinusoidally modulated exponential (Fig. \[Fig4\](e), $\mathrm{MSD}(I)$ is not proportional to Eq. (\[Brennerformula\]) and large values of the skewness occur as shown in (Figs. \[Fig4\](d,f)). Small deviations between the theoretical curves and the numerical data can be observed for intensities where the particle drift becomes large in comparison to the time-step ($\Delta t=2\, ms$); in our case this corresponds to a slope of the potential of about $1\,pN/\mu m$, which is close to the upper force limit of typical TIRM measurements. If necessary such deviations can be reduced employing shorter time-steps. ![Various intensity-distance relations and their effect on $\mathrm{MSD}(I)$ (black) and skewness $\mathrm{S}(I)$ (red). Both theoretical values (solid lines) and the results from the analysis of numerically simulated data (dots) using Eq. (\[I\_MSD\]) and Eq. (\[I\_Skewness\]) are presented ($R = 1.45\, \mu m$, $\rho_p = 1.053\, g/cm^3$, samples $10^6$, frequency $100\, Hz$). (a)-(b) Linear $I(z)$. (c)-(d) Exponential $I(z)$. (e)-(f) Exponential $I(z)$ modulated by a sinusoidal function.[]{data-label="Fig4"}](fig_4.eps){width="8cm"} Obtaining $I(z)$ from $I_t$ --------------------------- The correct $I(z)$ satisfies the conditions $$\label{conditions} \left\{ \begin{array}{ccc} \mathrm{MSD}(I(z)) & = & I'^2(z) \cdot 2 D_{\bot}(z) \\ \mathrm{S}(I(z)) & = &\frac{9}{2} \frac{I''(z)}{\|I'(z)\|} \cdot \sqrt{2 D_{\bot}(z)} \end{array} \right. \mbox{,}$$ where $\mathrm{MSD}(I)$ and $\mathrm{S}(I)$ are calculated from an experimental $I_t$. Thus, the problem of determining $I(z)$ can be regarded as a functional optimization problem, where Eqs. (\[conditions\]) have to be fulfilled. Analysis workflow ================= Here, we present a concrete analysis workflow to obtain $z_t$ from the experimental $I_t$ by finding the $I(z)$ that satisfies Eqs. (\[conditions\]). To do so, we will construct a series of approximations $I^{(i)}(z)$ indexed by $i$ converging to $I(z)$. \(1) As first guess, take $I^{(0)} (z) = I_0 \exp{(-z/\beta)}+b_s$, where $\beta$, $I_0$ and $b_s$ are parameters chosen to optimize Eqs. (\[conditions\]). Often some initial estimates are available from the experimental conditions: $\beta$ can be taken as the evanescent field penetration depth, $I_0$ as the scattering intensity at the wall, and $b_s$ as the background scattering in the absence of the Brownian particle. While $\beta$ is typically well known, $I_0$ and $b_s$ are prone to large experimental systematic errors and uncertainties. \(2) Take $I^{(1)} (z) = I^{(0)}(z) [1-G(I^{(0)}(z), \mu^{(1)}, \sigma^{(1)}, A^{(1)}) ]$, where $G(x, \mu, \sigma, A) = A \exp(-(x-\mu)^2/\sigma^2)$ is a gaussian, and the parameters $\mu^{(1)}$, $\sigma^{(1)}$, and $A^{(1)}$ optimize Eqs. (\[conditions\]). Gaussian functions were chosen because they have smooth derivatives and quickly tend to zero at infinite. Notice that the choice of a Gaussian is unessential for the working of the algorithm. \(3) Reiterate step (2), substituting $I^{(0)}$ with $I^{(i)}$ and $I^{(1)}$ with $I^{(i+1)}$, until Eqs. (\[conditions\]) are satisfied within the required precision. \(4) Invert $I^{(i+1)}(z)$, i.e. numerically construct $z^{(i+1)}(I)$. \(5) Take $z_t= z^{(i+1)}(I_t)$. Experimental case studies ========================= Validation of the technique --------------------------- We test our approach on experimental data (polystyrene particle with $R = 1.45\,\mu m$ near a glass-water interface kept in place by a vertically incident laser beam [@wal92]) for which the exponential $I(z)$ is justified ($\beta = 120\, nm$, $\lambda = 658 \, nm$) [@Hertlein2008TIRM]. As illustrated in Fig. \[Fig5\](a), there is indeed agreement between the measured (dots) and theoretical potential (solid line). In the inset, the measured diffusion coefficient (black dots) agrees with Eq. (\[Z\_Brenner\]) (black solid line) and the skewness (red dots) is negligible (small deviations in the region where the potential is steepest are due to the finite time-step). The criteria for $I(z)$ in Eqs. (\[conditions\]) are already fulfilled after $I_0$ and $b_s$ have been optimized in the first step of the analysis workflow in the previous section. As shown in Fig. \[Fig5\](b), the experimental $\mathrm{MSD}(I)$ (black dots) and skewness $\mathrm{S}(I)$ (red dots) fit Eqs. (\[I\_MSD\]) and (\[I\_Skewness\]) (solid lines). ![ TIRM with exponential intensity-distance relation. (a) The experimental (dots) and theoretical (solid line) potential. Inset: the diffusion coefficient on the position data (black dots) fits well Eq. (\[Brennerformula\]) (black solid line), while the absolute value of the Brownian motion skewness $\mathrm{S}(z)$ is small (red dots). (b) Experimental $\mathrm{MSD}(I)$ (black dots) and skewness $\mathrm{S}(I)$ (red dots) for a scattering intensity time-series (polystyrene particle, $R = 1.45\, \mu m$, $\rho_p = 1.053\, g/cm^3$, $n_p=1.59$ suspended in water $n_m = 1.33$ with $300\, \mu M$ NaCl background electrolyte, $\kappa^{-1} = 17\, nm$, near a glass surface $n_s=1.52$, acquisition time $1800\, s$, sampling frequency $500 \, Hz$) calculated using Eq. (\[I\_MSD\]) and Eq. (\[I\_Skewness\]). Given the short penetration depth ($\beta = 120\, nm$), the theoretical $\mathrm{MSD}(I)$ (black solid line) and $\mathrm{S}(I)$ (red solid line) for an exponential $I(z)$ fit the experimental ones and the conditions in Eqs. (\[conditions\]) are fulfilled. []{data-label="Fig5"}](fig_5.eps){width="8cm"} ![TIRM with large penetration depth. (a) The experimental potential (black dots) obtained using the fitted intensity-distance relation and the theoretical one (black solid line). The green dots represent the faulty potential obtained using the exponential $I(z)$. (b) The fitted $I(z)$ (black line) and the exponential one (green line) corresponding to the penetration depth $\beta = 720\, nm$. (c) Experimental intensity $\mathrm{MSD}(I)$ (black dots) and skewness $\mathrm{S}(I)$ (red dots) for a scattering intensity time-series (same particle and acquisition parameters as in Fig. \[Fig5\]) calculated using Eq. (\[I\_MSD\]) and Eq. (\[I\_Skewness\]). Due to the large penetration depth, the $I(z)$ diverges from an exponential; the theoretical $\mathrm{MSD}(I)$ (black solid line) and $\mathrm{S}(I)$ (red solid line) correspond to the fitted $I(z)$.[]{data-label="Fig6"}](fig_6.eps){width="8cm"} ![TIRM in front of a reflective surface. (a) The experimental potential (black dots) obtained using the fitted intensity-distance relation and theoretical one (solid black line). The green dots represent the faulty potential obtained using the exponential $I(z)$ with $\beta = 244\, nm$. (b) The fitted $I(z)$ (black line) and the exponential one (green line) corresponding to the evanescent field penetration depth $\beta = 244\, nm$. (c) Experimental intensity $\mathrm{MSD}(I)$ (black dots) and skewness $\mathrm{S}(I)$ (red dots) for a scattering intensity time-series (same particle and acquisition parameters as in Fig. \[Fig5\], except for background electrolyte $50\, \mu M$, $\kappa^{-1} = 42\, nm$) calculated using Eq. (\[I\_MSD\]) and Eq. (\[I\_Skewness\]). Due to the presence of a $20\, nm$-thick gold layer on the surface, the $I(z)$ deviates from an exponential; the theoretical $\mathrm{MSD}(I)$ (black solid line) and $\mathrm{S}(I)$ (red solid line) correspond to the fitted $I(z)$. []{data-label="Fig7"}](fig_7.eps){width="8cm"} TIRM with large penetration depth --------------------------------- We now apply our technique under conditions where an exponential $I(z)$ is not valid, i.e. for large penetration depth as mentioned above. Fig. \[Fig6\] shows the potential obtained for the same particle as in Fig. \[Fig5\] but for a penetration depth ($\beta = 720\, nm$). Note, that compared to \[Fig5\] the potential extends over a much larger distance range since the particle’s motion can be tracked from hundreds of nanometers to microns. The green data points show the faulty interaction potential that is obtained when assuming an exponential $I(z)$. Since the only difference is in the illumination, the same potential as in Fig. \[Fig5\] should be retrieved (solid line in Fig. \[Fig6\](a)). However, applying an exponential $I(z)$ (green line in Fig. \[Fig6\](b)) wiggles appear in the potential (green dots in Fig. \[Fig6\](a)). Their origin is due to multiple reflections between the particle and the wall as discussed in detail in [@Helden2006]. The correct $I(z)$ (black line in Fig. \[Fig6\](b)) is obtained with the algorithm proposed in the previous section: after 9 iterations the conditions in Eqs. (\[conditions\]) appear reasonably satisfied, as shown in Fig. \[Fig6\](c). With this $I(z)$, we reconstructed the potential represented by the black dots in Fig. \[Fig6\](a), in good agreement with the one in Fig. \[Fig5\]. It should be noticed that, even though the deviations of the correct $I(z)$ from an exponential function are quite small, this is enough to significantly alter the measurement of the potential. This again demonstrates the importance of obtaining the correct $I(z)$ for the analysis of TIRM experiments. TIRM in front of a reflective surface ------------------------------------- To demonstrate that our method is capable of correcting even more severe optical distortions, we performed measurements in front of a reflecting surface ($20\, nm$ gold-layer, reflectivity $\approx 60\%$, $\beta = 244\, nm$). The experimental conditions are similar to the previous experiments. Only the salt concentration was lowered to avoid sticking of the particle to the gold surface due to van der Waals forces, leading to a larger electrostatic particle-surface repulsion, and the optical trap was not used. Using an exponential $I(z)$ (green line in Fig. \[Fig7\](b)), we obtain the potential represented by the green dots in Fig. \[Fig7\](a), which clearly features unphysical artifacts, e.g. spurious potential minima. After 27 iterations of the data analysis algorithm, the black $I(z)$ in Fig. \[Fig7\](b) is obtained, which reasonably satisfies the criteria in Eqs. (\[conditions\]) (Fig. \[Fig7\](c)). The reconstructed potential (black dots in Fig. \[Fig7\](a)) fits well to theoretical predictions (solid line); in particular the unphysical minima disappear. Conclusions & Outlook ===================== TIRM is a technique which allows one to measure the interaction potentials between a colloidal particle and a wall with femtonewton resolution. So far, its applicability has been limited by the need for an [a priori]{} knowledge of the intensity-distance relation. $I(z) \propto \exp(-z/\beta)$ can safely be assumed only for short penetration depths of the evanescent field and transparent surfaces. This, however, poses considerable constraints to the experimental conditions and the range of forces where TIRM can be applied. Here, we have proposed a technique to determine $I(z)$ that relies only on the hydrodynamic particle-surface interaction (Eq. \[Brennerformula\]) and, differently from existing data evaluation schemes, makes no assumption on the functional form of $I(z)$ or on the wall-particle potential. This technique will particularly be beneficial for the extension of TIRM to new domains. Here, we have demonstrated TIRM with a very large penetration depth, which allows one to bridge the gap between surface measurements and bulk measurements, and TIRM in front of a reflecting (gold-coated) surface, which allows plasmonic and biological applications. This new technique only assumes the knowledge of the particle radius, which is usually known within an high accuracy and can also be measured [in situ*]{} [@Bevan2000], and the monotonicity of $I(z)$. Were $I(z)$ not monotonous, as it might happen for a metallic particle in front of a reflective surface, the technique can be adapted to use the information from two non-monotonous signals, e.g. the scattering from two evanescent fields with different wavelength [@Hertlein2008TIRM]. We notice that the technique encounters its natural limits when Eq. (\[Brennerformula\]) does not correctly describe the particle-wall hydrodynamic interactions. This may happen in situations when the stick boundary conditions do not apply or when the hydrodynamic interactions are otherwise altered, e.g. in a viscoelastic fluid. Since the conditions in Eqs. (\[conditions\]) are fulfilled only by the correct $I(z)$, they permit a self-consistency check on the data analysis. Even when an exponential $I(z)$ is justified, errors that arise from the estimation of some parameters (e.g. the zero-intensity $I_0$ and the background intensity $b_s$) can be easily avoided by checking the consistency of the analyzed data with the aforementioned criteria. In principle, the analysis of TIRM data can be completely automatized, possibly providing the missing link for a widespread application of TIRM to fields, such as biology, where automated analysis techniques are highly appreciated. The proposed technique can also be useful to determine the intensity-distance relation in all those situations where it is possible to rely on the knowledge of the system hydrodynamics, while the scattering is not accurately known. Often explicit formulas are available for the hydrodynamic interaction of an over-damped Brownian particle in a simple geometry, while complex numerical calculations are needed to determine its scattering. As a limiting case, this technique might also prove useful for the PFM technique working in bulk, where the diffusivity is constant. Indeed, under certain experimental conditions – e.g. using back-scattered light instead of the more usual forward-scattered light [@Volpe2007B] – the intensity-distance relation can be non-trivial and it can be necessary to determine it experimentally.
--- abstract: 'I develop a theoretical framework for inferring nonequilibrium equations of motion from incomplete experimental data. I focus on genuinely irreversible, Markovian processes, for which the incomplete data are given in the form of snapshots of the macrostate at different instances of the evolution, yet without any information about the timing of these snapshots. A reconstruction of the equation of motion must therefore be preceded by a reconstruction of time.' author: - Jochen Rau title: Reconstruction of Markovian dynamics from untimed data --- \[intro\]Introduction ===================== Markovian processes, both reversible and irreversible, are ubiquitous. Microscopic processes, described by the Schrödinger equation, are obviously Markovian; and so are many macroscopic processes that can be described by, say, a rate, master, or Boltzmann equation. All these processes share the common feature that they are local in time: The state of the system at any given time fully determines its future evolution, regardless of the system’s prior history; the system exhibits no memory. This renders Markovian processes reproducible, in the sense that preparation of the same initial state — no matter how — always entails the same subsequent evolution. The ubiquity of Markovian processes is linked to the existence of disparate time scales in many systems. The macroscopic observables whose dynamics one wishes to describe typically coincide with the slow observables, and hence they evolve on longer time scales; whereas any memory, which is due to interaction with the other, faster degrees of freedom, fades away on a much shorter time scale. There are situations in which this separation of time scales breaks down and memory effects do play a role [@PhysRev.124.983]; but in the present paper, I focus on Markovian dynamics. Much effort has been devoted to developing theoretical frameworks that allow one to [derive]{} Markovian transport equations from the underlying microscopic dynamics; and in turn, to deduce from these transport equations testable predictions for macroscopic experiments [@Nakajima01121958; @1.1731409; @Mori01031965; @PhysRev.144.151; @rau:physrep; @balian:physrep; @PhysRevE.56.6620; @PhysRevE.62.4720]. In contrast to this deductive approach, I start here from the opposite end: I ask how one can [infer]{} a Markovian transport equation from experimental data. In particular, I consider situations where the experimental data come without time information. Such data may stem from past processes (say, in the geological or astronomical realm) that have left visible traces, albeit without time information; or from processes that, again, leave visible traces but are so fast or delicate that they cannot be tracked with a clock. Under such circumstances, before inferring the pertinent Markovian equation of motion, one must first reconstruct “time.” The purpose of the present paper is to show that this is possible (up to an additive and multiplicative constant); and to furnish the necessary tools for doing so. The reconstruction of time and of the equation of motion presupposes rather detailed knowledge about the generic structure of Markovian dynamics. In particular, Markovian dynamics may be viewed geometrically as a flow on the manifold of macrostates. This manifold is endowed with a rich geometric structure, and there are certain consistency conditions that any Markovian flow must satisfy. These conditions, in conjunction with the experimental data, will turn out to constrain the form of the equation of motion just enough so that the reconstruction succeeds. Mirroring the importance of these a priori constraints, I start out with a comprehensive discussion of the geometry of macrostates (Sec. \[gibbs\]) and of generic Markovian dynamics (Sec. \[generic\]). These introductory sections draw on ideas from the projection-operator [@Nakajima01121958; @1.1731409; @Mori01031965; @PhysRev.144.151; @rau:physrep], geometric [@balian:physrep], and two-generator [@PhysRevE.56.6620; @PhysRevE.62.4720] approaches to nonequilibrium dynamics. Then I proceed to formulate the general prescription for reconstruction (Sec. \[reconstruction\]), which I later illustrate with a simple example (Sec. \[example\]). I conclude with a brief discussion and outlook on future work, Sec. \[discussion\]. \[gibbs\]Geometry of macrostates ================================ \[manifold\]Manifold and coordinates ------------------------------------ To describe the static or dynamic properties of a macroscopic system, typically only a few observables are deemed relevant – for example, the system’s constants of the motion (if static), slow observables (if dynamic), or observables pertaining to some subsystem of interest. These relevant observables, together with the unit operator, span the so-called level of description, a subspace within the linear space of observables [@rau:physrep]. For an arbitrary microstate $\rho$ and level of description ${\cal G}:=\mbox{span}\{I,G_a\}$, the associated macrostate is that state which, while yielding the same expectation values $\{g_a\}$ as $\rho$ for all relevant observables, comes closest to equidistribution. Closeness to equidistribution is measured in terms of the von Neumann entropy $$S[\mu]:=-\mbox{tr}(\mu\ln\mu) ;$$ so the macrostate – denoted by $\pi(\rho)$ – is determined by the maximization $$\pi(\rho):=\arg \max_{\mu\in g} S[\mu] ,$$ where $\mu\in g$ is short for the constraints $\langle G_a\rangle_\mu=g_a \forall a$. It has the Gibbs form $$\pi(\rho) = Z(\lambda)^{-1} \exp(-\lambda^a G_a) , \label{canonical}$$ with the partition function $$Z(\lambda):=\mbox{tr}\{\exp(-\lambda^a G_a)\}$$ ensuring state normalisation, the Lagrange parameters $\{\lambda^a\}$ adjusted such that $\langle G_a\rangle_{\pi(\rho)}=g_a$, and – for ease of notation – the Einstein convention that identical upper and lower indices are to be summed over. The operational meaning of “relevance” and how it leads to the Gibbs form have been discussed in Ref. [@PhysRevA.90.062114]. Let ${\cal S}$ denote the set of normalized (pure or mixed) states of a given physical system; this set constitutes a differentiable manifold. In this manifold, the macrostates with level of description ${\cal G}$ form a submanifold, $\pi({\cal S})$. This submanifold has dimension $(\dim {\cal G} -1)$, which equals the number of relevant observables (provided they are linearly independent). On $\pi({\cal S})$ there are two natural choices of coordinates: the relevant expectation values $\{g_a:=\langle G_a\rangle_{\pi(\rho)}\}$, or the Lagrange parameters $\{\lambda^a\}$. The former coordinates can be expressed in terms of the latter via $$g_a=-{\partial_a} \ln Z ,$$ where $\partial_a:=\partial/\partial \lambda^a$. Their respective gradients $\{dg_a\}$ and $\{d\lambda^a\}$ are related by $$dg_a = - C_{ab} d\lambda^b \ ,\ d\lambda^a = - (C^{-1})^{ab} dg_b , \label{gradients}$$ with the Jacobian (up to a sign) given by the correlation matrix $$C_{ab} := {\partial_a \partial_b} \ln Z . \label{correlationmatrix}$$ Associated with the expectation value coordinates $\{g_a\}$ is a local basis of tangent vectors $\{\partial^a:=\partial/\partial g_a\}$, related to their one-form duals $\{dg_a\}$ via $dg_b(\partial^a)=\delta^a_b$ [@schutz:book]; and likewise for the Lagrange parameter coordinates, with the associated local basis $\{\partial_a\}$ satisfying $d\lambda^b(\partial_a)=\delta^b_a$. Upon infinitesimal variation of a macrostate its von Neumann entropy changes by $$dS = dS(\partial^a) \, dg_a = \lambda^a \, dg_a ; \label{entropydifferential}$$ which in turn implies that the Hessian of the entropy yields (up to a sign) the inverse of the correlation matrix, $$(C^{-1})^{ab} = - \partial^a \partial^b S .$$ Expectation values $x:=\langle X\rangle_{\pi(\rho)}$ of arbitrary (not necessarily relevant) observables $X$ change by $$dx = dx (\partial^b) \, dg_b = - \langle\delta G_b;X\rangle d\lambda^b , \label{gradient_x}$$ where $\delta G_b:= G_b-g_b$, and $\langle;\rangle$ denotes the [canonical correlation function]{} $$\langle A;B\rangle:=\int_0^1 d\nu\,\mbox{tr}[{\pi(\rho)}^\nu A^\dagger {\pi(\rho)}^{1-\nu} B] . \label{canonicalcorrelation}$$ The latter constitutes a positive definite scalar product in the space of observables. Comparing Eq. (\[gradient\_x\]) for $x=g_a$ with Eq. (\[gradients\]), one finds that the correlation matrix can be expressed in terms of this scalar product, $$C_{ab}=\langle \delta G_a;\delta G_b\rangle . \label{correlationmatrix2}$$ \[coarse\]Coarse graining and projectors ---------------------------------------- The map $\pi:{\cal S}\to\pi({\cal S})$ constitutes a coarse graining. While retaining information about the relevant observables, it discards all information about the rest. Geometrically, it “projects” a microstate $\rho\in{\cal S}$ onto the submanifold of macrostates. Indeed, $\pi$ exhibits typical features of a projection operator: it is idempotent, $\pi\circ\pi = \pi$; successive coarse grainings with respect to smaller and smaller levels of description are equivalent to a one-step coarse graining with respect to the smallest level of description, $${\cal G} \subset {\cal F} \ \Leftrightarrow \ \pi_{\cal G}\circ\pi_{\cal F} = \pi_{\cal G} ;$$ and it is covariant under unitary transformations, $$\pi_{U{\cal G}U^\dagger}(U\rho U^\dagger) = U \pi_{\cal G}(\rho) U^\dagger . \label{covariance}$$ In contrast to an ordinary projection operator, however, this coarse graining map need not be linear. Mirroring the coarse graining of states, there is also a coarse graining (“super-”)operator ${\cal P}$ on the space of observables. The two are dual to each other in the sense that for arbitrary (not necessarily relevant) observables $X$ and arbitrary microstates $\rho$ it is $$\langle {\cal P} X \rangle_\rho = \langle X \rangle_{\pi(\rho)} \ \ \forall\ X,\rho .$$ The unique superoperator which satisfies this requirement, sometimes called the Robertson [@PhysRev.144.151] or Kawasaki-Gunton [@kawasaki+gunton] projector, is $${\cal P} X := x\, I + dx (\partial^a) \,\delta G_a .$$ It projects arbitrary observables onto the level of description, the projection being orthogonal with respect to the scalar product $\langle;\rangle$. As the latter is evaluated in the macrostate $\pi(\rho)$, and hence the notion of orthogonality varies with the macrostate, the projector, too, carries an implicit dependence on the macrostate: ${\cal P}\equiv{\cal P}[\pi(\rho)]$. Like the coarse graining operation $\pi$ on states, the projector is idempotent, ${\cal P}^2={\cal P}$; yet unlike $\pi$, it is always linear and hence a true projector. Its complement ${\cal Q}:={\cal I}-{\cal P}$ (with ${\cal I}$ being the unit superoperator, ${\cal I}X=X$) is also a projector and projects an arbitrary observable onto its “irrelevant” component. Table \[lod\_examples\] lists some examples for different levels of description and the associated coarse graining operations $\pi$ and ${\cal P}$. --------------------------------------------------------------------------------------------------------- system discard $\pi(\rho)$ ${\cal P}X$ ----------------------- -------------- -------------------------- --------------------------------------- single coherence $\sum_i P_i \rho P_i$ $\sum_i P_i X P_i$ \\[3pt\] $S\times E$ environment $\rho_S \otimes \iota_E$ $\mbox{tr}_E(\iota_E X)\otimes I_E$ \\[3pt\] $A\times B$ correlations $\rho_A\otimes \rho_B$ $ \begin{array}[t]{c} \mbox{tr}_B(\rho_B X)\otimes I_B \\ + I_A\otimes \mbox{tr}_A(\rho_A X) \\ - x\,I_A\otimes I_B \end{array} $ --------------------------------------------------------------------------------------------------------- : Three examples of levels of description and the associated coarse graining operations $\pi$ and ${\cal P}$. They refer to (i) a single quantum system where only classical probabilities are deemed relevant, and all information about coherence is discarded; (ii) a system $S$ coupled to an environment $E$ where only the system properties are deemed relevant, and all information about the environment (including system-environment correlations) is discarded; and (iii) a bipartite system where only single-particle properties are deemed relevant, and all correlations are discarded. The $\{P_i\}$ are projection operators (on Hilbert space) pertaining to some preferred orthonormal (“decoherence”) basis. $\rho_i$ denotes the reduced state of subsystem $i$. The state $\iota:=I/\mbox{tr}I$ is the totally mixed state. In the first two examples $\pi$ is linear, and ${\cal P}$ is state-independent. In contrast, in the third example $\pi$ is nonlinear, and ${\cal P}$ varies with the macrostate. \[lod\_examples\] \[riemannian\]Metric and covariant derivative --------------------------------------------- As the canonical correlation function $\langle;\rangle$ is a positive definite scalar product, Eq. (\[correlationmatrix2\]) implies that the correlation matrix is both symmetric and positive definite. Therefore, the symmetric $(0,2)$ tensor field $$C:= C_{ab}\, d\lambda^a \otimes d\lambda^b = (C^{-1})^{ab} dg_a \otimes dg_b \label{metrictensor}$$ constitutes a Riemannian metric on the manifold of macrostates. This particular metric is known as the Bogoliubov or Kubo-Mori metric [@10.1063/1.530611; @bengtsson:book]. Up to a sign, it relates the local basis in $g$-coordinates to the one-form duals in $\lambda$-coordinates and vice versa, $$C(\partial^a) = - d\lambda^a \quad , \quad C(\partial_a) = - dg_a . \label{metricandduals}$$ The Bogoliubov-Kubo-Mori (BKM) metric has an operational meaning. It quantifies the statistical distinguishability of nearby macrostates, in the following sense. Two states $\rho$ and $\mu$ can be distinguished statistically if measurements on a finite sample, taken from an i.i.d. source of one state, say, $\rho$, are highly unlikely to erroneously indicate the other state, $\mu$. The pertinent error probability is $$\begin{aligned} \lefteqn{ \mbox{prob}_{1-\epsilon}(\mu\|N,\rho) } && \nonumber \\ &:=& \inf_{\Gamma} \left.\left\{\mbox{prob}(\Gamma\|\rho^{\otimes N})\right\|\mbox{prob}(\Gamma\|\mu^{\otimes N})\geq 1-\epsilon\right\} , \quad \label{defprob}\end{aligned}$$ where $N$ denotes the size of the sample, and $\Gamma$ is a proxy for the measurement results, which asymptotically, i.e., to within an error probability $\epsilon$ ($0<\epsilon<1$) that does not depend on sample size, are compatible with the sample being in the state $\mu^{\otimes N}$. Asymptotically, this error probability decreases exponentially with sample size, $$\mbox{prob}_{1-\epsilon}(\mu\|N,\rho)\sim \exp[-N S(\mu\\|\rho)] , \label{quantumstein}$$ and no longer depends on the specific value of the error parameter $\epsilon$ (“quantum Stein lemma” [@hiai+petz; @ogawa+nagaoka]). The exponent features the relative entropy [@donald:cmp; @vedral:rmp], $$S(\mu\\|\rho):= \left\{ \begin{array}{ll} \mbox{tr}(\mu\ln\mu - \mu\ln\rho) & : \mbox{supp}\;\mu\subseteq\mbox{supp}\;\rho \\ +\infty & : \mbox{otherwise} \end{array} \right. , \label{relativeentropy}$$ which thus proves to be a natural distinguishability measure. For nearby macrostates $\pi(\rho)$ and $\pi(\rho+\delta\rho)$ connected by a distance vector $W$, this distinguishability measure is approximated to lowest (quadratic) order in the coordinate differentials by $$S(\pi(\rho)\\|\pi(\rho+\delta\rho)) \approx \textstyle\frac{1}{2} C(W,W) ; \label{relentropy_nearby}$$ and hence indeed, up to a numerical factor, by the length (squared) of the distance vector in the BKM metric. The above metric is also singled out by the fact that it is with respect to this metric that the “projection” $\pi$ onto the submanifold of Gibbs states is orthogonal. This can be seen as follows. Being the infinitesimal version of the relative entropy, and the latter being defined for arbitrary pairs of states (both macro and micro), the metric can be extended from the manifold of macrostates to full state space. Let $V$ denote an arbitrary vector field on the full state space that connects only states with identical expectation values for the relevant observables, $dg_a (V)=0$. Then with $dg_a=-C(\partial_a)$, it is $$C(\partial_a,V) = 0 \ \ \forall\ a ;$$ i.e., any such $V$ intersects $\pi({\cal S})$, which is generated by the basis vectors $\{\partial_a\}$, at a right angle. A Riemannian metric allows one to raise and lower indices of tensor fields, i.e., to map an $(n,m)$ tensor field to an $(n+1,m-1)$ or $(n-1,m+1)$ tensor field, respectively, thereby preserving its total rank, $n+m$. Henceforth I will not distinguish between tensor fields that differ only by raising or lowering of indices via the BKM metric; I will denote such fields by the same symbol and characterize them only by their total rank. Associated with the metric is a covariant derivative, $\nabla$. Given some vector field $V$, the covariant derivative along $V$, $\nabla_V$, maps an arbitrary rank-$r$ tensor field $A$ to another rank-$r$ tensor field, $\nabla_V A$. This map (i) obeys the sum rule, $\nabla_V (A+B)=(\nabla_V A) + (\nabla_V B)$; (ii) obeys the Leibniz rule for tensor products, $\nabla_V (A\otimes B)=(\nabla_V A)\otimes B + A\otimes (\nabla_V B)$; (iii) commutes with tensor contraction; (iv) is linear in the vector field, $\nabla_{fU+gV} A=f\nabla_U A + g\nabla_V A$, for arbitrary scalar functions $f,g$ and vector fields $U,V$. For this reason there exists a rank-$(r+1)$ tensor field, denoted $\nabla A$ and called the gradient of $A$, from which $\nabla_V A$ can be obtained via contraction with $V$; and (v) when applied to a scalar field $\phi$, the map coincides with the ordinary directional derivative, $\nabla_V\phi=d\phi(V)$. So in this special case the gradient coincides with the ordinary differential, $\nabla\phi=d\phi$. That the covariant derivative stems from the BKM metric is reflected in the fact that the gradient of the metric tensor vanishes, $$\nabla C = 0 .$$ In particular, there is no torsion, $$\nabla_U V - \nabla_V U = [U,V] \ \ \forall \ U,V ,$$ the bracket $[U,V]$ being the Lie bracket of the two vector fields $U$ and $V$ [@schutz:book]. The absence of torsion implies that the gradient of a one-form field $\alpha$ can be written as $$\nabla\alpha = \textstyle\frac{1}{2} [d\alpha + \pounds_{C^{-1}(\alpha)} C] , \label{gradient_oneform}$$ where $d$ denotes the exterior derivative and $\pounds$ the Lie derivative. The first term inside the square bracket is antisymmetric, whereas the second term is symmetric. Whenever $\alpha$ is itself a gradient, $\alpha=d\phi=\nabla\phi$, its exterior derivative vanishes, and hence it is $$\nabla\nabla\phi = \textstyle\frac{1}{2} \pounds_{C^{-1}(\nabla\phi)} C .$$ One scalar function that will play a special role in my subsequent argument is the modified entropy [@PhysRevA.84.012101] $$S_\sigma[\mu] := S[\sigma] - S(\mu\\|\sigma) \label{modifiedentropy}$$ with reference macrostate $\sigma\in\pi({\cal S})$. This modified entropy characterizes the closeness of $\mu$ to the reference macrostate; it reduces to the ordinary von Neumann entropy when the reference macrostate equals the totally mixed state. Upon infinitesimal variation of a macrostate (at fixed reference macrostate) the modified entropy changes in a manner similar to the ordinary entropy, Eq. (\[entropydifferential\]), only with $\lambda^a$ replaced by $(\lambda^a-\lambda^a_\sigma)$, $$dS_\sigma = (\lambda^a-\lambda^a_\sigma) dg_a ,$$ where the $\{\lambda^a_\sigma\}$ pertain to the reference macrostate $\sigma$. Taking the gradient of this differential yields, with the help of Eqs. (\[gradients\]) and (\[metrictensor\]), $$\nabla\nabla S_\sigma = -C + (\lambda^a-\lambda^a_\sigma) \nabla\nabla g_a . \label{Hessianandmetric}$$ So when evaluated at the reference macrostate, where $\lambda^a=\lambda^a_\sigma$, this covariant Hessian equals (up to a sign) the metric tensor, $$\nabla\nabla S_\sigma = -C \ \mbox{at}\ \sigma . \label{metricasgradient}$$ When macrostates are constrained to the hyperplane $\Sigma:=\{\mu\in\pi({\cal S})\|\langle\ln\sigma\rangle_\mu=\langle\ln\sigma\rangle_\sigma\}$, modified and ordinary entropies coincide, $S_\sigma\|_\Sigma=S\|_\Sigma$. Then Eq. (\[metricasgradient\]) carries over to the ordinary entropy, $$\nabla\nabla S\|_\Sigma = - C\|_\Sigma \ \mbox{at} \ \sigma . \label{metricasgradient_constrained}$$ As an example, $\sigma$ might be a canonical equilibrium state and hence $\ln\sigma$ proportional to the Hamiltonian. Then $\Sigma$ constitutes a hyperplane of constant energy. It comprises all macrostates that have the same energy as $\sigma$, including $\sigma$ itself. Whenever energy is conserved, evolution of the macrostate is constrained to such a hyperplane; so energy-conserving flows have $C\|_\Sigma$ as their relevant metric. At the equilibrium state $\sigma$ this constrained metric is given by the (negative) Hessian of ordinary entropy. Generic Markovian dynamics {#generic} ========================== Disparate time scales --------------------- In this paper I focus on the dynamics of an isolated quantum system with time-independent Hamiltonian $H$. (In principle, the description of an open system can be incorporated into this framework by enlarging it to include its environment.) On the microscopic level the dynamics of such a system is governed by the Liouville-von Neumann equation $$\dot{\rho}(t)=-i{\cal L}\rho(t) , \label{Liouville_vonNeumann}$$ where $\rho$ denotes the system’s microstate, and the [Liouvillian]{} ${\cal L}:=[H,\cdot]$ is a shorthand for the commutator with $H$. For simplicity, I set $\hbar=1$. On the macroscopic level one seeks to describe the dynamics of only certain selected expectation values $g_a(t):=\langle G_a\rangle_{\rho(t)}$ deemed “relevant”. Provided that initially, at $t=0$, these relevant expectation values suffice to determine the system’s microstate, i.e., $\rho(0)$ carries no information other than about $\{g_a(0)\}$ and hence has the Gibbs form $$\rho(0) \propto \exp(-\lambda^a(0) G_a) ,$$ their dynamics at $t\geq 0$ is governed by the Robertson equation [@PhysRev.144.151; @rau:physrep] $$\dot{g}_a(t) = \dot{g}_a^{(l)}(t) + \dot{g}_a^{(m)}(t) \label{robertson}$$ with the local term $$\dot{g}_a^{(l)}(t) = \langle i{\cal L} G_a\rangle_{\pi(\rho(t))}$$ and the memory term $$\dot{g}_a^{(m)}(t) = - \int_0^t dt'\,\langle {\cal L}{\cal Q}(t'){\cal T}(t',t){\cal Q}(t){\cal L} G_a \rangle_{\pi(\rho(t'))} . \label{memoryterm}$$ Here $\pi(\rho(t))$ is the [macrostate]{} at time $t$ as defined in Sec. \[manifold\]. The objects ${\cal Q}$ and ${\cal T}$ are, like the Liouvillian, superoperators acting on the space of observables. The former is a projector, ${\cal Q}^2={\cal Q}$, which projects any observable onto its “irrelevant” component; it is the complement of the Robertson projector defined in Sec. \[coarse\]. Like the Robertson projector, it may vary with the macrostate and thus may carry an implicit time dependence. The superoperator ${\cal T}$ effects the time evolution of irrelevant degrees of freedom, $$(\partial/\partial t') {\cal T}(t',t) = -i {\cal Q}(t') {\cal L} {\cal Q}(t') {\cal T}(t',t) ,$$ with initial condition ${\cal T}(t,t)={\cal I}$. All terms on the right-hand side of the Robertson equation depend on the macrostate and hence on relevant expectation values only; so the system of equations of motion for the $\{g_a(t)\}$ is indeed closed. Irrelevant degrees of freedom have been eliminated completely from the description of the macroscopic dynamics. The price to pay for this elimination is that in contrast to the microscopic Liouville-von Neumann equation, the Robertson equation can be both nonlocal in time and nonlinear. The former means that the change of relevant expectation values at any given time may depend not just on their current values but on their entire history since $t=0$, i.e., that the macroscopic dynamics has a [“memory”]{}. The latter – nonlinearity – may arise whenever the coarse graining operation $\pi$ is not linear. A simple example for such a nonlinear coarse graining was given in Table \[lod\_examples\]. Indeed, many well-known transport equations such as the Boltzmann or Navier-Stokes equations, which can be derived within the above framework or its classical counterpart, are nonlinear. The Robertson equation becomes Markovian, i.e., local in time, if and only if the physical system exhibits a clear separation of time scales, and it is the slow degrees of freedom which are chosen as the relevant ones. In this case the “memory time” $\tau_m$ – the time scale on which the integrand in Eq. (\[memoryterm\]) falls off towards the past – will be short compared to the time scale $\tau_r$ on which the relevant expectation values evolve. One may then replace $$\pi(\rho(t')) \to \pi(\rho(t)) \ ,\ {\cal Q}(t') \to {\cal Q}(t)$$ in both the Robertson equation and the differential equation for ${\cal T}$ (“Markovian approximation”). Moreover, provided there is a genuine gap between the two time scales, in the sense that there exists an intermediate scale $T$ such that $\tau_m\ll T \ll \tau_r$, and with the substitution $(t-t')\to\tau$ one may expand the integration range for $\tau$ from $[0,t]$ to $[0,T]$ and replace $$\int_0^t dt'\, {\cal T}(t',t) \ \to\ {\cal I}^{(+)} := \int_0^T d\tau\, \exp( i\tau{\cal Q}{\cal L}{\cal Q}) , \label{def_Iplus}$$ where for simplicity I omitted the dependence on $t$. Geometrically, the collected relevant expectation values at any given time can be represented as a point, and their time evolution as a curve, in the manifold of macrostates. The curve results from projecting the trajectory of the microstate in full state space onto the lower-dimensional submanifold of macrostates. In case the Robertson equation is Markovian (which I shall assume from now on), its general solution defines in the manifold of macrostates a congruence of curves. This congruence in turn gives rise to a vector field on the manifold of macrostates, $$V := \dot{g}_a \partial^a .$$ The correspondence being one-to-one, the Markovian dynamics may be characterized completely by the vector field $V$. In line with the split in Eq. (\[robertson\]), the vector field can be broken down into contributions from the local term and the memory term, $V=V^{(l)}+V^{(m)}$. Effective non-dissipative dynamics {#effectivedynamics} ---------------------------------- The effective non-dissipative dynamics of the relevant expectation values is described by the local term and the antisymmetric part of the memory term. As for the local term, one can use the general formula $$[X,\rho] = \int_0^1 d\nu \, \rho^{\nu} [X,\ln\rho] \rho^{1-\nu}$$ and define on the manifold of macrostates the antisymmetric $(2,0)$ tensor field $$K := \langle \textstyle\frac{1}{i} [G_a,G_b] ; (H-\langle H\rangle_{\pi(\rho)}) \rangle_{\pi(\rho)} \partial^b \otimes \partial^a \label{tensorK}$$ to cast it into the form $$V^{(l)} = K (dS,\cdot) .$$ As an immediate consequence of the antisymmetry of $K$, the local term preserves entropy, $$K(dS,dS) = 0 , \label{KpreservesEntropy}$$ and is thus indeed non-dissipative. As for the memory term, using the relation $$\langle {\cal L} X \rangle_{\pi(\rho)} = - \lambda^b \langle {\cal L} G_b ; X \rangle_{\pi(\rho)} \label{firsttermascorrelation}$$ for arbitrary (not necessarily relevant) $X$, the hermiticity of ${\cal Q}$ with respect to the canonical correlation function, as well as the fact that the memory term must be real, and defining the $(2,0)$ tensor field $$M:= \langle {\cal I}^{(+)} {\cal Q}{\cal L}G_a ; {\cal Q}{\cal L} G_b \rangle_{\pi(\rho)} \partial^b \otimes \partial^a ,$$ it can be cast into the form $$V^{(m)} = M(dS,\cdot) .$$ The tensor $M$ comprises an antisymmetric and a symmetric part whose respective components are given by $$M_{ab}^{(\pm)} := \textstyle\frac{1}{2} (M_{ab} \pm M_{ba}) .$$ The antisymmetric part conserves entropy, $$M^{(-)}(dS,dS) = 0 , \label{M-preservesEntropy}$$ and thus co-determines, together with the local term, the non-dissipative dynamics. Often neglected, this antisymmetric part of the memory term may contain interesting physics such as geometric or other effective forces [@PhysRevE.56.R1295], and it may play an important role in Hamiltonian renormalization [@PhysRevE.55.5147]. The most general Markovian dynamics is then described by the tensor $$T := (K+M^{(-)}) + M^{(+)} ,$$ where the first two, antisymmetric terms drive the non-dissipative dynamics, whereas the last, symmetric term drives the dissipative dynamics. With this tensor the complete equation of motion acquires the compact form $$V=T(dS,\cdot) . \label{generaldynamics}$$ It is possible to describe the non-dissipative dynamics by an effective Hamiltonian $H_{\rm eff}$ if and only if $$(K + M^{(-)})(dS,\cdot) = K_{\rm eff}(dS,\cdot) , \label{EffectiveHamiltonian}$$ where $K_{\rm eff}$ is defined as in Eq. (\[tensorK\]) but with $H$ replaced by $H_{\rm eff}$. In this case the non-dissipative time evolution of macrostates is unitary. Since unitary transformations leave relative entropies invariant and hence, thanks to Eq. (\[relentropy\_nearby\]), also the BKM metric, it is $$\pounds_{K_{\rm eff} (dS,\cdot)} C = 0 . \label{unitaritycondition}$$ Provided $H_{\rm eff}$ is itself a relevant observable, the non-dissipative dynamics can be written in the alternative form $$K_{\rm eff} (dS,\cdot) = L (dU,\cdot) ,$$ where $U:=\langle H_{\rm eff}\rangle_{\pi(\rho)}$ is the (effective) internal energy, and $L$ denotes another antisymmetric $(2,0)$ tensor field, $$L := \langle \textstyle\frac{1}{i}[G_a,G_b] \rangle_{\pi(\rho)} \partial^b \otimes \partial^a . \label{poisson}$$ In this formulation the conservation of entropy is reflected in the tensor property $$L(\cdot,dS)=0 .$$ In principle, the effective Hamilton operator may be influenced by the macrostate, $H_{\rm eff}\equiv H_{\rm eff}[\pi(\rho)]$, and thus, through the latter, depend on time. If, however, such an explicit time dependence is absent or may be neglected, the antisymmetry of $L$ ensures the conservation of internal energy, $$L (dU,dU) = 0 . \label{energyconservation}$$ In case the relevant observables form a Lie algebra, $(1/i)[{\cal G},{\cal G}]\subset{\cal G}$, the manifold of macrostates and their non-dissipative dynamics exhibit further structure. Such a Lie algebra property holds for many important choices of the level of description: for instance, when the relevant observables comprise (i) all constants of the motion; (ii) all observables pertaining to one or several subsystems of a composite system; or (iii) all block diagonal observables of the form $\sum_i P_i A P_i$, where $\{P_i\}$ is some set of mutually orthogonal projectors. In all these examples the commutator (times $1/i$) of two relevant observables is again a relevant observable. Then the bracket $$\{f,g\}:= L(df,dg)$$ of two functions $f$ and $g$, defined with the help of the tensor field $L$, possesses all properties of a Poisson bracket; it satisfies antisymmetry, linearity, Leibniz rule, and the Jacobi identity. Thus the manifold of macrostates is endowed with a Poisson structure [@vaisman:book]. According to the splitting theorem for Poisson manifolds [@weinstein1983] the manifold of macrostates can then be foliated into symplectic leaves, each with constant entropy. On every leaf one can define a symplectic two-form, i.e., a two-form which is antisymmetric, non-degenerate, and closed. Also on every leaf, the vector field associated with the non-dissipative dynamics, $L (dU,\cdot)$, becomes a Hamiltonian vector field, the pertinent Hamilton function being the internal energy $U$. One recovers thus the familiar structure of classical Hamiltonian mechanics [@arnold:book]. Dissipation ----------- Dissipation is described by the symmetric part of the memory term. This symmetric part is positive semidefinite, $$M^{(+)}\geq 0 , \label{positivity}$$ which can be understood as follows. One assumes that on short time scales smaller than $T$ the dynamics of the irrelevant degrees of freedom is (i) time translation invariant; and in particular, (ii) unaffected by the (slow) variation of the macrostate. Then it is $$\langle {\cal Q}{\cal L}G_a ; {\cal I}^{(+)} {\cal Q}{\cal L} G_b \rangle = \langle {\cal I}^{(-)} {\cal Q}{\cal L}G_a ; {\cal Q}{\cal L} G_b \rangle ,$$ where ${\cal I}^{(-)}$ is defined as in Eq. (\[def\_Iplus\]) but with a minus sign in the exponent. Considering moreover that the memory term must be real, and hence that the canonical correlation function (which is a scalar product) featuring in the definition of $M$ must be symmetric, the components of the symmetric and antisymmetric parts of $M$ are given by $$M_{ab}^{(\pm)} = \textstyle\frac{1}{2} \langle ({\cal I}^{(+)} \pm {\cal I}^{(-)}) {\cal Q}{\cal L}G_a ; {\cal Q}{\cal L} G_b \rangle .$$ For the symmetric part one can then write (invoking once more the time translation invariance of the irrelevant dynamics on short time scales) $$\begin{aligned} M_{ab}^{(+)} &=& \frac{1}{2} \int_{-T}^T d\tau\, \langle \exp(i \tau {\cal Q}{\cal L}{\cal Q}) {\cal Q}{\cal L}G_a ; {\cal Q}{\cal L} G_b \rangle \nonumber \\ &=& \frac{1}{2T} \int_0^T ds \int_{-T}^T d\tau\, \nonumber \\ && \times \langle \exp[i (\tau+s){\cal Q}{\cal L}{\cal Q}] {\cal Q}{\cal L}G_a ; \exp(i s{\cal Q}{\cal L}{\cal Q}) {\cal Q}{\cal L} G_b \rangle \nonumber \\ &=& \frac{1}{2T} \int_0^T ds \int_{-T+s}^{T+s} d\tau'\, \nonumber \\ && \times \langle \exp(i \tau'{\cal Q}{\cal L}{\cal Q}) {\cal Q}{\cal L}G_a ; \exp(i s{\cal Q}{\cal L}{\cal Q}) {\cal Q}{\cal L} G_b \rangle . \nonumber \\ && \end{aligned}$$ To the second integral only $\tau'\approx s \in [0,T]$ contribute significantly, and hence one may reduce its integration range $[-T+s,T+s] \to [0,T]$. This finally yields $$M_{ab}^{(+)} = ({1}/{2T}) \langle {\cal I}^{(+)} {\cal Q}{\cal L}G_a ; {\cal I}^{(+)} {\cal Q}{\cal L} G_b \rangle .$$ Since the canonical correlation function is a positive definite scalar product, this component matrix, and hence $M^{(+)}$ itself, must indeed be positive semidefinite. An immediate consequence of the above positivity is that the symmetric part of the memory term may only lead to an increase, but never to a decrease, of entropy, $$M^{(+)}(dS,dS) \geq 0 .$$ This embodies the $H$-theorem, originally formulated by Boltzmann for the dynamics of dilute gases [@boltzmann:h-theorem] but in fact valid for arbitrary Markovian processes. It marks the gradual approach of the macrostate towards equilibrium. One particularly simple type of dissipative dynamics is steepest descent towards the equilibrium state. Under this dynamics trajectories on the manifold of macrostates (or in case there are conservation laws: on the allowed submanifold) simply follow the entropy gradient [@rau:relaxation]. The only contribution to this type of dynamics stems from the symmetric part of the memory term, which has the simple form $$M^{(+)} \propto C^{-1} .$$ This is a rather natural ansatz: The metric $C$ measures the distance between two macrostates in the sense of their statistical distinguishability. The symmetric part of the memory term (or its inverse, respectively), being symmetric and positive semidefinite, is a metric, too; it measures the “dynamical” distance between two macrostates as mediated by interactions with the irrelevant degrees of freedom. In the absence of any specific information about the dynamics of the irrelevant degrees of freedom, in particular about any preferred direction on the manifold of macrostates, these two metrics are taken to be equal, up to some multiplicative constant. A direct consequence of this ansatz is that the gradient of $M^{(+)}$ vanishes everywhere, $\nabla M^{(+)} = 0$. This is not the case for other, more general forms of Markovian dynamics. The magnitude of this gradient can then be taken as a local, coordinate-independent measure for the deviation from steepest descent. When the overall dynamics drives the macrostate towards an equilibrium macrostate $\sigma\in\pi({\cal S})$ then its logarithm, $\ln\sigma$, which is a relevant observable, must be a constant of the motion. Hence in their approach towards $\sigma$, macrostates are constrained to the hyperplane $\Sigma$ defined in Sec. \[riemannian\]. Provided the equilibrium macrostate is a canonical state with the same effective Hamiltonian as that governing the non-dissipative dynamics, $\sigma\propto\exp(-\beta_\sigma H_{\rm eff})$, this hyperplane corresponds to fixed internal energy, $U=\mbox{const}$. The latter is conserved by the non-dissipative dynamics, Eq. (\[energyconservation\]), and so in order to be conserved overall, it must be conserved by the dissipative dynamics, too. This conservation of energy is implemented mathematically by imposing $$M^{(+)}(\cdot,dU)=0 .$$ Near equilibrium ---------------- Let $\sigma\in\pi({\cal S})$ be an equilibrium macrostate and $\Sigma\subset\pi({\cal S})$ the associated hyperplane, defined in Sec. \[riemannian\], to which macrostates are constrained as they approach $\sigma$. In the following, all mathematical objects —macrostates, functions, vector and tensor fields— and their relationships are meant to be constrained to this hyperplane, even if for simplicity I will omit the explicit notation “$\|_\Sigma$”. To begin with, taking the covariant derivative on both sides of the general equation of motion, Eq. (\[generaldynamics\]), and exploiting the relationship between the Hessian of the entropy and the metric tensor, Eq. (\[metricasgradient\_constrained\]), as well as $dS=0$ at equilibrium, yields an equation for $T$ at equilibrium, $$T = - \nabla V \ \mbox{at}\ \sigma ;$$ i.e., at equilibrium, the tensor $T$ which governs the Markovian dynamics can be gleaned from the observed vector field $V$ by taking (minus) the gradient. To first-order approximation, this relationship can be extended to the vicinity of equilibrium, $$T \approx - \nabla V ; \label{TnablaV}$$ which in turn, by the general equation of motion, implies $$V \approx - (\nabla V) (dS,\cdot) . \label{consistencycondition}$$ The latter equation imposes a consistency condition on the vector field $V$ near equilibrium. It goes beyond the trivial requirement that $V=0$ at $\sigma$. By symmetrizing both sides of Eq. (\[TnablaV\]) and exploiting the positivity of the symmetric part of $T$, Eq. (\[positivity\]), one finds that the symmetric part of the gradient $\nabla V$ must be negative semidefinite. This symmetric part is denoted by $[\nabla V]_+$ and may be called, in an analogy with fluid dynamics, the rate of strain tensor. In conjunction with Eq. (\[gradient\_oneform\]), its negativity leads to the inequality $$\pounds_V C = 2\,[\nabla V]_+ \leq 0 , \label{localconvergence}$$ again valid in the vicinity of equilibrium. Near equilibrium, therefore, macrostates converge (in terms of the BKM metric) not just towards equilibrium but also towards each other; as they evolve, their mutual distances, and hence their statistical distinguishability, decrease monotonically. This inequality may be viewed as a special case of Lindblad’s theorem [@lindblad:monotonicity], here applied to the linearized dynamics near equilibrium. When the dynamics is entirely non-dissipative, $M^{(+)}=0$, the above reasoning implies $\pounds_V C=0$. This is in keeping with the unitarity condition, Eq. (\[unitaritycondition\]). As the BKM metric is the infinitesimal version of relative entropy, Eq. (\[relentropy\_nearby\]), its conservation suggests that, moreover, arbitrary relative entropies are preserved. If this (unproven) conjecture were true then the conservation of the BKM metric, $\pounds_V C=0$, would in fact mandate, rather than merely allow, that the non-dissipative dynamics is unitary [@molnar1; @molnar2]. Reconstruction ============== The general framework laid out above offers a systematic route from the microscopic to the macroscopic realm. Starting from the microscopic Hamiltonian and the Liouville-von Neumann equation, Eq. (\[Liouville\_vonNeumann\]), it allows one, at least in principle, to calculate the tensor field $T$ that governs the Markovian macroscopic dynamics; and from there, via the general equation of motion, the vector field $V$, and hence the time evolution of macroscopic observables. Reconstruction starts from the other end: Having observed some macroscopic Markovian process experimentally, one aims to infer the dynamical law. Specifically, one endeavours to reconstruct the pertinent tensor field $T$. The generic experimental setting can be characterized as follows. On some coarse-grained level of description, one observes a hitherto unknown process whose only known feature is that it is reproducible and hence Markovian. The dynamical law governing this macroscopic process – here: the tensor field $T$ – is not known and yet to be inferred from the data. It is impossible to infer the dynamics from a single trajectory alone; rather, one has to study an entire family of trajectories pertaining to differently prepared test systems. More specifically, in order to collect sufficient data for such an inference, one must (i) observe multiple copies of the system of interest, or provided the system is large enough so that disturbances due to measurement may be neglected, observe one system undergo the same process several times; (ii) vary the initial macrostates of the different copies or sequential runs, respectively; (iii) in each run, measure the macrostate at different (ideally, closely spaced) instances of its evolution; and (iv) record the time between subsequent measurements. In combination, these data allow one to determine the vector field $V$, as illustrated schematically in Fig. \[figure1\](a). ![Schematic illustration of the basic reconstruction idea. The figures show a section of the hyperplane $\Sigma$ of macrostates. (a) Experimental data are given in the form of timed snapshots of the macrostate at different instances of the Markovian evolution. Crosses and dots represent data from two different runs of the experiment with varying initial conditions; in practice, there will be many more such runs and data points. In combination, these data allow one to construct the vector field $V$. (b) When time information is removed, it is no longer possible to construct the vector field $V$. (c) However, as long as the dynamics is genuinely irreversible, $\dot{S}>0$, at least the temporal order of the snapshots is easily established, and one may use entropy as a proxy for time. Here concentric circles around the equilibrium state $\sigma$ indicate hypersurfaces of constant entropy. With this proxy one can construct a vector field $W$, related to the true $V$ by $V=\dot{S} W$. The entropy production rate, $\dot{S}$, can in turn be estimated from the data, up to a multiplicative constant, if one assumes the existence of an effective Hamiltonian for the non-dissipative part of the dynamics (see text). []{data-label="figure1"}](figure1){width="4.8cm"} Having thus determined experimentally the vector field $V$, one proceeds to infer the macroscopic dynamical law, embodied in the tensor field $T$. Near equilibrium, this is a straightforward exercise: According to Eq. (\[TnablaV\]), $T$ is but the negative gradient of $V$. Calculating the gradient requires knowledge of the (generally non-Euclidean) geometry of the manifold of macrostates, which depends on the level of description but is otherwise independent of the experimental data. Further away from equilbrium, there may be corrections to Eq. (\[TnablaV\]). One will then end up with some system of first-order partial differential equations for the components of $T$. The difficulty of solving these differential equations will vary depending on the system at hand. For simplicity, I stick here to the vicinity of equilibrium and focus instead on another issue that may complicate the reconstruction of $T$. I consider the situation where one is given data points on the manifold of macrostates but without any information about time (Fig. \[figure1\](b)). On a large scale, this might be the case for, say, geological or astronomical data originating from different epochs of terrestrial or cosmic evolution, respectively, that are as yet undated but presumed to be connected via some hitherto unknown Markovian process (say, some yet-to-be-discovered form of stellar evolution). Or at the opposite extreme, it might apply to processes on a very small scale (say, in particle or molecular physics) that leave a visible trace but are too fast or delicate to be tracked with a clock. More exotically, such timeless data might be collected in a remote lab without access to an external reference clock and with all internal clocks broken, so that the very notion of time has to be reconstructed from scratch. In all these cases it is clearly impossible to construct the vector field $V$, and hence to infer the tensor $T$, in the straightforward manner outlined above. Nevertheless, with the help of assumptions to be discussed below, a reconstruction of the Markovian dynamics is feasible even under such adverse circumstances. One important prerequisite is that the dynamics is genuinely irreversible, $\dot{S}>0$. Then at least the temporal order of the snapshots is easily established; and the entropy being a strictly monotonic function of time, it may in a first step serve as a proxy for time. With the help of this proxy one can construct a vector field $W$, related to the true $V$ by $V=\dot{S} W$ (Fig. \[figure1\](c)). The reconstruction task thus reduces to the problem of finding the local entropy production rate, $\dot{S}$. In the following, this unknown entropy production rate shall be denoted by a separate letter, $\eta$. The true, yet to be determined vector field $V$ must satisfy a consistency condition, Eq. (\[consistencycondition\]). Replacing $V$ by $\eta W$ in this condition and exploiting $dS(W)=1$ leads to a first constraint on $\eta$, $$d\eta\cdot dS \approx -2\eta ,$$ where the scalar product $d\eta\cdot dS$ is short for $C^{-1}(d\eta,dS)$. I presume that the non-dissipative part of the dynamics is governed by some effective Hamiltonian, and that at least in the vicinity of equilibrium, this effective Hamiltonian does not vary with the macrostate. The macroscopic non-dissipative dynamics thus shares with the microscopic dynamics a common structure, namely, unitary evolution with a time-independent Hamiltonian. Such structural invariance is not guaranteed a priori; rather, it constitutes an extra assumption which, however, is justified for many systems. Mathematically, by Eqs. (\[EffectiveHamiltonian\]) and (\[TnablaV\]), and taking into account the change of sign when going from $\partial^a$ to $d\lambda^a$ via the metric, Eq. (\[metricandduals\]), this assumption means that there must exist an $H_{\rm eff}$ such that $$[\nabla V]_-(dS,\cdot) \approx \langle i{\cal L}_{\rm eff}G_a\rangle_{\pi(\rho)} d\lambda^a .$$ Here $[\nabla V]_-$ denotes the antisymmetrized gradient (“curl”) of $V$. Again replacing $V$ by $\eta W$, exploiting $dS(W)=1$, and using the previous constraint on $\eta$ leads then to the condition $$\textstyle\frac{1}{2} d\eta \approx \eta\omega - \langle i{\cal L}_{\rm eff}G_a\rangle_{\pi(\rho)} d\lambda^a . \label{etacondition}$$ On the right-hand side, the one-form $$\omega:= [\nabla W]_-(dS,\cdot) - W$$ is determined by the observed proxy field $W$. The effective Hamiltonian (up to an additive and multiplicative constant), and hence the effective Liouvillian and the expectation values $\langle i{\cal L}_{\rm eff}G_a\rangle$ (up to a multiplicative constant), can be inferred from the observed equilibrium state, $\sigma\propto\exp(-\beta_\sigma H_{\rm eff})$, or from the observed hyperplane $\Sigma$, respectively. Mirroring the indeterminacy of the effective Liouvillian, the entropy production rate $\eta$, too, is only determined up to a multiplicative constant. When the Liouvillian is rescaled by a constant factor $c$, ${\cal L}_{\rm eff}\to c {\cal L}_{\rm eff}$, then so is the entropy production rate, $\eta\to c \eta$. Up to this undetermined multiplicative constant, the above condition, Eq. (\[etacondition\]), specifies $\eta$ uniquely. This enables one to convert the proxy field $W$ to the true vector field $V$. From there, the reconstruction of the dynamics can proceed as before. Example ======= In this section, I apply the general framework to a simple example. In fact, the example is so simple that its dynamics might just as well be analyzed without the whole apparatus of differential geometry introduced above. However, it serves to illustrate the internal consistency of the approach, and has the advantage of being solvable analytically. To be specific, I consider an exchangeable assembly of qubits, possibly interacting with each other and with some unknown environment. (A physical realization might be a para- or ferromagnet.) The three Pauli operators $\sigma_x$, $\sigma_y$, and $\sigma_z$ constitute the relevant observables; the associated manifold of macrostates is thus the Bloch sphere. Measurements on samples taken from the assembly yield the expectation values of the relevant observables, $x:=\langle\sigma_x\rangle$, $y:=\langle\sigma_y\rangle$, and $z:=\langle\sigma_z\rangle$. Driven by some Markovian dynamics whose form is yet to be inferred, these expectation values change, tracing out orbits in the Bloch sphere. The geometric shape of these orbits is given but —for lack of time information— their parametrization is not. For the sake of concreteness, I presume that whenever $z=0$ initially, it is $z=0$ on the entire orbit; so orbits with this initial condition are constrained to the hyperplane $\Sigma=\{\mu\in\pi({\cal S})\|z=0\}$. Within this hyperplane, orbits are spiralling towards the equilibrium state, $\sigma$, which I take to be the totally mixed state, $\sigma=I/2$ (Fig. \[spiral\]). I assume that near equilibrium the orbits have the shape of a logarithmic spiral, $$\phi=\phi_0 + \ln r ,$$ where $r,\phi$ are polar coordinates in $\Sigma$, $$x=r\cos\phi \ ,\ y=r\sin\phi .$$ Given only this information, is it possible to reconstruct the dynamical law near equilibrium? ![Two-dimensional section ($z=0$) of the Bloch sphere. Near equilibrium, $\sigma=I/2$, orbits have the form of a logarithmic spiral. []{data-label="spiral"}](spiral){width="8cm"} Near equilibrium the entropy function is approximately quadratic, $$S\approx \ln 2 - r^2 / 2 .$$ With entropy as a proxy for time, the orbit can be given a parametric representation, $$r(S)=\sqrt{2\ln 2-2S} \ ,\ \phi(S)=\phi_0 + \textstyle\frac{1}{2}\ln(2\ln 2-2S) ,$$ which gives rise to the proxy field $$W= - \frac{1}{r}\frac{\partial}{\partial r} - \frac{1}{r^2}\frac{\partial}{\partial\phi} .$$ Mapping vectors to one-forms with the help of the metric tensor, given near equilibrium by $$C\approx dr\otimes dr + r^2 d\phi\otimes d\phi ,$$ yields the one-form $$\omega=\frac{1}{r}dr+d\phi .$$ From the orientation of the hyperplane $\Sigma$ one readily concludes that $H_{\rm eff}\propto\sigma_z$. Exploiting moreover that near equilibrium it is $\lambda^x\approx -x$ (and likewise for $\lambda^y$), one finds $$\langle i{\cal L}_{\rm eff}G_a\rangle_{\pi(\rho)} d\lambda^a \approx c\,r^2 d\phi ,$$ where $c$ is some undetermined multiplicative constant. Inserting these results into the equation for the entropy production rate, Eq. (\[etacondition\]), yields $$\frac{1}{2}d\eta \approx \frac{\eta}{r}dr + (\eta-c\,r^2)d\phi ,$$ which has the unique solution $$\eta=c\,r^2 .$$ This entropy production rate allows one to convert the proxy field $W$ to the “true” vector field $V$, $$V=-c \left[r\frac{\partial}{\partial r} + \frac{\partial}{\partial\phi}\right] .$$ Finally, by Eq. (\[TnablaV\]), this yields the tensor field $T$ governing the Markovian dynamics, with the antisymmetric, non-dissipative part $$T^{(-)}\approx \frac{c}{r}\left[ \frac{\partial}{\partial r}\otimes \frac{\partial}{\partial\phi} - \frac{\partial}{\partial\phi}\otimes \frac{\partial}{\partial r} \right]$$ and the symmetric, dissipative part $$T^{(+)}\approx c\,C^{-1} .$$ Discussion ========== In the preceding sections I showed how it is possible, in principle, to infer a Markovian equation of motion — with both its non-dissipative and dissipative parts — from experimental data even when the latter lack information about time. To achieve this inference, I exploited the Riemannian geometry of the manifold of macrostates, as well as the structure of generic Markovian dynamics. In addition, I made a number of assumptions: I presumed that (i) energy is conserved; (ii) the experimental data have been taken near equilibrium (limiting thus the validity of the inferred equation of motion to the vicinity of equilibrium, too); (iii) the dynamics is genuinely irreversible, $\dot{S}>0$; (iv) the equilibrium is canonical; and (v) the effective, time-independent Hamiltonian featuring in this canonical equilibrium state also governs the effective non-dissipative dynamics near equilibrium. The last assumption allows one to infer immediately the non-dissipative part of the Markovian dynamics (up to a multiplicative constant); one simply identifies the effective Hamiltonian with the logarithm of the equilibrium state or, equivalently, with the observable that is conserved in the hyperplane $\Sigma$. There remains then the task of inferring the dissipative part of the dynamics. This requires more effort but is feasible, too, with the tools provided above. The reconstruction scheme fixes time intervals only up to a multiplicative constant, and hence time itself only up to an affine transformation. This freedom ensures the compatibility of the reconstruction scheme with special relativity. Two observers, moving relative to each other, may see the same unparametrized orbits traced out by a Markovian process in the manifold of macrostates. In the example discussed in Sec. \[example\], one observer might be in the rest frame of the experiment, while the other observer moves relative to this frame in the $z$ direction; both observers see the same logarithmic spiral depicted in Fig. \[spiral\]. Their respective reconstructions of time are allowed to differ by an affine transformation. This provides, in particular, for the possibility that they differ by a Lorentz transformation, as demanded by relativity. The time reconstructed via the above scheme is a [macroscopic]{} time, for it is inferred from experimental data on some coarse-grained level of description. If the same process is observed on two different levels of description, and if on both levels the process is Markovian, then the above procedure can be applied to both sets of data, yielding respective macroscopic times. One may envision such a situation when one observes, say, a fluid on either the Boltzmann or the Navier-Stokes level of description; or when one observes a spin system on different coarse-grained length scales, corresponding to varying block spin sizes. As long as the assumptions spelt out above hold on all these levels of description, the various macroscopic times agree (up to affine transformations). The inferred time is thus invariant under a change of macroscopic scale, provided that on all scales the dynamics falls within the basic structure of generic Markovian dynamics; and provided that, moreover, on all scales there exists an effective Hamiltonian for its non-dissipative part. In this sense, the scale invariance of macroscopic time presupposes renormalizability [@PhysRevE.79.021124]. The results in the present paper show that, in principle, any Markovian process can serve as a clock. Absent an external reference clock, one may take an arbitrary, genuinely irreversible Markovian process, apply the above reconstruction scheme, and with the help of the inferred equation of motion, mark points on the orbits such that a segment between two successive points corresponds to some fixed time interval. When the Markovian process is repeated, the system evolves along one of the orbits; and whenever its macrostate reaches one of the designated points, the clock “ticks.” In contrast to everyday clocks, which are based on reversible, periodic motion, such a clock would be based on the irreversibility of the process involved — similar to, e.g., radiocarbon dating. Conceptually, one may wonder why a reconstruction of time is necessary at all, and why one does not simply stick to entropy as a proxy for time. In fact, any strictly monotonic function of time — like entropy in a genuinely irreversible process — serves the purpose of ordering events; and if such an alternative parameter were adopted universally, it would still be possible to keep appointments. What distinguishes physical time from all other conceivable parameters is that when expressed as a function of time, equations of motion become particularly [simple]{}. As Henri Poincaré put it succinctly [@poincare:time], > Time should be so defined that the equations of mechanics may be as simple as possible. In other words, there is not one way of measuring time more true than another; that which is generally adopted is only more convenient. Determining physical time is thus tantamount to finding the simplest possible parametrization for the widest possible range of processes; where “simplest” and “widest” must be suitably operationalized. On a speculative note, the reconstruction scheme expounded here may be viewed as a proposal for such an operationalization: “Simplicity” would be embodied in the assumptions summarized in the opening paragraph of this discussion; whereas “widest” would signify the scale invariance alluded to above. Physical time would then be singled out as being a fixed point under changes of scale. I see several avenues for further research. First, it will be important to apply the reconstruction scheme to real or simulated experimental data that pertain to more complex, real-world problems. Secondly, the framework should be extended to deal with processes further away from equilibrium. And finally, it might be worthwhile to explore in more detail the conceptual issues raised in this discussion, i.e., the relativistic covariance of the procedure, its invariance under a change of scale, and the possible import of these findings on the fundamental definition and meaning of time. I thank Hans Christian [Ö]{}ttinger for helpful discussions about the two-generator formulation of Markovian nonequilibrium dynamics. This work was supported by the EU Integrating Project SIQS, the EU STREP EQUAM, and the ERC Synergy grant BioQ. [31]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [**, ()](\doibase 10.1103/PhysRev.124.983) [, ()](\doibase 10.1143/PTP.20.948) [, ()](\doibase http://dx.doi.org/10.1063/1.1731409) [, ()](\doibase 10.1143/PTP.33.423) [, ()](\doibase 10.1103/PhysRev.144.151) [, ()](\doibase 10.1016/0370-1573(95)00077-1) [, ()](\doibase 10.1016/0370-1573(86)90005-0) [, ()](\doibase 10.1103/PhysRevE.56.6620) [, ()](\doibase 10.1103/PhysRevE.62.4720) [, ()](\doibase 10.1103/PhysRevA.90.062114) [](\doibase 10.2277/0521298873) (, ) [**, ()](\doibase 10.1103/PhysRevA.8.2048) [, ()](\doibase 10.1063/1.530611) [](\doibase 10.2277/0521814510) (, ) [**, ()](\doibase 10.1007/BF02100287) [, ()](\doibase 10.1109/18.887855) [, ()](\doibase 10.1007/BF01212339) [, ()](\doibase 10.1103/RevModPhys.74.197) [, ()](\doibase 10.1103/PhysRevA.84.012101) [, ()](\doibase 10.1103/PhysRevE.56.R1295) [, ()](\doibase 10.1103/PhysRevE.55.5147) @noop []{} (, ) [**, ()](http://projecteuclid.org/euclid.jdg/1214437787) @noop []{} (, ) @noop [**, ()]{} [, ()](\doibase 10.1016/j.physleta.2011.12.007) [, ()](\doibase 10.1007/BF01609396) [, ()](\doibase http://dx.doi.org/10.1063/1.2898693) [, ()](\doibase http://dx.doi.org/10.1016/j.laa.2010.01.025) [, ()](\doibase 10.1103/PhysRevE.79.021124) “,” in @noop []{} (, ) pp.
CALT-TH 2015-017, BRX-TH-6294 \ [ S. Deser$^{\rm a}$, A. Waldron$^{\rm b}$ and G. Zahariade$^{\rm c}$]{}\ * ${}^\mathfrak{\rm a}$ Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125; Physics Department, Brandeis University, Waltham, MA 02454.\ [[email protected]]{}\ ${}^{\rm b}\!$ Department of Mathematics University of California, Davis CA 95616, USA\ [[email protected]]{}* ${}^{\rm c}\!$ Department of Physics University of California, Davis CA 95616, USA\ [[email protected]]{}\ [Abstract]{}\ Massive gravity (mGR) describes a dynamical “metric” on a fiducial, background one. We investigate fluctuations of the dynamics about mGR solutions, that is about its “mean field theory”. Analyzing mean field massive gravity ([$\overline {\text m\hspace{-.1mm}}$GR ]{}) propagation characteristics is not only equivalent to studying those of the full non-linear theory, but also in direct correspondence with earlier analyses of charged higher spin systems, the oldest example being the charged, massive spin $3/2$ Rarita–Schwinger (RS) theory. The fiducial and mGR mean field background metrics in the [$\overline {\text m\hspace{-.1mm}}$GR ]{}model correspond to the RS Minkowski metric and external EM field. The common implications in both systems are that hyperbolicity holds only in a weak background-mean-field limit, immediately ruling both theories out as fundamental theories. Although both can still be considered as predictive effective models in the weak regime, their lower helicities exhibit superluminal behavior: lower helicity gravitons are superluminal as compared to photons propagating on either the fiducial or background metric. This “crystal-like” phenomenon of differing helicities having differing propagation speeds in both [$\overline {\text m\hspace{-.1mm}}$GR ]{}and mGR is a [peculiar]{} feature of these models. Introduction ============ Consistency is a powerful tool for studying field theories. Already classically, there are stringent conditions that are extremely difficult to fulfill for systems with spin $s>1$, the most important exception being ($s=2$, $m=0$) general relativity (GR). Key consistency requirements are 1. Correct degree of freedom (DoF) counts. 2. Non-ghost kinetic terms. 3. Predictability. 4. (Sub)luminal propagation. Requirements (i) and (ii) are closely related (as are (iii) and (iv)). Models whose constraints do not single out the correct propagating DoF suffer from relatively ghost kinetic terms: the relevant example here is the sixth ghost excitation that plagues generic massive gravity (mGR) theories [@BD]. The discovery that a class of mGR models satisfied requirements (i) and (ii) generated a revival of interest in massive spin 2 theories [@dRGT; @HR; @DMZ; @DW; @DSW; @DSWZ] even though failure of the propagation requirements (iii) and (iv) were long known to bedevil higher spin theories [@VZ; @KoS]. The predictability requirement is that initial data can be propagated to the future of spacetime hypersurfaces. In PDE terms, this means that the underlying equations must be hyperbolic [@CourantHilbert]. The final requirement, that signals cannot propagate faster than light, can be imposed once the hyperbolicity requirement is satisfied. The classic example of a model that obeys requirements (i) and (ii) as well as (iii) but only in a weak field region, is the charged, massive, $s=3/2$ RS theory. Curiously enough, the propagation problems of this model were first discovered in a quantum setting by Johnson and Sudarshan [@JS] who studied the model’s canonical field commutators (this is easy to understand in retrospect, because field commutators and propagators are directly related [@BjD]). The first detailed analysis of the model’s propagation characteristics was carried out by Velo and Zwanziger; our aim is to reproduce their RS results in [$\overline {\text m\hspace{-.1mm}}$GR ]{}, so we quote their 1971 abstract verbatim [@VZ]: > The Rarita–Schwinger equation in an external electromagnetic potential is shown to be equivalent to a hyperbolic system of partial differential equations supplemented by initial conditions. The wave fronts of the classical solutions are calculated and are found to propagate faster than light. Nevertheless, for sufficiently weak external potentials, a consistent quantum mechanics and quantum field theory may be established. These, however, violate the postulates of special relativity. In previous works we and other authors have shown that similar conclusions hold for the full non-linear mGR models [@DW; @DSW; @Izumi; @DeserOng; @DSWZ; @Gr]. These investigations rely on the method of characteristics, which amounts to studying leading kinetic terms and is thus essentially equivalent to an analysis of linear fluctuations around a mean field background. Since this mean field massive gravity ([$\overline {\text m\hspace{-.1mm}}$GR ]{}) fluctuation model depends both on a background and a fiducial metric, it is in direct correspondence with the charged RS model. Hence, without any computation at all, one can readily predict that: (a) mGR loses hyperbolicity in some strong field regime and (b) in the weak field hyperbolic regime where predictability is restored, lower helicity modes have propagation characteristics differing from maximal helicity $\pm 2$; thus superluminality with respect to (luminal) photons is inevitable. Apart from confirming earlier conclusions in a very simple setting, our results give a precise description of mGR’s effective, weak field, regime. Massive Gravity =============== At its genesis, the first known non-linear mGR model of [@Zumino] was originally formulated in terms of dynamical and fiducial vierbeine $e^m$ and $f^m$. It took some forty years for researchers—independently in an effective field theory-inspired metric formulation—to discover that this model was one of a three-parameter family [@dRGT] that avoided the sixth, ghost-like excitation of [@BD]. The action describing these fiducial mGR models is given by[^1] [S]{}\_[mGR]{}\[e,;f\]=- \_[mnrs]{}e\^m &{ e\^n .\ & -. m\^2 } . The parameter $\beta_0$ governs a standard cosmological term; this is required to obtain the Fierz–Pauli (FP) linearized limit when both the fiducial and mGR backgrounds are Minkowski. When both the fiducial and mGR backgrounds are Einstein with cosmological constant $\bar \Lambda$, the model’s parameters must obey $\frac{\bar \Lambda}{3!} = m^2 \left(\beta_0+\beta_1+\beta_2+\beta_3\right)$ and the linearized theory is FP with mass $ m_{\rm FP}^2:=m^2(\beta_1+2\beta_2+3\beta_3)$. Varying the model’s dynamical fields $(e^m, \omega^{mn})$ gives equations of motion \[mGReoms\] e\^[m]{}0G\_[m]{}-m\^[2]{} t\_m , where $t_m:=\epsilon_{mnrs}\big[\beta_0 e^n e^r e^s + \beta_1 e^n e^r f^s + \beta_2 e^n f^r f^s + \beta_3 f^n f^r f^s \big] $. Also, the Einstein three-form is defined by $G_{m}:=\frac{1}{2}\epsilon_{mnrs}e^{n}{{}} R^{rs}$ and $R^{mn}:=d\omega^{mn}+\omega^{m}{}_{r}{{}}\omega^{rn}$ is the Riemann curvature; $\nabla$ is the connection of $\omega^{mn}$. The forty equations above are subject to thirty constraints that are spelled out in detail in [@DSWZ]. In particular, these include the covariant algebraic relations[^2] $$e^m f_m\approx 0 \approx K_{mn} e^m f^n\approx \epsilon_{mnrs} M^{mn}K^{rs}\, ,$$ where the tensor $K^{mn}:=\omega^{mn}-\chi^{mn}$ denotes the contorsion and $M^{mn}:=\beta_1 e^m e^n + 2 \beta_2 e^{[m}f^{n]} + 3 \beta_3 f^m f^n$. Mean field massive gravity {#cca} ========================== Consider mGR propagating in an arbitrary fiducial (pseudo-)Riemannian manifold $(M,\bar g_{\m\n})$ with corresponding vierbeine and spin connections $(f^m,\chi^{mn})$. Now let $(e^m,\omega^{mn})$ be a solution to the mGR equations of motion . We wish to study fluctuations $(\varepsilon^m,\lambda^{mn})$ about this configuration: $$\tilde e^m = e^m+\varepsilon^m\, ,\quad \tilde \omega^{mn}=\omega^{mn}+\lambda^{mn}\, .$$ The action governing these is the quadratic part of $S_{\rm mGR}[\tilde e,\tilde \omega;f]-S_{\rm mGR}[e,\omega;f]$, namely $$\begin{split} S[h,\lambda;e,f]:=-\frac12 \int \epsilon_{mnrs}\Big[e^m\varepsilon^n\nabla \lambda^{rs} &+\, \, \frac12\, \, \big(e^m e^n \lambda^r{}_t \lambda^{ts}+R^{mn} \varepsilon^r \varepsilon^s\big)\\ &-m^2\big(3\beta_0 e^m e^n \varepsilon^r \varepsilon^s +2\beta_1 e^m f^n \varepsilon^r \varepsilon^s +\beta_2 f^m f^n \varepsilon^r \varepsilon^s\big) \Big]\, . \end{split}$$ The mean field model is a theory of forty dynamical fields $(\varepsilon^m,\lambda^{mn})$. In the above, $\nabla$ is the Levi-Civita connection of $e^m$, and $R^{mn}$ its Riemann tensor; we stress that henceforth the fiducial field $\big(f^m,\chi^{mn}(f),\bar g_{\mu\nu}(f)\big)$ and mGR background fields $\big(e^m, \omega^{mn}(e), g_{\mu\nu}(e)\big)$ are [non-dynamical]{}; all index manipulations will be carried out using the mGR background metric and vierbein. The [$\overline {\text m\hspace{-.1mm}}$GR ]{}equations of motion are $$\begin{aligned} \label{mbGR} {\mathcal T}^m&:=& \nabla \varepsilon^m +\lambda^{mn} e_n \approx 0\, ,\nn\\[1mm] {\mathcal G}_m&:=& \frac12\epsilon_{mnrs}\big[ e^n \nabla \lambda^{rs}+\varepsilon^n R^{rs}\big] -m^2 \, \tau_m \approx 0\, ,\end{aligned}$$ where $ \tau_m:=\epsilon_{mnrs}\Big[3\beta_0\, e^n e^r \varepsilon^s + 2 \beta_1 \, e^n f^r \varepsilon^s + \beta_2 \, f^n f^r \varepsilon^s\Big] $. Mean field degrees of freedom ============================= In principle, since we are describing the linearization of a model whose constraints have been completely analyzed in [@DSWZ], we know [a priori]{} that [$\overline {\text m\hspace{-.1mm}}$GR ]{}describes five propagating degrees of freedom. However for completeness and our causality study, we reanalyze its constraints. The first step is to introduce a putative choice of time coordinate $t$, which for now need not rely in any way on either the fiducial or background metric, and use this to decompose any $p$-form $\mbox{\resizebox{2.5mm}{2.85mm}{$\theta$}}$ (with $p<4$) as \[ring\] := + , where $\mbox{\resizebox{2.5mm}{4.1mm}{$\ring\theta$}}\wedge dt=0$. Thus $\bm \theta$ is the purely spatial part of the form . Hence for any on-shell relation ${\cal P}\approx 0$ polynomial in $(\nabla,\varepsilon,\lambda)$, its spatial $\bm {\mathcal P}\approx 0$ part is a constraint because it contains no $t$-derivatives. Thus we immediately find [sixteen]{} primary constraints: $${\bm {\mathcal T}}^{m}={\bm \nabla}{\bm \varepsilon}^m+{\bm \lambda}^{mn}{\bm e}_n\approx 0\approx {\bm {\mathcal G}}_{m}= \frac12\epsilon_{mnrs}\big[ {\bm e}^n {\bm \nabla \bm \lambda}^{rs}+{\bm \varepsilon}^n {\bm R}^{rs}\big] -m^2\, {\bm \tau}_m\, .$$ There are [ten]{} secondary constraints in total: The first six of these follow from the integrability condition $e^{[m} \nabla {\mathcal T}^{n]}\approx 0$ which yields the so-called [symmetry constraint]{} $$e_{[m} \tau_{n]}+\varepsilon_{[m} t_{n]}\approx 0\, .$$ As mentioned above, we assume that the set of two forms $\{M^{mn}\}$ is a basis for the space of two-forms, so the symmetry constraint yields $$\varepsilon^{m}f_{m}\approx 0\ .$$ The remaining four secondary constraints come from the covariant curl $\nabla{\mathcal G}_m\approx 0$ and give the [vector constraint]{} $$\nabla \tau_m+\lambda_m{}^n t_n\approx 0\, .$$ Employing the equation of motion ${\mathcal T}^m\approx 0$, this implies $$\epsilon_{mnrs} \big[M^{mn} \lambda^{rs} + 2 (\beta_1 e^m +\beta_2 f^m)\, \varepsilon^n K^{rs}\big]\approx 0\, .$$ Finally there are [four]{} tertiary constraints stemming from covariant curls of the secondaries: the temporal part of the [curled symmetry constraint]{} $K_{mn}\varepsilon^{m}f^{n}+\lambda_{mn}e^{m}f^{n}\approx 0$, [i.e.]{} $$\ring K_{mn} \bm\varepsilon^m \bm f^n+\bm K_{mn} \ring \varepsilon^m \bm f^n+\bm K_{mn} \bm \varepsilon^m \ring f^n +\ring \lambda_{mn} \bm e^m \bm f^n+\bm \lambda_{mn} \ring e^m \bm f^n+\bm \lambda_{mn} \bm e^m \ring f^n \approx 0\ ,$$ and the [scalar constraint]{} &&\_[mnrs]{}(\_[1]{}(\^[m]{}e\^[t]{}+e\^[m]{}\^[t]{})-2\_[2]{}\^[(m]{}f\^[t)]{})K\^[nr]{}K\^[s]{}\_[t]{}\ &&+ \_[mnrs]{}(\_[1]{}e\^[m]{}e\^[t]{}-2\_[2]{}e\^[(m]{}f\^[t)]{}-3\_[3]{}f\^[m]{}f\^[t]{})(\^[nr]{}K\^[s]{}\_[t]{}+K\^[nr]{}\^[s]{}\_[t]{})\ &&+ 2 m\^[2]{} \_[1]{}\^[m]{}t\_[m]{} + 2 m\^[2]{}( \_[1]{}e\^[m]{}+2\_[2]{}f\^[m]{})\_[m]{}+3 \_[mnrs]{}\_[3]{}f\^[m]{}f\^[n]{}\^[rs]{}\ &&- 4 \_[2]{}\^[m]{}\|[G]{}\_[m]{}-2\_[mnrs]{}\_[1]{}\^[m]{}e\^[n]{}\|[R]{}\^[rs]{}0 . The $\nabla\lambda^{rs}$ term seems to indicate that the above display is not a constraint for $\beta_3\neq 0$, however as shown in [@DSWZ], this quantity (weakly) equals one without time derivatives of fields. In summary, the model describes forty fields subject to thirty constraints and thus propagates five[^3] DoF. Characteristic matrix ===================== We now study whether [$\overline {\text m\hspace{-.1mm}}$GR ]{}can propagate initial data off a given hypersurface $\Sigma$. This amounts to asking if derivatives normal to $\Sigma$ are determined by the equations of motion . For that, we simply replace all derivatives in the equations of motion and gradients of their constraints by the normal covector $\xi_\mu$ to $\Sigma$ multiplied by the normal derivative of the corresponding field: $$\partial_\mu \varepsilon^m\big\|_\Sigma = \xi_\mu \partial_{\rm n} \varepsilon^m \ \mbox{ and }\ \partial_\mu \lambda^{mn}\big\|_\Sigma = \xi_\mu \partial_{\rm n}\lambda^{mn} \, .$$ We will also, for reasons of simplicity alone, restrict to the parameter choices $\beta_2=\beta_3=0$ (the model’s characteristic matrix for its entire parameter range has been computed in [@DSWZ]). In particular we must focus on the question whether the linear system of equations for the normal derivatives ${\partial_{\rm n} \varphi}:= ({{\partial_{\rm n} \varepsilon},{\partial_{\rm n} \lambda}})$ implied by the equations of motion along $\Sigma$ is invertible. This amounts to a matrix problem encoded by the theory’s characteristic matrix ${\mathcal C}$. In what follows we compute the system of equations given by the homogeneous linear system ${\mathcal C} \cdot {\partial_{\rm n} \varphi} = 0$. Starting with the equations of motion we find $$\begin{aligned} \label{cheom} &\xi\wedge \partial_{\rm n} \varepsilon^m = 0 \, , \\[1mm] &\epsilon_{mnrs} e^n\wedge \xi \wedge \partial_{\rm n} \lambda^{rs}=0\, .\nn\end{aligned}$$ The gradients of the secondary and tertiary constraints then imply $$\begin{aligned} \label{2and3} &\xi_\mu\, f_m \wedge \partial_{\rm n} \varepsilon^m = 0 = \xi_{\mu}\, \epsilon_{mnrs} e^m \wedge \big[e^n \wedge \partial_{\rm n} \lambda^{rs}-2 K^{nr}\wedge \partial_{\rm n} \varepsilon^s\big]\, , &\\[1mm] &\xi_\mu\, f^m\wedge \big[ K_{mn} \wedge \partial_{\rm n}\varepsilon^n + e^n \wedge \partial_{\rm n} \lambda_{mn} \big]=0\, ,& \nn \\[1mm] &\xi_\mu\, \epsilon_{mnrs}e^{m}\!\wedge\!\big[e^{n}\!\wedge\! K^{rt}\!\wedge\!\partial_{\rm n}\lambda_{t}{}^{s}+\big(K^{nt}\!\wedge\! K_{t}{}^{r}-\bar{R}^{nr}+m^{2}(4\beta_{0}e^{n}\!\wedge\! e^{r}+3\beta_{1}e^{n}\!\wedge\! f^{r})\big)\!\wedge\!\partial_{\rm n}\varepsilon^{s}\big] = 0 \ . \!\!\!\!\!\!\!& \nn\end{aligned}$$ In the above the prefactor $\xi_\mu$ was included to indicate the origin of these equations but can be removed with impunity. To handle Equation we decompose form-valued normal derivatives as earlier in Equation , and find[^4] $$\partial_{\rm n} {\bm \varepsilon}^{m}=0=\partial_{\rm n} {\bm \lambda}^{mn}\, .$$ Supposing that the one-form $\xi = dt$, for some evolution coordinate $t$, we now use a shorthand notation $\partial_n \varepsilon^{m} = dt \, \dot \varepsilon^m_t$ and $\partial_n \lambda^{mn}= dt \, \dot \lambda^{mn}_t$. We thus have the [reduced characteristic system]{} $$\label{reduce} \begin{pmatrix} {\bm f}_m & 0\\[1mm] 2\epsilon_{mnrs} {\bm e}^n\times {\bm K}^{rs}& \epsilon_{mnrs} {\bm e}^r\times {\bm e}^s\\[1mm] {\bm f}^n\times {\bm K}_{nm}& {\bm f}_m\times {\bm e}_n\\[1mm] {\mathcal R}_m&{\mathcal K}_{mn} \end{pmatrix} \begin{pmatrix} \dot\varepsilon_t^m \\[2mm] \dot\lambda_t^{mn} \end{pmatrix}=0\, .$$ In the above square matrix, $\times$ denotes the standard three-dimensional cross product while the spatial densities on its last line can be read off from and are simple for flat fiducial metrics. Vanishing of the determinant of the above $10\times 10$ matrix completely characterizes the boundary of the model’s predictive hyperbolic regime (modulo the restriction explained in Footnote \[fineprint\]). As we shall see, the reduced characteristic system describes the propagation of superluminal lower helicity modes: In the next section, we specialize to flat fiducial spaces and show how to analyze this determinant in direct analogy with the RS system. Analogy with Rarita–Schwinger ============================= The charged, spin $3/2$, Rarita–Schwinger (RS) equation of motion reads $$\gamma^{\mu\nu\rho} \big(\nabla_\nu + ie A_\nu + \frac m2\, \gamma_\nu\big)\psi_\rho=0\, .$$ Here $\nabla$ is the Levi-Civita connection of the fiducial spacetime and $A$ is the background EM potential. These are analogous to the [$\overline {\text m\hspace{-.1mm}}$GR ]{}fiducial and background-mean-field metrics. The RS characteristic matrix was computed in [@VZ] for flat fiducial metrics and spacelike hypersurfaces $\Sigma$ and found to have zero determinant when the magnetic field $\bm B$ obeyed[^5] $$\label{R} 1-\Big(\frac{2e}{3m^2}\Big)^{\!2} {\bm B}^2 =0\, .$$ The condition $\frac{2e\|B\|}{3m^2}<1$ thus determines the weak field, hyperbolic, regime. Our aim now is to develop the analogous statement for [$\overline {\text m\hspace{-.1mm}}$GR ]{}and inherit the conclusions of [@VZ]. We begin with a short calculation. Consider now a flat fiducial metric so $f^m=\delta^m_\mu dx^\mu$ and $d\bar s^2=-dt^2 + d{\bm x}^{ 2}$. This simplifies the reduced characteristic system considerably. Firstly the equation ${\bm f}_m \dot\varepsilon^m_t=0$ of implies $\dot\varepsilon^a_t=0$, where we have decomposed the Lorentz index $m=(0,a)$. Let us introduce an EM-like notation $$\epsilon_{abc} {\bm K}^{bc}=:{\bm B}_a\, ,\quad {\bm K}^{0a}=:{\bm E}^a \, .$$ For pure simplicity reasons only, we now restrict to the case where $\bm e^{0}=\bm E^{a}=\bm 0\ .$ Thus using $\epsilon^0{}_{abc}=\epsilon_{abc}$ the second equation of implies $$\dot\lambda_t^{0a}=-\frac12 \epsilon^{abc} {\bm B}_b\cdot\tilde {\bm e}_c \, \dot\varepsilon_t^0 \, .$$ Here the 3-vectors $\tilde {\bm e}_a$ form the 3-inverse of ${\bm e}^a$ so that $\tilde {\bm e}_a\cdot {\bm e}^b=\delta^b_a$. The third equation of then gives $$\bm f_{[a}\times\bm e_{b]} \dot\lambda_{t}^{ab}=0\ .$$ This equation generically allows $\dot\lambda_{t}^{ab}$ to be expressed as a function of $\dot\varepsilon_{t}^{0}$ but this requires a non-trivial condition on $\bm e^{a}$. Under the hypothesis that the eigenvalue spectra of the matrices $\bm f^{a}\cdot\tilde{\bm e}_{b}$ and $-\bm f^{a}\cdot\tilde{\bm e}_{b}$ do not intersect, the above equation implies that $\dot\lambda_{t}^{ab}=0$. In this framework, the last equation of gives the single VZ-type condition $$\begin{aligned} \label{m} \left[m_{\text{FP}}^{2}\big(4-(\bm f^{a}\cdot\tilde{\bm e}_{a})\big)-\frac16\Big( (\bm B^{a}\cdot\tilde{\bm e}^{b})(\bm B_{[a}\cdot\tilde{\bm e}_{b]})-\frac12(\bm B^{a}\cdot\tilde{\bm e}_{[a})(\bm B^{b}\cdot\tilde{\bm e}_{b]})\Big)\right]\dot\varepsilon_{t}^{0}=0\ .\end{aligned}$$ In the weak field limit where the mean field approaches Minkowski space, the coefficient of $m_{\rm FP}^2$ approaches unity but can change sign in a strong-field, large $\tilde {\bm e}$ limit. Hence there are certainly strong field configurations where the model loses hyperbolicity [@CourantHilbert] and closed causal curves are unavoidable. (This signals the onset of strong coupling in an effective field theory.) Now, comparing Equations and , we see that we have reduced [$\overline {\text m\hspace{-.1mm}}$GR ]{}’s weak field propagation analysis to a previous—well understood—case. Finally, we note that the same analogy and characteristic method can also be applied to the bimetric theory by treating the two background metrics as a (fiducial,background) pair [@prep]. Conclusions =========== mGR is not a fundamental theory but rather an effective one with a range of validity determined by requiring hyperbolicity in a weak field regime. Excepting the further caveats explained in the text, the reduced characteristic matrix of Equation completely determines this allowed regime. Even in the weak regime, modes exhibit a crystal structure with differing maximal propagation speeds. For mGR to give a useful effective theory for physical applications, one must couple to matter (or at least photons) and require consistent causal cones for all modes. Once these couplings are decided upon, the characteristic method will determine their effective range of validity, if any. There is also the logical possibility that there exists a causal, luminal, UV completion of mGR analogous to that for QED in curved space [@Hollowood] (see however the more general discussion of UV completions [@Dubovsky]). Acknowledgements {#acknowledgements .unnumbered} ================ We thank C. Deffayet, S. Dubovsky, K. Hinterbichler, K. Izumi, M. Porrati and Y.C. Ong for discussions. A.W. and G.Z. thank the Perimeter Institute for an illuminating “Superluminality in Effective Field Theories for Cosmology” workshop. S.D. was supported in part by grants NSF PHY-1266107 and DOE \# de-sc0011632. A.W. was supported in part by a Simons Foundation Collaboration Grant for Mathematicians. 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--- abstract: \| How much adversarial noise can protocols for interactive communication tolerate? This question was examined by Braverman and Rao (IEEE Trans. Inf. Theory, 2014) for the case of “robust” protocols, where each party sends messages only in fixed and predetermined rounds. We consider a new class of non-robust protocols for Interactive Communication, which we call [adaptive]{} protocols. Such protocols adapt structurally to the noise induced by the channel in the sense that both the order of speaking, and the length of the protocol may vary depending on observed noise.=-1 We define models that capture adaptive protocols and study upper and lower bounds on the permissible noise rate in these models. When the length of the protocol may adaptively change according to the noise, we demonstrate a protocol that tolerates noise rates up to $1/3$. When the order of speaking may adaptively change as well, we demonstrate a protocol that tolerates noise rates up to $2/3$. Hence, adaptivity circumvents an impossibility result of $1/4$ on the fraction of tolerable noise (Braverman and Rao, 2014). author: - Shweta Agrawal - Ran Gelles - Amit Sahai bibliography: - 'coding-short.bib' title: Adaptive Protocols for Interactive Communication --- Introduction {#sec:intro} ============ One of the fundamental questions considered by Computer Science is “What is the best way to encode information in order to recover from channel noise”? This question was studied most notably by Shannon, in a pioneering work [@shannon48] which laid the foundation of the rich area of information theory. Shannon considered this question in the context of one way communication, where one party wants to transmit a message “once and for all” to another. More recently, in a series of beautiful papers, Schulman [@schulman92; @schulman93; @schulman96] generalized this question to subsume interactive communication, i.e. the scenario where two remote parties perform some distributed computation by “conversing” with each another in an interactive manner, so that each subsequent message depends on all messages exchanged thus far. Surprisingly, Schulman showed that, analogous to the case of one way communication, it is indeed possible to embed any interactive protocol $\pi$ within a larger protocol $\pi'$ so that $\pi'$ computes the same function as $\pi$ but additionally provides the requisite error correction to tolerate noise introduced by the channel. The noise in the channel may be stochastic, in which error occurs with some probability, or adversarial, in which the channel may be viewed as a malicious party Eve who disrupts communication by injecting errors in the worst possible way. In this work we focus on adversarial noise. Schulman [@schulman93; @schulman96] provided a construction that turns a protocol $\pi$ with communication complexity $T$, to a noise-resilient $\pi'$ which communicates at most $O(T)$ symbols, and can recover from an adversarial (bit) noise rate of at most $1/240$. This result was later improved by Braverman and Rao [@BR11; @BR14], who provided a protocol that can recover from a (symbol) noise rate up to $1/4 -{\varepsilon}$ and also communicates at most $O(T)$ symbols. Both the above constructions assume robust protocols. Intuitively speaking, robust protocols are synchronized protocols that have a fixed length and a predetermined “order of communication”. In this class of protocols, each party knows at every time step whose turn it is to speak and whether the protocol has terminated, since these properties are fixed in advance and independent of the noise introduced by the adversary. However, one can imagine more powerful, general protocols where the end point of the protocol or the order of speaking are not predetermined but rather depend on the observed transcript, that is, on the observed noise. While Braverman and Rao show that for any robust protocol $1/4$ is an upper bound on the tolerable noise rate, they explicitly leave open the question of whether non robust protocols admit a larger amount of noise. We address this question by considering two types of non-robust protocols, that allow for greater adaptivity in the behavior of the participants. First, we allow the length of the protocol to be adaptively specified during the protocol by its participants. Next, we consider even greater adaptivity and allow the party that speaks next in the protocol to be adaptively chosen by the participants of the protocol. In both these situations, we show that increasing adaptivity allows for a dramatic increase in the noise resilience of protocols. We draw attention of the reader to the fact that while for robust protocols, Yao’s [@Yao79] model is almost universally accepted as natural and meaningful, it is far less obvious what is the right way to model non-robust protocols, or even if there is a unique choice. Defining models to capture adaptivity is subtle, and several choices must be made, for example in how adversarial noise is budgeted and in how to model rounds in which there is no consensus regarding who the speaker is. Different modeling choices lead to different protocol capabilities and we believe it is important to explore the domian of this very young area in order to find settings that are both natural and admit protocols with higher noise resilience. In a recent work, Ghaffari, Haeupler, and Sudan [@GHS14; @GH14], proposed one natural set of choices to model adaptivity, and provided efficient protocols in that model which resist noise rates of up to $2/7$, surpassing the maximal resilience of the non-robust case. In this paper we make a different, but arguably just as natural, set of choices, which lead to adaptive protocols with even higher noise resilience. We proceed to summarize the most salient differences in our modeling choices and the ones of [@GHS14]. First, the model in [@GHS14] does not permit adaptive modification of the length of the protocol, while our model does. To the best of our knowledge, our work is the first to consider varying length interactive protocols and their noise resilience. The second main difference is that in [@GHS14] the channel may be used to communicate only in one direction at each round. Specifically in [@GHS14], at each round, each party decides either to only talk or to only listen: if both parties talk at the same round, a collision occurs and no symbol is transferred, and if both listen at the same round, they receive some adversarial symbol not counted towards the adversary’s budget. In our model, on the other hand, both parties may talk at the same round without causing any collision (similar to the case of robust protocols [@schulman96; @BR14]). The adaptivity stems from the parties’ ability to individually choose at each round, whether they talk or not. The two different modeling choices taken by [@GHS14] and by us lead to different bounds on the noise an adaptive protocol can handle. For instance, while the protocols of [@GHS14] can handle up to a relative noise of $2/7$, our protocols can resists a higher noise rate of $1/3$ if the length of the protocol may adaptively change, or noise rate of up to $2/3$ when both the length and the order of speaking adaptively vary. We now give more details about our adaptive model and the noise rate our protocols can resist. Our Results: Adaptive Length ---------------------------- We begin by considering adaptive protocols in which the length of the protocol may vary as a function of the noise, however the order of speaking is still predetermined. Specifically, each party individually decides whether to continue participating in the protocol, or terminate and give an output. We denote the class of such protocols as ${{{\mathscr{M}}_\text{term}}}$ (see formal definition in Section \[sec:mterm-model\]). Intuitively, changing the length of the protocol is useful for two reasons. First, the parties may realize that they still did not complete the computation, and communicate more information in order to complete the task. On the other hand, the parties may see that the noise level is so high that there is no hope to correctly complete the protocol. In this case the parties should abort the computation, since for such a high noise level, the protocol is not required to be correct anyway. The difficult part for the parties is, however, to be able to distinguish between the first case and the second one in a coordinated way and despite the adversarial noise. If the length of the protocol is not fixed (and subsequently, its communication complexity), the noise rate must be defined with care. Generalizing the case of fixed-length protocols, we consider the ratio of corrupted symbols out of all the symbols that were communicated in that instance, and call this quantity the relative noise rate. We emphasize that both the numerator and the denominator of this ratio vary in adaptive protocols. Our main result for this type of adaptivity is a protocol that resists relative noise rates of up to $1/3$ (Theorem \[thm:protocol-third\]). The protocol works in two steps: in the first step Alice communicates her input to Bob using some standard error correction code; in the second step Bob estimates the noise that occurred during the first step, and then he communicates his input to Alice using an error correction code with parameters that depend on his noise estimation. In general, the more noise Bob sees during the first step, the less redundant his reply to Alice would be—if there was a lot of noise during the first part, the adversary has less budget for the second part, and the code Bob uses can be weaker. The communication complexity of our protocol above is a constant factor (where the constant depends on the channel quality) times the input lengths of Alice and Bob. However, our protocol requires the parties to communicate their inputs, even in cases where the lengths of the inputs may be very long with respect to the communication complexity of the best noiseless protocol; thus, the rate of this coding strategy (the length of the noiseless protocol divided by the length of the resilient protocol) can be vanishing when the length of the noiseless protocol tends to infinity. Nevertheless, our coding protocol serves as an important proof of concept for the strength of this model: Indeed, in the non-adaptive setting, the rate of the coding scheme has no effect on the noise resilience. E.g., an upper bound of $1/4$ holds for coding schemes even when their rate is vanishing [@BR14]. In addition to our schemes, we show an upper (impossibility) bound of $1/2$ on the tolerable noise in that model (Theorem \[thm:mterm-half\]). We emphasize that previous impossibility proofs (i.e., [@BR14]) crucially use the property of robustness: in robust protocols there always exists a party that speaks at most half of the symbols, whose identity is known in advance, making it a convenient target for adversarial attack. Contrarily, in adaptive protocols the party that speaks less may depend on the noise and vary throughout the protocol. We provide a new impossibility bound by devising an attack that corrupts both parties with rate $1/2$, and carefully arguing that at least one of the parties must terminate before it learns the correct output. Model Lower Bound $\alpha$$\quad$ Upper Bound $\beta$$\quad$ Ref. --------------------------------- --------------------------------- -------------------------------- ---------------- $({\text{non-adaptive}})$ $1/4$ $1/4$ [@BR14] ${{{\mathscr{M}}_\text{term}}}$ $1/3$ $1/2$ §\[sec:mterm\] : Summary of our bounds for the ${{{\mathscr{M}}_\text{term}}}$ model, compared to the non-adaptive model. $\alpha$ and $\beta$ are the lower (existence) and upper (impossibility) bounds on the allowed noise rate: for any function there exists a protocol that withstands noise rate $c$ if $c<\alpha$. Yet, there exists a function for which no protocol withstands noise rate $\beta$. \[tab:res-mterm\] Our Results: Adaptive Order of Speaking --------------------------------------- Next, we define the ${{{\mathscr{M}}_\text{adp}}}$ model in which we allow the order of speaking to depend on the noise (see formal definition in Section \[sec:mmain-model\]). Specifically, at each round each party decides whether it sends the next symbol or it keeps silent; the other side, respectively, either learns the symbol that was sent, or receives “silence”.[^1] We stress that silent rounds, i.e. when no message is delivered, are not counted towards the communication, or otherwise the model becomes equivalent to the ${{{\mathscr{M}}_\text{term}}}$ model. We note that this type of adaptivity also implies a varying length of the protocol, e.g., in order to terminate, a party simply keeps silent and disregards any incoming communication. Similar to the ${{{\mathscr{M}}_\text{term}}}$ model, the adversary is allowed to corrupt any transmission, and we measure the noise rate as the ratio of corrupted transmissions to the communicated (non-silent) symbols. It is important to emphasize that the adversary is not limited to only corrupting symbols, but it can also create a symbol when a party decides to keep silent, or remove a transmitted symbol leading the receiving side to believe the other side is silent. This makes a much stronger adversary[^2] that may induce relative noise rates that exceed 1.=-1 Here we construct an adaptive protocol which crucially uses both the ability to remain silent as well as the ability to vary the length of the protocol, to withstand noise rates $<2/3$ (Theorem \[thm:ProtocolTwoThirds\]). The protocol behaves quite similar to the ${{{\mathscr{M}}_\text{term}}}$ protocol that achieves noise rates up to $1/3$ with an additional layer of encoding that takes advantage of being able to remain silent, and provides another factor of $2$ in resisting noise. We name this new layer of code silence encoding. The idea behind this layer is that using $k$ non-silent transmissions, one can obtain a code with distance $2k$. Then, in order to cause a decoding error, the adversary must invest $\tfrac12 2k +1$ corruptions, or otherwise either the correct symbol can be decoded, or the symbol becomes an erasures (which is easier to handle than an error). Note that for the special case of $k=1$, two corruptions are required to cause decoding of an incorrect symbol (or otherwise, the adversary only causes an erasure). The main drawback of the above protocol is that its length (i.e., its round complexity) may be very large with respect to the length of the optimal noiseless protocol; thus it has a vanishing rate. Next, we restrict the discussion only to adaptive protocols whose length is linear in the length of the optimal noiseless protocol (thus their rate is a positive constant and not vanishing), and show a protocol with non-vanishing rate that tolerates noise rates of up to $1/2$ (Theorem \[thm:ProtocolHalf\]). The protocol is based on the optimal (non-robust) protocol of [@BR14] with an additional layer of of silence encoding which effectively forces the adversary to “pay twice” for each error it wishes to make. This way the protocol can withstand twice the number of errors than [@BR14]. Model Noise Resilience $\quad$ Non-Vanishing Rate $\quad$ Ref. ------------------------------------------------------ ------------------------------ -------------------------------- -------------------------------------------- ${\text{non-adaptive}}$ $1/4$ $\surd$ [@BR14] ${{{\mathscr{M}}_\text{adp}}}$ $2/3$ §\[sec:dsa\], §\[app:freepos\] ${{{\mathscr{M}}_\text{adp}}}$ $1/2$ $\surd$ §\[app:mainCRprotocol\] ${{{\mathscr{M}}_\text{adp}}}$ (shared randomness) $1$ $\surd$ §\[app:protocolSharedRand\] ${{{\mathscr{M}}_\text{adp}}}$ over erasure channels $1$ $\surd$ §\[sec:dsa\] (§\[app:protocolSharedRand\]) : Summary of the noise resilience of our protocols in the ${{\mathscr{M}}_\text{adp}}$ model. For any function $f$, and for any constant $c$ less than the resilience, there exists a protocol that correctly computes $f$ over any channel with relative noise rate $c$. Note that $1$ is a trivial impossibility bound for the ${{{\mathscr{M}}_\text{adp}}}$ model, as the adversary can delete the entire communication. \[tab:res-madp\] If we relax the model to permit the parties to share some randomness unknown to the adversary, then we can construct a protocol that withstands an optimal $1 - {\varepsilon}$ fraction of errors (Theorem \[thm:main-shared\]) and also achieves non-vanishing rate. The key technique here is to adaptively repeat transmissions that were corrupted by the adversary: each symbol is sent multiple times until the other side indicates that the symbol was received correctly. However, now the adversary can corrupt this “feedback” and falsely indicate that a symbol was received correctly by the other side. To prevent such an attack we use the shared randomness to add a layer of error-detection (via the so called Blueberry code [@FGOS15]). The adversary, without knowing the randomness, has a small probability to corrupt a symbol so it passes the error-detection layer, and corrupts the sensitive “feedback” symbols with only a negligible probability. An interesting observation is that we can apply our methods to the setting of erasure channels and obtain a protocol with linear round complexity (i.e., with a non-vanishing rate) and erasure resilience of $1-{\varepsilon}$ without the need for a shared randomness (Corollary \[cor:erasure\]). We note that for non-adaptive protocols over erasure channels, $1/2$ is a tight bound on the noise: a noise of $1/2-{\varepsilon}$ is achievable via the Braverman-Rao protocol (see [@FGOS15]) or via the simple protocol of Efremenko, Gelles and Haeupler [@EGH15]; on the other hand, a noise rate of $1/2$ is enough to erase the entire communication of a single party, thus disallowing any interaction [@FGOS15]. Our protocol for adaptive settings hints that adaptivity can double the resilience to noise (similar to the effect of possessing preshared private randomness [@FGOS15], etc.). Our bounds for the ${{{{\mathscr{M}}_\text{adp}}}}$ model are summarized in Table \[tab:res-madp\].=-1 Related Work. ------------- As mentioned in the introduction, the study of coding for interactive communication was initiated by Schulman [@schulman92; @schulman93; @schulman96] who provided protocols for interactive communication using [tree codes]{} (see Appendix \[app:mainCRprotocol\] for a definition and related works). In his work, Schulman considered both the stochastic as well as adversarial noise model, and for the latter provided a protocol that resists (bit) noise rate $1/240$. Braverman and Rao [@BR11; @BR14] improved this bound to $1/4$ by constructing a different tree-code based protocol (which is efficient except for the generation of tree codes). Braverman and Efremenko [@BE14] considered the case where $\alpha$ fraction of the symbols from Alice to Bob are corrupted and $\beta$ fraction of the symbols in the other direction are corrupted. For any point $(\alpha,\beta)\in [0,1]$ they determine whether or not interactive communication (with non-vanishing rate) is possible. This gives a complete characterization of the noise bounds for the non-adaptive case. Over the last years, there has been great interest in interactive protocols, considering various properties of such protocols such as their efficiency [@GMS11; @GMS14] (stochastic noise), [@BK12; @BN13; @GH14; @BKN14] (adversarial noise), their noise resilience under different assumptions and models [@FGOS13; @BNTTU14; @EGH15; @FGOS15], their information rate [@KR13; @Pankratov13; @Haeupler14; @GH15] and other properties, such as privacy [@CPT13; @GSW14] or list-decoding [@GHS14; @GH14; @BE14]. We stress that all the works prior to this work (and to the independent work [@GHS14; @GH14]), assume the robust, non-adaptive setting. The only other work that studies adaptive protocols is the abovementioned work of Ghaffari, Haeupler, and Sudan [@GHS14], which makes different modeling decision than our work. Ghaffari et al. show that in their adaptive model, $2/7$ is a tight bound on fraction of permissible noise. The length of the protocol obtained in [@GHS14] is quadratic in the length of the noiseless protocol, thus its rate is vanishing. However Ghaffari and Haeupler [@GH14] later improve the length to be linear while still tolerating the optimal $2/7$ noise of that model. Allowing the parties to preshare randomness increases the admissible noise to $2/3$. We stress again that the setting of [@GHS14] and ours are incomparable. Indeed, the tight $2/7$ bound of [@GHS14] does not hold in our model and we can resist relative noise rates of up to $1/3$ or $2/3$ in the ${{{\mathscr{M}}_\text{term}}}$ and ${{{\mathscr{M}}_\text{adp}}}$ models respectively. Similarly, while $2/3$ is the bound on noise when parties are allowed to share randomness in [@GHS14], in our model, the relative noise resilience for this setting is $1$. We note that interactive communication can also be extended to the multiparty case, following the more simple two party case, see e.g. [@RS94; @JKL15; @HS14; @ABEGH15]. The adaptive setting is particularly relevant to asynchronous multiparty settings (as in [@JKL15]) which is closely related to the ${{{\mathscr{M}}_\text{adp}}}$ model we present here. Interactive (noiseless) communication in a model where parties are allowed to remain silent (similar to the case of the ${{{\mathscr{M}}_\text{adp}}}$ model), was introduced by Dhulipala, Fragouli, and Orlitsky [@DFO10], who consider the communication complexity of computing symmetric functions in the multiparty setting. In their general setting, each symbol $\sigma$ in the channel’s alphabet has some weight $w_\sigma \in [0,1]$ and the weighted communication complexity, both in the average and worst case, is analyzed for a specific class of functions. Remaining silent can be thought of sending a special “silence” symbol, whose weight is usually 0. Impagliazzo and Williams [@IW10] also consider communication complexity given a special silence symbol for the two-party case. They establish a tradeoff between the communication complexity and the round complexity. Additionally, they relate these two measures to the “standard” communication complexity, i.e., without using a silence symbol. Protocols with an Adaptive Length {#sec:mterm} ================================= In this section, we study the ${{{\mathscr{M}}_\text{term}}}$ model in which parties adaptively determine the length of the protocol by (locally) terminating at will. First, let us formally define the model. The ${{{\mathscr{M}}_\text{term}}}$ model {#sec:mterm-model} ----------------------------------------- We assume Alice and Bob wish to compute some function $f: {\mathcal{X}}\times{\mathcal{Y}}\to {\mathcal{Z}}$ where Alice holds some input $x\in {\mathcal{X}}$ and Bob holds $y\in {\mathcal{Y}}$. The sets ${\mathcal{X}},{\mathcal{Y}}$ and ${\mathcal{Z}}$ are assumed to be of finite size. We assume Alice and Bob run a protocol $\pi=(\pi_A,\pi_B)$, over a channel controlled by a malicious Eve. At every step of the protocol, $\pi$ defines a message over some alphabet $\Sigma$ of finite size (which may depend only on the targeted noise resilience) to be transmitted by each party as a function of the party’s input, and the received messages so far. In this model, each party sends symbols according to a predetermined order. Let $I_A,I_B \subseteq \mathbb{N}$ be the round indices in which Alice and Bob talk, respectively. Note that $I_A$ and $I_B$ may be overlapping but we may assume without loss of generality that there are no “gaps” in the protocol, i.e. $I_A \cup I_B = \mathbb{N}$. The channel expects an input from party $P\in \{A,B\}$ only during rounds in $I_P$. The behavior of Alice in the protocol is as follows (Bob’s behavior is symmetric): - In a given round $i \in I_A$, if Alice has not terminated, she transmits a symbol $a_i \in \Sigma \cup \{{\emptyset}\}$, where $\Sigma$ is the channel’s alphabet and ${\emptyset}$ is a special symbol we call silence. - At the beginning of any round $i\in \mathbb{N}$, Alice may decide to terminate. In that case she outputs some value, sets ${\mathsf{TER}}_A = i$ and stops participating in the protocol. This is an irreversible decision. - In every round $i \in I_A$ where $i \ge {\mathsf{TER}}_A$, Alice’s input to the channel is the special silence symbol $a_i = {\emptyset}$. - Eve may corrupt any symbol, including silence, transmitted by either party. Thus, she acts upon transmitted symbols via the function ${\mathsf{Ch}}: \Sigma \cup \{{\emptyset}\} \rightarrow \Sigma \cup \{{\emptyset}\}$, conditioned on the parties input, Eve’s random coins and the transcript so far. Note that even after Alice has terminated, $a_i={\emptyset}$ is sent over the channel and still might be corrupted by Eve. Next we formally define some important measures of a protocol. For a specific instance of the protocol we define the Noise Pattern $E\in (\Sigma\cup \{\bot\})^$ incurred in that instance in the following way. Assume that Alice sends $(a_1,a_2,\ldots)$ and Bob sends $(b_1,b_2,\ldots)$, then $E=((e_{a_1},e_{a_2}, \ldots), (e_{b_1},e_{b_2}, \ldots))$ so that $e_{a_i}=\bot$ if ${\mathsf{Ch}}(a_i)=a_i$ and otherwise, $e_{a_i}={\mathsf{Ch}}(a_i)$, and similarly for $e_{b_i}$. For any protocol $\pi$ in the ${{{\mathscr{M}}_\text{term}}}$ model, we define the following measures for any given instance of $\pi$ running on inputs $(x,y)$ suffering the noise pattern $E$: 1. Communication Complexity: $ {\mathsf{CC}}^{\text{term}}_\pi(x,y,E) {\overset{\Delta}{=}}\left\lvert[{\mathsf{TER}}_A-1] \cap I_A\right\rvert + \left\lvert[{\mathsf{TER}}_B-1] \cap I_B\right\rvert, $ where $[n]$ is defined as the set $\{1,2, \ldots, n\}$. 2. Round Complexity: $ {{\mathsf{RC}}}^{\text{term}}_\pi(x,y,E) {\overset{\Delta}{=}}\max({\mathsf{TER}}_A, {\mathsf{TER}}_B). $ 3. Noise Complexity: $ {\mathsf{NC}}^{\text{term}}_\pi(x,y,E) {\overset{\Delta}{=}}\\ \big\|\{ i\in I_A \mid i < {{\mathsf{RC}}}^{\text{term}}_\pi \; ,\; {\mathsf{Ch}}(a_i)\neq a_i\} \big\| + \big\|\{ i\in I_B \mid i < {{\mathsf{RC}}}^{\text{term}}_\pi \; ,\; {\mathsf{Ch}}(b_i)\neq b_i\} \big\|. $ 4. Relative Noise Rate: $ {\mathsf{NR}}^{\text{term}}_\pi(x,y,E) {\overset{\Delta}{=}}{{\mathsf{NC}}^{\text{term}}_\pi (x,y,E)}/{{\mathsf{CC}}^{\text{term}}_\pi (x,y,E)}. $ In order to avoid protocols that never halt, we assume there exists a global constant ${R_{\text{max}}}$ and that for any input and noise pattern ${{\mathsf{RC}}}\le {R_{\text{max}}}$. Finally, we say that a protocol is correct* if both parties output $f(x,y)$. We say that a protocol resists ${\varepsilon}$-fraction of noise (or, resists noise rate ${\varepsilon}$), if the protocol is correct (on any input) whenever the relative noise rate induced by the adversary is at most ${\varepsilon}$. Note that if a protocol resists noise rate of ${\varepsilon}$, and the relative noise in a specific instance is higher than ${\varepsilon}$, there is no guarantee on the output of the parties. Tolerating noise rates up to $1/3$ {#sec:mterm-achieve} ---------------------------------- In this section we show how to use the power of adaptive termination in order to circumvent the $1/4$ bound on the noise of [@BR14]. Below, we provide a protocol that resists noise rate $1/3-{\varepsilon}$ in the ${{{\mathscr{M}}_\text{term}}}$ model. \[thm:protocol-third\] For any function $f$ and any ${\varepsilon}>0$, there exist a protocol $\pi$ for in the ${{{\mathscr{M}}_\text{term}}}$ model, that resist a noise rate of $1/3-{\varepsilon}$. We assume parties’ inputs are in $\{0,1\}^n$. We will use a family of good error correcting codes ${{\mathsf{ECC}}}_i: \{0,1\}^n \to \Sigma^{c_in}$ with $i=1,\ldots, i_{\max}$. Each such code corrects up to $1/2-{\varepsilon}$ fraction of errors while having a constant rate $1/c_i$ and using a constant alphabet $\Sigma$, both of which depend on ${\varepsilon}$. The redundancy of each code increases with $i$, i.e., $c_{i+1} > c_{i}$. Moreover, these codes will have the property that for any $x$, ${{\mathsf{ECC}}}_i(x)$ is a prefix of ${{\mathsf{ECC}}}_{j}(x)$ for any $j>i$. This can easily be done with random linear codes, e.g., by randomly choosing a large generating matrix of size $n\times c_{i_{\max}}n$ and encoding ${{\mathsf{ECC}}}_i$ by using a truncated matrix using only the first $c_i$ columns. Formally, for any $n$ and ${\varepsilon}>0$, let $\{{{\mathsf{ECC}}}_i\}$ be a family of error correcting codes as described above and let $j$ be such that $ c_j\cdot 4{\varepsilon}\ge c_1$. Set $I_A =\{1, \ldots, c_jn \}$ and $I_B =\{ c_jn+1,\ c_jn +2, \ldots\}$. 1. Alice encodes her input using ${{\mathsf{ECC}}}_j$, and sends the codeword over to Bob in the first $c_jn$ rounds of the protocol. 2. After $c_jn$ rounds, Bob decodes Alice’s transmission to obtain $\tilde x$. Let $t$ be the Hamming distance between the codeword Bob receives and ${{\mathsf{ECC}}}_j(\tilde x)$. 3. Bob continues in an adaptive manner: 1. if $t <(1/2-{\varepsilon})c_jn$ Bob encodes his input using a code ${{\mathsf{ECC}}}: \{0,1\}^n \to \{0,1\}^{2c_jn-4t}$. Note that the maximal value $t$ can get is $(1/2-{\varepsilon})c_jn$ which makes $2c_jn-4t > 4{\varepsilon}c_jn \ge c_1n$, so a suitable code can always be found. 2. \[step:bobabort\] otherwise, Bob aborts. 4. After completing his transmission, Bob terminates and outputs $f(\tilde x,y)$. 5. Alice waits until round $3c_jn$ and then decodes Bob’s transmission to obtain $\tilde y$ and outputs $f(x,\tilde y)$. Suppose an instance of the protocol that is not correct, and let us analyze the noise rate in that given failed instance. First, note that if Bob aborts at step \[step:bobabort\], the noise rate is clearly larger than $1/3$. Next, assume Bob decodes a wrong value $\tilde x\ne x$. Note, that the minimal distance of the code is $1-2{\varepsilon}$, thus given that Bob measures Hamming distance $t$, Eve must have made at least $(1-2{\varepsilon})c_jn-t$ corruptions. The total communication in this scenario is $c_jn+2(c_jn-2t)$ which yields a relative noise rate $\frac{(1-2{\varepsilon})c_jn-t}{3c_jn-4t}$, with a minimum of $1/3-O({\varepsilon})$. On the other hand, if Bob decodes the correct value $\tilde x = x$, and measures Hamming distance $t$, Eve must have made $t$ corruptions at Alice’s side. To corrupt Bob’s codeword, she must perform at least $({1/2-{\varepsilon}})(2c_jn-4t)$ additional corruptions, yielding a relative noise rate at least $\frac{t+ (1/2-{\varepsilon})(2c_jn-4t)}{3c_jn-4t}= \frac{(1-2{\varepsilon})c_jn-(1-4{\varepsilon}) t}{3c_jn-4t}$ which also obtains a minimal value of $1/3-O({\varepsilon})$. There is still a remaining subtlety of how Alice knows the right code to decode. Surely, if there is no noise, Bob’s transmission is delimited by silence. However, if Eve turns the last few symbols transmitted by Bob into silence, she might cause Alice to decode with the wrong parameters. This is where we need the prefix property of the code, which keeps a truncated codeword a valid encoding of Bob’s input for smaller parameters. Eve has no advantage in shortening the codeword: if Eve tries to shorten a codeword of ${{\mathsf{ECC}}}_i$ into ${{\mathsf{ECC}}}_j$ with $j<i$ and then corrupt the shorter codeword, she will have to corrupt $(c_i-c_j + (1/2-{\varepsilon})c_j)n \ge (1/2-{\varepsilon})c_in$ symbols, which only increases her noise rate. Similarly, if she tries to enlarge ${{\mathsf{ECC}}}_i$ into ${{\mathsf{ECC}}}_j$ with $j>i$, in order to cause Alice to decode the longer codeword incorrectly, Eve will have to perform at least $(1/2-{\varepsilon})c_j$ corruptions which is again more than needed to corrupt the original message sent by Bob. Impossibility bound ------------------- Next, we show that in the general ${{{\mathscr{M}}_\text{term}}}$ case, no interactive protocol resists a noise rate of $1/2$ or more. At a high level, the attack proceeds by changing $1/2$ of [both]{} Alice and Bob’s messages so that whoever terminates first is completely confused about their partner’s input. One must exercise some care to ensure that the attack is well defined, but this high level idea can be formalized, as shown below. \[thm:mterm-half\] There exists a function $f$, such that any adaptive protocol $\pi$ for $f$ in the ${{{\mathscr{M}}_\text{term}}}$ setting, cannot resist noise rate of $1/2$. Assume $f$ is the identity function on input space $\{0,1\}^n\times \{0,1\}^n$, and consider an adaptive protocol $\pi$ that computes $f$. We show an attack that causes a relative noise rate of at most $1/2$ and causes at least one of the parties to output the wrong value. Fix two distinct inputs $(x,y)$ and $(x',y')$. Given any input $\xi$ out of the set $\{ (x,y), (x',y), (x,y'), (x',y')\}$ we can define an attack on $\pi(\xi)$. The attack will change both parties’ transmissions in the following way: Alice’s messages will be changed to the “middle point” between what she should send given that her input is $x$ and what she should send given that her input is $x'$ (i.e., to a string which has the same Hamming distance from what Alice sends on $x$ and on $x'$). At the same time, Bob’s messages are changed to the middle point between what he should send given that his input is $y$ and what he should send given that his input is $y'$. Specifically, at each time step, Eve considers the next transmission of Alice on $x$ and on $x'$ (given the transcript so far). If Alice sends the same symbol in both cases, Eve doesn’t do anything. Otherwise, Eve alternates between sending a symbol from Alice’s transcript on $x$ and on $x'$. Note that the attack is well defined even if Alice has already terminated on input $x$ but not on $x'$,[^3] although we only use the attack until Alice aborts on one of the inputs. Corrupting Bob’s transmissions is done in a similar way. Next, consider the termination time of the attack on inputs $\{ (x,y), (x',y), (x,y'), (x',y')\}$. There exists an input whose termination time is minimal. Denote this input by $\xi^$ and assume, without loss of generality, that Alice is the party that terminates first when the attack is employed on $\pi(\xi^)$. It follows that when employing the above attack on any of the other three inputs in $\{ (x,y), (x',y), (x,y'), (x',y')\}\setminus\{\xi^\}$, the termination time of the parties are not smaller than Alice’s termination time in the instance $\pi(\xi^)$ under the same attack. Without loss of generality, assume $\xi^=(x,y)$. Finally note that when Alice terminates, she cannot tell whether Bob holds $y$ or $y'$. Indeed, up to the point she terminates, the attack on $\xi^=(x,y)$ and the attack on $(x,y')$ look exactly the same from Alice’s point of view. This is because Bob does not terminate before Alice (for both his inputs!), and our attack changes Bob’s messages in both instances in a similar way. Thus Alice’s view is identical for both Bob’s inputs, and she must be wrong at least on one of them. Note that such an attack causes at most $1/2$ noise in each direction up to the point where Alice terminates (there’s no need to continue in the attack after that point). Thus, the total corruption rate is at most $1/2$. One subtlety that arises from the above proof, is the ability of a party to convey some amount of information by the specific time it terminates. In order to better understand the power of termination yet without allowing the parties to convey information solely by their time of termination, we define the ${{{{\mathscr{M}}_\text{term}}}^{\dagger}}$ model, which is exactly the same as ${{{\mathscr{M}}_\text{term}}}$ defined in Section \[sec:mterm-model\] above, except that if $\min\{{\mathsf{TER}}_A,{\mathsf{TER}}_B\} \ne {R_{\text{max}}}$ then the parties’ output is defined as an invalid output $\bot$. In Appendix \[app:mqwh\] we analyze protocols in the ${{{{\mathscr{M}}_\text{term}}}^{\dagger}}$ model which are fully utilized: at every round both parties send a single symbol over the channel. We show that $1/4$ is a tight bound on the noise in that case. While the protocol of [@BR14] is enough to resist such a noise level (even without using the adaptivity), an impossibility bound of noise $\ge1/4$ is not implied by previous work. In the appendix we prove the following, \[thm:impAbort\] There exists a function $f:\{0,1\}^n\times\{0,1\}^n\to \{0,1\}^{2n}$ such that for any fully utilized adaptive protocol $\pi$ for $f$ in the ${{{{\mathscr{M}}_\text{term}}}^{\dagger}}$ model, $\pi$ does not resist a noise rate of $1/4$. Protocols with an Adaptive Order of Communication {#sec:dsa} ================================================= In this section we extend the power of protocols to adaptively determine the order of speaking as a function of the observed transcript and noise. To this end, at every round each party decides whether to send an additional symbol, or to remain silent. We begin by defining the ${{{\mathscr{M}}_\text{adp}}}$ model. The ${{{\mathscr{M}}_\text{adp}}}$ model {#sec:mmain-model} ---------------------------------------- Similar to the ${{{\mathscr{M}}_\text{term}}}$ model, we assume Alice and Bob wish to compute some function $f: {\mathcal{X}}\times{\mathcal{Y}}\to {\mathcal{Z}}$ where Alice holds some input $x\in {\mathcal{X}}$ and Bob holds $y\in {\mathcal{Y}}$. The sets ${\mathcal{X}},{\mathcal{Y}}$ and ${\mathcal{Z}}$ are assumed to be of finite size. We assume a channel with a finite alphabet $\Sigma$ that can be used by either of the parties at any round. Parties in this model behave as follows (described for Alice, Bob’s behavior is symmetric): - In a given round $i$, Alice decides whether to speak or remain silent. If Alice speaks, she sends a message $a_i \in \Sigma$; if Alice is silent, $a_i={\emptyset}$. - Eve may corrupt any symbol, including the silence symbol, transmitted by either party. Thus, Eve acts upon transmitted symbols via the function ${\mathsf{Ch}}: \Sigma \cup \{{\emptyset}\} \rightarrow \Sigma \cup \{{\emptyset}\}$, conditioned on the parties’ input, Eve’s random coins and the transcript so far. - The corresponding symbol received by Bob is $\tilde{a}_i={\mathsf{Ch}}(a_i)$. - We assume the protocol terminates after a finite time. There exists a number ${R_{\text{max}}}$ at which both parties terminate and output a value as a function of their input and the communication. For a specific instance of the protocol we denote the messages sent by the parties $M=(a_1,b_1,a_2,b_2,\ldots)$ in that instance, and the Noise Pattern $E=(e_{a_1},e_{b_1},\ldots)$ so that $e_{a_i}=\bot$ if ${\mathsf{Ch}}(a_i)=a_i$ and otherwise, $e_{a_i}={\mathsf{Ch}}(a_i)$, and similarly for $e_{b_i}$. We will treat $E$ and $M$ as strings of length $2R_{\max}$ and refer to their $i$-th character as $E_i$ and $M_i$. For any protocol $\pi$ in the ${{{\mathscr{M}}_\text{adp}}}$ model, and for any specific instance of the protocol on inputs $(x,y)$ with noise pattern $E$ we define: 1. Communication Complexity: $ {\mathsf{CC}}^{\text{adp}}_\pi(x,y,E) {\overset{\Delta}{=}}\|\{ i \le 2{R_{\text{max}}}\mid M_i \not= {\emptyset}\}\|, $ where $M$ is the message string observed when running $\pi$ on inputs $(x,y)$ with noise pattern $E$. 2. Noise Complexity: $ {\mathsf{NC}}^{\text{adp}}_\pi(x,y,E) {\overset{\Delta}{=}}\|\{ i \le 2{R_{\text{max}}}\mid E_i \ne \bot \}\|. $ 3. Relative Noise Rate: $ {\mathsf{NR}}^{\text{adp}}_\pi(x,y,E) {\overset{\Delta}{=}}{{\mathsf{NC}}^{\text{adp}}_\pi(x,y,E) }/{{\mathsf{CC}}^{\text{adp}}_\pi(x,y,E) }. $ As before, the protocol is correct if both parties output $f(x,y)$. The protocol is said to resist ${\varepsilon}$-fraction of noise (or, a noise rate of ${\varepsilon}$) if the protocol is correct (on any input) whenever the relative noise rate is at most ${\varepsilon}$. Note that the relative noise rate may exceed $1$. Resilient Protocols in the ${{{\mathscr{M}}_\text{adp}}}$ model --------------------------------------------------------------- In this section, we study several protocols in the ${{{\mathscr{M}}_\text{adp}}}$ model that achieve better noise resilience than in the robust case. The main result of this section is a protocol that tolerates relative noise rates of up to $2/3$. \[thm:ProtocolTwoThirds\] Let ${\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}}$ be some finite sets. For any function $f : {\mathcal{X}}\times {\mathcal{Y}}\to {\mathcal{Z}}$ there exists an adaptive protocol $\pi$ for $f$ in the ${{{\mathscr{M}}_\text{adp}}}$ model that resists noise rates below $2/3$. This protocol builds upon the protocol constructed in Theorem \[thm:protocol-third\] but additionally adds a layer of coding that takes advantage of the partially-utilizing nature of message delivery in this model, which we call silence encoding. More formally, \[def:SE1\] Let $X=\{x_1, x_2, ..., x_n\}$ be some finite, totally-ordered set. The silence encoding is a code $\textit{SE}_1: X\to (\Sigma\cup\{{\emptyset}\})^n $ that encodes $x_i$ into a string $y_1,...,y_n$ where $\forall j\ne i$, $y_j={\emptyset}$ and $y_i\ne {\emptyset}$.\ Intuitively, such an encoding has the property that two transmissions must be corrupted in order to make the receiver decode an incorrect message, while only a single symbol is transmitted. This, along with the technique that tolerates relative noise rates of up to $1/3$ when only the length of the protocol is adaptive, yields the claimed result. See Appendix \[app:freepos\] for full details and proof of Theorem \[thm:ProtocolTwoThirds\]. While the protocol of Theorem \[thm:ProtocolTwoThirds\] obtains noise rate resilience of $2/3$ and very small communication complexity, it has double exponential round complexity with respect to the round complexity of the best noiseless protocol. Our next theorem limits the round complexity to be linear, thus yielding a coding scheme with a non-vanishing rate. Specifically, it shows that for any ${\varepsilon}>0$ we can emulate any protocol $\pi$ of length $T$ (defined in the noiseless model) by a protocol $\Pi$ in the ${{{\mathscr{M}}_\text{adp}}}$ model, which takes at most $O(T)$ rounds and resists noise rate of $1/2-{\varepsilon}$. \[thm:ProtocolHalf\] For any constant ${\varepsilon}>0$ and for any function $f$, there exists an interactive protocol in the ${{{\mathscr{M}}_\text{adp}}}$ model with round complexity $O({\mathsf{CC}}_f)$, that resists a relative noise rate of $1/2-{\varepsilon}$. The protocol follows the emulation technique set forth by Braverman and Rao [@BR14], and requires a generalized analysis for channels with errors and erasures as performed in [@FGOS15] albeit for a completely different setting. The key insight is that silence encoding forces the adversary to pay twice for making an error (or otherwise to cause “only” an erasure). This allows doubling the maximal noise rate the protocol resists. The proof appears in Appendix \[app:mainCRprotocol\]. Finally, we extend the model by allowing the parties to share a random string, unknown to the adversary. We show that shared randomness setup allows the relative noise rate to go as high as $1-{\varepsilon}$. Formally, \[thm:main-shared\] For any small enough constants ${\varepsilon}>0$ and for any function $f$, there exists an interactive protocol in the ${{{\mathscr{M}}_\text{adp}}}$ model with round complexity $O({\mathsf{CC}}_f)$ such that, if the adversarial relative corruption rate is at most $1-{\varepsilon}$, the protocol correctly computes $f$ with overwhelming success probability over the choice of the shared random string. The proof appears in Appendix \[app:protocolSharedRand\]. At a high level, the main idea is to adaptively repeat transmissions that were corrupted by the adversary. This turns each transmission into a varying-length message whose length (i.e., the number of repetitions) is determined by the relative noise at that message. This forces Eve to spend more and more of her budget in order to corrupt a single transmission, since she needs to corrupt all the repetitions that appear in a single message. The shared randomness is used as a means of detecting corruptions (similar to [@FGOS15]), converting most of Eve’s noise into easy to handle erasures. Each detected corruption is replaced with an erasure mark and treated accordingly. It is immediate then, that the same resilience of $1-{\varepsilon}$ holds for protocols over the erasure channel, even when no preshared randomness is available: such channels can only make “erasures” to begin with, so there is no need for preshared randomness in order to detect corruptions. \[cor:erasure\] For any small enough constant ${\varepsilon}>0$ and for any function $f$, there exists an interactive protocol in the ${{{\mathscr{M}}_\text{adp}}}$ model over an erasure channel that has round complexity $O({\mathsf{CC}}_f)$ and that correctly computes $f$ as long as the adversarial relative erasure rate is at most $1-{\varepsilon}$. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Hemanta Maji and Klim Efremenko for useful discussions. Research supported in part from a DARPA/ONR PROCEED award, NSF Frontier Award 1413955, NSF grants 1228984, 1136174, 1118096, and 1065276, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant. This material is based upon work supported by the Defense Advanced Research Projects Agency through the U.S. Office of Naval Research under Contract N00014-11-1-0389. The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, or the U.S. Government. Appendix {#appendix .unnumbered} ======== The ${{{{\mathscr{M}}_\text{term}}}^{\dagger}}$ model: impossibility bounds {#app:mqwh} =========================================================================== In this section we study upper (impossibility) bounds on the admissible noise in the ${{{{\mathscr{M}}_\text{term}}}^{\dagger}}$ model. We consider fully utilized protocols in which both parties send a symbol at every round (i.e., $I_A=I_B=\mathbb{N}$). In this setting we show an impossibility bound of $1/4$ on the amount of tolerable noise, matching the achievable resilience of protocols in this setting [@BR14]. Specifically, we provide an adversarial strategy that always wins with error rate $<1/4$. Note that Braverman and Rao [@BR14] showed a similar result for non-adaptive protocols. Informally speaking, their proof goes along the following lines: Eve picks the player, say Bob, who speaks for fewer slots, and changes half his messages so that the first half corresponds to input $y$ while the second half corresponds to $y'$. Now, Eve’s noise rate is at most $1/4$, and Alice cannot tell whether Bob’s input is $y$ or $y'$ and cannot output the correct value. The above strategy does not carry over to the ${{{{\mathscr{M}}_\text{term}}}^{\dagger}}$ model. Specifically, the above attack is not well defined. Indeed, Eve can inject messages in the first half of the attack, by running Bob’s part of $\pi$ on the input $y$. However, when Eve wishes to switch to $y'$, she now needs to run $\pi$ on input $y'$ given the transcript so far, say, given $\mathsf{tr}(y)$. It is possible that $\pi(\cdot,y')$ conditioned on $\mathsf{tr}(y)$ is not defined, for example upon occurrence of $\mathsf{tr}(y)$ given input $y$, Bob may have already terminated and Eve cannot conduct the second part of the attack. We address this issue by demonstrating a more sophisticated attack that does not abruptly switch $y$ to $y'$ after half the messages, but rather gradually moves from $y$ towards $y'$. That is, at any time during the protocol the adversary’s relative noise rate is at most $1/4$, therefore the parties’ ability to prematurely terminate doesn’t give them any power.[^4] Recall the statement of Theorem \[thm:impAbort\], Theorem \[thm:impAbort\]. There exists a function $f:\{0,1\}^n\times\{0,1\}^n\to \{0,1\}^{2n}$ such that for any adaptive protocol $\pi$ for $f$ in the fully utilized ${{{{\mathscr{M}}_\text{term}}}^{\dagger}}$ model, $\pi$ does not resist a noise rate of $1/4$. Before we prove the theorem we show the following technical lemma, which is the main idea of our proof. Denote the Hamming distance of two strings by ${{\Delta}}(\cdot,\cdot)$. In order to cause ambiguity when decoding a codeword from $\{x,y\}$, one needs to corrupt at most $({{\Delta}}(x,y)+1)/2$ symbols, and this can be done in a “rolling” manner. Formally, \[lem:rollingChange\] Assume $\mathbb{F}$ is some finite field. For any two strings $x,y\in \mathbb{F}^n$ there exists a string $z\in \mathbb{F}^n$ such that $${{\Delta}}(x+z,x) \ge {{\Delta}}(x+z,y)$$ and for any $j\le n$, $w(z_1,\ldots,z_j) \le \frac{j+1}2$, where $w(\cdot)$ is the Hamming weight function. We begin by proving that when ${{\Delta}}(x,y)$ is even, a more restricted form of the lemma holds, namely, that for any $j\le n$, $w(z_1,\ldots,z_j) \le \frac{j}2$. We prove this by induction on the hamming distance $d={{\Delta}}(x,y)$. The case of $d=2$ is easily obtained by setting $z$ to be all zero except for the second index where $x$ and $y$ differ. Now assume the hypothesis holds for an even $d$ and consider $d+2$. Split $x=x_1x_2$ and $y=y_1y_2$ such that $\|x_1\|=\|y_1\|$ and ${{\Delta}}(x_1,y_1)=d$ (thus ${{\Delta}}(x_2,y_2)=2$). Let $u,v$ be the strings guaranteed by the induction hypothesis for $x_1,y_1$ and $x_2,y_2$ respectively, and set $z=uv$. By the way we construct $z$, it holds $ {{\Delta}}(x+z,x) \ge {{\Delta}}(x+z,y). $ Moreover, for any $j<\|x_1\|$ we know that $w(z_1,\ldots,z_j) \le \frac{j}2$, by the induction hypothesis. Note that $w(v)$ is at most $1$, and that $v_1=0$ by the construction of the base case. Then it is clear that the claim holds for $j=\|x_1\|+1$; for any $j>\|x_1\|+1$ we get $w(z_1,\ldots,z_j) = w(u)+w(v_1,...,v_{j-\|x_1\|+1}) \le \frac{\|x_1\|}{2} + 1 \le \frac{j}{2}$. Completing the proof of the original lemma (where $d$ can be odd and the weight is $\le \frac{j+1}2$) is immediate. If $d$ is odd we construct $z$ by using the induction lemma (of the even case) over the prefix with hamming distance $d-1$ and change at most a single additional index, which is located after that prefix. Assume that the prefix is of length $n_{\rm{prefix}}$. The claim holds for any $j \le n_{\rm{prefix}}$ due to the induction hypothesis. For any $j> n_{\rm{prefix}}$ it holds that $$w(z_1,\ldots,z_{j})= w(z_1,\ldots,z_{n_{\rm{prefix}}})+w(z_{n_{\rm{prefix}}+1},\ldots, z_j) \le \frac{n_{\rm{prefix}}}2+1 \le \frac{j+1}{2}.$$ We now continue to proving that $1/4$ is an upper bound of the permissible noise rate. (Theorem \[thm:impAbort\].) Let $f$ be such that for any $y,y'$, $f(x,y)\ne f(x,y')$, for instance, the identity function $f(x,y)=(x,y)$, and let $\pi$ be any adaptive protocol for $f$. Consider the transcripts of $\pi$ up to round 10.[^5] By the pigeon-hole principle for large enough $n$, there must be $y,y'$ that for some $x$ produce the same transcript up to round 10. Let $m = \min \left\{ TER_A(x,y), TER_A(x,y')\right\}$. The basic idea is the following. Assuming no noise, let $t$ be Bob’s messages in $\pi(x,y)$ up to round $m$ and $t'$ be Bob’s messages in $\pi(x,y')$ up to round $m$. Using Lemma \[lem:rollingChange\] Eve can change $t$ into $t+z$ (starting from round 10), so that ${{\Delta}}(t+z,t)\ge {{\Delta}}(t+z,t')$ and Eve’s relative noise rate never exceeds $1/4$. Furthermore, Eve can change $t'$ into $t'+z'=t+z$ and also in this case Eve’s relative noise rate never exceeds $1/4$: the string $z'_{11},...,z'_{m}$ must satisfy, for any index $10< j \le m$, that $w(z'_{11},\ldots,z'_{j})\le \frac{j+1}2$ (this follows from the way we construct $z$ and the fact that ${{\Delta}}(t'+z',t)={{\Delta}}(t+z,t)\ge {{\Delta}}(t+z,t')={{\Delta}}(t'+z',t')$). Thus, the relative noise rate made by Eve up to round $j$ is at most $\frac{(j-10+1)/2}{2j} <1/4$. The same argument should be repeated until we reach the bound on the round complexity ${\mathsf{TER}}_\pi$, which we formally prove in Lemma \[lem:adaptiveBob\], yet before getting to that we should more carefully examine the actions of both parties during this attack. Consider Alice actions when the messages she receives are $t+z=t'+z'$. She can either (i) abort (output $\bot$), (ii) output $f(x,y)$ or (iii) output $f(x,y')$, however, her actions are independent of Bob’s input (since her view is independent of Bob’s input). Assuming Eve indeed never goes beyond $1/4$, it is clear that Eve always wins in case (i). For case (ii) Eve wins on input $(x,y')$ and for case (iii), Eve wins on input $(x,y)$. However, while in the above analysis Alice’s actions must be the same between the two cases of $t\to t+z$ and $t'\to t'+z'$, this is not the case for Bob. We must be more careful and consider Bob’s possible adaptive reaction to errors made by Eve. In other words, Bob, noticing Alice’s replies, may either abort, or send totally different messages so that his transcript is neither $t$ nor $t'$. We now show that even in this case Eve has a way to construct $z,z'$ and never exceed a relative noise rate of $1/4$. \[lem:adaptiveBob\] Assume $\pi$ takes $k$ rounds. Eve always has a way to change (only) messages sent by Bob, so that Alice’s view is identical between an instance of $\pi(x,y)$ and of $\pi(x,y')$, while Eve corrupts no message up to round $10$ and at most $(k-9)/2$ messages between rounds $11$ and $k$ (incl.). We prove by induction. The base case where $k\le 10$ is trivial. Assume the lemma holds for some even $k$, and we prove for $k+1$ and $k+2$. By the induction hypothesis, Eve can cause the run of $\pi(x,y)$ and $\pi(x,y')$ look identical in Alice’s eyes while corrupting at most $(k-9)/2$ messages after round $10$. Let $t$ denote the next two messages (rounds $k+1,k+2$) sent by Bob in the instance of $\pi(x,y)$ and $t'$ in the instance of $\pi(x,y')$.[^6] There are strings $z,z'$ such that $w(z),w(z')\le 1$ and $t+z=t'+z'$. Assume we construct $z$ via the the construction of Lemma \[lem:rollingChange\] then also $z_1=0$ and $z'_2=0$. Also note that $t,t'$ are independent of errors made in rounds $k+1, k+2$ (Bob ‘sees’ that his message at $k+1$ has been changed at round $k+2$ at the earliest, thus this information can affect only his messages at rounds $>k+2$). At round $k+1$ the amount of corrupt messages (in both cases) is at most $$\left\lfloor \frac{k-9}2\right\rfloor + 1 \underset{k\text{ is even}}{=} \frac{k-10}{2}+1 \le \frac{(k+1) -9}{2}$$ And the same holds for round $k+2$ (for both cases). With the above lemma, Eve can always cause Alice to be confused between an instance of $\pi(x,y)$ and $\pi(x,y')$ by inducing, at any point of the protocol, a relative noise rate of at most $$\frac{\frac{k-9}2}{2k} < \frac14.$$ Therefore, unless one of the parties aborts[^7], Alice outputs a wrong output. In all these cases the protocol is incorrect while the noise rate is at most $1/4$. Attempts to extend the above proof to work for the fully utilized ${{{\mathscr{M}}_\text{term}}}$ model runs into a hurdle created by parties’ ability to communicate information about their inputs by the time of aborting. Indeed, in the above attack Alice learns Bob’s inputs (since they were never corrupted), and Bob might be able to distinguish $x$ from $x'$ by whether or not Alice has prematurely aborted (i.e., according to the number of silence symbol implicitly communicated by the channel after Alice terminates). Proof of Theorem \[thm:ProtocolTwoThirds\] {#app:freepos} ========================================== We now show that every function can be computed by an ${{{\mathscr{M}}_\text{adp}}}$ protocol that can suffer noise rates less than $2/3$. The main technique used in this section is a simple code that takes advantage of the ‘silence’ symbols, which we call silence encoding defined in Definition \[def:SE1\] for a simple special case, and below for the general case: Let $X=\{x_1, x_2, ..., x_n\}$ be some finite, totally-ordered set. The $k$-silence encoding is a code $\textit{SE}_k: X\to (\Sigma\cup\{{\emptyset}\})^{kn}$ that encodes $x_i$ into a string $y_1,...,y_{kn}$ where all $y_j={\emptyset}$ except for the $k$ indices $y_{(i-1)k+1}, \ldots, y_{ik}\in \Sigma^k$. Decoding a $k$-silence-encoded codeword is straightforward. The receiver tries to find a message $x_i$ whose encoding minimizes the Hamming distance to the received codeword. If the string that minimizes the distance is not unique, the decoder marks this event as an erasure and outputs $\bot$. The event where the decoder decodes $x_j\ne x_i$ is called an error. Both encoding and decoding can be done efficiently. We note the following interesting property of $k$-silence encoding: in order to cause ambiguity in the decoding (i.e., an erasure), the adversary must change at least $k$ indices in the codeword. Moreover, in order to make the decoder output an incorrect value (i.e., an error), the adversary must make at least $k+1$ changes to the codeword. Specifically for $k=1$, a single corruption always causes an erasure (i.e., ambiguity), while in order to make a decoding error, at least 2 transmissions must have been changed. We are now ready to prove our main theorem of this section. Theorem \[thm:ProtocolTwoThirds\]. Let ${\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}}$ be some finite sets. For any function $f : {\mathcal{X}}\times {\mathcal{Y}}\to {\mathcal{Z}}$ there exists an adaptive protocol $\pi$ for $f$ in the ${{{\mathscr{M}}_\text{adp}}}$ model that resists noise rates below $2/3$. The protocol is composed of two parts, similar to the protocol of Theorem \[thm:protocol-third\]: in the first part Alice communicates her input to Bob and in the second part Bob communicates his input to Alice. After the first part, Bob estimates the error injected and proceeds to the second part only if the noise-rate is low enough to correctly complete the protocol, or is high enough so that the adversary will surely exceed its budget by the time the protocol ends (as these two cases are indistinguishable). In addition, Bob’s message crucially depends on the amount of error Eve introduced in the channel. Assume the channel is defined over some alphabet $\Sigma$ and denote one of the alphabet’s symbols by ‘$\sigma$’. For any $k\in \mathbb{N}$ define $\pi$ on inputs $x_i,y_j \in {\mathcal{X}}\times {\mathcal{Y}}$ in the following way: 1. Alice communicates a $k$-silence encoding of her input, namely, she waits $k\cdot (i-1)$ rounds and then sends the symbol $\sigma$ for $k$ consecutive rounds. 2. Bob waits until round $k\|{\mathcal{X}}\|$ and decodes the codeword sent by Alice. Bob adaptively chooses his actions according to the following cases: 1. if there is ambiguity regarding what $x_i$ is, Bob aborts. 2. otherwise, Bob decodes some $x_{i'}$. Let $t$ be the difference between the number of $\sigma$ symbols Bob received during those rounds that “belong” to $x_{i'}$ and the number of $\sigma$’s received during the rounds that “belong” to a value $x_{i''}$, where $x_{i''}$ is the 2nd best decoding of the received codeword (when decoding by minimizing Hamming distance) Bob communicates his input $y_j$ using the following $2t$-silence encoding: he waits $2k\cdot(j-1)$ rounds and then sends the symbol $\sigma$ for $2t$ consecutive rounds.\ Then, Bob outputs $f(x_{i'},y_j)$ and terminates. 3. Alice waits until round $R_{\max} \triangleq k\|{\mathcal{X}}\|+2k\|{\mathcal{Y}}\|$, and decodes the codeword sent by Bob. If there is ambiguity regarding the value of $y_j$, Alice aborts. Otherwise, she obtains some $y_{j'}$. Alice then outputs $f(x_i, y_{j'})$ and terminates. Let us analyze what happens at round $k\|{\mathcal{X}}\|$. As mentioned above, in order to cause ambiguity at that round, Eve must change at least $k$ transmissions. In this case Bob aborts at round $k\|{\mathcal{X}}\|$; observe that neither of the parties communicates any symbol after round $k\|{\mathcal{X}}\|$, thus their total communication for this instance is $k$ symbols. This implies noise rate of at least $1$. If, on the other hand, at round $k\|{\mathcal{X}}\|$ there was no ambiguity, one of two things must have happened: either Bob correctly decodes Alice’s input, or he decodes a wrong input. First assume the latter, which implies that at least $k+1$ corruptions were done. Since there is no ambiguity, we know that $t>0$ and it must hold that Eve made $e\ge k+t$ corruptions. Then, by the end of the protocol, the relative noise rate is at least $\frac{e}{k+2(e-k)}$. This value decreases as $e$ increases, up till the point where $e=2k$ at which it gets a minimal value of $2/3$. Eve has no incentive to perform more than $e=2k$ corruptions, this will only increase the relative noise rate without changing the actions of Bob. Now assume Bob decodes the correct value, thus Eve wins only if Alice decodes a wrong value from Bob or aborts. We consider two cases. (i) If Eve has corrupted $e<k$ symbols by round $k\|{\mathcal{X}}\|$, then Bob will send his input via $2t$-silence encoding, where $t\ge k-e$. Thus, in order for Alice to decode a wrong value (or abort), Eve must perform at least additional $2t$ corruptions, yielding a relative noise rate of at least $\frac{e+2t}{k+2t}$. Under the constraints that $0 \le e \le k-1$ and $k-e\le t \le k$, it is easy to verify that $$\frac{e+2t}{k+2t} \ge 1- \frac{t}{k+2t} \ge \frac23.$$ (ii) If Eve has made $e\ge k$ corruptions by round $k\|{\mathcal{X}}\|$, yet Bob decoded the correct value, Eve will have to corrupt additional $2t$ symbols to to cause confusion at Alice’s side. This implies a relative noise rate of at least $$\frac{e+2t}{k+2t} \ge \frac{k+2t}{k+2t} =1.$$ Proof of Theorem \[thm:ProtocolHalf\] {#app:mainCRprotocol} ===================================== In this section we prove Theorem \[thm:ProtocolHalf\], and show a protocol with resilience $1/2-{\varepsilon}$ and non-vanishing rate in the ${{{\mathscr{M}}_\text{adp}}}$ Model. First, let us recall some primitives and notations that will be used in our proof. We denote the set $\{1,2,\ldots,n\}$ by $[n]$, and for a finite set $\Sigma$ we denote by $\Sigma^{\le n}$ the set $\cup_{k=1}^n \Sigma^k$. The Hamming distance ${{\Delta}}(x,y)$ of two strings $x,y \in \Sigma^n$ is the number of indices $i$ for which $x_i \ne y_i$, and the Hamming weight of some string, is its distance from the all-zero string, $w(x) = {{\Delta}}(x,0^n)$. Unless otherwise written, $\log()$ denotes the binary logarithm (base 2). A $d$-ary tree-code [@schulman96] over alphabet $\Sigma$ is a rooted $d$-regular tree of arbitrary depth $N$ whose edges are labeled with elements of $\Sigma$. For any string $x\in [d]^{\le N}$, a $d$-ary tree-code ${{\cal T}}$ implies an encoding of $x$, ${\mathsf{TCenc}}(x)=w_1w_2..w_{\|x\|}$ with $w_i\in \Sigma$, defined by concatenating the labels along the path defined by $x$, i.e., the path that begins at the root and whose $i$-th node is the $x_i$-th child of the $(i\!-\!1)$-st node. For any two paths (strings) $x,y\in [d]^{\le N}$ of the same length $n$, let $\ell$ be the longest common prefix of both $x$ and $y$. Denote by ${anc}(x,y)= n-\|\ell\|$ the distance from the $n$-th level to the least common ancestor of paths $x$ and $y$. A tree code has distance $\alpha$ if for any $k\in [N]$ and any distinct $x,y\in [d]^{k}$, the Hamming distance of ${\mathsf{TCenc}}(x)$ and ${\mathsf{TCenc}}(y)$ is at least $\alpha \cdot {anc}(x,y)$. For a string $w \in \Sigma^n$, decoding $w$ using the tree code ${{\cal T}}$ means returning the string $x\in [d]^n$ whose encoding minimizes the Hamming distance to the received word, namely, $${\mathsf{TCdec}}(w) = \operatorname{\arg\!\min}_{x\in [d]^n} \Delta( {\mathsf{TCenc}}(x), w)\text{.}$$ A theorem by Schulman [@schulman96] proves that for any $d$ and $\alpha<1$ there exists a $d$-ary tree code of unbounded depth and distance $\alpha$ over alphabet of size $d^{O(1/(1-\alpha))}$. However, no efficient construction of such a tree is yet known. For a given depth $N$, Peczarski [@peczarski06] gives a randomized construction for a tree code with $\alpha=1/2$ that succeeds with probability at least $1-\epsilon$, and requires alphabet of size at least $d^{O(\sqrt{\log\epsilon^{-1}})}$. Braverman [@braverman12] gives a sub-exponential (in $N$) construction of a tree-code, and Gelles, Moitra and Sahai [@GMS11; @GMS14] provide an efficient construction of a randomized relaxation of a tree-code of depth $N$, namely a potent tree code, which is powerful enough as a substitute for a tree code in most applications. Finally, Moore and Schulman [@MS14] suggested an efficient construction which is based on a conjecture on some exponential sums. We now prove Theorem \[thm:ProtocolHalf\]. For any function $f$ and any constant ${\varepsilon}>0$, we construct a protocol that correctly computes $f$ as long as the relative noise rate does not exceeds $1/2-{\varepsilon}$. Let ${\varepsilon}>0$ be fixed, and let $\pi$ be an interactive protocol in the noiseless model for $f$, in which the parties exchange bits with each other for up to $T$ rounds. We begin by turning $\pi$ into a resilient version $\pi_{BR}$ which resist noise rate of up to $1/4-{\varepsilon}$, using techniques from [@BR14]. The protocol takes $N=O(T)$ rounds in each of which both parties send a message over some finite alphabet $\Sigma$ For every ${\varepsilon}$ there is an alphabet $\Sigma$ of size $O_{\varepsilon}(1)$ such that any binary protocol $\pi$ can be compiled to a protocol $\pi_{BR}$ of $N=O_{\varepsilon}(\|\pi\|)$ rounds in each of which both parties send a symbol from $\Sigma$. For any input $x,y$, both parties output $\pi(x,y)$ if the fraction of errors is at most $1/4-{\varepsilon}$. The conversion is described in [@BR14]; We give more details about this construction in the proof of Lemma \[lem:Nhigh\]. Next, we construct a protocol $\Pi$ that withstands noise rate of $1/2-{\varepsilon}$. The parties run $\pi_{BR}$, yet each symbol from $\Sigma$ is silence-encoded. That is, every round of $\pi_{BR}$ in which a party sends some symbol $a\in \Sigma$ is expanded into $\|\Sigma\|$ rounds of $\Pi$ in which a single symbol ‘$\sigma$’ is sent at a timing that corresponds to the index of $a$ in the total ordering of $\Sigma$. The channel alphabet used in $\Pi$ is thus unary. Decoding is performed by minimizing Hamming distance and the decoder obtains either a symbol of $\Sigma$ or an erasure mark ${\bot}$. From this point and on, we regard only rounds of $\pi_{BR}$ protocol, ignoring the fact that each such ‘round’ is composed of $\|\Sigma\|$ mini-rounds. Denote by ${{\cal N}}(i,j)$ the ‘effective’ noise-rate between rounds $i$ and $j$, for which an erasure is counted as a single error and decoding the wrong symbol of $\Sigma$ is counted as two errors. Formally, assume that at time $n$, Alice sends a symbol $a_n\in \Sigma$, and Bob receives $\tilde{a}_n\in \Sigma \cup \{\bot\}$, possibly with added noise or an erasure mark (similarly, Bob sends $b_n \in \Gamma$, and Alice receives $\tilde{b}_n$). \[def:N\] Let the effective noise in Alice’s transmissions be $${\cal N}_A(i,j) = \| \{ k\mid i\le k \le j, \tilde{a}_k=\bot \}\| + 2\| \{ k \mid i \le k \le j, \tilde{a}_k \notin \{ a_k, \bot\} \}\| \text{,}$$ and similarly define ${\cal N}_B(i,j)$ for the effective noise in Bob’s transmissions. The effective number of corruptions* in the interval $[i,j]$ is ${\cal N}(i,j) = {\cal N}_A(i,j) + {\cal N}_B(i,j) $. The following lemma states that if the $\pi_{BR}$ fails, then ${{\cal N}}$ must be high. \[lem:Nhigh\] Let ${\varepsilon}>0$ be fixed and let $\|\pi_{BR}\|=N$. If $\pi_{BR}$ fails, then $${{\cal N}}(1,N) \ge (1-{\varepsilon})^2 N.$$ With this lemma, the proof of the theorem is immediate: recall that with silence encoding, causing an erasure costs at least one corruption and causing an error costs at least two corruptions. Observe that ${{\mathsf{CC}}^{\text{adp}}}_{\Pi}={{\mathsf{CC}}^{\text{adp}}}_{\pi_{BR}}=2N$, then if the amount of corruptions is limited to $1/2-{\varepsilon}$, $$\max\ {\cal N}(1,N) = (1-2{\varepsilon})N \;\; < (1-{\varepsilon})^2N$$ where the maximum is over all possible noise-patterns of at most $(1/2-{\varepsilon})\cdot 2N$ corruptions. Finally, we give the proof for Lemma \[lem:Nhigh\]. Parts of this analysis were taken as-is from [@FGOS15] and we re-iterate them here (with the authors’ kind permission) for self containment. (Lemma \[lem:Nhigh\].) Let us recall how to construct a constant (non-vanishing) rate protocol $\pi_{BR}$ for computing $f(x,y)$ over a noisy channel out of an interactive protocol $\pi$ for the same task that assumes a noiseless channel [@BR14]. We assume that $\pi$ consists of $T$ rounds in which Alice and Bob send a single bit according to their input and previous transmissions. Without loss on generality, we assume that Alice sends her bits at odd rounds while Bob transmits at even rounds. We can view the computation of $\pi$ as a root-leaf walk along a binary tree in which odd levels correspond to Alice’s messages and even levels to Bob’s, see Figure \[fig:pitree\]. (z)[root]{} child\[thin\] [ child[ child[ edge from parent node \[left\] [0]{} ]{} child[ edge from parent \[dash\] node \[right\] [1]{} ]{} edge from parent \[reg\] node \[left\] [0]{} ]{} child[ child[ edge from parent \[reg\] node \[left\] [0]{} ]{} child[ edge from parent \[tik\] node \[right\] [1]{} ]{} edge from parent \[tik\] node \[right\] [1]{} ]{} edge from parent \[tik\] node \[left\] [0]{} ]{} child [ child[ child[ edge from parent \[reg\] node \[left\] [0]{} ]{} child[ edge from parent \[densely dotted,->\] node \[right\] [1]{} ]{} edge from parent \[dash\] node \[left\] [0]{} ]{} child[ child[ edge from parent \[densely dotted,->\] node \[left\] [0]{} ]{} child[ edge from parent node \[right\] [1]{} ]{} edge from parent node \[right\] [1]{} ]{} edge from parent node \[right\] [1]{} ]{} ; at (-5,-0.7) [Alice]{}; at (-5,-2) [Bob]{}; at (-5,-3.5) [Alice]{}; at (5,0) ; In order to obtain a protocol that withstands (a low rate of) channel noise, Alice and Bob simulate the construction of path $P$ along the $\pi$-tree. The users transmit edges of $P$ one by one, where each user transmits the next edge that extends the partial path transmitted so far. This process is repeated for $N=O_{\varepsilon}(T)$ times. In [@BR14] it is shown that unless the noise rate exceeds $1/4$, after $N$ rounds both parties will decode the entire path $P$. We refer the reader to [@BR14] for a full description of the protocol and correctness proof. We now extend the analysis for the case of channels with errors and erasures. To simplify the explanation, assume that the players wish to exchange, at each round, a transmission over $\Gamma'=\{0,\ldots,N\} \times \{0,1\}^{\le2}$. Intuitively, the transmission $(e,s)\in\Gamma'$ means “extend the path $P$ by taking at most two steps defined by $s$ starting at the child of the edge I have transmitted at transmission number $e$”. Since $\Gamma$ is not of constant size, the symbol $(e,s)$ is not communicated directly over the channel, but is encoded in the following manner. Let $\Gamma = \{ <, 0, 1, >, \}$ and encode each $(e,s)$ into a string $< z>\,\in \Gamma^{\le \log N+2}$ where $z$ is the binary representation of $(e,s)$. Furthermore, assume that $\|\!<z>\!\| \le c\log(e)$ for some constant $c$ we can pick later. Next, each symbol of $\Gamma$ is encoded via a $\|\Gamma\|$-ary tree-code with distance parameter $1-{\varepsilon}$ and label alphabet $\Sigma=O_{\varepsilon}(\|\Gamma\|)$.[^8] At time $n$ Alice sends $a_n\in \Sigma$, the last symbol of ${\mathsf{TCenc}}((e,s)_1,\ldots,(e,s)_n)=a_1a_2\cdots a_n$, and Bob receives $\tilde{a}_n\in \Gamma \cup \{\bot\}$, possibly with added noise or an erasure mark (similarly, Bob sends $b_n \in \Sigma$, and Alice receives $\tilde{b}_n$). Let ${\mathsf{TCdec}}(\tilde{a}_1,\ldots, \tilde{a}_n)$ denote the string Bob decodes at time $n$ (similarly, Alice decodes ${\mathsf{TCdec}}(\tilde{b}_1,\ldots, \tilde{b}_n)$). For every $i>0$, we denote with $m(i)$ the largest number such that the first $m(i)$ symbols of ${\mathsf{TCdec}}(\tilde{a}_1, \ldots, \tilde{a}_i)$ equal to $a_1, \ldots, a_{m(i)}$ and the first $m(i)$ symbols of ${\mathsf{TCdec}}(\tilde{b}_1, \ldots, \tilde{b}_i)$ equal to $b_1, \ldots, b_{m(i)}$. Let ${{\cal N}}$ be as defined in Definition \[def:N\]. We begin by showing that if $m(i) < i$ then many corruptions must have happened in the interval $[m(i)+1, i]$. \[lem:br1\] ${{\cal N}}(m(i)+1,i) \ge (1-{\varepsilon})(i-m(i))$. Assume that at time $i$ Bob decodes the string $a'_1, \ldots, a'_i$. By the definition of $m(i)$, $a'_1,\ldots, a'_{m(i)} = a_1,\ldots, a_{m(i)}$, and assume without loss of generality that $a'_{m(i)+1} \ne a_{m(i)+1}$. Note that the Hamming distance between ${\mathsf{TCenc}}(a_1, \ldots, a_i)$ and ${\mathsf{TCenc}}(a'_1, \ldots, a'_i)$ must be at least ${(1-{\varepsilon})}{(i-m(i))}$. It is immediate that for Bob to make such a decoding error, ${{\cal N}}_A \ge {(1-{\varepsilon})}{(i-m(i))}$. Next, we demonstrate that if some party didn’t announce the $k$-th edge by round $i+1$, it must be that the $(k-1)$-th edge wasn’t correctly decoded early enough to allow completing the transmission of the $k$-th edge. \[lem:br2\] Let $t(i)$ be the earliest time such that both users announced the first $i$ edges of $P$ within their transmissions. For $i\ge 0$, $k \ge 1$, if $t(k) > i+1$, then either $t(k-1) > i-c\log(i-(t(k-1))$, or there exists $j$ such that $t(k-1)>m(j)$ and $i-c\log(i-t(k-1)) < j \le i$. \[The proof is taken from [@BR14], as this claim is independent of the definition of ${{\cal N}}$.\] Without loss of generality, assume that the $k$-th edge of $P$ describes Alice’s move. Suppose that for any $j$ that satisfies $i-c\log(i-t(k-1))<j \le i$ both $t(k-1) \le m(j)$ and $t(k-1) \le i-c\log(i-(t(k-1))$. Then it must be the case that the first $k-1$ edges of $P$ have already been announced, and correctly decoded by Alice for any $j$ in the last $c\log(i-t(k-1))$ rounds, yet the $k$th edge has not. However, by the protocol definition, Alice should announce this edge, and this takes her at most $c\log(i-t(k-1))$ rounds, thus by round $i+1$ she has completed announcing it, in contradiction to our assumption that $t(k) > i+1$. Finally, we relate the effective noise rate with the progress of the protocol. \[lem:br3\] For $i\ge -1$, $k\ge 0$, if $t(k) > i+1$, then there exist numbers $\ell_1, \ldots, \ell_k \ge 0$ such that $\sum_{s=1}^k \ell_s \le i+1$ and ${{\cal N}}(1,i)\ge (1-{\varepsilon})(i-k+1-\sum_{s=1}^kc\log(\ell_s+2))$. We prove by induction. The claim trivially holds for $k=1$ and for $i\le0$ by choosing $\ell_s=0$. Otherwise, by Lemma \[lem:br2\] there are two cases. The first case is that $t(k-1) > i-c\log(i-t(k-1))$. Let $i'=t(k-1)-1$ and $k'=k-1$, thus by the induction hypothesis (on $i',k'$), there exist $\ell_1, \ldots, \ell_{k-1} \ge 0$ with $\sum_{s=1}^{k-1}c\log( \ell_s) \le t(k+1)$ such that $$\begin{aligned} {{\cal N}}(1,i) \ge {{\cal N}}(1,'i)& \ge (1-{\varepsilon})\left( \left(t(k-1)-1\right)-(k-1)+1-\sum_{s=1}^{k-1} c\log(\ell_s +2)\right) \\ & = (1-{\varepsilon})\left( i-k+1-\sum_{s=1}^{k-1} c\log(\ell_s +2) - (i-t(k-1))\right)\end{aligned}$$ Set $\ell_k= i-t(k+1)$ to complete this case. In the other case there exists $j$ such that $m(j)< t(k-1)$ and $i-c\log(i-t(k-1)) < j \le i$. In this case we can write $${{\cal N}}(1,i) = {{\cal N}}(1,m(j)) + {{\cal N}}(m(j)+1,i).$$ The second term is lower bounded by ${{\cal N}}(m(j)+1,j)$, which by Lemma \[lem:br1\] is lower bounded by $(1-{\varepsilon})(j-m(j))$. We use the induction hypothesis to bound the first term (with $i'=m(j)-1$ and $k'=k-1$) to get $$\begin{aligned} {{\cal N}}(1,m(j)) \ge {{\cal N}}(1,m(j)-1) &\ge (1-{\varepsilon})\left( m(j)-1 -(k-1)+1- \sum_{s=1}^{k-1}c\log(\ell_s+2) \right) \\ & = (1-{\varepsilon})\left(j -k+1- \sum_{s=1}^{k-1}c\log(\ell_s+2) -j +m(j) \right)\end{aligned}$$ for $\ell_1, \ldots, \ell_{k-1} \ge 0$ such that $\sum_{s=1}^{k-1}\ell_s < m(j)$. Take $\ell_k=i-t(k-1)$. Since $t(k-1)\ge m(j)$ we get that $\sum_{s=1}^{k}\ell_s < m_j + (i-m(j)) < i +1$ and $$\begin{aligned} {{\cal N}}(1,i) &\ge {{\cal N}}(1,m(j)) + {{\cal N}}(m(j)+1,i) \\ & \ge (1-{\varepsilon}) \left( j-k+1 - \sum_{s=1}^{k-1}c\log(\ell_s+2) \right) \\ & \ge (1-{\varepsilon}) \left( i-k+1 - \sum_{s=1}^{k-1}c\log(\ell_s+2) -i+j \right)\end{aligned}$$ Which completes the proof since for this case $i-j < c\log(i-t(k-1))=c\log(\ell_k)$. We can now complete the proof of Lemma \[lem:Nhigh\]. Suppose the protocol $\pi_{BR}$ has failed, thus $m(N) <t(T)$. By Lemma \[lem:br3\] we have $\ell_1,\ldots,\ell_T \ge 0$ that satisfy $\sum_{s=1}^T \ell_s < m(N) < N$ and $$\begin{aligned} {{\cal N}}(1,N) &\ge {{\cal N}}(1,m(N)-1) + {{\cal N}}(m(N)+1,N) \\ & \ge (1-{\varepsilon}) (m(N)-T-\sum_{s=1}^T c\log(\ell_s+2)) + (1-{\varepsilon})(N-m(N)) \\ & \ge (1-{\varepsilon})\left(N - T - cT\log\left(\frac1T\sum_{s=1}^T (\ell_s+2)\right)\right) \\ &\ge (1-{\varepsilon})\left(N - T - cT\log\left(\frac{m(N)}T+2\right)\right),\end{aligned}$$ where the second transition is due to the concavity of the $\log$ function. Setting, for instance, $N=T\frac{c^2}{{\varepsilon}}\log({\varepsilon}^{-1})$ gives $$\begin{aligned} {{\cal N}}(1,N) & \ge (1-{\varepsilon})\left(N - {\varepsilon}N / c^2\log({\varepsilon}^{-1}) - {\varepsilon}N \log(3N/T)/c\log({\varepsilon}^{-1})\right) \\ &= (1-{\varepsilon})\left (N - {\varepsilon}N / c^2\log({\varepsilon}^{-1}) - {\varepsilon}\log(3c^2{\varepsilon}^{-1}\log({\varepsilon}^{-1}))/c\log({\varepsilon}^{-1}) \right) \\ &= (1-{\varepsilon})\left (1- {\varepsilon}\frac{1+\log({\varepsilon}^{-1}) + c\log(3c^2\log({\varepsilon}^{-1}))}{c^2\log({\varepsilon}^{-1})} \right) N \\ &> (1-{\varepsilon})^2N ,\end{aligned}$$ for a large enough constant $c$. Proof of Theorem \[thm:main-shared\] {#app:protocolSharedRand} ==================================== In this section we provide the detailed proof for Theorem \[thm:main-shared\]. For convenience, we re-state the theorem below. Theorem \[thm:main-shared\]. For any small enough constants ${\varepsilon}>0$ and for any function $f$, there exists an interactive protocol in the ${{{\mathscr{M}}_\text{adp}}}$ model with round complexity $O({\mathsf{CC}}_f)$ such that, if the adversarial relative corruption rate is at most $1-{\varepsilon}$, the protocol correctly computes $f$ with overwhelming success probability over the choice of the shared random string. We begin with a short motivation for our construction. Our starting point is the protocol of Theorem \[thm:ProtocolHalf\], i.e., concatenating [@BR14] with silence encoding. We need to deal with two issues: deletion of labels (erasures) and altering labels (errors). First we take care of the errors, which is done by the technique of the so called Blueberry code [@FGOS15]. A [Blueberry code]{}with parameter $q$ encodes each symbol in $\Sigma$ into a random symbol in $\Gamma$, where $\|\Sigma\|/\|\Gamma\| < q$. Since the mapping $\Sigma\to\Gamma$ is unknown to the adversary, any change to the coded label will be detected with probability $1-q$ and the transmission will be considered as an erasure. By choosing $q$ to be small enough (as a function of ${\varepsilon}$), we can guarantee that the adversary cannot do to much harm by changing symbols. Next, we need to deal with the more problematic issue of erasures. The problem is that the [@BR14] protocol is symmetric, that is, Alice and Bob speak the same amount of symbols. Thus, a successful corruption strategy with relative noise rate $1/2$ just deletes all Alice’s symbols. To overcome this issue we need to “break” the symmetry. We will do that by sending indication of deleted labels: If Alice’s label was deleted, Bob will tell her so, and she will send more copies of the deleted label. If all of these repeated transmissions are deleted again, Bob will indicate so and Alice will send again more and more copies of that label. This continues until the total amount of re-transmissions surpasses the amount of transmissions in the noiseless scenario. This breaks the symmetry: if Eve wishes to delete all the copies she will end up causing Alice to speak more, which forces Eve to delete the additional communication as well, which in turn forces her into increasing the average relative noise rate she introduces. The remaining issue is to prevent Eve from causing the parties to communicate many symbols without her making many corruptions, e.g., by forging Bob’s feedback to make Alice send unnecessary copies of her labels. This is prevented by the [Blueberry code]{}: such an attack succeeds with very small probability that makes it unaffordable. (Theorem \[thm:main-shared\].) The protocol is based on the protocol of Theorem \[thm:ProtocolHalf\] (i.e., on the scheme of [@BR14]), yet replacing each label transmission with an adaptive subprotocol that allows retransmissions of deleted symbols. Furthermore, each transmission is encoded via a [Blueberry code]{} [@FGOS15], which allows the parties to notice Eve’s attack most of the times. Fix ${\varepsilon}>0$ to be some small enough constant. For any noiseless protocol $\pi$ of length $T$ we will simulate $\pi$ in the ${{{\mathscr{M}}_\text{adp}}}$ model using the following procedure, defined with parameters $k={\varepsilon}^{-1}, t=k{\varepsilon}^{-1}, q < (kt)^{-2}$. 1. The parties perform the scheme [@BR14] for $N=O(T/{\varepsilon})$ rounds. 2. In each round, each label is transmitted via the following process: 1. The sender encodes the label via a [Blueberry code]{}with parameter $q$, and sends it encoded with a $k$-silence encoding. 2. Repeat for $t$ times: - if the receiver hasn’t received a valid label, he sends back a “repeat-request” encoded using the [Blueberry code]{}and a $1$-silence encoding. \[Otherwise, he does nothing.\] - each time the sender gets a valid repeat-request, he sends the original message transmission again encoded with a (fresh instance of) [Blueberry code]{}, and $k$-silence encoding. 3. The receiver sets that round’s output to be the first valid label he received during step (b), or $\bot$ if all the $t$ repetitions are invalid (either erased, or marked invalid by the [Blueberry code]{}). First we note that indeed the protocol takes at most $O(N)=O(T)$ rounds, since the BR protocol takes $O(N)$ rounds, each of which is expended by at most $O(kt\|\Gamma\|)=O_{\varepsilon}(1)$ rounds. We now show that the above protocol achieves noise rate up to $1-O({\varepsilon})$. We split the protocols into epochs , where each epoch corresponds to a single label transmission of the [@BR14] simulation (i.e., the label and its repetitions are a single epoch). Next, we divide the epochs into two disjoint sets: deleted and undeleted epochs. The former consists of any epoch in which the receiver outputs $\bot$. The set of undeleted epochs is split again into two disjoint sets: “lucky” and “non-lucky”. A lucky epoch is any epoch in which Eve’s corruption is not detected by the [Blueberry code]{}. For each epoch $e$ we define $c(e)$ as the communication made by the parties in this epoch, and $r(e)$ as the rate of noise made by Eve in this epoch, that is, the number of corruptions Eve makes in the epoch is $r(e)c(e)$. The global noise rate is a weighted average of the noise rate per epoch, where each epoch is weighted by the communication in that epoch. Fix a noise pattern $E$ for the adversary, and assume that the simulation process fails with noise pattern $E$. Lemma \[lem:Nhigh\] tells us that the number of deleted epochs plus twice the number of incorrect epochs (where the receiver outputs a wrong label) must be at least $(1-{\varepsilon})^2N$. Note that incorrect epoch must be lucky, and the probability for an epoch to be lucky is at most $q\cdot 2t$, since there are at most $2t$ messages in each epoch and a probability at most $q$ to break the [Blueberry code]{}for a single message. Hence, for our choice of $q$, the number of deleted epochs is at least $(1-3{\varepsilon})N$ with overwhelming probability. First, we analyze deleted epochs. The following is immediate. In each deleted epoch $e$, the noise rate is $$r(e) \ge \frac{k(t+1)}{(k+1)t+k} = 1-\frac{t}{kt+t +k} \ge 1-{\varepsilon}$$ while the communication is $c(e) \ge k+t > {\varepsilon}^{-2}+{\varepsilon}^{-1}$. Next we analyze the non-deleted epochs. We note that $c(e)$ in this case ranges between ${\varepsilon}^{-1}$ to ${\varepsilon}^{-3}$, and we now relate $r(e)$ to $c(e)$. It is crucial that whenever $c(e)$ exceeds ${\varepsilon}^{-2}$, the amount of noise will be high enough, to maintain a global noise rate of almost one. First, we deal with “lucky” epochs. For simplification, we assume that if $e$ is “lucky”, then the noise rate is 0, and the communication is the maximal possible $kt+t+1$. We will choose $q$ to satisfy $2qt\cdot (kt+t+1) \ll 1$, so that the effective noise added by such epochs is negligible. Then, we need to relate $c(e)$ and $r(e)$ for the rest of the epochs. If $e$ is a non-lucky non-deleted epoch, then $$r(e)c(e) \ge \max \big ( 0\, ,\, c(e) - 2k -1 \big ).$$ If the epoch is not lucky, then it must have concluded correctly (Bob has eventually received the correct label). Thus the only attack Eve can perform in order to increase $c(e)$, is to delete Bob’s reply-request and Alice’s answers up to some point (in addition to corrupting Alice original label). Thus, Eve must block the first $k$ symbols sent by Alice (as long as $c(e)>k$), but she must not block the last $k$ symbols made by Alice. Assume Eve only blocks Bob’s reply-requests. Then she can block $x \le t-1$ such requests and make $x$ corruptions out of total communication $x+2k+1$. Another possible attack is to let $y$ of Bob’s repeat-request go through (as long $y+x \le t-1$) but delete Alice’s replies (again, except for the last one). This will cause $yk+x$ corruptions out of communication $(y+2)k+x+1$. The relative noise rate caused by any noise pattern $E$ that results in a failed instance of the simulation is thus bounded by $$\begin{aligned} {\mathsf{NR}}^{\text{adp}}(E) &\ge \frac{\sum_{e:\text{ deleted}}r(e)c(e)+\sum_{e: \text{ lucky}}0 + \sum_{e:\text{ correct non-lucky}}\max(0 , c(e)-2k-1)}{\sum_e c(e) } \\ &\ge \frac{(1-{\varepsilon})\sum_{e: \text{ deleted}}c(e) + \sum_{e:\text{ correct non-lucky}}\max(0 , c(e)-2k-1)}{ \sum_{e: \text{ deleted}}c(e) + \sum_{e: \text{ lucky}}(kt+t+1)+\sum_{e:\text{ correct non-lucky}}c(e)}.\end{aligned}$$ Recall that with very high probability $>(1-3{\varepsilon}) N$ epochs are deleted and at most $2qtN$ epochs are lucky. Thus $ \sum_{e: \text{ lucky}}(kt+t+1)$ is upper bounded by $2qt(kt+t+1)N$ with high probability. We take $q \ll 1/2t(kt+t+1)$ and neglect this term in the denominator. Now split the correct non-lucky epochs to two sets: $B_0=\{ e \text{} \mid c(e) \le {\varepsilon}^{-1.5}\}$ contains epochs with “low” communication and $B_1$ contains all the other correct non-lucky epochs (with “high” communication). For small enough ${\varepsilon}$, $$\begin{aligned} {\mathsf{NR}}^{\text{adp}}(E)&\gtrapprox \frac{(1-{\varepsilon})\sum_{e: \text{ deleted}}c(e) + \sum_{B_1}(c(e)-2{\varepsilon}^{-1})} {\sum_{e: \text{ deleted}}c(e) +\|B_0\|{\varepsilon}^{-1.5} + \sum_{B_1}c(e)} \\ & \ge \frac{(1-{\varepsilon})\sum_{e: \text{ deleted}}c(e) - \|B_1\|2{\varepsilon}^{-1}} {\sum_{e: \text{ deleted}}c(e) +\|B_0\|{\varepsilon}^{-1.5} } \\ &\ge 1-O({\varepsilon}),\end{aligned}$$ since $\sum_{e: \text{ deleted}}c(e) \ge (1-3{\varepsilon})({\varepsilon}^{-2}+{\varepsilon}^{-1})N$ and $\|B_0\|+\|B_1\| \le (1+3{\varepsilon})N$, with very high probability. [^1]: A similar notion of party keeping silent was used in interactive protocols over noiseless channels, by [@DFO10; @IW10]. [^2]: It is easy to show that unless we give the adversary the power to insert and delete symbols, the model is too strong and the question of resisting noise becomes trivial: in that case the protocol can encode a ‘0’ as a silent transmission, and a ‘1’ as a non-silent transmission, thus perfectly resisting any possible noise. [^3]: Recall that once Alice terminates, we assume the symbol $\emptyset$ is being transmitted by the channel. [^4]: In fact, the bound we obtain is $1/4-O(1/k)$ where $k$ is the round complexity of $\pi$. Therefore, the only hope to obtain protocols that resist any noise rate strictly less than $1/4$ is having infinite protocols. This is however beyond the scope of this work, and is left as an open question. [^5]: 10 is obviously arbitrary, and has the sole purpose of avoiding the edge case in which Eve corrupts the first couple of rounds, possibly causing (a relative) noise rate higher than $1/4$. [^6]: Note that $t,t'$ are conditioned on the noise Eve has introduced throughout round $k$. [^7]: As before, Eve needs not corrupt any message after one of the parties aborts, since she is always within her budget. [^8]: On top of the tree-code encoding, we implicitly perform silence encoding of every symbol in $\Sigma$.
--- abstract: 'We study unstable epitaxy on singular surfaces using continuum equations with a prescribed slope-dependent surface current. We derive scaling relations for the late stage of growth, where power law coarsening of the mound morphology is observed. For the lateral size of mounds we obtain $\xi \sim t^{1/z}$ with $z \! \geq \! 4$. An analytic treatment within a self–consistent mean–field approximation predicts multiscaling of the height–height correlation function, while the direct numerical solution of the continuum equation shows conventional scaling with $z=4$, independent of the shape of the surface current.' author: - \| Martin Rost[^1] and Joachim Krug\ [Fachbereich Physik]{}\ [Universität GH Essen]{}\ [D-45117 Essen, Germany]{} title: Coarsening of Surface Structures in Unstable Epitaxial Growth --- Introduction ============ On many crystal surfaces step edge barriers are observed which prevent interlayer (downward) hopping of diffusing adatoms [@Ehrlich; @Schwoebel]. In homoepitaxy from a molecular beam this leads to a growth instability which can be understood on a basic level: Adatoms form islands on the initial substrate and matter deposited on top of them is caught there by the step edge barrier. Thus a pyramid structure of islands on top of islands develops. At late stages of growth pyramids coalesce and form large “mounds”. Their lateral size $\xi$ is found experimentally to increase according to a power law in time, $\xi \sim t^{1/z}$ with $z \simeq 2.5$ – 6 depending on the material and, possibly, deposition conditions used. A second characteristic is the slope of the mounds’ hillsides $s$, which is observed to either approach a constant (often referred to as a “magic slope” since it does not necessarily coincide with a high symmetry plane) or to increase with time as $s \sim t^{\alpha}$ [@exps; @advances]. The surface width (or the height of the mounds) then grows as $w \sim s \xi \sim t^\beta$ with $\beta = 1/z + \alpha$, where $\alpha = 0$ for the case of magic slopes. On a macroscopic level these instabilities can be understood in terms of a growth-induced, slope-dependent surface current [@Villain; @KPS]. Since diffusing adatoms preferably attach to steps from the terrace [below]{}, rather than from [above]{}, the current is uphill and destabilizing. The concentration of diffusing adatoms is maintained by the incoming particle flux; thus, the surface current is a nonequilibrium effect. The macroscopic view is quantified in a continuum growth equation, which has been proposed and studied by several groups [@Johnson; @Stroscio; @Siegert1; @Siegert2; @Sander; @masch; @paveljoachim]. The goal of the present contribution is to obtain analytic estimates for the scaling exponents and scaling functions of this continuum theory. To give an outline of the article: In the next section we briefly introduce the continuum equations of interest. A simple scaling ansatz, presented in Section 3, leads to scaling relations and inequalities for the exponents $1 \! / \! z, \alpha$ and $\beta$. In Section 4 we present a solvable mean–field model for the dynamics of the height–height correlation function. Up to logarithmic corrections, the relations of Section 3 are corroborated. Finally, in the concluding Section 5 the mean–field correlation functions are compared to numerical simulations of the full growth equation, and the special character of the mean–field approximation is pointed out. Continuum equation for MBE ========================== Under conditions typical of molecular beam epitaxy (MBE), evaporation and the formation of bulk defects can be neglected. The height $H({\bf x},t)$ of the surface above the substrate plane then satisfies a continuity equation, $$\label{cont1} \partial_t H + \; \; \nabla \! \! \cdot \! \! {\bf J}_{\mbox{surface}}\{ H \} = F,$$ where $F$ is the incident mass flux out of the molecular beam. Since we are interested in large scale features we neglect fluctuations in $F$ (“shot noise”) and in the surface current (“diffusion noise”). In general, the systematic current ${\bf J}_{\mbox{surface}}$ depends on the whole surface configuration. Keeping only the most important terms in a gradient expansion[^2], subtracting the mean height $H \! = \! Ft$, and using appropriately rescaled units of height, distance and time [@paveljoachim], Eq. (\[cont1\]) attains the dimensionless form $$\label{cont2} \partial_t h = - (\nabla^2)^2 h - \nabla \cdot f(\nabla h^2) \; \nabla h.$$ The linear term describes relaxation through adatom diffusion driven by the surface free energy [@Mullins], while the second nonlinear term models the nonequilibrium current [@Villain; @KPS]. Assuming in-plane symmetry, it follows that the nonequilibrium current is (anti)parallel to the local tilt $\nabla h$, with a magnitude $f(\nabla h^2)$ depending only on the magnitude of the tilt. We consider two different forms for the function $f(\nabla h^2)$: \(i) Within a Burton-Cabrera-Frank-type theory [@advances; @masch; @Politi], for small tilts the current is proportional to $\| \nabla h \|$, and in the opposite limit it is proportional to $\| \nabla h\|^{-1}$. This suggests the interpolation formula [@Johnson] $f(s^2) = 1/(1+s^2)$. Since we are interested in probing the dependence on the asymptotic decay of the current for large slopes, we consider the generalization $$\label{model(i)} f(s^2) = 1/( 1 + \|s\|^{1 + \gamma}) \;\;\; {\rm [model \; (i)]}.$$ Since $\gamma = 1$ also in the extreme case of complete suppression of interlayer transport [@masch; @wedding], physically reasonable values of $\gamma$ are restricted to $\gamma \geq 1$. \(ii) Magic slopes can be incorporated into the continuum description by letting the nonequilibrium current change sign at some nonzero tilt [@KPS; @Siegert1; @Siegert2]. A simple choice, which places the magic slope at $s^2=1$, is $$\label{model(ii)} f(s^2) = 1 - s^2 \;\;\; {\rm [model \; (ii)]};$$ a microscopic calculation of the surface current for a model exhibiting magic slopes has been reported by Amar and Family [@amar]. The stability properties of a surface with uniform slope ${\bf m}$ are obtained by inserting the ansatz $h({\bf x},t) = {\bf m \! \cdot \! x} + \epsilon({\bf x},t)$ into (\[cont2\]) and expanding to linear order in $\epsilon$. One obtains $$\label{linstab} \partial_t \epsilon = \bigl[ \nu_\\| \partial_\\|^2 + \nu_\perp \partial_\perp^2 - (\nabla^2)^2 \bigr] \epsilon,$$ where $\partial_\\| (\partial_\perp)$ denotes the partial derivative parallel (perpendicular) to the tilt [m]{}. The coefficients are $\nu_\\| = -(d/d\|{\bf m}\|) \|{\bf m}\|f({\bf m}^2)$ and $\nu_\perp = - f({\bf m}^2)$. If one of them is negative, the surface is unstable to fluctuations varying in the corresponding direction: Variations perpendicular to [m]{} will grow when the current is uphill (when $f > 0$), while variations in the direction of [m]{} grow when the current is an increasing function of the tilt. Both models have a change in the sign of $\nu_\\|$, model (i) at $\|{\bf m}\| = \gamma^{-1/(1+\gamma)}$, model (ii) at $\|{\bf m}\| = 1/\sqrt{3}$. For model (i) $\nu_\perp < 0$ always, corresponding to the step meandering instability of Bales and Zangwill [@paveljoachim; @Bales]. In contrast, for model (ii) the current is downhill for slopes $\|{\bf m}\| > 1$, and these surfaces are absolutely stable. In this work we focus on singular surfaces, ${\bf m} \! = \! 0$, which are unstable in both models; coarsening behavior of vicinal surfaces has been studied elsewhere [@paveljoachim]. The situation envisioned in the rest of this article is the following: For solutions of the PDE (\[cont2\]) we choose a flat surface with small random fluctuations $\epsilon({\bf x})$ as initial condition. Mostly the initial fluctuations will be uncorrelated in space, though the effect of long range initial correlations is briefly addressed in Section \[meanfield\]. The fluctuations are amplified by the linear instability, and eventually the surface enters the late time coarsening regime that we wish to investigate. Scaling Relations and Exponent Inequalities {#scalingansatz} =========================================== In this section we assume that in the late time regime the solution of (\[cont2\]) is described by a scaling form, namely that the surface $h({\bf x},t)$ at time $t$ has the same (statistical) properties as the rescaled surface $\tau^{-\beta} \; h({\bf x}/\tau^{1/z},\tau t)$ at time $\tau t$. The equal time height–height correlation function $G({\bf x},t) \equiv \langle h({\bf x},t) \; h(0,t) \rangle$ then has a scaling form $$G({\bf x},t) = w(t)^2 g(\|{\bf x}\|/\xi(t)),$$ where the relevant lengthscales are the surface width $w(t) \! = \! \langle h({\bf x},t)^2 \rangle^{1/2} \! \sim \! t^{\beta}$, i.e. the typical height of the mounds, and their lateral size $\xi(t) \! \sim \! t^{1/z}$, given by the first zero of $G$. These choices correspond to $g(0) \! = \! 1$ and $g(1) \! = \! 0$. Moreover they lead to a definition of the typical slope of mounds as $ s \equiv w/\xi \sim t^\alpha $ with $\alpha = \beta - 1/z$. We start our reasoning with the time dependence of the width $$\label{widthdyn} \frac{1}{2} \partial_t w^2(t) = - \langle \left( \Delta h({\bf x},t) \right)^2 \rangle + \langle \nabla h({\bf x},t)^2 f( \nabla h({\bf x},t)^2 ) \rangle \equiv - I_1 + I_2.$$ Clearly $I_1 \geq 0$. Since we expect the width to increase with time, we obtain the inequalities $$\label{ineqa} 0 \leq \frac{1}{2} \partial_t w^2(t) \leq I_2$$ and $$\label{ineqb} I_1 \leq I_2.$$ The first conclusion can be drawn even without the scaling assumption: For model (ii) and model (i) with $\gamma \geq 1$, $(\nabla h)^2 f(\nabla h^2)$ has an upper bound, and so has $I_2$. Therefore $\partial_t w^2 \leq const$. We conclude that the increase of the width $w(t)$ cannot be faster than $t^{1/2}$ if it is caused by a destabilizing nonequilibrium current on a surface with step edge barriers. Assuming scaling we estimate $I_1 \sim (s/\xi)^2$ and $\partial_t w^2 \sim w^2/t \sim (s \xi)^2/t$. For model (i) we further have $I_2 \sim s^2 f(s^2) \sim s^{1-\gamma}$. In terms of the scaling exponents $\alpha$ and $1/z$ inequality (\[ineqa\]) yields $2(\alpha + 1/z) -1 \leq \alpha (1 - \gamma)$, while the second inequality (\[ineqb\]) leads to $2 \alpha - 2/z \leq \alpha ( 1 - \gamma)$. Combining both inequalities we have $$\label{ineq1} \frac{1 + \gamma}{2} \; \alpha \; \; \; \leq \; \; \; \frac{1}{z} \; \; \; \leq \; \; \; \frac{1}{2} - \frac{1 + \gamma}{2} \; \alpha.$$ To proceed we note that an upper bound on the lateral mound size $\xi$ can be obtained from the requirement that the mounds should be stable against the Bales-Zangwill step meandering instability [@paveljoachim; @Bales]: Otherwise they would break up into smaller mounds. >From (\[linstab\]) it is easy to see that, for the large slopes of interest here, fluctuations of a wavelength exceeding $2 \pi / \sqrt{\|\nu_\perp \|}$ are unstable. Since $- \nu_\perp = f(s^2) \sim s^{-(1+\gamma)}$, we impose the condition $\xi \leq 2 \pi / \sqrt{\|\nu_\perp \|} \sim m^{(1 + \gamma)/2}$ or, in terms of scaling exponents, $$\label{ineq2} \frac{1}{z} \; \; \; \leq \; \; \; \frac{1 + \gamma}{2} \; \alpha.$$ Hence the first relation in (\[ineq1\]) becomes an equality (which was previously derived for the one-dimensional case [@advances]), and the second relation yields $$\label{scalineq} z \geq 4, \; \; \; \alpha \; = \; \frac{2}{z(1 + \gamma)} \; \leq \; \frac{1}{2(1 + \gamma)}, \; \; \; \beta \; \leq \; \frac{3 + \gamma}{4(1 + \gamma)}. \;\;\; [{\rm model \; (i)}]$$ For model (ii) we assume that the slope $s$ approaches its stable value $s=1$ as $s \sim 1 - t^{-\alpha'}$ with $\alpha' > 0$. The estimate of the last term in (\[widthdyn\]) then becomes $I_2 \sim s^2 (1-s^2) \simeq 1 - s^2 \sim t^{-\alpha'}$. Thus inequality (\[ineqa\]) yields $2/z - 1 \leq - \alpha'$, and from (\[ineqb\]) it follows follows that $-2/z \leq - \alpha'$. As for model (i) the next estimation uses $\xi \leq 2 \pi/\sqrt{\|\nu_\perp\|}$ with $\|\nu_\perp\| = 1 - s^2 \sim t^{-\alpha'}$. Again we obtain the inverse of the second of the above inequalities, [viz.]{} $1/z \leq \alpha' /2$. Altogether this yields $$\label{scalineq2} \frac{1}{z} = \frac{\alpha'}{2} = \beta \leq \frac{1}{4}. \;\;\; [{\rm model \; (ii)}]$$ To summarize the general results obtained in this section: In addition to the bound on the temporal increase of the surface width, $w(t) < const. \; t^{1/2}$, the scaling ansatz yields an upper bound on the increase of the lateral length scale, $\xi(t) < const. \; t^{1/4}$, valid for both models. A more elaborate approximation, to be presented in the next section, predicts the above inequalitites (\[scalineq\],\[scalineq2\]) to hold as equalities (up to logarithmic corrections). Spherical Approximation {#meanfield} ======================= We consider the time dependence of the equal time height-height correlation function defined above: $$\label{corrdyn} \partial_t G({\bf x},t) = - 2 \; \Delta^2 G({\bf x},t) - 2 \; \nabla \cdot \langle h(0,t) \; f(\nabla h({\bf x},t)^2) \; \nabla h({\bf x},t) \rangle,$$ where $\Delta = \nabla^2$ is the Laplace operator. In order to obtain a closed equation for $G({\bf x},t)$ we replace $f(\nabla h^2)$ by $f(\langle \nabla h^2 \rangle)$ in the second term on the right hand side. This approach is inspired by the spherical ‘large $n$’ limit of phase ordering kinetics [@bray], and will be referred to as the spherical approximation. The argument of $f$ is then easily expressed in terms of $G$: $$\langle \nabla h({\bf x},t)^2 \rangle = - \Delta \langle h(x,t)^2 \rangle = - \Delta G(0,t),$$ and the closure of (\[corrdyn\]) reads $$\label{cdmf} \partial_t G({\bf x},t) = - 2 \; \Delta^2 G({\bf x},t) - 2 \; f(\|\Delta G(0,t)\|) \; \Delta G({\bf x},t).$$ Since we consider dynamics which are isotropic in substrate space, and also isotropic distributions of initial conditions, $G({\bf x},t)$ will only depend on $\|{\bf x}\|$ and $t$. Consequently we consider the structure factor $S(k,t)$ as a function of $k = \|{\bf k}\|$ and $t$, which satisfies $$\label{Seq} \partial_t S(k,t) = -2 \left[ k^4 - f(a(t)) k^2 \right] S(k,t).$$ Here we have defined the function $a(t)$ through $$\label{a(t)} a(t) = (2\pi^{d/2}/\Gamma(d/2)) \! \int_0^{\infty} \! \! dk \; k^{d+1} S(k,t),$$ and $d$ denotes the surface dimensionality ($d=2$ for real surfaces). The formal solution of (\[Seq\]) then reads $$\label{formal} S(k,t) = S_0(k) \exp \left[ -2 t k^4 + 2 k^2 \! \int_0^t \! \! ds \; f(a(s)) \right].$$ The initial condition $S_0(k)$ reflects the disorder in the initial configuration of Eq. (\[cont1\]). It consists of fluctuations at early times, i.e. the first nucleated islands, from which mounds will later develop. Simulations of microscopic models for MBE on singular surfaces at submonolayer coverages [@harald] indicate the following shape of $S_0(k)$: From a hump at some finite wavenumber, corresponding to the typical distance $\ell_D$ between island nuclei, it falls off to zero for $k \to \infty$. For $k \to 0$ it goes down to a [finite]{} value $c > 0$. At late times the hump in $S(k,t)$ persists, situated at some $k_{max}$ near the maximum of the exponential in (\[formal\]). It belongs to a lateral lengthscale $\xi$, denoting the typical distance of neighboring mounds. For late times $k_{max}$ will go to zero, so we need only consider $S_0(k)$ near $k = 0$. In fact for the leading contribution to $k_{max}$ (and the leading power in $\xi$) we only need $S_0(k) \equiv c$. More detailed remarks on the case $\lim_{k \to 0}S_0(k)=0$ and on the presence of long range correlations in the initial stage can be found at the end of this section. The particular value of $c$ has no influence on the coarsening exponent, so we take $S_0(k) = (2 \pi)^{-d/2}$ which corresponds to $G({\bf x},t \! = \! 0) = \delta({\bf x})$. To follow the analysis, note that $a(t)$ is a functional (\[a(t)\]) of $S(k,t)$, and on the other hand it is used for the calculation of $S(k,t)$. This imposes a condition of self consistency on the solution, which we write as follows $$\label{selfc} \frac{db}{dt} = f \left( 2/(2^{d/2}\Gamma(d/2)) \int_0^{\infty} \! \! dk \; k^{d+1} \exp \left(-2tk^4 + 2k^2 b(t) \right) \right).$$ We used the initial conditions motivated above, and the shorthand $b(t) \! = \! \int_0^t ds \; f(a(s))$. The integral can be evaluated, yielding for $b(t)$ the differential equation $$\label{dgl} \frac{db}{dt} = f \left( \frac{d}{2} (4t)^{-(d+2)/4} \; 2^{-d/2} \; D_{ \! -\frac{d+2}{2}} ( -b/\sqrt{t}) \; \exp \frac{b^2}{4t} \right),$$ where $D$ denotes a parabolic cylinder function [@gradstein]. Equation (\[dgl\]) cannot be solved explicitly, but for the late time behavior we can use an asymptotic approximation for $D$, since its argument $b/\sqrt{t} \to \infty$ for $t \to \infty$. To see this, note that (\[dgl\]) is of the form $$\frac{db}{dt} = f \left( t^{-(d+2)/4} \; F \! \left( \! \frac{b}{\sqrt{t}} \! \right) \right).$$ Therefore, if $b/\sqrt{t}$ remained bounded, for large $t$ the argument of $f$ in Eq. (\[dgl\]) would be close to 0, and (\[dgl\]) would approximately be $db/dt \simeq f(0) = 1$. This is in contradiction to the assumption $b <$ const.$\times \sqrt{t}$ which therefore cannot be true. For large $t$ (and large $b/\sqrt{t}$) we then approximate (\[dgl\]) by $$\frac{db}{dt} = f \left( \sqrt{\frac{\pi}{2^{d-1}}} \; (4t)^{-(d+2)/4} /\Gamma(d/2) \; \left( \frac{b}{\sqrt{t}} \right)^{d/2} \exp \frac{b^2}{2t} \right).$$ This shows that $b/\sqrt{t}$ must grow more slowly than any power of $t$: If $b \sim t^{1/2 + \epsilon}$ then $db/dt \sim t^{\epsilon - 1/2}$, whereas the argument in $f$ would increase exponentially, dominated by a term $\exp t^{2 \epsilon}$. For both choices (\[model(i)\]) and (\[model(ii)\]) of $f$ the right hand side of (\[dgl\]) would decrease much faster than the left or even become negative. Depending on the choice of the current function $f$ we get different asymptotic behaviors in the leading logarithmic increase of $B = b^2/(2t)$. We first consider the case (ii), where $f(s^2) = 1 - s^2$. Here Eq. (\[dgl\]) reads $$t \; \frac{d B}{dt} = - B + \sqrt{2 B t} \left( 1 - C_{\rm (ii)} \; B^{d/4} \; t^{-(d+2)/4} \; \exp B \right),$$ where $C_{\rm (ii)}$ is a constant without any interest. None of the terms must increase with time as a power of $t$. Hence asymptotically the term in brackets must vanish, which requires that $\exp B \sim t^{(d+2)/4}$. The leading behavior of $B$ is therefore $$\label{betaii} B \simeq \frac {d+2}{4} \; \log t.$$ Similarly we treat case (i), using the asymptotic behavior $f(s^2) \simeq (s^2)^{-\gamma/2}$ for large $s^2$. Equation (\[dgl\]) then becomes $$t \; \frac{d B}{dt} = - B + C_{\rm (i)} \; t^{(\gamma/2)(d+2)/4 + 1/2} \; B^{-(\gamma/2)d/4 + 1/2} \exp -(\gamma/2) B.$$ Again the powers of $t$ in the last term must cancel, yielding $$\label{betaiii} B \simeq \left( \frac {d+2}{4} + \frac{1}{\gamma} \right) \; \log t.$$ There is a noteworthy correspondence between models (i) and (ii): The solution of (ii) is the limit $\gamma \to \infty$ of the solution of (i). In this sense, a current which vanishes at a finite slope is equivalent to a positive shape function $f(s^2)$ decreasing faster than any power of $s$. The same correspondence applies also on the level of the inequalities derived in Section \[scalingansatz\], as can be seen by letting $\gamma \to \infty$ in (\[scalineq\]) and comparing to (\[scalineq2\]). The asymptotic form of $b(t)$ gives us the following time dependence of the coarsening surface structure: Inserting $b(t)$ into the expression for the structure factor $S(k,t)$ (\[formal\]) we obtain for each time $t$ a wavenumber $$\label{km} k_{m}(t) = \left( \frac{1}{2} \left( \frac{d+2}{8} + \frac{1}{\gamma} \right) \; \; \frac{\log t}{t} \right)^{1/4},$$ which has the maximal contribution to $S(k,t)$. It can be interpreted as the inverse of a typical lateral lengthscale $\xi \sim (t/\log t)^{1/4}$. Up to a logarithmic factor, we obtain lateral coarsening with a power $1/4$ for both choices of $f(s^2)$. This corresponds to $z=4$, which saturates the bound derived in Section \[scalingansatz\]. It is however important to note that the resulting structure factor [cannot]{} be written in a simple scaling form $S(k,t) = w^2 k_m^{-d} {\cal S}(k/k_m)$, as would be expected if $k_m^{-1}$ were the only scale in the problem [@bray]. Rather, one obtains the [multiscaling form]{} [@coniglio] $$\label{multi} S(k,t) = L(t)^{\varphi(k/k_m \! (t))},$$ where $\varphi(x) = 2x^2 - x^4$, and $L(t) \sim t^{(\frac{d+2}{4} + \frac{1}{\gamma})/d}$ is a second lengthscale in the system. In contrast to $k_m^{-1}$, the exponent describing the temporal increase of $L(t)$ [does]{} depend on the shape of the current function $f$. Next we discuss the behavior of the typical slope of the coarsening mounds, given by $a(t) = \langle (\nabla h)^2 \rangle$. This is obtained directly from (\[selfc\]). For model (ii) $a(t)$ approaches the stable value (“magic slope”) $s^2 \! = \! 1$, with a leading correction $$\label{magic} a(t) = 1 - \dot b(t) \simeq 1 - \frac{1}{2} \sqrt{\frac{d+2}{2}} \left( \frac{\log t}{t} \right)^{1/2}.$$ Note that the approach to the magic slope is very slow, a possible explanation for the common difficulty of deciding whether $a(t)$ attains a final value or grows indefinitely in numerical simulations [@pavel]. We further remark that, up to a logarithmic factor, the inequality $\alpha' \leq 1/2$ derived in (\[scalineq2\]) for the exponent describing the approach to the magic slope becomes an equality within the spherical approximation. For model (i) the typical slopes diverge as $$\label{steep} a(t) \simeq \dot b^{-2/(1+\gamma)} \simeq \left( \frac{8}{\frac{d+2}{4}+\frac{1}{\gamma}} \right)^{1/(1+\gamma)} \left( \frac{t}{\log t} \right)^{1/(1+\gamma)},$$ consistent with the value $\alpha = 1/(2 + 2 \gamma)$ derived as a bound in (\[scalineq\]). In the limit $\gamma \to \infty$ the slope does not increase at all, which again is comparable to the presence of a stable slope. To close this section we briefly comment on the the shape of the structure factor $S(k,t)$ and the correlation function $G({\bf x},t)$ obtained within the spherical approximation. Assuming initial correlations as used above, $S_0(k) \equiv c$, the structure factor is analytical at any time $t$, as can be seen in equation (\[formal\]). The corresponding correlation function therefore decays faster than any power of $\|{\bf x}\|$, modulated with oscillations of wavenumber $k_{max}(t)$. We can also predict the further evolution of long range correlations, assuming that they were initially present. A power law decay of $G({\bf x},t \! = \!0)$ corresponds to a singularity in $S_0(k)$. Suppose the singularity is located at some point $k_0 > 0$ (the power decay of $G$ is then modulated by oscillations). Then the singularity will remain present in $S(k,t)$, but it will be suppressed as $\exp -t k_0^4$ for late times. This implies that $G({\bf x},t)$ has a very weak power law tail for very large $\|{\bf x}\|$, but up to some $x_0$ (which increases with time) it decays faster than any power. However a singularity in $S_0(k)$ will not be suppressed if it lies at the origin $k_0 = 0$, since then in Eq.(\[formal\]) it is multiplied by unity. In real space, this implies that a power law decay of correlations without oscillations will remain present. Even if $S_0(k)$ is singular at $k=0$, the scaling laws derived above remain valid. Suppose for example that $S_0(k) \sim k^{\sigma}$ for $k \to 0$. In transforming (\[formal\]) back to real space, such a power law singularity can be absorbed into the phase space factor $k^{d-1}$ involved in the ${\bf k}$-integration. The result is simply a shift in the dimensionality, $d \to d + \sigma$, which affects the prefactors of the scaling laws (\[km\]), (\[magic\]) and (\[steep\]) for $k_m(t)$ and $a(t)$ but not the powers of $t/\log t$. Conclusions =========== We have presented two approximate ways to predict the late stage of mound coarsening in homoepitaxial growth. To our knowledge this is the first theoretical calculation of coarsening exponents for this problem. Although we have made heavy use of concepts developed in phase–ordering kinetics [@bray], our results cannot be directly inferred from the existing theories in that field. As was explained in detail by Siegert [@Siegert2], equation (\[cont2\]) rewritten for the slope ${\bf u} \equiv \nabla h$ has the form of a relaxation dynamics driven by a generalized free energy, $\dot {\bf u} = \nabla \nabla \cdot \delta {\cal F}({\bf u})/ \delta {\bf u}$. Phase ordering with a conserved vector order parameter ${\bf m}$ is described by a similar form, $\dot {\bf m} = \nabla \cdot \nabla \delta {\cal F}({\bf m})/ \delta {\bf m}$, however it appears that the interchange of the order of the differential operators, from $\nabla^2$ to $\nabla \nabla \cdot$, may lead to a qualitatively different behavior [@Siegert2]. Nevertheless, the results obtained so far must be refined. Ideally, one would like to derive [equalities]{} for the exponents using the scaling ansatz of Section \[scalingansatz\]. More modestly, it would be desirable to extend the approach so that the effects of current functions without in-plane isotropy [@Siegert1; @Siegert2] and of contributions proportional to $\nabla (\nabla h)^2$ [@Stroscio; @Politi] on the scaling behavior can be assessed. The main drawback of the spherical approximation in Section \[meanfield\] is that it does not predict conventional scaling. The experience from phase–ordering kinetics in the $O(n)$ model suggests that the multiscaling behavior obtained above may be an artefact of the spherical approximation [@bray; @coniglio]. To address this issue, we have carried out a numerical integration of Equation (\[cont2\]), with weak uncorrelated noise as initial condition. The results indicate conventional scaling behavior in the late stage of growth, with exponents $z=4$, $\alpha = 1/(2 + 2 \gamma)$, which saturate the bounds of Section \[scalingansatz\]. Figure \[fig:scalplot\] shows a scaling plot of $G(\|{\bf x}\|,t)$ of model (i) with $\gamma = 1$ for times $t = 500, 600, \dots, 10000$ obtained from the numerical integration of (\[cont2\]). It is compared at times $t = 1000, 1100, \dots, 10000$ to the spherical approximation. The first zero and the width at $\|{\bf x}\| \! = \! 0$ are rescaled to $1$. Initial conditions of the approximation were chosen to coincide with the full dynamics at $t \! = \! 100$. The spherical approximation of $G$ takes a slightly different shape – its oscillations are more pronounced for larger $\|{\bf x}\|$. We obtained a similar scaling plot for $\gamma \! = \! 3$ and for model (ii). The inset shows the evolution of the first zero of $G$: In the full dynamics it is best approximated by a power law $\xi \sim \tau^{1/z}$ with $z \! = \! 3.85$ for the full dynamics, where $\tau \! = \! t \! - \! t_0$. The spherical approximation deviates from a power law. Here for late times the best fit is $\xi \sim (\tau / \log \tau)^{1/z}$ with $z \! = \! 3.87$. The beginning of the time integration $t \! = \! 0$ does not coincide with the exptrapolated zero of the power laws $t_0$, because the mounds take a finite time to develop out of the initial growth instability. This introduces the additional fitting parameter $t_0$. The steepening of the mounds (not shown in the graph) develops with the power $\alpha \! = \! 0.26$. For $\gamma \! = \! 3$ in model (i) we obtain $z \! = \! 4.18$ and $\alpha \! = \! 0.126$. For model (ii) we refer to the intergrations of Siegert [@Siegert2], which indicate $z \! = \! 4$. Note that the multiscaling behavior of $G$ in the spherical approximation is very weak, in the sense that the curves at different times do not differ in shape too much. A more sensitive test of the scaling behavior of Equation (\[cont1\]), in order to pin down the difference to the spherical approximation, would be desirable and can be achieved by extracting the function $\varphi$ (see Eq.(\[multi\])) from the data of the numerical integration. Conventional scaling yields $\varphi \! \equiv \! const$. Work in this direction is currently in progress [@claudio]. [Acknowledgements]{} We thank P. Šmilauer, F. Rojas Iníguez, A. J. Bray and C. Castellano for many helpful hints and discussions. This work was supported by DFG within SFB 237 [Unordnung und grosse Fluktuationen]{}. [99]{} G. Ehrlich and F.G. Hudda, J. Chem. Phys. [44]{}, 1039 (1966). R.L. Schwoebel and E.J. Shipsey, J. Appl. Phys., 3682 (1966); R.L. Schwoebel, J. Appl. Phys. [40]{}, 614 (1969). Some selected experiments are H.-J. Ernst, F. Fabre, R. Folkerts and J. Lapujoulade, [Phys. Rev. Lett.]{} 72:112 (1994); C. Orme, M.D. Johnson, K.-T. Leung, B.G. Orr, P. Šmilauer and D. Vvedensky, [J. Cryst. Growth]{} 150:128 (1995); J.E. Van Nostrand, S.J. Chey, M.-A. Hasan, D.G. Cahill and J.E. Greene, [Phys. Rev. Lett.]{} 74:1127 (1995); K. Thürmer, R. Koch, M. Weber and K.H. Rieder, [Phys. Rev. Lett.]{} 75:1767 (1995); as well as Refs.[@Johnson; @Stroscio]. A review is given in J. Krug, Origins of scale invariance in growth processes, [Advances in Physics]{} (in press). J. Villain, J. de Physique I [1]{},1 (1991). J. Krug, M. Plischke and M. Siegert, Phys. Rev. Lett. [70]{}, 3271 (1993). M.D. Johnson, C. Orme, A.W. Hunt, D. Graff, J. Sudijono, L.M. Sander and B.G. Orr, Phys. Rev. Lett. [72]{}, 116 (1994). J.A. Stroscio, D.T. Pierce, M. Stiles, A. Zangwill and L.M. Sander, Phys. Rev. Lett. [75]{}, 4246 (1995). M. Siegert and M. Plischke, Phys. Rev. Lett. [73]{}, 1517 (1994). M. Siegert, [in:]{} “Scale Invariance, Interfaces and Non-Equilibrium Dynamics", A.J. McKane, M. Droz, J. Vannimenus and D.E. Wolf, eds., Plenum Press, New York (1995), p.165. A.W. Hunt, C. Orme, D.R.M. Williams, B.G. Orr and L.M. Sander, Europhys. Lett. [27]{}, 611 (1994). J. Krug and M. Schimschak, J. Phys. I France [5]{}, 1065 (1995). M. Rost, J. Krug and P. Šmilauer, Surface Science, in press. P. Politi and J. Villain, Phys. Rev. [54]{}, 5114 (1996). W.W. Mullins, J. Appl. Phys. [30]{}, 77 (1959). J. Krug, On the shape of wedding cakes (to appear in Journal of Statistical Physics). J.G. Amar and F. Family, Phys. Rev. B (in press). G.S. Bales and A. Zangwill, Phys. Rev. B [41]{}, 5500 (1990). A.J. Bray, Theory of phase-ordering kinetics, Advances in Physics, [43]{}, 357 (1994). H. Kallabis and P. Šmilauer, private communications. I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Section [3.462]{}, Academic Press (1980). A. Coniglio and M. Zannetti, Europhys. Lett., 575 (1989). P. Šmilauer and D.D. Vvedensky, Phys. Rev. B [52]{}, 14263 (1995). C. Castellano, private communication. [^1]: Email: [email protected]. Fax: +49-201-183 2120. [^2]: We follow the common practice and disregard contributions to the current which are [even]{} in $h$, such as $\nabla (\nabla h)^2$, though they may well be relevant for the coarsening behavior of the surface [@Stroscio; @Politi].
--- abstract: 'Energy transport equations are derived directly from full molecular dynamics models as coarse-grained description. With the local energy chosen as the coarse-grained variables, we apply the Mori-Zwanzig formalism to derive a reduced model, in the form of a generalized Langevin equation. A Markovian embedding technique is then introduced to eliminate the history dependence. In sharp contrast to conventional energy transport models, this derivation yields [stochastic]{} dynamics models for the spatially averaged energy. We discuss the approximation of the random force using both additive and multiplicative noises, to ensure the correct statistics of the solution.' author: - Weiqi Chu - Xiantao Li bibliography: - 'Mori-Heat.bib' title: 'The Mori-Zwanzig formalism for the derivation of a fluctuating heat conduction model from molecular dynamics' --- Introduction ============ During the past two decades, there has been rapidly growing interest in modeling heat transport at the microscopic scale. Such renewed interest has been driven by the progress in designing and manufacturing micro mechanical and electrical devices, for which thermal conduction properties have great influences on the performance and reliability. As the size of electrical and mechanical devices is decreased to the micron and sub-micron scales, they often exhibit heat conduction properties that are quite different from the observations familiar at macroscopic level. For example, extraordinarily large heat conductivity for carbon nano-tubes has been reported by many groups [@Blandin-2011; @KiShMaMc-2001; @VoZh-2012], the conductivity of two-dimensional systems shows a strong dependence on the system size [@WaHuLi12], which indicates the failure of the Fourier’s law, and heat pulses are observed in experiments [@tzou1995experimental], which are typical behavior of wave equations, etc. From a modeling viewpoint, a natural approach to incorporate some of the observed behavior is to modify the traditional heat equation, , by introducing nonlocal terms [@tzou1995experimental; @Tzou-2001]. This approach is simple, but quite ad hoc. On the other hand, one may rely on the more fundamental description of phonons, which are the energy carriers in solids. The distribution of phonons is governed by the Peierles-Boltzmann (PB) equation [@Callaway59; @Peierles55; @Ziman01], or simplified PB equations [@Chen01; @JoMa93; @Majumdar93; @SvJuGo01; @XuLi12], where the cubic collision term is replaced by a relaxation term, similar to the Bhatnagar–Gross–Krook model [@bhatnagar1954model] in kinetic theory. In general, however, the computation of the full PB model is still an open challenge due to the high dimensionality. Thus far, the most popular approach to study heat conduction is direct molecular dynamics (MD) simulations, which often mimics the experimental setup. Given an interatomic potential $V$, either empirically constructed or derived from more fundamental considerations, a MD model is typically expressed in terms of the Newton’s equations of motion. There are many well established techniques for MD simulations [@AlTi89; @FrSm02]. In particular, many interatomic potentials have been developed, which have shown great promises in predicting defect structures and phase configurations. Of particular interest to the present paper is that the phonon spectrum computed from some of the models is in good agreement with results from experiments or first-principle calculations (, see [@SiPaSa97; @WiMiHa2006]). In spite of the many constributions that have recently appeared to report the studies of heat conduction processes (, [@gill2006rapid; @jolley2009modelling; @Che2000; @ChZhLi10; @DoGa09; @HeCh08; @McKa06; @SchPhKe02; @VoCh00; @Wang07; @WaHuLi12; @McKa2003; @PoMa-2006; @VoZh-2012]), direct MD models have several serious limitations when applied to heat conduction problems. The first obvious limitation is the computational cost. Typical quantities of interest are expressed as ensemble averages or two-point correlations. For a non-equilibrium process such as the transient heat conduction process, the ensemble averages may not be replaced by time averages, at least very little theory exists to support such a practice. Therefore, many copies, typically tens of thousands, need to be created to average out statistical fluctuations. In addition, the size of the system (and time scale) that can be modeled by direct MD simulations is small, often comparable or smaller than the mean free path of phonons. Most current MD studies are restricted to quasi-one-dimensional systems, , nanowires [@DoGa09; @li2003thermal; @volz1999molecular; @dames2004theoretical; @wang2009thermal; @donadio2009atomistic; @lu2002size; @chen2005effect; @chen2004molecular], nanotubes [@ChOk-2008; @Fujii-etal-2005; @KiShMaMc-2001; @PoMa-2006; @VoZh-2012; @hone1999thermal; @maruyama2002molecular; @yu2005thermal; @maruyama2003molecular; @osman2001temperature], and nanoribbons [@hu2009thermal; @savin2010suppression; @evans2010thermal]. They have motivated a lot of recent effort to understand the origin and limitations of Fourier’s Law [@lepri1997heat; @lepri2003thermal; @lepri1998anomalous; @bonetto2000fourier; @garrido2001simple; @bernardin2005fourier]. Furthermore, most of the studies have been focused on the dependence of the heat conductivity on the geometry, length and temperature of the system. A typical setup is to connect the boundaries to two heat bath with different temperature, modeled by stochastic (Langevin) or deterministic (Nosè-Hoover [@nose1984molecular]) thermostats. The MD equations are solved to drive the system to a steady state, at which point the heat flux can be measured to estimate the heat conductivity. More general transient heat conduction problems, however, would require more substantial effort. Secondly, it is often straightforward to incorporate quantities such as displacement, velocity, temperature and pressure into MD simulations as constraints. However, temperature gradient is very difficult to impose. The temperature gradient that can be imposed is usually on the order of $10^9 - 10^8 K/m$, which is too large to model realistic systems. It is unclear whether results obtained this way can be appropriately extrapolated to the correct regime. Another way to interpret this is that a realistic temperature gradient, when applied to molecular systems, is too small to be incorporated accurately by the numerical methods. One is often confronted with the issue of small signal-to-noise ratio, a problem that also arises in fluid mechanics problems [@evans1984nonlinear; @EvMo08; @morriss1987application]. This paper is strongly motivated by the above-mentioned issues, and the purpose is to present a coarse-grained (CG) model to alleviate these fundamental modeling difficulties. The CG procedure drastically reduces the number of degrees of freedom and offers a practical alternative. Coarse-graining methodologies have found enormous application in material science problems and biological problems [@RuBr05; @RuBr98; @SiLe07; @ChSt05; @IzVo06; @KaMaVl03; @CoHoSh90; @Li2009c; @baaden_coarse-grain_2013; @gohlke_natural_2006; @golubkov_generalized_2006; @gramada_coarse-graining_2011; @nielsen_recent_2010; @noid_perspective:_2013; @noid_multiscale_2008; @poulain_insights_2008; @praprotnik_multiscale_2008; @riniker_developing_2012; @rudzinski_role_2012; @shi_coarse-graining_2008; @stepanova_dynamics_2007; @zhang_systematic_2008]. Many CG models have been developed and they have shown great promise in reducing the computational cost and efficiently capturing the primary quantities of interest. However, almost all the existing CG molecular models are focused on finding the effective potentials, known as the potential of mean forces, at a [constant]{} temperature. These existing CG models are in the similar form as the MD models with possible addition of damping terms or random forces. They typically describe only the time evolution of the averaged position and momentum. The current approach works with the local energy and aims at the energy transport process. Starting with a locally averaged internal energy as CG variables, we use the Mori-Zwanzig formalism [@Mori65; @Zwanzig73] to first derive an [exact]{} equation for these variables. In particular, we choose Mori’s orthogonal projection [@Mori65] to project the equations to the subspace spanned by the CG variables. For such CG variables, this projection yields a memory term, which exhibits a simple form of a convolution in time. To alleviate the effort to compute the memory term at every step, we use the Markovian embedding techniques, recently developed in [@ceriotti2010colored; @lei2016data; @ma2016derivation], to approximate the memory using an extended system of differential equations with [no]{} memory. The coefficients for these approximations can be determined based on the statistics of the CG variables. In principle, the noise term can be averaged out by simply taking the average of every term in the CG model. This is a particular advantage of the Mori’s projection [@ChSt05]. However, motivated by the crucial observation that many mechanical systems at the micron scale or smaller are subject to strong fluctuations, we will forgo such an averaging step, and work with the models [with]{} the random noise. This results in an energy transport model [with fluctuation]{}, represented by a system of stochastic differential equations (SDE). Consequently, the solutions are expected to be stochastic in nature. An important issue naturally arises: How does one guarantee that the corresponding solution has the correct statistics? We first consider an approximation of the random force by an additive white noise, in which case the solution should have Gaussian statistics. Unfortunately, by examining a one-dimensional chain model, we have found that the correct statistics behaves more like a Gamma distribution. In particular, the energy must have a lower bound. Although the approximation by additive noise yields reasonable approximations to the time correlations, the probability density function (PDF) of the solution is incorrect. To ensure that correct PDF is obtained, we propose to approximate the noise by a [multiplicative]{} noise. In this case, the diffusion constant depends on the solution itself. We determine the diffusion coefficient by solving the steady-state Fokker-Planck equation. We are able to find a diagonal matrix for the diffusion coefficient matrix such that the Gamma distribution is an equilibrium probability density. As a further extension, we introduce a higher order approximation where both the CG energy variables and their time derivatives have the correct PDF. This leads to a Langevin type of equation with multiplicative noise. We point out that one existing stochastic heat conduction model has been proposed by Ripoll et al in [@ripoll1998dissipative], as an extension of the dissipative particle dynamics (DPD) [@espanol1995statistical]. The model was postulated, rather than derived from MD. The rest of the paper is organized as follows: In section \[math\], we discuss the mathematical derivation, and examine the general properties of the generalized Langevin equations derived from the Mori’s projection. We introduce the Markovian embedding technique for the approximation of the memory term. Then in section \[1d\], we present the approximation of the random noise using a one-dimensional system as a example. Mathematical derivation {#math} ======================= The general projection formalism -------------------------------- Our starting point is an all-atom description, which embodies the detailed interactions among all the atoms in the system. More specifically, let $x$ and $v$ be displacements and velocities of the atoms respectively; $x, v \in \mathbb{R}^{dN}$ with $d$ being the space dimension and $N$ being the total number of atoms. The dynamics follows the Newton’s second law, $$\left\{ \begin{aligned} \dot{x} &= v, \quad & x(0)=x_{0},\\ m\dot{v} &= f(x)=-\frac{\partial V(x)}{\partial x},\quad & v(0)=v_{0}, \\ \end{aligned} \right. \label{eq: newton2}$$ where $V(x)$ is the potential energy of the system. Solutions $\{x(t),v(t)\| t\ge 0\}$ can be viewed as trajectories in the phase space with an ensemble of initial states $(x_0,v_0).$ Following the notations in [@EvMo08], we use $\Gamma=\mathbb{R}^{2dN}$ for the phase space and define the propagating operator $\mathcal{L}$ on $\Gamma$ as, $$\mathcal{L} := v_{0}\cdot\frac{\partial } {\partial x_{0}} +\frac{f(x_{0})}{m}\cdot \frac{\partial}{\partial v_{0}}.$$ We define a Hilbert space $\cal{H}$ equipped with an inner product weighted by a probability density $\rho_0$. This corresponds to the initial preparation of the system. For any two $n$-dimensional functions, $F, G: \Gamma \to \mathbb{R}^n,$ we define the average and the correlation matrix as follows, $$\begin{aligned} &\left<F\right>_{i} \overset{\Delta}{=} \int_{\Gamma} F_i(z) \rho_0(z) dz, \quad 1 \le i \le n, \\ &\left<F, G\right>_{ij} \overset{\Delta}{=} \int_{\Gamma} F_i(z) G_j(z) \rho_0(z) dz, \quad 1 \le i, j \le n. \label{eq: minnerproduct} \end{aligned}$$ Suppose ${a}: \Gamma \to \mathbb{R}^{n}$ is a quantity of interest (QOI) and depends only on phase space variables $x$ and $v$ explicitly. For convenience, we work with a somewhat abused notation, $${a}(t) \overset{\Delta}{=} {a}\big(x(t),v(t)\big) \text{ and } {a} \overset{\Delta}{=} {a}(0).$$ For the last part of our definition, we have followed the convention in [@EvMo08]. Our goal is to derive a reduced equation for the QOI $a(t)$, also known as coarse-grain (CG) variables. For this purpose, we follow the Mori-Zwanzig (MZ) procedure [@ChHaKu02; @ChKaKu98; @ChSt05; @Mori65; @Zwanzig73]. A key step in the MZ formulation is a projection operator $\mathcal{P}$ that maps functions of $\Gamma$ to those of $a$. We adopt the orthogonal projection suggested by Mori [@Mori65]. More specifically, for any function $G:\Gamma\to\mathbb{R}^{n}$, we define, $$\mathcal{P}G\overset{\Delta}{=} \left< G,a \right>M^{-1}a,$$ where $M^{-1}$ is the inverse of matrix $M=\left< a,a\right>$, and the inner product of $G$ and $a$ is defined according to . We also define $\mathcal{Q}$ as the complementary operator of $\mathcal{P}$, i.e. $\mathcal{Q}=\mathcal{I}-\mathcal{P}$. Note that the covariance matrix $M$ only involves the one-point statistics of $a$ and can be guaranteed to be non-singular by carefully selecting the CG variables. In practice, this corresponds to the appropriate choice of $a$ so that the CG variables are not redundant. Even in the case when the CG variables are redundant, e.g., when there is energy conservation and the matrix $M$ becomes singular, the projection can still be well defined by interpreting $M^{-1}$ as the pseudo-inverse. Once the projection operator is in place, the Mori-Zwanzig formalism can be invoked, and the following generalized Langevin equation (GLE) can be derived [@Mori65], $$\dot{a}(t) = \Omega a(t) - \int_{0}^{t} \theta(t-s)a(s) ds + F(t), \label{eq: GLE}$$ where $$\label{mz} \Omega=\left<\mathcal{L}a,a\right>M^{-1},\; F(t) = e^{t\mathcal{QL}}\mathcal{QL}a, {\text{ and }}\; \theta(t)=-\left<\mathcal{L}F(t),a\right>M^{-1}.$$ For the choice of $\rho_0,$ we pick an equilibrium probability density. More specifically, let $\mathcal{L}^{}$ be the $L^{2}$-adjoint operator of $\mathcal{L}$. Then for any equilibrium density $\rho_{eq}$ that satisfies $\mathcal{L}^{}\rho_{eq}=0$ we can set $\rho_0=\rho_{eq}.$ When the system is near equilibrium, this serves as the first approximation. Further corrections can be made using the linear response approach [@Toda-Kubo-2]. For a Hamiltonian dynamics like , we have $\mathcal{L}^{}=-\mathcal{L}$ [@EvMo08]. We pick the canonical ensemble for $\rho_{eq},$ $$\rho_{eq}= \frac{1}{Z} e^{-\beta H}.$$ In principle, other forms of the probability density, especially those obtained from the maximum entropy principle [@Zwanzigbook; @Zwan80], can be used as well. Several properties can be deduced from the derivation. They are summarized in the following theorem. Assuming that $\left<a\right>=0$, then the following properties hold, $$\label{eq: prop-F} \begin{aligned} \left<F(t)\right> &=0, \qquad \forall \; t\ge 0,\\ \theta(t_1 -t_2) &= \left<F(t_1),F(t_2) \right>M^{-1}, \quad \forall \; t_{1},t_{2}\ge 0 \text{ and } t_{1}\ge t_{2},\\ \left<F(t),a\right> &=0, \qquad \forall \; t\ge 0.\\ \end{aligned}$$ Note that with the inner product defined above, the adjoint operator of $\mathcal{L}$ is $-\mathcal{L}$, and $\mathcal{P}$ and $\mathcal{Q}$ are self-adjoint, i.e., $\left< \mathcal{L}b,c\right>=-\left< b,\mathcal{L}c\right>$ and $\left< \mathcal{P}b,c\right>=\left< b,\mathcal{P}c\right>$ for any $b,c: \Gamma \to \mathbb{R}^n$. Now for the first property, we proceed as follows, $$\begin{aligned} \left<F(t)\right> &= \left<\mathcal{QL}e^{t\mathcal{QL}}a\right> = \left< \mathcal{L}e^{t\mathcal{QL}}a\right> - \left<\mathcal{PL}e^{t\mathcal{QL}}a\right> \\ &= -\left< e^{t\mathcal{QL}}a\mathcal{L}1\right> - \left<\mathcal{L}e^{t\mathcal{QL}}a,a\right>M^{-1}\left< a\right> = 0. \end{aligned}$$ For the second property, we start with the second equation in and we get, $$\begin{aligned} \theta(t_{1}-t_{2}) &=-\left<\mathcal{L}F(t_{1}-t_{2}),a\right>M^{-1} = \left<e^{(t_{1}-t_{2})\mathcal{QL}}\mathcal{QL}a,\mathcal{L}a\right>M^{-1}\\ &=\left<F(t_{1}),e^{t_{2}\mathcal{LQ}}\mathcal{L}a\right>M^{-1}=\left<F(t_{1}),\mathcal{Q}e^{t_{2}\mathcal{LQ}}\mathcal{L}a\right>M^{-1} \\ &= \left<F(t_1),F(t_2) \right>M^{-1}. \end{aligned}$$ Finally, since $\mathcal{P}a =a$ and $\mathcal{Q}F(t)=F(t)$, one can easily verify that, $$\left< F(t),a\right> = \left< \mathcal{Q}F(t),\mathcal{P}a\right> = \left< F(t),\mathcal{QP}a\right> = 0.$$ The first two equations imply that the random process, $F(t),$ is a stationary random process in the wide sense [@chorin2009stochastic], and it also satisfies the second fluctuation-dissipation theorem (FDT). It is a necessary condition for the solution to have the correct variance [@Kubo66]. The last condition suggests that the random force and the initial value of $a$ are uncorrelated. The last two properties have also been discussed in the Mori’s original paper [@mori1965b]. Using the first property, one can take an average of the GLE , and arrive at a deterministic system, $$\dot{\left<a\right>}(t) = \Omega \left<a\right>(t) - \int_{0}^{t} \theta(t-s)\left<a\right>(s) ds, \label{eq: GLE-avg}$$ which is a set of integral-differential equations describing the time evolution of the average of the quantity of interest [@ChKaKu98]. This is often seen as a particular advantage of Mori’s projection. However, in this paper, we are more concerned with the quantity $a$ with fluctuation. An explicit representation of the random noise ---------------------------------------------- In the general MZ formalism, the random noise has been expressed in a quite abstract form. The practical implementation is rather difficult in general. Here, we provide a more detailed characterization of the random noise, by embedding it in an infinite system of ordinary differential equations. Important properties can also be deducted from these differential equations. Suppose that $\{\mathcal{L}^{j}a\}_{j\ge 0}$ are linearly independent, by inspecting the first few terms in the exponential operator, one finds that the random force term can be written as, $$\label{eq: exp-F} F(t) = \sum_{j\geq 0} C_{j}(t)\mathcal{L}^{j}a.$$ Together with the orthogonal dynamics, $\dot{F}(t)=\mathcal{QL}F(t)$, we can derive a set of equations for the coefficients, $$\begin{aligned} \dot{C}_j(t) = &C_{j-1}(t), \quad j \ge 1, \\ \dot{C}_0(t) =&- \sum_{j\ge0} C_j (t)M_{j+1}M^{-1}, \end{aligned} \label{eq: odeC}$$ where $M_{j}$ are referred to as the [moments]{} associated with the statistics of $a$, defined as follows, $$M_j \overset{\Delta}{=} \left<\mathcal{L}^ja,a\right>=\left< \frac{d^j}{dt^j}a,a\right>.$$ The initial condition is given by, $$C_{0}(0)=-M_{1}M^{-1}, \; C_{1}(0)=I \text{ and } C_j(0)=0, \; \forall j\ge 2.$$ Therefore, the random noise can be characterized via an infinite set of ordinary differential equations. Properties of the kernel function --------------------------------- Thanks to the explicit representation of the random force and the FDT , certain values of the memory kernel can be reconstructed or approximated using the equilibrium properties. In this section, we will explain how the connections can be made. Direct calculations yield, $$\theta(t) = -\sum_{j\ge 0}C_{j}(t) M_{j+1}M^{-1},$$ where $C_{j}(t)$ are the coefficients of $F(t)$ in the expansion and are given by the solution of . The derivatives of $\theta(t)$ at $t=0$ can be written out explicitly, as shown by the following theorem, which can be proved by direct substitutions. $C_{0}^{(k)}(0)$ can be computed recursively, and they satisfy, $$C_{0}^{(k)}(0) = -\sum_{j=-1}^{k-1}C_{0}^{(j)}(0)\overline{M}_{k-j-1}, \;\; k\ge 0.$$ Furthermore, the derivatives of the memory kernel are given by, $$\begin{aligned} \theta^{(k)}(0) = -\sum_{j=-1}^{k} C_{0}^{(j)}(0)\overline{M}_{k-j+1}, \;\; k\ge 0, \label{eq: dtheta} \end{aligned}$$ where we defined $C_{0}^{(-1)}\overset{\Delta}{=}C_{1}$ and $\overline{M}_{j}\overset{\Delta}={M}_{j}M^{-1}$. Based on the theorem, one can write down the first few derivatives of $\theta(0)$, $$\begin{aligned} \theta(0)=& -\overline{M}_2+\overline{M}_{1}^{2} ,\\ \theta'(0)=& -\overline{M}_3+\overline{M}_{2}\overline{M}_{1}+\overline{M}_{1}\overline{M}_{2}-\overline{M}_{1}^{3},\\ \theta''(0)=& -\overline{M}_{4}+\overline{M}_{3}\overline{M}_{1}+\overline{M}_{2}^{2}+\overline{M}_{1}\overline{M}_{3}-\overline{M}_{2}\overline{M}_{1}^{2}-\overline{M}_{1}\overline{M}_{2}\overline{M}_{1}-\overline{M}_{1}^{2}\overline{M}_{2}+\overline{M}_{1}^{4}, \\ \cdots \\ \end{aligned}$$ This routine allows us to express the values of the kernel function at $t=0$ in terms of the [equilibrium]{} statistics of the CG variables. The approximation scheme in the next section takes advantage of these properties. Markovian embedding – a systematic approximation of the memory term ------------------------------------------------------------------- A well known practical issue associated with the solution of the GLE is that computation of the memory term. Clearly, a direct evaluation of the integral requires the storage of the solutions from all previous steps, and such evaluations have to be carried out at every time step. To alleviate such effort, we will use the Markovian embedded technique and approximate the memory term via an extended system of equations [@lei2016data]. The idea is to incorporate the aforementioned values of the kernel function into the Laplace transform of $\theta$. More specifically, we define, $$\Theta(\lambda)= \int_0^{+\infty} \theta(t) e^{-t/\lambda} dt.$$ As $\lambda \to 0+,$ using integration by parts repeatedly, we find that $$\Theta(\lambda) = \lambda \theta(0) + \lambda^2 \theta'(0) + \lambda^3 \theta''(0) + \lambda^4 \theta'''(0) + \cdots.$$ We now turn to the limit as $\lambda \to +\infty$, which embodies long-time behavior of the kernel function. For this calculation, we start with the GLE , multiply both side by $a^{\intercal},$ and take the average. Let $M(t)=\left<a(t),a\right>,$ then we have, $$\dot{M}(t)= \Omega M(t) - \int_0^t \theta(t-s) M(s)ds. \label{eq: odeM}$$ Let ${\widetilde{M}}(\lambda)$ be the Laplace transform of $M(t)$. Taking the Laplace transform of , we find, $$\frac{1}{\lambda} {\widetilde{M}}(\lambda) - M= \Omega {\widetilde{M}}(\lambda) - \Theta(\lambda) {\widetilde{M}}(\lambda),$$ which yields, $$\label{eq: th-inf} \Theta(+\infty)= \Omega + M \Big[ \int_0^{+\infty} M(t) dt\Big]^{-1}.$$ Again this is related to the statistics of $a$. We now incorporate the values of $\Theta$ from both regimes: $\lambda \to 0+$ and $\lambda \to +\infty.$ Such two-sided approximations, which are similar to the Hermite interpolation problems, have demonstrated promising accuracy over both short and long time scales [@lei2016data]. The idea of the Markovian embedding is to approximate the memory term by rational functions in terms of the Laplace transform. In general, we can consider a rational function in the following form, $$R_{k,k}= \big[I-\lambda B_1 - \cdots - \lambda^k B_{k}\big]^{-1}\big[A_{0}+\lambda A_1 + \lambda^2 A_2 + \cdots + \lambda^k A_{k}\big].$$ The coefficients in the rational function can be determined based on the values of the kernel functions, e.g., those presented in the previous section. When $k=0,$ we are led to a constant function, $R_{0,0}=\Gamma$, which, we choose to be given by : $$\Gamma= \Theta(+\infty).$$ In the time domain, this amounts to approximating the kernel function by a delta function: $$\int_{0}^{t} \theta(t-s)a(s) ds \approx \Gamma a(t).$$ This is often referred to as the Markovian approximation [@hijon2006markovian; @kauzlaric2012markovian]. When $k=1,$ we have, $$R_{1,1}(\lambda) = \big[I - \lambda B_1]^{-1} \big[ A_0+\lambda A_1].$$ To determine the coefficients, we match the following values, $$\label{eq: R11} R_{1,1}(0)= \Theta(0), \; R_{1,1}'(0)= \Theta'(0)\; \text{ and } R_{1,1}(+\infty)= \Theta(+\infty),$$ which yield, $$A_{0}=0,\; A_1= \theta(0) \; \text{ and } B_1= -A_1\Theta(+\infty)^{-1}. \label{eq: A1B1}$$ In the time-domain, this corresponds to an approximation of the kernel function by an matrix exponential $e^{B_{1}t}A_{1}$. Meanwhile, if we define the memory term as $z,$ $$\label{eq: z} z= \int_{0}^{t} \theta(t-s)a(s) ds,$$ we can write down an auxiliary equation, $$\dot{z}= A_1 a + B_1 z.$$ This way, the memory term is embedded in an extended dynamical system [without]{} memory. As the order of the approximation $k$ increases, we obtain an hierarchy of approximations for the memory term, which can be written as a larger extended system of equations [@lei2016data; @ma2016derivation]. We will not discuss the higher order approximations in this paper. It remains to approximate the random noise term. This will be discussed in the next section, along with a specific example of the MD model. Application to energy transport {#1d} =============================== A one-dimensional example ------------------------- Let’s consider a one-dimensional isolated chain model of $N$ atoms and they are evenly divided into $n$ blocks, each of which contains $\ell$ atoms, as shown in Figure \[fig: 1dchain\]; $N=n\ell $. We will focus on the study of energy transport between these blocks. ![1-D chain of particles. Every $\ell$ atoms are grouped into one block. \[fig: 1dchain\][]{data-label="fig: const"}](1dchain.eps) Let $x$ and $v$ be the displacements and velocities respectively, satisfying . Periodic boundary conditions are imposed. Let $S_{I}$ is the index set of $I$-th block, labeled as, $S_{I}=\{\ell (I-1)+1,\cdots,\ell I\}$, $I=1,2,\cdots,n$. Let $\phi(x_{i}-x_{j})$ be the pairwise potential coming from interactions between the $i$th and $j$th atoms. Here, we use the Fermi-Pasta-Ulam (FPU) potential, $$\phi(r) = \frac{r^{4}}{4}+\frac{r^{2}}{2}.$$ If we only consider the nearest neighbor interactions, the potential energy of this 1-d chain is given by, $$V(x) = \sum_{i=1}^{N}\phi(x_{i-1}-x_{i}).$$ We define the local energy associated with the $I$-th block as follows, $$E_{I}(t) = \sum_{i\in S_{I}} \frac{1}{2}mv_{i}^{2} + \frac12\phi(x_{i}-x_{i-1})+\frac12\phi(x_{i+1}-x_{i}). \label{eq: edef}$$ The rate of change of the local energy can be attributed to the energy flux $J$, $$\dot{E}_{I}(t) = J_{I+\frac12}(t) - J_{I-\frac12}(t),$$ where $J_{I+\frac{1}{2}}$ is the energy flux between the $I$th and $(I+1)$th blocks. Direct calculation yields, $$J_{I+\frac12} = \frac12 \phi'(x_{\ell I+1}-x_{\ell I}) (v_{\ell I+1} + v_{\ell I}).$$ Notice that the energy flux only depends on the atoms next to the interfaces between two adjacent blocks. The generalized Langevin equation for the energy ------------------------------------------------ As an application of the Mori’s projection method, we define the quantity of interest $a$ as the shifted energy of blocks, $$a_{I}(t) \overset{\Delta}{=} E_{I}(t) - \left< E_{I}(0)\right>. \label{eq: adef}$$ The subtraction of the average is to ensure that $\left<a\right>=0.$ If $a(t)$ is as defined in and , the odd moments of $a$ vanish, i.e., $$M_{2j+1}=0, \quad \forall \;j \geq 0.$$ \[thm: modd\] The following lemma can be easily established, as a preparation to prove this theorem, If $J(x,v)=G(x)F(v)$ is a scalar, separable function, then there exist $m\ge 1$, $G_{i}$ and $F_{i}$ s.t. $$\dot{J}(x(t),v(t)) \overset{\Delta}{=} \frac{d}{dt} J(x(t),v(t)) = \sum_{i=1}^{m} G_{i}(x(t))F_{i}(v(t)).$$ Furthermore, if $F(v)$ is an odd function w.r.t. $v$, then $\dot{J}(x,v)$ is an even function w.r.t $v$. If $F(v)$ is even w.r.t. $v$, then $\dot{J}$ is odd. \[lemma: gf\] Now let’s turn to the proof of Theorem \[thm: modd\]. Write $ \overline{E} \overset{\Delta}{=} \langle E(0) \rangle$, and consider $j\geq 0$, $$\begin{aligned} M_{2j+1} &= \langle \mathcal{L}^{2j+1}a,a \rangle \\ &= \langle \mathcal{L}^{2j+1}E(0), E(0) -\overline{E} \rangle - \langle \mathcal{L}^{2j+1}\overline{E}, E(0) -\overline{E} \rangle \\ &= \langle \mathcal{L}^{2j+1}E(0), E(0) \rangle + \langle \mathcal{L}^{2j}E(0), \mathcal{L}\overline{E} \rangle \\ &= \left< \frac{d^{2j+1}}{dt^{2j+1}}E(0), E(0) \right>. \end{aligned}$$ By induction and Lemma \[lemma: gf\], one can verify that $\frac{d^{2j+1}}{dt^{2j+1}}E(0)$ is an odd function w.r.t $v$. $E(0)$ is even w.r.t $v$, so when we integrate the product over the velocity domain weighted by a Gaussian distribution, we get a zero average. With this result, we can see that the Markovian term in the GLE must be zero, i.e., $\Omega=0$. Furthermore, we have The derivatives of the memory function at $t=0$ are given by, $$\begin{aligned} &C_{0}^{2j}(0) = 0, \quad j\geq 0, \\ &\theta^{(2j)}(0)= -\sum_{i=0}^{j}C_{0}^{(2i-1)}\overline{M}_{2(j+1-i)}, \quad j\geq 0, \\ &\theta^{(2j+1)}(0) = 0, \quad j\geq 0.\\ \end{aligned}$$ For instance, the first few derivatives of $\theta(0)$ in even orders are listed below, $$\begin{aligned} &\theta(0)= -\overline{M}_2, \\ &\theta''(0)= -\overline{M}_{4}+\overline{M}_{2}^{2}, \\ &\theta^{(4)}(0) = -\overline{M}_{6}+\overline{M}_{4}\overline{M}_{2}+\overline{M}_{2}\overline{M}_{4}+\overline{M}_{2}^{3},\\ & \cdots \cdots \\ \end{aligned}$$ Before we discuss the approximation of the random noise, we first start with a full MD simulation, from which we can obtain the time series of $a(t)$ and $F(t)$. Our numerical test simulates an equilibrium system containing $500$ atoms which are evenly divided into $n=50$ blocks. The histograms of one entry of $a(t)$ and $F(t)$ at equilibrium are shown in Figure \[fig: afpdf\], which can be assumed to be the exact stationary distributions. Interestingly, both quantities exhibit non-Gaussian statistics. The PDF of $a(t)$ fits perfectly to a shifted Gamma distribution, and the PDF of the random noise follows a Laplace distribution. See Figure \[fig: afpdf\]. ![The left figure shows the histogram of $a_{1}(t)$ obtained from direct simulations. The data fits well to a shifted Gamma distribution $\Gamma_{\alpha,\beta}(x) \sim (x+\mu)^{\alpha-1}\exp(-\beta(x+\mu)), \; x\ge -\mu,$ with $\alpha=10.0596, \; \beta=0.7825$ and $\mu=\alpha/\beta$. The right figure shows the histogram of $F_{1}(t)$ and it fits a Laplace distribution $\text{Lap}_{b}(x)\sim\exp(-b\|x\|)$ with $b=1.0801$. []{data-label="fig: afpdf"}](apdf.eps "fig:") ![The left figure shows the histogram of $a_{1}(t)$ obtained from direct simulations. The data fits well to a shifted Gamma distribution $\Gamma_{\alpha,\beta}(x) \sim (x+\mu)^{\alpha-1}\exp(-\beta(x+\mu)), \; x\ge -\mu,$ with $\alpha=10.0596, \; \beta=0.7825$ and $\mu=\alpha/\beta$. The right figure shows the histogram of $F_{1}(t)$ and it fits a Laplace distribution $\text{Lap}_{b}(x)\sim\exp(-b\|x\|)$ with $b=1.0801$. []{data-label="fig: afpdf"}](fpdf.eps "fig:") In the following section, we will focus on the approximation of the noise term so that the solution of the reduced models gives consistent statistics of the CG energy variables. Approximation of the noise -------------------------- In the previous section, we have discussed how the memory term can be approximated using the rational approximation in terms of the Laplace transforms. In particular, it gives rise to deterministic (or drift) terms in the resulting approximate models. Here, we discuss the approximation of the noise. We will consider both additive and multiplicative noises. ### Approximations by additive noise A natural (and most widely used) approximation is by a Gaussian white noise. For instance, for the first order approximation, we are led to a linear SDE, $$\dot{{a}}(t) = -\Gamma {a}(t) + \sigma \zeta(t), \label{eq: adsde0}$$ where $\zeta(t)$ is the standard Gaussian-white noise, $$\left< \zeta(t_1), \zeta(t_2)^\intercal\right> = \delta(t_1-t_2).$$ In order for the solution $a$ to have the correct covariance $M$, the parameter $\sigma$ has to satisfy the Lyapunov equation, [@risken1989fpe] $$\Sigma \overset{\Delta}{=} \sigma \sigma^\intercal = \Gamma M + M \Gamma^\intercal.$$ On the other hand, with the rational approximation of the kernel function, we may introduce noise via the second equation. Namely, $$\left\{ \begin{aligned} \dot{{a}}(t) = & - z(t), \\ \dot{z}(t) = & B_1 z(t) + A_1 {a}(t) + \sigma\zeta(t). \end{aligned} \right. \label{eq: adsde1}$$ It is clear that the second equation can be solved explicitly and substituted into the first equation, which would yield a similar equation to the GLE . By choosing the initial condition $z(0)$ and $\Sigma$ appropriately, the approximations to the memory kernel and the random noise can be made consistent, in terms of the second FDT . Assuming the covariance of $z(0)$ is $A_1$, and $$B_1 A_1 + A_1 B_1^\intercal + \Sigma=0,$$ then, the extended system is equivalent to approximating the kernel function by $\theta_1(t)=e^{tB_1}A_1$, and the approximate noise, denoted by $F_1(t)$, to $F(t)$ satisfies the second FDT exactly. Namely, $$\theta_1(t -t') = \left<F_1(t),F_1(t') \right>M^{-1}, \quad \forall\; t,t'\ge0.$$ The proof of this theorem can be found in [@ma2016derivation]. The approximation by additive noises inevitably leads to a Gaussian distribution for $a(t)$ [@risken1989fpe]. To check the validity of this assumption, we ran a direct simulation based on both the zeroth order model and the first order model . The solutions are then compared to the true statistics, and the results are displayed in Figure \[fig: addcorr\]. ![The left figure shows the true PDF of $a_1$ and approximate PDF from approximations with Gaussian additive noise. The right panel shows the time correlation of approximate models.[]{data-label="fig: addcorr"}](addpdf.eps "fig:"){width="45.00000%"} ![The left figure shows the true PDF of $a_1$ and approximate PDF from approximations with Gaussian additive noise. The right panel shows the time correlation of approximate models.[]{data-label="fig: addcorr"}](addcorr.eps "fig:"){width="45.00000%"} From Figure \[fig: addcorr\], we observe that although the time correlation of the CG energy is well captured, the PDF deviates from the true distribution. ### Approximations using multiplicative noise As alluded to in the previous section, the approximate model driven by Gaussian additive white noise may not capture the correct PDF. In this section, we consider multiplicative noise, with the objective of enforcing the correct equilibrium statistics for the solution of the SDEs. We start with a further observation that the energy of each block is almost independent to each other. Figure \[fig: pdfs\] shows the agreement between the joint histogram of the energy of two adjacent blocks and the true marginal PDFs. It is interesting that same observations have been made for biomolecules [@faure2017entropy]. It is clearly difficult to prove the exact independence theoretically. Therefore, we keep this as our main assumption, and postulate the stationary PDF ($\rho(a)$) of the energy as a shifted multi-Gamma distribution with parameters $\alpha_i$ and $\beta_i$, $$\label{multi-gamma} \rho(a) \propto \prod_{i=1}^{n} \left({a}_{i}-\frac{\alpha_{i}}{\beta_{i}}\right)^{\alpha_{i}-1}\!\!\!\!\!e^{-\beta_{i}\left({a}_{i}+\alpha_{i}/\beta_{i}\right)}.$$ ![The left figure shows the normalized joint histogram of $a_{1}$ and $a_{2}$ and the right panel shows the product of two normalized individual histogram of $a_{1}$ and $a_{2}$. []{data-label="fig: pdfs"}](jointpdf.eps "fig:") ![The left figure shows the normalized joint histogram of $a_{1}$ and $a_{2}$ and the right panel shows the product of two normalized individual histogram of $a_{1}$ and $a_{2}$. []{data-label="fig: pdfs"}](marpdf.eps "fig:") Now we reconsider the zeroth order approximation, the Markovian approximation by $R_{0,0}=\Theta(+\infty)=\Gamma.$ With a multiplicative noise, we are solving the following SDE, $$\dot{a}(t) = -\Gamma a(t) + \sigma(a(t)) \xi(t), \label{eq: osde}$$ where $\xi(t)$ is the again standard Gaussian white noise. The SDE is interpreted in the Itô sense. To derive a simple formula, we seek $\sigma$ in a diagonal form. We aim to construct $D\overset{\Delta}{=}2\sigma\sigma^{\intercal}$ to ensure the desired PDF given by . By simplifying the Fokker-Planck equation (FPE) that corresponds to , we obtain, $$\frac{\partial D_{ii}\rho}{\partial a_{i}}= -\rho\left( \Gamma_{ii}a_{i}+\sum_{j=1,j\neq i}^{n} \Gamma_{ij}a_{j}\right). \label{eq: Dii1}$$ By directly solving these differential equations, we obtained an explicit formula for the matrix $D$, as summarized in the following theorem, which can be proved by direct integration of . If $\Gamma$ is a Z-matrix, i.e., the off-diagonal entries are non-positve, and $\Gamma$ is semi-positive definite, then there exists a diagonal matrix $D$ for which the multi-Gamma distribution is a steady state solution of the Fokker-Planck equation. The diagonals of $D$ are given by, $$D_{ii} = \frac{\Gamma_{ii}}{\beta_{i}}\left( a_{i}+\frac{\alpha_{i}}{\beta_{i}}\right) -\sum_{j=1, j\neq i}^{m}\Gamma_{ij} \frac{a_{j}\int_{-\frac{\alpha_{i}}{\beta_{i}}}^{a_{i}}\rho_{i}(x)dx+ \alpha_{j}/\beta_{j}}{\rho_{i}(a_{i})} \geq 0,$$ where $\rho_{i}$ is the marginal PDF of $a_{i}$ and $\rho_{i}$ is propontional to $\left({a}_{i}+\frac{\alpha_{i}}{\beta_{i}}\right)^{\alpha_{i}-1}\!\!\!\!\!e^{-\beta_{i}\left({a}_{i}+\alpha_{i}/\beta_{i}\right)}$. As a verification, we solved the SDEs with coefficients determined by the above formula. Figure \[fig: multi\] shows the PDF of the first component along with its time correlation. Due to the uniform partition of the system, we expect the statistics to be the same for all the components of $a$. So we will only show the properties of the first component $a_1.$ It is clear that they are both consistent with the truth. It is worthwhile to point out that the SDE contains a diffusion coefficient $\sigma$ which is unbounded. This can be viewed as a mechanism for the energy to stay above a lower bound. However, this introduces a stiff problem for the numerical approximation. To resolve this numerical issue, we applied the implicit Taylor method [@Tian2001implicit]. ![The left figure shows the approximate PDF of $a_{1}$ from the multiplicative model , along with the exact PDF. The right panel indicates the time correlation function.[]{data-label="fig: multi"}](multipdf.eps "fig:") ![The left figure shows the approximate PDF of $a_{1}$ from the multiplicative model , along with the exact PDF. The right panel indicates the time correlation function.[]{data-label="fig: multi"}](multicorr.eps "fig:") Finally let’s turn to the model obtained by the first order approximation of the memory term. With Gaussian multiplicative noise, the first order model can be written formally as follows, $$\left\{ \begin{aligned} \dot{{a}}(t) = &- z(t), \\ \dot{z}(t) = & A{a}(t)+Bz(t) + \sigma(a(t),z(t))\xi(t), \end{aligned} \right. \label{eq: 1stmul}$$ where $A=A_{1}$ and $B=B_{1}$ are given in . This is a Langevin equation. But notice that in the multiplicative noise term, we allowed the diffusion coefficient to depend on both $a$ and $z.$ Again from the results of direct simulations, we observed that the components of $a$ and $z$ are independent. So we assume all these components are independent and its joint distribution function is written as, $$\rho(a,z) \propto \prod_{i=1}^{n} \left({a}_{i}-\frac{\alpha_{i}}{\beta_{i}}\right)^{\alpha_{i}-1}\!\!\!\!\!e^{-\beta_{i}\left({a}_{i}+\alpha_{i}/\beta_{i}\right)}e^{-\gamma_{i}\|z_{i}\|}. \label{eq: azpdf}$$ Similarly, we assume $\sigma$ to be diagonal and work with the steady state solution of the FPE, which can be written as, $$\frac{\partial D_{ii}\rho}{\partial z_{i}} = \left( \sum_{j=1}^{n}A_{ij}a_{j} + \sum_{j=1,j\neq i}^{n} B_{ij}z_{j} - \frac{1}{\gamma_{i}^{2}}\frac{\partial W(a)}{\partial a_{i}} \right) \rho + B_{ii}z_{i}\rho - \frac{1}{\gamma_{i}}\frac{\partial W(a)}{\partial a_{i}}\|z_{i}\|\rho. \label{eq: Diirho}$$ By integrating this equation, we have, In , suppose $B$ has non-positive diagonal entries. We have for $i=1,2,\cdots,n$, $$\begin{aligned} D_{ii} = &- \frac{sgn(z_{i})}{\gamma_{i}} \left( 1-e^{\gamma_{i}\|z_{i}\|} \right)\left( \sum_{j=1}^{n}A_{ij}a_{j} + \sum_{j=1,j\neq i}^{n} B_{ij}z_{j} - \frac{2}{\gamma_{i}^{2}}\frac{\partial W(a)}{\partial a_{i}} \right)+ f_{1}e^{\gamma_{i}\|z_{i}\|}\\ &+\frac{1}{\gamma_{i}^{2}}\frac{\partial W(a)}{\partial a_{i}}z_i + f_{2}e^{\gamma_{i}\|z_{i}\|} + B_{ii}\left( -\frac{1}{\gamma_{i}}\|z_{i}\| - \frac{1}{\gamma_{i}^{2}} +\frac{1}{\gamma_{i}^{2}}e^{\gamma_{i}\|z_{i}\|}\right)+f_{3}e^{\gamma_{i}\|z_{i}\|}, \end{aligned} \label{eq: Dii}$$ where $$\begin{aligned} f_{1} &= \frac{1}{\gamma_{i}}\left( \sum_{j=1}^{n}\|A_{ij}\|\|a_{j}\| + \sum_{j=1,j\neq i}^{n} \|B_{ij}\|\|z_{j}\| + \frac{2}{\gamma_{i}^{2}}\left(\beta_{i}+\frac{\alpha_{i}-1}{a_{i}+\mu_{i}}\right) \right) ,\\ f_{2} &= \frac{1}{\gamma_{i}^{3}e}\left( \beta_{i} + \frac{\alpha_{i}-1}{a_{i}+\mu_{i}} \right),\\ f_{3} &= -B_{ii}\frac{1}{\gamma_{i}^{2}}, \end{aligned}$$ Further, $D_{ii}$ are positive and $\rho$ given in is a steady state solution of the FPE, if $\sigma$ is a diagonal matrix and $2\sigma_{ii}^{2}=D_{ii}$. From Figure \[fig: mul1\], we see that the first order model is able to capture the statistics of both energy $a$ and $z$. Similar to the zeroth order model, the SDEs are stiff, and the implicit Taylor method [@Tian2001implicit] with stepsize $\Delta t=5\times 10^{-4} $ is used to generate a long time series to obtain the statistics. ![The left figure shows the comparison of the true PDF of $a_1$ and its approximation obtained from solving the first order SDE model using the implicit Taylor method. The right panel shows the PDFs of $z_1$.[]{data-label="fig: mul1"}](mul1-pdf.eps "fig:"){width="45.00000%"} ![The left figure shows the comparison of the true PDF of $a_1$ and its approximation obtained from solving the first order SDE model using the implicit Taylor method. The right panel shows the PDFs of $z_1$.[]{data-label="fig: mul1"}](zpdf.eps "fig:"){width="45.00000%"} Summary and discussions ======================= This work is concerned with a coarse-grained energy model directly obtained from the full molecular dynamics model. The goal is to find a more efficient model so that the heat conduction process can be simulated with a model that is a lot cheaper than non-equilibrium molecular dynamics simulations. Our focus has been placed on the equilibrium statistics of such models, which conceptually, is often a good starting point to develop a stochastic model. Unlike many of the coarse-grained molecular models, in which the coarse-grained velocity is expected to have Gaussian statistics, we found that the coarse-grained energy follows non-Gaussian statistics. We proposed to introduce multiplicative noise, within the Markovian embedding framework for the memory term, to ensure that the solution of the coarse-grained models has the correct equilibrium statistics. Although we only considered a one-dimensional model as the example, the framework is applicable to more general systems. In particular, none of the theorems assumed the space dimensionality. The applications to nano-mechanical systems is currently underway. This work suggests a new paradigms for modeling nano-scale transport phenomena in general: Rather than relying on traditional deterministic models, e.g., the Fourier’s Law and the Fick’s Law, one may derive from first-principle a [stochastic and nonlocal]{} constitutive relation for the mass or heat current. The extension of the current effort to general diffusion processes will be in the scope of our future work. [100]{} M. P. Allen and D. J. Tildesley. . Oxford University Press, 1989. M. Baaden and S. J. Marrink. 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--- abstract: \| The effectiveness of carbon supported polyaniline as anode catalyst in a fuel cell (FC) with direct formic acid electrooxidation is experimentally demonstrated. A prototype FC with such a platinum-free composite anode exhibited a maximum room-temperature specific power of about 5 $mW/cm^2$ PACS: 82.47.Gh 82.65.+r 82.45.Fk author: - \| A. G. Zabrodskii, M. E. Kompan, V. G. Malyshkin[^1]\ Ioffe Physico-Technical Institute, RAS\ St. Petersburg, 194021 Russia\ I. Yu. Sapurina[^2]\ Institute of Macromolecular Compounds, RAS\ St. Petersburg, 199004 Russia date: 'April 11, 2006' title: \| Carbon Supported Polyaniline\ as Anode Catalyst:\ Pathway to Platinum-Free Fuel Cells --- 1\. The main advantage of fuel cells (FC) employing liquid organic fuels as compared to those using hydrogen is the simplicity of storage and recharging. Among the FC with liquid fuels, the most thoroughly studied are FC with direct methanol electrooxidation (DMFC) \[1\]. The other potential liquid organic fuels include formic acid. In FC employing direct formic acid electrooxidation (DFAFC) \[2\], a somewhat lower energy capacity (as compared to that of DMFC) is compensated by other advantages, including relatively simple reaction nature (and, accordingly, facilitated problem of catalyst selection) and lower extent of fuel crossover through the proton-conducting (nafion) membrane. As is known, direct electrochemical oxidation of organic fuels on the FC anode leads to the formation of $CO_2$ (complete oxidation). In the case of DFAFC, the process is: $$\begin{aligned} HCOOH&\rightarrow&CO_2 +2H^{+} +2e^{-} \label{rand}\end{aligned}$$ The active evolution of carbon dioxide is a reliable manifestation of this reaction. The process on the cathode consists in the reduction of molecular oxygen $O_2$ with the formation of water$H_2O$ : $$\begin{aligned} \frac{1}{2} O_2 + 2H^{+} +2e^{-} &\rightarrow& H_2O \label{rcat}\end{aligned}$$ It should be noted that cathodic reaction (2) is common to all FC with proton-conducting membranes, which use oxygen as the oxidizer. The main problems inherent in all FC with direct electrochemical oxidation of organic fuels are the following: \(a) The output power yield is limited by the rate of anodic reaction and usually varies at room temperature within 5—20 $mW/cm^2$ \[3–5\]. \(b) The fuel crossover through a nafion membrane to cathode and its oxidation on the FC cathode surface is equivalent to an opposite emf \[4, 6\], that is, to a decrease in output power and in the FC efficiency. \(c) Some organic fuels produce degradation of nafion membrane in the course of FC operation. The anode and cathode of FCs are electrodes with catalytic coatings, which are usually based on platinum (Pt). The limited resource of this metal on the Earth stimulates the interest in creating nonplatinum catalysts. For cathodic reaction (2), catalysts containing no metallic platinum have been developed based on macrocyclic complexes (see \[3, 7\] and references therein), biological materials \[8\], and transition metal compounds \[3, 6, 7\]. It was also reported that polyaniline (PANI) exhibited catalytic activity in the reaction of oxygen reduction in air–metal cells \[9\]. All these systems exhibit a lower catalytic activity than platinum, but their development and practical use nevertheless good have prospects. In recent years, FC units have been developed with extensive use of conducting polymers (synthetic metals) such as PANI, polypyrrole, and polythiophene \[10–12\]. Some data \[3, 13\] and our previous results \[14, 15\] indicate that polymers with electron–proton conductivity (such as PANI) can increase the efficacy of platinum group metal catalysts \[16\]. As for anodic reaction (1), no catalysts free of platinum group metals have yet been reported, although there were communications on the catalytic activity of PANI in the reactions of anodic oxidation of hydrogen \[17\], methanol \[18\], and ascorbic acid \[19\]. As is known, PANI combines a high level of electron (hole) conductivity ($1$ — $10$ S/cm \[11, 12\]) with proton conductivity (up to $10^{-2}$ S/cm) \[15, 20, 21\]. The mixed conductivity type of this polymer is of key importance in electrochemical processes involving simultaneous transport of both protons and electrons. It should also be noted that PANI has a variable structure; this polymer contains benzoid and quinoid fragments linked via nitrogen atoms occurring in various oxidation and protonation states. The ratio of the fragments of various types can change in a reversible manner depending at least on two parameters of the reaction medium: oxidation potential and acidity \[11\]. This circumstance provides broad possibilities for controlled modification of the properties of PANI. 2\. This preprint reports the process of electrochemical oxidation of formic acid in an FC with the anode made of a carbon material coated with a nanolayer (100 nm) of PANI. PANI in an emeraldine form was obtained via aniline oxidation with ammonium peroxydisulfate immediately on the surface of fibers of a porous carbon material \[22\]. The resulting PANI was strongly adsorbed on the carbon substrate. The polymer nanolayer encapsulated carbon fibers in the entire volume of the porous carbon matrix, so that the total weight fraction of PANI in the resulting composite reached 20%. The electro- chemical cell schematically depicted in Fig. 1. In order to study the anodic reaction, we used a cathode half–MEA (membrane–electrode assembly) \[23\] comprising a Nafion 117 membrane with the standard DMFC cathode containing $\rm 4mg/cm^2$ of platinum. The cathode operated in air under natural convection conditions. The anode was made of Torray TGPH-060 carbon cardboard covered with PANI nanolayer as described above. The cardboard was pressed against the nafion membrane with a thicker cardboard (free of PANI coating), which simultaneously served as electrode and gasdiffusion medium. The fuel was a 5% solution of formic acid (HCOOH) in a 0.5M aqueous $H_2SO_4$ solution. Since the experiments were performed with a fuel possessing ionic conductivity, the problem of contact quality in the anodic region was not so important, since protons could readily pass from one medium to another via the ion-containing fuel. Figure 2 shows the experimental loading characteristics measured using a prototype FC describe above. Plots of the FC output voltage versus current density in the membrane had the typical form, beginning at about 0.7V (at a nearly zero current density) and rapidly decaying at current densities below 5 $mA/cm^2$ . Using the subsequent less steep decrease, the internal resistance of the FC prototype was estimated at about 10 $\Omega cm^2$. The maximum specific output power reached in our experiments at room temperature was about 5 $mW/cm^2$ . This value corresponds to the output power of the typical DMFCs operating at room temperature (20C) \[3, 4\]. In the region of high current densities (10–40 $mA/cm^2$), the curves were poorly reproduced in different experimental runs. As was noted above, the active gas evolution on the FC anode operating on formic acid fuel is a reliable criterion for reaction (1) to actually take place. Indeed, when the FC was connected to a low-ohmic load and a relatively large current passed through the cell, intensive evolution of $CO_2$ bubbles (Fig. 3) was observed. The volume of liberated gas was measured and compared to the amount of charge transported vie the FC circuit. In this calculation, we assumed that the oxidation of the HCOOH molecule yields a charge of $2\times 1.6 \cdot 10^{-19} C$. The volume of $CO_2$ calculated for the transferred charge was 1.8 times the volume evolved in the experiment. The reason for so large a discrepancy is still unclear. One possible explanation is offered by the following mechanism: a fraction of current could be, in principle, related to the additional oxidation of PANI with the formation of pernigraniline. However, calculations showed that an additional charge provided by complete oxidation of PANI present on the anode was two orders of magnitude lower than the total charge transported in the experiment. Thus, the electrochemical oxidation of formic acid is the only process that can be responsible for the liberation of energy in the prototype FC studied. Moreover, if the observed current were related to PANI oxidation, the output current would unavoidably drop from one run to another, which was not observed in our experiments. The experiments showed no systematic decrease in the output power during the first two days. In the family of loading curves presented in Fig. 2, the curves corresponding to the maximum current and density were obtained in the last experimental run of the series. The current and output power exhibited a reversible decrease as a result of the fuel consumption in the FC and were restored on the initial level upon adding a new portion of the fuel. However, the experiments performed in the following days showed a decrease in the current and approximately proportional decrease in the FC open-circuit voltage (down to 0.4V), and even to a lower level in the subsequent week. We explain this behavior by the diffusion (crossover) of formic acid in the membrane, which results in the appearance of fuel on the oxidizer side and is equivalent to the opposite emf operation \[4, 6\]; an additional detrimental factor is degradation of the nafion membrane surface in contact with formic acid. It is necessary to emphasize the stability of results. The electrochemical oxidation of formic acid is not characteristic of the given type of carbon material. Specific output power on a level of $3-5 mW/cm^2$ was also obtained in prototype FCs where PANI was supported (instead of Torray TGPH-060) on carbon materials of the Kinol ACC-10-20 or Busofit T-1-55 types. Thus, the results of our experiments convincingly demonstrated the catalytic activity of PANI in the anodic reaction of formic acid oxidation. 3\. We can only suggest some notions concerning the nature of the catalytic activity of PANI. One possible mechanism is the reduction of PANI from the emeraldine to leuco-emeraldine form, which is accompanied by the oxidation of formic acid and is followed by leuco-emeraldine oxidation to emeraldine and electron transfer to the anode. We believe that the redox transition of PANI from the emeraldine form to a lower oxidation state (leuco-emeraldine) mediates in the electron transfer and accelerates the oxidation of formic acid. Another special question concerns the possible role of the carbon-PANI interface in the enhancement of the catalytic activity. We suggest that, since PANI is a p-type conductor \[24\] and carbon materials possess metallic conductivity, this interface features a potential barrier of the Schottky type. Since PANI is a medium permeable to liquids (in this case, to formic acid), HCOOH molecules occur in a strong electric field near the proton-conducting membrane and the field induces their polarization, which can in principle lead to a decrease in the dissociation energy. The close phenomenon of the polarization and subsequent ionization of shallow impurity states in the electric field is well known in semiconductor physics \[25\]. 4\. Thus, we have experimentally demonstrated that carbon supported PANI exhibits high catalytic activity in the reaction of anodic oxidation (1) of formic acid in fuel cells of the DFAFC type. This activity was observed in a working prototype FC with nonplatinum composite anode of the carbon supported PANI type, which ensured a stable specific output power of about 5 $mW/cm^2$ over a long period of time. 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[ http://www.poly-aniline.com/ ]( http://www.poly-aniline.com/ ) ![\[figcell\] Schematic diagram of the electrochemical cell involving a cathode half-MEA: (1) liquid fuel container; (2) membrane–electrode assembly (see the text for details); (3) plate electrodes with holes; (4) case with air channels. Spacers and screws are not shown. ](fig_1x){width="8cm"} ![\[figvi\] Plots of voltage U (left scale) and specific power (right scale) versus current density measured in a prototype FC. Curves 1–3 refer to three sequential experimental runs performed with a 10-min time interval. ](fig_2x){width="16cm"} ![\[figpuzyri\] $CO_2$ evolution on the anode of operating FC. ](fig_3x){width="15cm"} [^1]: [email protected] [^2]: [email protected]
--- abstract: 'In this paper, we present a real-time successive convexification algorithm for a generalized free-final-time 6-degree-of-freedom powered descent guidance problem. We build on our previous work by introducing the following contributions: (i) a free-ignition-time modification that allows the algorithm to determine the optimal engine ignition time, (ii) a tractable aerodynamics formulation that models both lift and drag, and (iii) a continuous state-triggered constraint formulation that emulates conditionally enforced constraints. In particular, contribution (iii) effectively allows constraints to be enabled or disabled by if-statements conditioned on the solution variables of the parent continuous optimization problem. To the best of our knowledge, this represents a novel formulation in the optimal control literature, and enables a number of interesting applications, including velocity-triggered angle of attack constraints and range-triggered line of sight constraints. Our algorithm converts the resulting generalized powered descent guidance problem from a non-convex free-final-time optimal control problem into a sequence of tractable convex second-order cone programming subproblems. With the aid of virtual control and trust region modifications, these subproblems are solved in succession until convergence is attained. Simulations using a third-party solver demonstrate the real-time capabilities of the proposed algorithm, with a maximum execution time of less than $0.7$ seconds over a multitude of problem feature combinations.' author: - 'Michael Szmuk[^1], Taylor P. Reynolds[^2], and Behçet Aç[i]{}kmeşe[^3]' bibliography: - 'bibliography.bib' title: 'Successive Convexification for Real-Time 6-DoF Powered Descent Guidance with State-Triggered Constraints' --- Introduction {#sec:intro} ============ paper presents a real-time guidance algorithm that solves a generalized free-final-time 6-degree-of-freedom (DoF) powered descent guidance problem with free ignition time, aerodynamic effects, and conditionally enforced constraints. Real-time optimal guidance algorithms are an enabling technology for future manned and unmanned planetary missions that require autonomous precision landing capabilities. Such algorithms allow for the explicit inclusion of operational and mission constraints, and are able to compute trajectories that are (locally) optimal with respect to key metrics, such as propellant consumption, burn time, or miss distance. Consequently, these algorithms enhance a lander’s ability to recover from a wider range of dispersions encountered during the entry, descent, and landing phase, and to react to obstructions on the surface that become apparent only as the vehicle approaches the landing site. The constraint satisfaction afforded by these algorithms allows designers to select more ambitious and scientifically interesting landing sites, enhances the lander’s ability to handle uncertainties, and ultimately increases the likelihood of mission success. Moreover, the optimality may be leveraged to improve the scientific return of planetary missions by reducing the propellant mass fraction [@Scharf15]. The relevance of real-time optimal guidance algorithms been demonstrated in the (now routine) landings of orbital-class vertical-takeoff-vertical-landing reusable launch vehicles [@larsNAE]. Solving the 6-DoF powered descent guidance problem in real-time is challenging for several reasons. First, the problem consists of nonlinear dynamics and non-convex state and control constraints, and does not yet have an analytical solution. Second, since the problem must be solved using numerical methods, the validity of the solution is depends heavily on the discretization scheme. Third, the non-convex nature of the problem makes it difficult to select a suitable reference trajectory to initialize an iterative solution process. The successive convexification methodology used in this paper addresses these issues, and is able to solve the problem quickly and reliably over a wide range of conditions. Related Work ------------ The powered descent guidance literature can be separated into works that consider 3-DoF translation dynamics, and ones that consider more general 6-DoF rigid body dynamics. Work on 3-DoF powered descent guidance began many years ago during the Apollo program, with several authors approaching the problem using optimal control theory [@Meditch1964; @Lawden1963] and the calculus of variations [@Marec1979]. These early works noted that the solution to the fuel-optimal powered descent guidance problem exhibited bang-bang behavior, where the thrust assumed either the minimum or maximum allowable value everywhere along the optimal trajectory. While these results offered important insights into the powered descent guidance problem, they were not incorporated into the Apollo flight code, since the polynomial-based guidance methods used to perform the landings were deemed sufficiently close to optimal, and were far simpler to design [@klumpp]. In the years following Apollo, researchers continued searching for analytical solutions to the 3-DoF landing problem [@Kornhauser1972; @Breakwell1975; @Azimov1996; @dsouza97] that would be conducive to on-board implementation for future missions. Unmanned missions to the Martian surface in the early the 2000s renewed interest in using direct methods to compute solutions to the powered descent guidance problem. In 2005, Topcu, Casoliva, and Mease presented results for the 3-DoF problem that resembled a modern take on [@Lawden1963], adding numerical simulations to reinforce and demonstrate theoretical results [@Topcu2005; @topcu_pdg]. Around the same time, Aç[i]{}kmeşe and Ploen published work on a convex programming approach to the problem [@Acikmese2005; @ploenaiaa06; @behcetjgcd07]. Using Pontryagin’s maximum principle, they showed that the non-convex thrust magnitude lower bound constraint could be losslessly convexified by introducing a relaxation that rendered the optimal solution of the (now convex) relaxed problem identical to that of the original non-convex problem. Hence, the difficult task of solving the non-convex 3-DoF powered descent guidance problem was converted into the far simpler, but equivalent, task of solving a convex second-order cone programming problem. Subsequently, lossless convexification was extended to include non-convex pointing constraints, and to encompass more general optimal control problems [@behcetaut11; @harris_acc; @lars_12scl; @behcet_aut11; @larssys12; @pointing2013; @matt_aut1]. This methodology was demonstrated in a sequence of flight experiments in the early 2010s [@dueri2016customized; @Scharf14]. More recently, researchers have considered the 6-DoF powered descent guidance problem. In 2012, Lee and Mesbahi formulated the powered descent guidance problem using dual quaternions [@Lee2012]. This work revealed that certain coupled rotational-translational constraints are convex when using this parameterization [@Lee2015]. In [@Lee2017], a piecewise-affine approximation of the dynamics was used to design a model predictive controller that generated thrust and torque commands. However, since the accuracy of the equations of motion relied heavily on the temporal resolution of the approximation, the approach resulted in either short prediction horizons or prohibitively large optimization problems. Work on powered descent guidance and atmospheric entry problems turned to Sequential Convex Programming (SCP) methods in order to handle more general non-convexities [@jordithesis; @liu2014solving; @liu2015entry; @wang2016constrained]. These methods solve non-convex problems by iteratively solving a sequence of local convex approximations obtained via linearization. The first order approach used in SCP methods guarantees that the approximations are convex, and lies in stark contrast to the second order approach used in Sequential Quadratic Programming (SQP) methods [@Boggs1995], which may expend significant computational effort to ensure the convexity of each approximation. In this paper, we solve a generalized free-final-time 6-DoF powered descent guidance problem using the successive convexification framework. Successive convexification can be classified as an SCP method that uses virtual control and trust region modifications to facilitate convergence. The algorithms detailed in [@SCvx_cdc16; @mao2016successive; @SCvx_2017arXiv] use an exact penalty method in conjunction with hard trust regions, whereas those used in this paper and in [@szmuk2016successive; @szmuk2017successive; @szmuk2018successive] employ soft trust regions. Statement of Contribution ------------------------- This paper presents three primary contributions: (i) a free-ignition-time modification that allows the algorithm to determine the optimal engine ignition time, (ii) a tractable aerodynamics formulation that models both lift and drag, and (iii) a continuous state-triggered constraint formulation that emulates conditionally enforced constraints. In particular, contribution (iii) effectively allows constraints to be enabled or disabled by if-statements in a continuous optimization framework. To the best of our knowledge, state-triggered constraints bear the most resemblance to two existing approaches: mixed-integer programming, and complementarity constraints. The former approach implements discrete decisions explicitly using integer variables, whereas the latter formulates such decisions implicitly using continuous variables. Despite the existence of efficient branch and bound techniques, mixed-integer formulations can suffer from poor complexity [@Richards2015], and are not conducive to solving the generalized powered descent guidance problem in real-time. Like state-triggered constraints, complementarity constraints [@Cottle1992; @Heemels2000] permit completely continuous formulations that can be more efficient than mixed-integer approaches (see Section 2.3 in [@biegler2014]). However, complementarity constraints represent bi-directional if-and-only-if-statements, whereas state-triggered constraints represent more general uni-directional if-statements. Therefore, we argue that the proposed continuous state-triggered constraints are a key building block that enable the formulation of a broader set of constraints. The secondary contributions of this paper are an improved description of the successive convexification methodology used in [@szmuk2018successive], and timing results that demonstrate the real-time capabilities of the algorithm. This paper regards the powered descent guidance problem as a feedforward trajectory generation problem, and does not address the topic of feedback control or issues arising from trajectory re-computation. Outline ------- In , we present the primary contributions of this paper in the context of a generalized powered descent guidance problem. In , we detail the successive convexification procedure and algorithm. In , we present simulation results that highlight the paper’s contributions, and timing results that demonstrate the real-time capabilities of the proposed algorithm. Lastly, provides concluding remarks. Problem Statement {#sec:problem_statement} ================= In this section, we present a 6-DoF formulation for a generalized powered descent guidance problem in the presence of atmospheric effects. This section is organized as follows. In and , we introduce the assumptions and notation used in our formulation. In , we present a baseline problem formulation. In , , and , we discuss the primary contributions of the paper, namely the free-ignition-time modification, the aerodynamic models, and state-triggered constraints. Lastly, provides a statement of the non-convex generalized powered descent guidance problem. Assumptions {#sec2:assumptions} ----------- Most powered descent maneuvers commence at speeds substantially below orbital velocities and within only a few kilometers of the landing site. Hence, we neglect the effects of planetary rotation and assume a uniform gravitational field. We assume that the vehicle is equipped with a single rocket engine that can be gimbaled symmetrically about two axes up to a maximum gimbal angle, but stress that other thruster configurations can be readily accommodated. Further, we assume that the engine can be throttled between fixed minimum and maximum thrusts, and that once the engine is ignited it remains on until the terminal condition is reached. To tailor our treatment to applications with non-negligible atmospheric effects, we assume that the ambient atmospheric density and pressure are constant, that the aerodynamic forces are governed by the simplified models detailed in , and that the center-of-pressure is fixed with respect to a body-fixed reference frame. Further, we neglect the effects of winds, noting that constant uniform wind profiles can be readily incorporated into our formulation, and account for thrust reduction induced by atmospheric back-pressure by assuming the affine mass depletion dynamics in [@szmuk2018successive]. Lastly, to make the problem tractable, we neglect higher order phenomena such as elastic structural modes and fuel slosh, and model the vehicle as a rigid body with a constant body-fixed center-of-mass and inertia. Notation {#sec2:notation} -------- We begin by denoting time as $t\in\real$, and define the initial time $\tin$ as the time at which the optimal control problem begins, the ignition time $\tig$ as the time at which the engine turns on, and the final time $\tf$ as the time at which the vehicle reaches the terminal condition. These epochs are defined such that $\tin\leq\tig < \tf$, and their subscripts are used to denote problem parameters associated with the respective time epochs. Further, we define coast time as $\tc\definedas\tig-\tin$ and burn time as $\tb\definedas\tf-\tig$. During the coast phase, the vehicle’s states passively evolve according to its engine-off dynamics, whereas during the burn phase, the vehicle actively maneuvers in order to achieve its landing objective. Without loss of generality, we define the ignition time epoch as the time at which $t=0$, where it follows that $\tc = -\tin$ and that $\tb = \tf$. This timeline is illustrated in Figure \[fig:traj\_timeline\]. The subscripts $\inertial$ and $\body$ are used to denote problem parameters expressed in the inertial and body-fixed reference frames $\iframe$ and $\bframe$, respectively. We define $\iframe$ as a surface-fixed Up-East-North coordinate frame whose origin coincides with the landing site. Likewise, we define $\bframe$ such that its origin coincides with the vehicle’s center-of-mass, its $x$-axis points along the vertical axis of the vehicle, its $y$-axis points out the side of the vehicle, and its $z$-axis completes the right-handed system. We use $\m(t)\in\real_{++}$, $\pos(t)\in\real^3$, and $\vel(t)\in\real^3$ to respectively denote the mass, position, and velocity states, and $\thrust(t)\in\real^3$ and $\aero(t)\in\real^3$ to respectively denote the thrust vector and aerodynamic force. We denote the unit quaternion that parameterizes the transformation from $\iframe$ to $\bframe$ by $\qIB(t)\in\sthree\subset\real^{4}$, and its corresponding direction cosine matrix by $\cIB(t)\definedas\cIB\big(\qIB(t)\big)\in\sothree$ [@Hanson2006]. The conjugate of $\qIB(t)$ is denoted by $\qIB^{}(t)$, where it follows that $\cBI(t)\definedas\cIB^T(t) = \cIB\big(\qIB^(t)\big)$. Quaternion multiplication and the identity quaternion are denoted by $\otimes$ and $\qidentity$, respectively. The angular velocity of $\bframe$ relative to $\iframe$ is denoted by $\omegaB(t)\in\real^3$, and is expressed in body-fixed coordinates. Lastly, we use $\bvec{e}_i$ to denote the $\ith{i}{th}$ basis vector of $\real^n$, $\crossprod{}{}$ to denote the vector cross product, and $\;\dotprod{}{}\;$ to denote the vector dot product. Baseline Problem {#sec2:6dof} ---------------- ### Dynamics As stated in , the mass-depletion dynamics are assumed to be an affine function of the thrust magnitude, and are given by $$\label{eq:mass_dyn} \mdot(t) = -\mdotalpha\twonorm{\TB(t)}-\mdotbeta\,,$$ where $\mdotalpha\definedas1/\Isp\gstd$ and $\mdotbeta\definedas\mdotalpha\Pamb\,\Anoz$, $\Isp$ is the vacuum specific impulse of the engine, $\gstd$ is standard Earth gravity, $\Anoz$ is the nozzle exit area of the engine, and $\Pamb$ is ambient atmospheric pressure. The second term in represents the specific impulse reduction incurred by atmospheric back-pressure, and assumes that the nozzle remains choked over the allowable throttle range. The translational states are governed by the following dynamics \[eq:trans\_dynamics\] $$\begin{aligned} \rIdot(t) &= \vI(t)\,, \label{eq:trans_dynamics_a} \\ \vIdot(t) &= \frac{1}{\m(t)}\FI(t)+\gI, \label{eq:trans_dynamics_b} \end{aligned}$$ where $\gI\in\real^3$ and $\FI(t)\definedas\cBI(t)\TB(t) + \aeroI(t)\in\real^{3}$ respectively denote the constant gravitational acceleration and net propulsive and aerodynamic force acting on the vehicle. The thrust vector is expressed in $\bframe$ coordinates to simplify the attitude dynamics and control constraints that follow. The aerodynamic term $\aeroI(t)$ is defined . The attitude states are governed by the following rigid-body attitude dynamics \[eq:att\_dyn\] $$\begin{aligned} \qIBdot(t) &= \frac{1}{2}\skewsymbig\big(\omegaB(t)\big)\qIB(t)\,, \label{eq:att_dyn_a} \\ \inertia\omegaBdot(t) &= \MB(t) - \crossprod{\omegaB(t)}{\inertia\omegaB(t)}\,, \label{eq:att_dyn_b} \end{aligned}$$ where $\skewsymbig(\cdot)$ is a skew-symmetric matrix defined such that the quaternion kinematics in hold [@Hanson2006], $\inertia\in\spd{3}$ denotes the body-fixed constant inertia tensor of the vehicle about its center of mass, and $\MB(t)\definedas\displaystyle{\crossprod{\rTB}{\TB(t)} + \crossprod{\rCPB}{\aeroB(t)}}\in\real^{3}$ denotes the net propulsive and aerodynamic torque acting on the vehicle. The vectors $\rTB\in\real^3$ and $\rCPB\in\real^3$ give the constant positions of the engine gimbal pivot point and the aerodynamic center of pressure, respectively. The aerodynamic term $\aeroB(t)$ is defined in . ### State Constraints We impose four state constraints in our baseline formulation. First, we constrain the mass of the vehicle to values greater than a minimum dry mass $\mdry\in\real_{++}$ by enforcing $$\label{eq:minmass} \m(t) \geq \mdry.$$ Second, we constrain the inertial position to lie inside of a glide-slope cone with half-angle $\glideslope\in\intei{0\dg}{90\dg}$ and vertex at the origin of $\iframe$ by enforcing $$\label{eq:glide_slope} \dotprod{\ex}{\rI(t)} \geq \tan \glideslope \twonorm{\Hgs \rI(t)},$$ where $\Hgs\definedas[\ey\;\,\ez]^T\in\real^{2 \times 3}$. Third, we define the tilt angle of the vehicle as the angle between the $x$-axes of $\iframe$ and $\bframe$, and constrain it to be less than a maximum tilt angle $\tiltmax\in\intie{0\dg}{90\dg}$ by enforcing $$\label{eq:max_tilt} \cos\tiltmax \leq 1-2\twonorm{\Htilt \qIB(t)},$$ where $\Htilt\definedas[\bvec{e}_3\;\,\bvec{e}_4]^T\in\real^{2 \times 4}$ if a scalar-first quaternion convention is used. Fourth, we limit the angular velocity to a maximum value of $\omegamax\in\real_{++}$ by enforcing $$\label{eq:wmax} \twonorm{\omegaB(t)} \leq \omegamax.$$ ### Control Constraints We impose two control constraints in our baseline formulation. First, we impose lower and upper bounds on the thrust magnitude such that $$\label{eq:thrust_bounds} 0 < \Tmin \leq \twonorm{\TB(t)} \leq \Tmax\,,$$ where $\Tmin$ and $\Tmax$ are the minimum and maximum allowable thrust magnitudes, respectively. Second, we constrain the thrust vector to lie within a prescribed maximum gimbal angle $\gimbalmax\in\intii{0\dg}{90\dg}$ relative to the $x$-axis of $\bframe$ by enforcing $$\label{eq:max_gimbal} \cos \gimbalmax \twonorm{\TB(t)} \leq \dotprod{\ex}{\TB(t)}.$$ ### Boundary Conditions We now present a notional set of boundary conditions, with the understanding that they may be modified to accommodate different scenarios. The conditions at the ignition time epoch are given by $$\label{eq:bcs_initial} m(\tig) = \mig\,, \quad \rI(\tig) = \rIig\,, \quad \vI(\tig) = \vIig\,, \quad \omegaB(\tig) = \omegaBig\,,$$ where $\mig\in\real_{++}$, $\rIig\in\real^3$, and $\vIig\in\real^3$ are the prescribed mass, position, and velocity at ignition time $\tig$, respectively. We assume that $\mig>\mdry$. The conditions at the final time epoch are given by $$\label{eq:bcs_final} \rI(\tf) = \rIf\,, \quad \vI(\tf) = \vIf\,, \quad \qIB(\tf) = \qIBf\,, \quad \omegaB(\tf) = \omegaBf\,,$$ where $\vdes\in\real_{+}$ is the prescribed terminal vertical descent speed. Free Ignition Time {#sec2:fit} ------------------ In the baseline problem formulation presented in , $\rI(\tig)$ and $\vI(\tig)$ are restricted to prescribed points at the time of ignition. We now introduce the free-ignition-time modification to , such that $\rI(\tig)$ and $\vI(\tig)$ are constrained to a prescribed curve as follows $$\m(\tig) = \mig\,, \quad \rI(\tig) = \prig(\tc)\,, \quad \vI(\tig) = \pvig(\tc)\,, \quad \omegaB(\tig) = \omegaBig\,, \label{eq:bcs_initial_fit}$$ where the coast time $\tc\in\intee{0}{\tcmax}$ is included as a solution variable, and $\prig: \real\rightarrow\real^3$ and $\pvig: \real\rightarrow\real^3$ are predetermined vector valued polynomials describing an engine-off trajectory. We choose these polynomials to represent an aerodynamics-free trajectory using $$\label{eq:bcs_fit_polynomials} \prig(\xi) \definedas \rIin + \vIin\,\xi + \frac{1}{2}\gI\xi^2, \quad \pvig(\xi) \definedas \vIin + \gI\xi,$$ where $\rIin$ and $\vIin$ are prescribed position and velocity vectors at the initial time epoch (see Figure \[fig:traj\_timeline\]). Higher order effects (e.g. aerodynamics) can be embedded in $\prig(\cdot)$ and $\pvig(\cdot)$ by using higher fidelity models to propagate the vehicle state over a prediction horizon of length $\tcmax$, and fitting polynomials to the resulting path. Aerodynamic Model {#sec2:aero} ----------------- [0.5]{} [0.5]{} We now introduce a tractable aerodynamic model that approximates the relationship between the aerodynamic force and the velocity vector. The model expresses the aerodynamic force in $\bframe$ coordinates as follows $$\label{eq:dragB} \aeroB(t) = - \frac{1}{2}\density V(t)\Sa\Ca\cBI(t)\vI(t)\,,\quad\Ca\in\spd{3}\,,$$ where $\density$ is the ambient atmospheric density, $V(t)\definedas\twonorm{\vI(t)}$, and $\Sa\in\real_{++}$ is a constant reference area. We refer to $\Ca$ as the aerodynamic coefficient matrix, and emphasize that it is a symmetric-positive-definite matrix that does not conform to the standard scalar definition. Our definition of $\Ca$ ensures that for any $V\in\real_{+}$ the set $\setaero\definedas\big\{ \aeroB : \aeroB = -\frac{1}{2}\density V^2\Sa\Ca\hat{\bvec{v}}_\body\,,\; \hat{\bvec{v}}_\body\in\mathcal{S}^2 \,\big\}$ defines a fixed ellipsoid in $\bframe$ coordinates. Since most rocket-powered vehicles are approximately axisymmetric, we assume $\Ca = \diag{[c_{a,x},\; c_{a,yz},\; c_{a,yz}]}$, where $c_{a,x}$ and $c_{a,yz}$ are positive scalars. This assumption aligns the principal axes of $\setaero$ with the axes of $\bframe$. If $c_{a,x} = c_{a,yz}$, then $\aeroB(t)$ is always anti-parallel to $\vB(t)$. In this case, $\aeroB(t)$ may be interpreted as a pure drag force, and the model recovers the aerodynamic drag model used in [@szmuk2016successive]. Since the set $\setaero$ corresponding to this choice of $\Ca$ defines a sphere, we refer to the corresponding model as the spherical aerodynamic model, illustrated in Figure \[fig:aerodynamic\_model\_a\]. Under the assumptions of the spherical model, the product $\cBI(t) \Ca \cIB(t)$ simplifies to $c_{a,x} I_{3 \times 3}$, thus rendering $\aeroI(t) = \cBI(t)\aeroB(t)$ independent of attitude. Alternatively, if $c_{a_x} \neq c_{a,yz}$, then $\aeroB(t)$ can also have components orthogonal to $\vB(t)$. In this case, $\aeroB(t)$ may be interpreted as the vector sum of a drag and lift force. Furthermore, if we assume that $c_{a,x} < c_{a,yz}$, we ensure that the vehicle experiences minimum drag when $\vB$ is aligned with the $x$-axis of $\bframe$, and that the lift component of $\aeroB$ points in the correct direction. Since the set $\setaero$ corresponding to this choice of $\Ca$ defines an ellipsoid, we refer to the corresponding model as the ellipsoidal aerodynamic model, illustrated in Figure \[fig:aerodynamic\_model\_b\]. State-Triggered Constraints {#sec2:stc} --------------------------- In this section, we introduce the most important contribution of this paper: a continuous formulation of state-triggered constraints (STCs). The most common type of constraints seen in the optimal control literature are enforced over predetermined time intervals; we refer to such constraints as temporally-scheduled constraints. In contrast, an STC is enforced only when a state-dependent condition is satisfied, and emulates a constraint gated by an if-statement conditioned on the solution variables. Thus, an optimal control problem containing an STC determines its solution variables with a simultaneous understanding of how the constraint affects the optimization, and of how the optimization enables or disables the constraint. While the resulting continuous formulation is still non-convex, we have found that it is readily handled by the successive convexification framework [@SCvx_2016arXiv; @SCvx_2017arXiv; @SCvx_cdc16], as demonstrated in . ### Formal Definition of State-Triggered Constraints {#sec2:stc_formal_def} Formally, we define a state-triggered constraint as an equality constraint that is enforced conditionally according to the following logical statement $$\label{eq:stc_orig} \stcg(\stcx) < 0 \;\Rightarrow\; \stcf(\stcx) = 0\,,$$ where $\stcx\in\real^\stcnx$ represents the optimization variable of the parent problem, $\stcg(\cdot) : \real^\stcnx \rightarrow \real$ is a piecewise continuously differentiable function called the trigger function, and $\stcf(\cdot) : \real^\stcnx \rightarrow \real$ is a piecewise continuously differentiable function called the constraint function. Accordingly, we refer to $\stcg(\stcx)<0$ as the trigger condition, and $\stcf(\stcx) = 0$ as the constraint condition. The logical implication in states that if the trigger condition is satisfied, then the constraint condition is enforced. By De Morgan’s Law, also implies satisfaction of the contrapositive. However, we emphasize that the satisfaction of the constraint condition does not imply satisfaction of the trigger condition (see ). \[rem:stc\_eq\_ineq\] The constraint condition in can be used to represent an inequality constraint by augmenting $\stcx$ with a non-negative slack variable and modifying $\stcf(\stcx)$ accordingly [@BoydConvex]. ### Continuous State-Triggered Constraints {#sec2:stc_cont_form} The STC expressed in represents a binary decision that is not readily incorporated into a continuous optimization framework. We address this issue by introducing continuous state-triggered constraints (cSTCs), which represent the logical implication in using the auxiliary variable $\stcs\in\real_{+}$ and the system of equations \[eq:stc\_cont\] $$\begin{aligned} \stcg(\stcx) + \stcs &\geq 0\,, \label{eq:stc_cont_a} \\ \stcs &\geq 0\,, \label{eq:stc_cont_b} \\ \stcs\cdot\stcf(\stcx) &= 0. \label{eq:stc_cont_c} \end{aligned}$$ The geometry of the cSTC in is shown on the left side of Figure \[fig:cstc\_geom\], where the lower-left axes show the feasible set of the STC in . The formulation in ensures that $\stcs$ is strictly positive if the trigger condition is satisfied. Since $\stcs>0$ implies that holds if and only if $\stcf(\stcx) = 0$, and are logically equivalent. Subject to mild assumptions on $\stcg(\cdot)$ and $\stcf(\cdot)$, admits a solution for any $\stcx\in\real^{\stcnx}$. ### Improved Formulation Using Linear Complementarity {#sec2:stc_lcp} As illustrated in Figure \[fig:cstc\_geom\], does not admit a unique $\stcs$ given $\stcx$. To resolve this ambiguity, we augment and to form a complementarity constraint and obtain the following improved cSTC formulation \[eq:lcp\_ad\] $$\begin{aligned} 0 \leq \stcs &\perp \big(\stcg(\stcx)+\stcs\big) \geq 0\,, \label{eq:lcp_a} \\ \stcs &\perp \stcf(\stcx)\,,\label{eq:lcp_d} \end{aligned}$$ where $a \perp b$ is used to denote $a \cdot b = 0$. For a given $\stcx$, forms a linear complementarity problem (LCP) in $\stcs$ [@Cottle1992]. This problem has a unique solution $\stcshat$ that varies continuously in $\stcg(\stcx)$ [@Cottle1992], and therefore in $\stcx$. The analytical solution to the LCP is given by $$\stcshat(\stcx) \definedas -\min\big(\stcg(\stcx)\,,0\big). \label{eq:lcp_sol}$$ Substituting into guarantees satisfaction of , resulting in the following equality constraint $$\label{eq:stc_lcp} \stch(\stcx) \definedas -\min\big(\stcg(\stcx)\,,0\big)\cdot\stcf(\stcx) = 0\,,$$ where the negative sign is retained to allow to be formulated as an inequality (see Remark \[rem:stc\_eq\_ineq\]). Thus, the ambiguity in $\stcs$ is resolved by replacing the constraints in with the logically equivalent constraint given in , which complies with the feasible set of the corresponding STC defined in . This can be seen in the two rightmost axes of Figure \[fig:cstc\_geom\]. Subsequent sections use the improved cSTC formulation in lieu of the original one. ### Example Application {#sec2:stc_ex_app} [0.5]{} [0.5]{} We now present an example application whose formulation within a continuous optimization framework is enabled by the state-triggered constraints introduced in -. Consider the problem of limiting the aerodynamic loads on a vehicle during a powered descent maneuver. On ascent, aerodynamic loads are often limited by imposing what are known as limits, where $q$ refers to dynamic pressure, and $\aoa$ refers to angle of attack. These constraints are typically valid only for small angles of attack, where aerodynamic loads are relatively easy to model. Unlike an ascent trajectory, a powered descent maneuver can exhibit a wider range of angles of attack, possibly exceeding $90\dg$ in cases where “hopping” maneuvers are permitted. Measures must therefore be taken to ensure that the vehicle does not operate in flight regimes with large model uncertainty. Specifically, we propose a simplified limit that enforces an angle of attack constraint only at high dynamic pressures. Formally, we express this constraint using the following STC $$\label{eq:stc_ex_aoa_orig} \twonorm{\vB(t)} > \Vaoa \;\Rightarrow\; -\dotprod{\bvec{e}_1}{\vB(t)} \geq \cos{\aoamax}\twonorm{\vB(t)},$$ where $\Vaoa\in\real_{++}$ is a speed above which the angle of attack is limited to $\aoa(t) \in\intee{0}{\aoamax}$. This STC is illustrated in Figure \[fig:stc\_ex\_aoa\]. Noting Remark \[rem:stc\_eq\_ineq\], we convert the STC in into the following cSTC \[eq:stc\_ex\_aoa\_imp\] $$\begin{aligned} \haoa{\vI(t)}{\qIB(t)} &\definedas -\min\Big(\gaoa{\vI(t)}\,,0 \Big) \cdot \faoa{\vI(t)}{\qIB(t)} \leq 0\,, \label{eq:stc_ex_aoa_imp_a} \\ \gaoa{\vI(t)} &\definedas \Vaoa-\twonorm{\vI(t)}, \\ \faoa{\vI(t)}{\qIB(t)} &\definedas \cos{\aoamax}\twonorm{\vI(t)}+\dotprod{\bvec{e}_1}{\cIB\big(\qIB(t)\big)\vI(t)}. \end{aligned}$$ A range-triggered field of view constraint imposed between a body-fixed downward-looking camera and the landing site may be obtained by replacing $\vI(t)$, $\Vaoa\,$, and $\aoamax$ in with $\rI(t)$, $\Rfov\,$, and $\losmax\,$, respectively (see Figure \[fig:stc\_ex\_fov\]). Such a constraint limits the line of sight angle to $\lambda(t)\in\intee{0}{\losmax}$ when the vehicle is at distances greater than $\Rfov$ away from the landing site. To conclude this section, we briefly discuss the shortcomings of three non-combinatorial alternatives to the cSTC proposed in : Consider a naive temporally-scheduled implementation, in which the problem is first solved without , and whose solution is used to determine the set of times $\mathcal{T}$ over which the trigger condition in is satisfied. The problem is then solved a second time with the constraint condition enforced over $t\in\mathcal{T}$. This new solution now satisfies the constraint condition over $t\in\mathcal{T}$, but does not necessarily satisfy the trigger condition for all $t\in\mathcal{T}$. In fact, this solution may satisfy the trigger condition for times where $t\notin\mathcal{T}$, thus necessitating a redefinition of $\mathcal{T}$. Unlike cSTCs, this approach does not convey the relationship between the trigger and constraint conditions to the optimization, thus allowing this situation to persist. Next, consider an implementation that replaces the STC in with $-\dotprod{\ex}{\vB(t)}/\twonorm{\vB(t)} \geq \cos{f_{\aoa}(t)}$, where $f_{\aoa}(t)\definedas f_{\aoa}\big(\twonorm{\vB(t)}\big)$ is a nonlinear scalar valued function that relates the maximum allowable angle of attack to the speed of the vehicle. This approach has two disadvantages: (i) it is less intuitive since the relationship between $\aoa$ and $q$ is embedded in $f_{\aoa}(t)$, and (ii) obtaining a satisfactory $f_{\aoa}(t)$ with proper numerical scaling may be difficult. Lastly, consider an implementation using two phases: the first with an angle of attack constraint and no velocity constraint, and the second with a velocity constraint and no angle of attack constraint. Further, assume that the terminal condition of the first phase is equated to the initial condition of the second, and that both phases are solved simultaneously. Such a multi-phase optimization approach ensures satisfaction of , and is well suited for applications where the temporal ordering and quantity of phases are known a priori. However, since the formulation presupposes a specific phase structure, this approach is not capable of introducing additional phases. In contrast, cSTCs can do so in order to achieve feasibility or improve optimality. Non-Convex Problem Statement {#sec2:nonconvex} ---------------------------- We now summarize the problem developed throughout this section. We assume a minimum-fuel objective function, but note that other objective functions can be readily used (e.g. a minimum-time problem would minimize $\tb$). The non-convex generalized powered descent guidance problem is summarized in Problem \[prob:ncvx\]. Convex Formulation {#sec:convex_formulation} ================== The successive convexification algorithm described in this section is designed to solve Problem \[prob:ncvx\] such that the converged solution (i) exactly satisfies the original nonlinear dynamics, (ii) approximates the state and control constraints by enforcing them only at a finite number of temporal nodes, and (iii) conservatively approximates the optimality and feasibility of the problem by using a finite-dimensional representation of the infinite-dimensional control signal. The proposed algorithm works by iteratively solving a sequence of subproblems until a converged solution is attained. Each iteration consists of two steps: a propagation step responsible for obtaining a subproblem, followed by a solve step responsible for solving said subproblem to full optimality. Each subproblem is a convex approximation of Problem \[prob:ncvx\], and each solve step results in a state and control trajectory, or iterate. The propagation step in the first iteration is computed using a user-defined initialization trajectory (see ), whereas subsequent iterations perform said approximation about the trajectory obtained by the previous iteration’s solve step. Since the solve step is executed using well understood algorithms (e.g. IPMs [@nocedal2006numerical]), this section primarily focuses on the propagation step. This section is organized as follows. In , we outline a procedure to convert a free-final-time nonlinear continuous-time optimal control problem into a general implementable fixed-final-time linear-time-varying discrete-time convex parameter optimization subproblem [@szmuk2018successive]. Specifically, - discuss three analytical steps that comprise the propagation step, and introduces the virtual control and trust region modifications used to aid convergence. In , we specialize the general subproblem to Problem \[prob:ncvx\]. Lastly, in , we describe two initialization strategies, and conclude by summarizing the proposed algorithm. General Implementable Convex Subproblem {#sec3:gen_subproblem} --------------------------------------- This section assumes the following general optimal control problem \[eq:gen\_ocp\] $$\begin{aligned} &\underset{\tc,\,\tb,\,\uu(t)}{\text{minimize}} \;\; \Jxu{\tb}{\xx(\tf)}, \hspace{-2.2cm}& & \label{eq:gen_ocp_a} \\ \text{s.t.} \quad &\dot{\xx}(t) = \fxu{\xx(t)}{\uu(t)}\,, \hspace{-2.2cm}& \forall&\,t\in\intee{\tig}{\tf}\,, \label{eq:gen_ocp_b} \\ &\hi{\zz(t)}{i} = 0\,, \hspace{-2.2cm}& \forall&\,t\in\intee{\tig}{\tf}\,,\; \forall\,i\in\setcvx\cup\setncvx\cup\setcstc, \label{eq:gen_ocp_c} \end{aligned}$$ where $\xx(t)\in\real^{\ns}$ and $\uu(t)\in\real^{\nc}$ denote respectively the state and control vectors, $\zz(t)\definedas \big[\tc\;\, \tb\;\, \xx^T(t) \;\, \uu^T(t)\big]^T\in\real^{\nz}$ (see for definitions of $\tc$ and $\tb$), $\fcn{\fcnJ}{\real_{++}\times\real^{\ns}}{\real}$ is a Mayer objective function [@berkovitz_opt_75] that is convex in its arguments, $\fcn{\fcnf}{\real^{\ns}\times\real^{\nc}}{\real^{\ns}}$ represents the continuous-time nonlinear dynamics, and $\fcn{\fcncons_i}{\real^{\nz}}{\real}$ represent equality constraints imposed on the trajectory. The equality constraints in are assumed to be convex for $i\in\setcvx\definedas\{1,\,\ldots\,,\np\}$, and non-convex for $i\in\setncvx\definedas\{\np+1,\,\ldots\,,\np+\nq\}$ and $i\in\setcstc\definedas\{\np+\nq+1,\,\ldots\,,\np+\nq+\nr\}$. The functions $\fcnJ$ and $\fcnf$ are assumed to be continuously differentiable, whereas each $\fcncons_i$ is assumed to be at least once differentiable almost everywhere. ### Normalization {#sec3:norm} The normalization step converts the free-final-time nonlinear continuous-time optimal control problem in into an equivalent fixed-final-time nonlinear continuous-time problem. This is achieved by temporally normalizing the burn phase from $t\in\intee{\tig}{\tf}$ to $\tau\in\intee{0}{1}$, where $\tnorm$ is the normalized burn phase time. Using the chain rule, the nonlinear dynamics in can be rewritten as $$\label{eq:tnorm_dyn} \xx'(t) \definedas \dd{}{\tnorm}\xx(t) = \dd{t}{\tnorm}\dd{}{t}\xx(t) = \left(\dd{t}{\tnorm}\right)\dot{\xx}(t).$$ Defining the dilation factor $\dilation\definedas dt / d\tnorm\in\real_{++}$ and replacing $\dot{\xx}(t)$ with the right-hand side of , the temporally-normalized dynamics are expressed as $$\label{eq:normalized_dyn} \xx'(\tnorm) = \fxus{\xx(\tnorm)}{\uu(\tnorm)}{\dilation} \definedas \dilation\cdot\fxu{\xx(\tnorm)}{\uu(\tnorm)}.$$ Since $\tnorm\in\intee{0}{1}$, it follows that $\dilation = \tb$. Thus, the temporal normalization of and is achieved by replacing the first argument of the cost function with $\dilation$, and all subsequent $\tf$ and $t$ arguments with $1$ and $\tnorm$, respectively. ### Linearization {#sec3:lin} The linearization step converts the fixed-final-time nonlinear continuous-time problem obtained in into a fixed-final-time linear-time-varying continuous-time problem. By approximating non-convexities to first-order, the linearization step guarantees convexity of the subproblem. The right-hand side of is approximated by a first-order Taylor series, evaluated about a reference trajectory denoted by $\zzo(\tnorm)\definedas\big[ \tco\;\, \sso\;\, \xxo^T(\tnorm)\;\, \uuo^T(\tnorm)\big]^T$ for all $\tnorm\in\intee{0}{1}$. The resulting linear-time-varying dynamics are given by \[eq:dynamics\_lin\] $$\begin{aligned} \xx'(\tnorm) &\approx \Ac(\tnorm)\xx(\tnorm) + \Bc(\tnorm)\uu(\tnorm) + \Sc(\tnorm)\dilation + \rr(\tnorm), \\[2ex] \Ac(\tnorm) &\definedas \dfsdx\,,\\ \Bc(\tnorm) &\definedas \dfsdu\,,\\ \Sc(\tnorm) &\definedas \dfsds\,,\\ \rr(\tnorm) &\definedas - \Ac(\tnorm)\xxo(\tnorm) - \Bc(\tnorm)\uuo(\tnorm). \end{aligned}$$ Using the assumption that the functions $\stch_{i}(\cdot)$ are at least once differentiable almost everywhere, we define $\delta\zz(\tau)\definedas\zz(\tau)-\zzo(\tau)$ and approximate the constraints for each $i\in\setncvx$ in using a first-order Taylor series: $$\begin{aligned} \hi{\zz(\tnorm)}{i} \approx \hi{\zzo(\tnorm)}{i} &+ \dhidz \delta\zz(\tnorm)\,.\quad \label{eq:constraint_ncvx_lin}\end{aligned}$$ Similarly, since the trigger and constraint functions of each cSTC are assumed to be piecewise continuously differentiable, it follows that each $\stch_{i}$ is at least once differentiable almost everywhere for all $i\in\setcstc$. However, due to the $\min(\cdot)$ function in , the partial $\partial\stch_{i}/\partial\zz$ is not well defined when $\stcg_{i}(\cdot)=0$. To ensure that the constraint condition is not enforced when $\stcg_{i}\big(\zzo(\tnorm)\big) = 0$, we define $\partial\stch_{i}/\partial\zz$ to hold the same value as when $\stcg_{i}\big(\zzo(\tnorm)\big) > 0$. Thus, the approximation is given as follows: $$\begin{aligned} \hi{\zz(\tnorm)}{i} \approx \left\{\begin{aligned} \hi{\zzo(\tnorm)}{i}+\dhidz \delta\zz(\tnorm) &\,,\quad\text{if}\;\,\stcg_{i}\big(\zzo(\tnorm)\big) < 0\,, \\ 0 &\,,\quad\text{otherwise.} \\ \end{aligned}\right. \label{eq:constraint_cstc_lin}\end{aligned}$$ ### Discretization {#sec3:disc} The discretization step converts the fixed-final-time linear-time-varying continuous-time problem obtained in into a fixed-final-time linear-time-varying discrete-time parameter optimization problem. This step is critical in ensuring that the converged solution exactly adheres to the prescribed nonlinear dynamics. We begin by introducing $\KK$ evenly spaced temporal nodes that divide the burn phase into $\KK-1$ subintervals, and define the sets $\setK\definedas\{1,\,2,\,\ldots\,,\,\KK\}$ and $\setKm\definedas\{1,\,2,\,\ldots\,,\,\KK-1\}$. Each temporal node is associated with an index $k\in\setK$, and corresponding normalized time $\tauk = (k-1)/(\KK-1)$. To proceed, the control signal must be projected from the infinite-dimensional space it inhabits to a finite-dimensional space suitable for numerical optimization. This can be done in numerous ways, including zero-order-hold (ZOH) and first-order-hold (FOH) interpolation. Alternatively, the state may be projected to a finite-dimensional space directly using pseudospectral methods [@betts1998; @DM_fahroo]. We have found that, compared to pseudospectral methods, ZOH and FOH interpolation yield sparsity patterns that noticeably decrease solve time. Our approach utilizes FOH interpolation because it (i) provides a noticeable increase in optimality when compared ZOH interpolation, and (ii) ensures that when the discrete-time control variables satisfy convex control constraints, the interpolated values follow suit. Formally, FOH interpolation represents the control signal over each subinterval $k\in\setKm$ as \[eq:foh\_def\] $$\begin{gathered} \uu(\tnorm) = \lambdalk{\tnorm}\uuk+\lambdark{\tnorm}\uukp\,,\quad\forall\tnorm\in\intee{\tauk}{\taukp}\,,\\[2ex] \lambdalk{\tnorm}\definedas\frac{\taukp-\tnorm}{\taukp-\tauk}\,,\quad\lambdark{\tnorm}\definedas\frac{\tnorm-\tauk}{\taukp-\tauk}, \end{gathered}$$ where $\uuk\definedas\uu(\tauk)$. Substituting into , we obtain the following for each subinterval $k\in\setKm$: $$\xx'(\tnorm) = \Ac(\tnorm)\xx(\tnorm) + \lambdalk{\tnorm}\Bc(\tnorm)\uuk + \lambdark{\tnorm}\Bc(\tnorm)\uukp + \Sc(\tnorm)\dilation + \rr(\tnorm)\,,\quad\forall\,\tnorm\in\intee{\tauk}{\taukp}. \label{eq:norm_lti_dyn}$$ The state transition matrix $\stm{\xi}{\tauk}$ for $\xi\in\intee{\tauk}{\taukp}$ associated with is given by $$\label{eq:stm_dyn} \stm{\xi}{\tauk} = I_{\ns\times\,\ns} + \int_{\tauk}^{\xi}{A(\zeta)\,\stm{\zeta}{\tauk}d\zeta}\,.$$ Denoting the discrete-time state vectors by $\xxk\definedas\xx(\tauk)$, the inverse and transitive properties of $\stm{\cdot}{\cdot}$ [@Antsaklis2007] are used to obtain the following discrete-time solution to for each $k\in\setKm$: \[eq:ltv\_disc\] $$\begin{aligned} \xxkp &= \Ad\xxk+\Bdm\uuk+\Bdp\uukp+\Sd\dilation+\rd, \label{eq:ltv_disc_a} \\[2ex] \Ad &\definedas \stm{\taukp}{\tauk}, \label{eq:ltv_disc_b} \\ \Bdm &\definedas \Ad\int_{\tauk}^{\taukp}{\stminv{\xi}{\tauk}\Bc(\xi)\lambdalk{\xi}d\xi}, \label{eq:ltv_disc_c} \\ \Bdp &\definedas \Ad\int_{\tauk}^{\taukp}{\stminv{\xi}{\tauk}\Bc(\xi)\lambdark{\xi}d\xi}, \label{eq:ltv_disc_d} \\ \Sd &\definedas \Ad\int_{\tauk}^{\taukp}{\stminv{\xi}{\tauk}\Sc(\xi)d\xi}, \label{eq:ltv_disc_e} \\ \rd &\definedas \Ad\int_{\tauk}^{\taukp}{\stminv{\xi}{\tauk}\rr(\xi)d\xi}. \label{eq:ltv_disc_f} \end{aligned}$$ In implementation, the previous iteration’s solve step generates $\tco$, $\sso$, $\xxko\definedas\xxo(\tauk)$, and $\uuko\definedas\uuo(\tauk)$ for all $k\in\setK$. The $\uuko$’s and are used to obtain $\uuo(\tnorm)$ over $\tau\in\intee{0}{1}$, and , , and the integrands in - are computed simultaneously for each $k\in\setKm$ using the intermediate quantity $\stm{\xi}{\tauk}$. When the propagation reaches $\taukp$, the quantity in can be left multiplied against the integrands of - to obtain the final values of $\Ad$, $\Bdm$, $\Bdp$, $\Sd$, and $\rd$. The integration of is initialized with $\xxko$, and is analogous to a multiple shooting method. We have observed that this multiple shooting strategy improves the convergence behavior of the algorithm by keeping $\zzo(\tnorm)$ closer to the path obtained by the constrained optimization problem. In contrast, since the dynamics are nonlinear and may be unstable, a single shooting method is more susceptible to poor initializations, since the dynamics have more time to evolve away from feasibility. Thus, the propagation and solve steps are designed to play complementary roles, whereby the former relates the discrete-time optimization problem to the continuous-time physics in the absence of constraints, and the latter helps reset the shooting method at more frequent temporal intervals while taking into account the constraints. The initial condition of the $\ith{k}{th}$ subinterval depends only on $\zzko$ obtained by the solve step of the previous iteration. It depend on the propagation of the $\ith{(k-1)}{th}$ subinterval. Therefore, the propagation step can be parallelized such that all subintervals $k\in\setKm$ are computed simultaneously. The state and control constraints are enforced at the temporal nodes $\tauk$ for all $k\in\setK$. This discretization choice is adopted for simplicity, and does not guarantee the absence of inter-node state constraint and non-convex control constraint violations. We conclude by noting that the number of temporal nodes and interpolation scheme do not affect the accuracy of the converged solution with respect to the nonlinear dynamics. Instead, these choices affect the optimality and, in extreme cases, the feasibility of the solution. Simply put, a coarser or less expressive interpolation results in a finite-dimensional control signal with fewer degrees of freedom. This yields a problem that is implicitly more constrained, and thus a solution that is generally less optimal. ### Virtual Control & Trust Region Modifications {#sec3:vctrl} The process outlined in - results in an implementable convex parameter optimization subproblem that may suffer from artificial infeasibility and artificial unboundedness. These issues are addressed by the virtual control and trust region modifications, respectively. Consider linearizing and discretizing Problem \[prob:ncvx\] about a burn time $\sso=0$ (or some small value). From , , and , it follows that does not admit a solution for arbitrary $\xxkp\neq\xxk$. This remains true even if Problem \[prob:ncvx\] admits a feasible solution, resulting in a condition we term artificial infeasibility. To mitigate artificial infeasibility, we augment with a virtual control term $\nuk\in\real^{\ns}$ for all $k\in\setKm$, $$\label{eq:ltv_disc_relaxed} \xxkp = \Ad\xxk+\Bdm\uuk+\Bdp\uukp+\Sd\dilation+\rd+\nuk\,,$$ and add the following penalty term to , $$\label{eq:J_vc} \Jvc{\nubig} \definedas \wvc\sum_{k\in\setKm}{\onenorm{\nuk}}\,,$$ where $\wvc\in\real_{++}$ is a large weight. The virtual control modification guarantees that each subproblem has a non-empty feasible set, and thus ensures that the convergence process is not obstructed. A 1-norm minimization is employed in to encourage sparsity in the vectors $\nuk$. Upon successful convergence, the virtual control terms are zero, and is equivalent to . The second issue that may arise is that of artificial unboundedness, which occurs when the linearized constraints permit the cost of a subproblem to be minimized indefinitely. This issue is mitigated by adding the following soft quadratic trust region to the cost function, $$\label{eq:J_tr} \Jtr{\zzo}{\zz} \definedas \sum_{k\in\setK}{\delta\zzk^T\,\Wtr\,\delta\zzk}\,,$$ where $\delta\zzk\definedas\delta\zz(\tauk)$ for all $k\in\setK$, and $\Wtr\in\spd{\nz}$ is a symmetric positive definite weighting matrix. The trust region modification also serves to ensure that the solve step does not venture excessively far from the reference trajectory used in the propagation step. Specialized Implementable Convex Subproblem {#sec3:spec_subproblem} ------------------------------------------- To specialize the subproblem developed in to Problem \[prob:ncvx\], we define the objective function to be the minimum-fuel objective $\Jxu{\dilation}{\xxkf} \definedas -\mkf$. We define $\xx(t) \definedas \big[\m(t)\;\, \rI^T(t)\;\, \vI^T(t)\;\, \qIB^T(t)\;\, \omegaB^T(t)\big]^T$ and $\uu(t) \definedas \TB(t)$, and concatenate - to form the corresponding dynamics. In accordance with , the thrust lower bound constraint in is approximated using the following form for each $k\in\setK$ $$\label{eq:lin_thrust_lower_bound} \ftlbk + \Htlbk\,\delta\zzk \leq 0.$$ Similarly, in accordance with , the cSTC in is approximated using the following form for each $k\in\setK$ $$\label{eq:lin_aoa_cstc} \haoak + \Haoak\,\delta\zzk \leq 0.$$ Problem \[prob:cvx\] presents a summary of the specialized convex parameter optimization subproblem used in our algorithm. This problem is primarily concerned with solving for the states $\xxk$, controls $\uuk$, and time dilation $\dilation$ associated with the burn phase of the trajectory. The only aspect of the coast phase optimized by Problem \[prob:cvx\] is the coast time $\tc$, which in turn determines the ignition time position $\rIki$ and velocity $\vIki$ via and . Successive Convexification Algorithm {#sec3:scvx_alg} ------------------------------------ ### Initialization {#sec3:init} We consider two initialization approaches: straight-line initialization and 3-DoF initialization. Both approaches assume $\tco=0$ and a user-specified initial guess for $\sso$. The former is equivalent to assuming $\tin=\tig$, whereas the latter is equivalent to guessing the burn time $\tb$ (see Figure \[fig:traj\_timeline\]). The nature of the powered descent guidance problem is such that $\sso$ can typically be guessed accurately as a function of distance to the landing site, initial velocity, and available thrust. In our experience, the proposed algorithm is able to handle a wide range of initialization values for $\sso$, although poor guesses may lead to increased convergence time. The straight-line initialization approach constructs the discrete-time state trajectory $\xxko$ by linearly interpolating the state at each temporal node between the ignition and final states. The control trajectory $\uuko$ is assumed to oppose the gravitational force at each temporal node. The initialization assumes an initial attitude $\qidentity$ and final mass $\mdry$, since these quantities are not known a priori. Formally, the state and control for each $k\in\setK$ is computed as follows $$\begin{aligned} \xxko = &\left(\dfrac{\KK-k}{\KK-1}\right)\xxoig+\left(\dfrac{k-1}{\KK-1}\right)\xxof, \\ \uuko = -&\left(\dfrac{\KK-k}{\KK-1}\right)\mig\gI-\left(\dfrac{k-1}{\KK-1}\right)\mdry\gI, \end{aligned}$$ where $\xxoig\definedas\big[\mig\;\,\rIig^T\;\,\vIig^T\;\,\qidentity^T\;\,\omegaBig^T\big]^T$, and $\xxof\definedas\big[\mdry\;\,\rIf^T \vIf^T\;\,\qIBf^T\;\,\omegaBf^T\big]^T$. If a value other than $\qidentity$ is assumed for the initial attitude, this approach can be improved by interpolating the quaternion states using Spherical Linear Interpolation (SLERP) [@Hanson2006]. The 3-DoF initialization approach constructs the state and control trajectories using a solution obtained from a convex 3-DoF guidance problem [@behcetjgcd07]. The 3-DoF problem is solved using the same number of temporal nodes, $\KK$, as in the 6-DoF problem. This problem may be solved once using a user-defined burn time, or in conjunction with a line-search that optimizes the burn time, and thus generates $\sso$. The mass, position, and velocity components of $\xxko$ and the controls $\uuko$ are obtained directly from the 3-DoF solution. The attitude is computed such that the vertical axis of the vehicle is aligned with $\uuko$, and the angular velocity is obtained by inverting . Unsurprisingly, the initialization approach can have a significant impact on the converged solution attained by the algorithm. In our work, we have found that neither approach offers a clear and consistent advantage over the other, (see ). We ultimately regard the initialization approach as a design choice. ### Algorithm {#sec3:alg} Initialize $\{\tco,\sso,\xxo,\uuo\}$ Compute $\{\Ad,\Bdp,\Bdm,\Sd,\rd\}\;\;\forall\,k\in\setKm$ according to Solve Problem \[prob:cvx\] to obtain $\{\tc,\dilation,\xx,\uu,\nubig\}$ converged $\{\tco,\sso,\xxo,\uuo\} \gets \{\tc,\dilation,\xx,\uu\}$ \[euclidendwhile\] return $\{\tc,\dilation,\xx,\uu\}$ The algorithm is initialized using one of the two approaches discussed in . For each iteration, the algorithm performs a propagation step to compute $\Ad$, $\Bdp$, $\Bdm$, $\Sd$, and $\rd$ for all $k\in\setKm$, followed by a solve step that solves the convex second-order cone programming subproblem summarized in Problem \[prob:cvx\]. The process terminates when $\Jvc{\nubig}<\epsvc$ and $\Jtr{\zz}{\zzo}<\epstr$, where $\epsvc,\,\epstr\in\real_{++}$ are user-specified convergence tolerances. Satisfaction of the convergence criteria ensures that the attained solution eliminates the first order terms $\delta\zzk$ generated by the approximation without using virtual control. If an iterate satisfies the convergence criteria, then feasibility of the corresponding subproblem implies that the iterate exactly satisfies the nonlinear dynamics of Problem \[prob:ncvx\] for all $t\in\intee{\tig}{\tf}$, and satisfies the state and control constraints of Problem \[prob:ncvx\] at each temporal node. However, failure to converge does not necessarily imply that Problem \[prob:ncvx\] is infeasible. A summary of the algorithm is provided in Algorithm \[alg:scvx\]. We note that prior to convergence, the iterates may not be feasible with respect to Problem \[prob:ncvx\] due to the linearization used in the propagation step. This statement holds true even though each convex subproblem is designed to be feasible through the use of virtual control. Finally, we highlight that no convergence guarantees are presented in this paper. However, we have found that Algorithm \[alg:scvx\] works well in practice, and note its similarity to [@mao2016successive], which does guarantee convergence to a local optima of the original problem when the converged solution requires no virtual control. Numerical Results {#sec:numerical_results} ================= In this section, we present simulation results that demonstrate the proposed successive convexification algorithm, while highlighting the principal contributions of this paper. In , , and we present case studies that respectively illustrate the effects of the aerodynamic models introduced in , the state-triggered constraints introduced in , and the free-ignition-time modification introduced in . In , we provide performance and timing results. To present the results, we introduce the problem feature labels given in Table \[tab:prob\_names\]. The simulations are designed around a notional non-dimensionalized scenario with time, length, and mass units $\TU$, $\LU$, and $\MU$. Each scenario is defined by the problem parameters in Tables \[tab:params\_ex\] and the initial position and velocity vectors defined in each subsequent subsection. For the sake of illustration, the initial conditions used in the case studies in - define in-plane maneuvers, whereas those used to generate the timing results in define more computationally intensive out-of-plane maneuvers. Feature Section Description --------------------- --------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- (B) Baseline problem setup with no aerodynamics and straight-line initialization. (FI) Includes the free-ignition-time modification shown in Figure \[fig:traj\_timeline\], where initial position and velocity are restricted to a free fall trajectory prior to engine ignition. (SA) Includes the spherical aerodynamic model from Figure \[fig:aerodynamic\_model\_a\]. (EA) Includes the ellipsoidal aerodynamic model from Figure \[fig:aerodynamic\_model\_b\]. (ST) Includes the state-triggered constraint from Figure \[fig:stc\_ex\_aoa\], and introduced in . (3I) Uses the 3-DoF initialization in lieu of straight-line initialization. Straight-line initialization is implied in the absence of this feature. \[tab:prob\_names\] : Generalized Powered Descent Guidance Problem Features -------------------- ----- ------------------------------------------------ --------------------- -- ------------------------- ----- ------------------------- ----------- Parameter Value Units Parameter Value Units $\gI$ $-$ $\ex$ $\LU/\TU^2$ $\Vaoa$ $2.0$ $\LU/\TU$ $\density$ $1.0$ $\MU/\LU^3$ $\aoamax$ $3.0$ $\dg$ $\inertia$ $0.168\cdot\diag{\big[2\text{e-}2\;1\;1\big]}$ $\MU\cdot\LU^2$ $\mdry$ $1.0$ $\MU$ $\Pamb$ $0.0$ $\MU/\TU^2/\LU$ $\mig$ $2.0$ $\MU$ $\Anoz$ $0.0$ $\LU^2$ $\rIkf$ $\bvec{0}_{3 \times 1}$ $\LU$ $\rCPB$ $0.05\cdot\ex$ $\LU$ $\vIkf$ $-$ $0.1\cdot\ex$ $\LU/\TU$ $\rTB$ $-$ $0.25\cdot\ex$ $\LU$ $\omegaBki,\,\omegaBkf$ $\bvec{0}_{3 \times 1}$ $\dg/\TU$ $\Isp$ $30.0$ $\TU$ $\qIBkf$ $\qidentity$ - $\tiltmax$ $90.0$ $\dg$ $\KK$ $20$ - $\omegamax$ $28.6$ $\dg/\TU$ $\wvc$ $1\text{e+}4$ - $\glideslope$ $75.0$ $\dg$ $\Wtr$ $0.5$ - $\gimbalmax$ $20.0$ $\dg$ $\epsvc$ $1\text{e-}8$ - $\Tmin$ $1.5$ $\MU\cdot\LU/\TU^2$ $\epstr$ $5\text{e-}4$ - $\Tmax$ $6.5$ $\MU\cdot\LU/\TU^2$ $\sso$ $5.0$ - \[tab:params\_ex\] -------------------- ----- ------------------------------------------------ --------------------- -- ------------------------- ----- ------------------------- ----------- : Problem Parameters Aerodynamic Models Case Study {#sec4:exI} ----------------------------- [0.5]{} [0.5]{} \ [0.5]{} [0.5]{} In this case study, we solve three otherwise identical powered descent guidance problems, assuming no aerodynamic effects, a spherical aerodynamic model, and an ellipsoidal aerodynamic model. These problems are labeled using (B), (B)+(SA) and (B)+(EA), respectively. In each problem, the vehicle begins above and east of the landing pad, traveling west at a shallow flight path angle. These initial conditions are given by $\rI(\tin) = [4.33\ 3.5\ 0.0]^T\; \LU$ and $\vI(\tin) = [-0.5\ -2.5\ 0.0]^T \; \LU/\TU$. To land successfully, the vehicle must shed significant horizontal momentum while ensuring an upright final attitude. Figure \[fig:exI\_trjs\] shows the converged trajectory for each of the three cases. To reduce clutter, only $11$ of the $20$ temporal nodes are shown. The insets (i) and (ii) represent free body diagrams of the forces acting on the vehicle at the same two temporal nodes for each case. Figure \[fig:exI\_nostc\_nodrag\] shows the trajectory for the case without aerodynamic effects. From the insets, it is clear that the maneuver resembles a reverse gravity turn, with the thrust pointed nearly anti-parallel to the velocity vector. Figure \[fig:exI\_nostc\_sphere\] shows the trajectory for the case with the spherical aerodynamic model. The corresponding insets show aerodynamic forces that are anti-parallel to the velocity vector, and may therefore be interpreted as drag forces. The thrust directions, however, remain consistent with the no-aerodynamic case. Thus, the trajectory in Figure \[fig:exI\_nostc\_sphere\] may be interpreted as the atmospheric counterpart to the reverse gravity turn observed in case (B). Figure \[fig:exI\_nostc\_elip\] shows the trajectory for the case with the ellipsoidal aerodynamic model. In this case, the aerodynamic force vectors are seen to have a component orthogonal to the velocity vector, and may therefore be interpreted as a composition of a drag and lift force (recall Figure \[fig:aerodynamic\_model\]). The vehicle is observed to exploit the lift force to bend the trajectory downwards by adjusting its attitude to control the angle of attack. As a consequence, the thrust is no longer aligned with the velocity as it was in the previous two cases, and is instead gimbaled such that the vehicle remains trimmed at an angle of attack that applies the lift force in a desirable direction. State-Triggered Constraints Case Study {#sec4:exII} -------------------------------------- [0.5]{} [0.5]{} [0.5]{} ![State-Triggered Constraint Case Study: Trajectories and speed-angle of attack histories for cases (B)+(EA) and (B)+(EA)+(ST). This case study uses $\rI(\tin)=[5.33\ 4.5\ 0.0]^T$ $\LU$ and $\vI(\tin)=[-0.5\ -2.5\ 0.0]^T$ $\LU/\TU$. Case (B)+(EA)+(ST) implements the state-triggered constraint introduced in , which limits the angle of attack to $3.0\dg$ when the speed is greater than $2.0$ $\LU/\TU$. The horizontal dashed red lines in (c) and (d) represent the velocity limit $\Vaoa$ in the top set of axes, and the angle of attack limit $\aoamax$ in the bottom set of axes. Note that that these limits are not enforced in (c), and are shown only for reference. Refer to the legend in Figure \[fig:exI\_trjs\] for additional definitions.[]{data-label="fig:exII_trjs"}](jgcd_nostc_nofit_vandaoa "fig:"){width="\textwidth"} [0.5]{} ![State-Triggered Constraint Case Study: Trajectories and speed-angle of attack histories for cases (B)+(EA) and (B)+(EA)+(ST). This case study uses $\rI(\tin)=[5.33\ 4.5\ 0.0]^T$ $\LU$ and $\vI(\tin)=[-0.5\ -2.5\ 0.0]^T$ $\LU/\TU$. Case (B)+(EA)+(ST) implements the state-triggered constraint introduced in , which limits the angle of attack to $3.0\dg$ when the speed is greater than $2.0$ $\LU/\TU$. The horizontal dashed red lines in (c) and (d) represent the velocity limit $\Vaoa$ in the top set of axes, and the angle of attack limit $\aoamax$ in the bottom set of axes. Note that that these limits are not enforced in (c), and are shown only for reference. Refer to the legend in Figure \[fig:exI\_trjs\] for additional definitions.[]{data-label="fig:exII_trjs"}](jgcd_stc_nofit_vandaoa "fig:"){width="\textwidth"} This case study highlights the effects of the state-triggered constraint introduced in on a landing scenario with non-negligible atmospheric effects. We consider the cases (B)+(EA) and (B)+(EA)+(ST), with initial position and velocity vectors $\rI(\tin)=[5.33\ 4.5\ 0.0]^T$ $\LU$ and $\vI(\tin)=[-0.5\ -2.5\ 0.0]^T$ $\LU/\TU$, respectively. The resulting trajectories are shown in Figures \[fig:stc\_noaoa\]-\[fig:stc\_aoa\], with the corresponding speed and angle of attack profiles provided in Figures \[fig:exII\_nostc\_nofit\_vandaoa\]-\[fig:exII\_stc\_nofit\_vandaoa\]. In Figures \[fig:stc\_noaoa\]-\[fig:stc\_aoa\], inset (i) shows the initial condition where the speed is greater than $\Vaoa$, while inset (ii) shows the last temporal node at which the STC is active in case (B)+(EA)+(ST). The lift-to-drag ratios displayed in inset (i) of Figures \[fig:stc\_noaoa\]-\[fig:stc\_aoa\] indicate that the aerodynamic loading in case (B)+(EA)+(ST) is reduced when the compared to case (B)+(EA) due to the inclusion of the constraint. The same effect is apparent to a lesser extent in inset (ii), where the STCs trigger condition is still satisfied for case (B)+(EA)+(ST). In Figure \[fig:stc\_aoa\], the temporal nodes after inset (ii) clearly exhibit larger angles of attack, indicating that the vehicle’s speed has dropped below the trigger limit and that the constraint has been disabled. Figures \[fig:exII\_nostc\_nofit\_vandaoa\]-\[fig:exII\_stc\_nofit\_vandaoa\] show the corresponding speed and angle of attack time histories. Since case (B)+(EA) does not implement the STC, the angle of attack is seen to violate the $3.0\dg$ limit when the speed is greater than $2.0$ $\LU/\TU$. In contrast, the angle of attack in case (B)+(EA)+(ST) remains below the prescribed $3.0\dg$ limit until $t\approx 2.0$ $\TU$, where the speed drops below the prescribed $2.0$ $\LU/\TU$ speed limit. Lastly, notice that Figure \[fig:exII\_stc\_nofit\_vandaoa\] shows that case (B)+(EA)+(ST) rides the speed constraint for $t\in\intee{2.0}{3.0}$. Although the trajectory may gain optimality by traveling faster over this time interval, the angle of attack is above the specified $3.0\dg$ limit over this time interval. Consequently, we observe the enforcement of the contrapositive of (i.e. an angle of attack greater than $3.0\dg$ implies that the speed must be less than $2.0$ $\LU/\TU$). Further, certain scenarios may allow the angle of attack to drop below the prescribed limit towards the end of the trajectory, thereby allowing the speed to increase above the STC trigger limit. Such an eventuality would cause the optimization algorithm to add a new constrained phase without a priori input, per the discussion in Alternative 3 in . Free-Ignition-Time Case Study {#sec4:exIII} ----------------------------- The third case study considers how the free-ignition time modification affects the trajectory by comparing the cases (B)+(EA) and (B)+(EA)+(FI). The initial position and velocity are identical to those used in . Figure \[fig:exIII\_trjs\] depicts the converged trajectories of both cases. Inset (i) on each figure corresponds to the ignition time epoch $\tig$, while inset (ii) corresponds to a temporal node in the middle of the maneuver. Case (B)+(EA)+(FI) selects a coast time of $\tc = 0.96$ $\TU$ and as a result observes a reduction in both burn time and fuel cost (see ). [0.5]{} [0.5]{} Trajectory and Computational Performance {#sec4:timing} ---------------------------------------- The trajectory and computational performance data presented in this section were generated for the cases listed in the leftmost column of Table \[tab:performance\]. The results were obtained by executing a batch of 10 runs for each case using the initial position and velocity vectors $\rI(\tin)=[5.33\ 4.5\ 0.0]^T$ $\LU$ and $\vI(\tin)=[-0.5\ -2.5\ 0.25]^T$ $\LU/\TU$. These initial conditions generate three-dimensional out-of-plane trajectories that render the problem less sparse and thus more computationally intensive than their their planar counterparts presented in -. Trajectory performance metrics are given in Table \[tab:performance\]. For each case, the entries represent the median values of the metrics generated in the batch. Due to the determinism of the proposed algorithm, the standard deviations of the trajectory performance metrics was zero. To validate each solution against the nonlinear dynamics, we integrate the nonlinear equations of motion using a piecewise linear interpolation of the controls, given in . The errors between the integrated trajectory and the discrete states generated by the optimization process are then used as a measure of the solution’s feasibility. The position and attitude errors are defined as $e_{\textit{pos}} \definedas \max_{k \in K}\;\\| \rI(t_k) - \rIk \\|_2$ and $e_{\textit{att}} \definedas \max_{k \in K} \; 2 \cos^{-1} \big[\qIB^(t_k) \otimes \bvec{q}_{\body \leftarrow \inertial,k} \big]$, where $\bvec{q}_{\body \leftarrow \inertial,k}$ and $\rIk$ are the discrete solution values, while $\qIB(t_k)$ and $\rI(t_k)$ are the corresponding integrated values. The computational performance results for the solve step is given in Table \[tab:timing\], and were generated on a 2014 MacBook Pro with a 2.2 GHz Intel Core i7 processor and 16 GB of RAM. The propagation step was implemented in C++ using the Eigen matrix library [@eigenweb], and was omitted from Table \[tab:timing\] since the maximum propagation time per run was on the order of $10$ milliseconds. The solve time results were generated in MATLAB using ECOS [@Domahidi2013ecos] and CVX [@cvx]. For each case, the solve times were obtained by totaling the fifth argument reported by the `cvx_toc` function over all ECOS calls in a run, and computing the statistics over the entire batch. [lccccc]{} \[Case]{} & Burn & Fuel & Successive & $e_{\textit{pos}}$ & $e_{\textit{att}}$\ & Time ($\TU$) & Consumed (%) & Iterations & ($\LU$) & ($\dg$)\ (B) & $4.42$ & $3.17$ & $4$ & $1.1\text{e-}3$ & $8.0\text{e-}2$\ (B) + (SA) & $4.27$ & $3.72$ & $4$ & $2.2\text{e-}2$ & $1.3\text{e-}1$\ (B) + (EA) & $4.67$ & $3.55$ & $4$ & $7.8\text{e-}2$ & $1.0\text{e-}0$\ (B) + (EA) + (ST) & $4.06$ & $3.40$ & $7$ & $3.3\text{e-}3$ & $3.7\text{e-}1$\ (B) + (ST) + (FI) & $3.10$ & $3.03$ & $7$ & $3.2\text{e-}5$ & $1.1\text{e-}2$\ (B) + (EA) + (FI) & $2.56$ & $3.13$ & $5$ & $1.4\text{e-}2$ & $\leq1.0\text{e-}6$\ (B) + (EA) + (3I) & $4.89$ & $3.61$ & $4$ & $2.7\text{e-}2$ & $3.2\text{e-}1$\ (B) + (EA) + (3I) + (FI) & $4.41$ & $3.58$ & $4$ & $2.2\text{e-}2$ & $1.1\text{e-}1$\ (B) + (EA) + (ST) + (FI) & $2.69$ & $3.21$ & $7$ & $2.1\text{e-}3$ & $\leq1.0\text{e-}6$\ (B) + (EA) + (ST) + (FI) + (3I) & $3.21$ & $3.27$ & $7$ & $8.0\text{e-}3$ & $\leq1.0\text{e-}6$\ \[tab:performance\] [lccccc]{} \*[Case]{} &\ & Min & Max & Median & Std\ (B) & $0.164$ & $0.287$ & $0.164$ & $0.007$\ (B) + (SA) & $0.170$ & $0.210$ & $0.174$ & $0.012$\ (B) + (EA) & $0.271$ & $0.312$ & $0.276$ & $0.012$\ (B) + (EA) + (ST) & $0.496$ & $0.565$ & $0.504$ & $0.020$\ (B) + (ST) + (FI) & $0.643$ & $0.682$ & $0.654$ & $0.011$\ (B) + (EA) + (FI) & $0.233$ & $0.263$ & $0.241$ & $0.008$\ (B) + (EA) + (3I) & $0.299$ & $0.329$ & $0.312$ & $0.011$\ (B) + (EA) + (3I) + (FI) & $0.224$ & $0.253$ & $0.229$ & $0.008$\ (B) + (EA) + (ST) + (FI) & $0.494$ & $0.537$ & $0.505$ & $0.012$\ (B) + (EA) + (ST) + (FI) + (3I) & $0.602$ & $0.632$ & $0.618$ & $0.009$\ \[tab:timing\] We conclude with three final observations. First, in the cases presented, the inclusion of feature (ST) yielded a decrease in both burn time and fuel consumption. This result is counter intuitive since one would expect the optimal cost of a problem to remain the same or increase when additional constraints are added. Since Problem \[prob:ncvx\] may have multiple local minima, we posit that the inclusion of the state-triggered constraint forced the iterative solution process towards a more optimal local minima. On the other hand, the inclusion of feature (FI) reduced the fuel cost. This result agrees with intuition since the free-ignition-time modification effectively adds a degree of freedom to the problem. Second, we note that the 3-DoF initialization approach (i.e. the inclusion of feature (3I)) did not yield a clear improvement in optimality or computational performance. In fact, cases (B)+(EA)+(3I), (B)+(EA)+(3I)+(FI), and (B)+(EA)+(ST)+(FI)+(3I) all resulted in less optimal trajectories, while cases (B)+(EA)+(3I) and (B)+(EA)+(ST) +(FI)+(3I) also increased the solve time. We conclude that the straight-line initialization approach initializes the algorithm in a more favorable region of attraction, but stress that this may not be the case for different scenarios. Lastly, the timing results presented in Table \[tab:timing\] show a maximum solve time of $0.7$ seconds and a standard deviation on the order of milliseconds and were all obtained using the same algorithm parameters (e.g. $\wvc$, $\Wtr$, $\KK$). We argue that these results are an important step in demonstrating the efficacy of the successive convexification methodology for real time autonomous applications. Ultimately, results obtained on representative flight hardware will be crucial in accurately assessing the viability of the proposed methodology for on-board computation in real-world applications. Concluding Remarks {#sec:conclusion} ================== This paper presents a real-time successive convexification algorithm for a generalized free-final-time 6-DoF powered descent guidance problem, and introduces three primary contributions: (i) a free-ignition-time modification that allows the algorithm to determine when to begin the burn phase, (ii) an ellipsoidal aerodynamics model that provides a computationally tractable way to model lift and drag forces, and (iii) a continuous formulation for state-triggered constraints. Contribution (iii) allows continuous optimization problems to be formulated using conditionally enforced constraints, and was motivated by landing scenarios that necessitate velocity-triggered angle of attack and range-triggered line of sight constraints. Three simulation case studies are presented, each illustrating one of the primary contributions of this paper. The corresponding trajectory and computational performance results show that the proposed algorithm can successfully compute trajectories in under $0.7$ seconds for the problem features considered. While additional work is required to provide convergence guarantees and to quantify the optimality of the computed trajectories, we argue that our results demonstrate the efficacy of the successive convexification approach for real-time powered descent guidance applications. [^1]: PhD Candidate, UW Aero. and Astro., [email protected], AIAA Member. [^2]: PhD Candidate, UW Aero. and Astro., [email protected], AIAA Member. [^3]: Associate Professor, UW Aero. and Astro., [email protected], AIAA Member.
--- abstract: 'In physics we encounter particles in one of two ways. Either as fundamental constituents of the theory or as emergent excitations. These two ways differ by how the particle relates to the background. It either sits on the background, or it is an excitation of the background. We argue that by choosing the former to construct our fundamental theories we have made a costly mistake. Instead we should think of particles as excitations of a background. We show that this point of view sheds new light on the cosmological constant problem and even leads to observable consequences by giving a natural explanation for the appearance of MOND-like behavior. In this context it also becomes clear why there are numerical coincidences between the MOND acceleration parameter $a_0$, the cosmological constant $\Lambda$ and the Hubble parameter $H_0$.' author: - Olaf DREYER title: 'Not on but of.' --- Which of our basic physical assumptions are wrong? {#sec:intro} ================================================== In theoretical physics we encounter particles in two different ways. We either encounter them as fundamental constituents of a theory or as emergent entities. A good example of the first kind is a scalar field. Its dynamics is given by the Lagrangian $$\int d^4x \left( \frac{1}{2}(\partial\phi)^2 - m^2\phi^2 + \ldots \right).$$ An example of an emergent particle is a spin-wave in a spin-chain. It given by $$\vert k \rangle = \sum_{n=1}^{N} \exp{\left(2\pi i \frac{n k}{N}\right)} \vert n \rangle$$ If $\vert 0 \rangle$ is the gound state of the spin-chain then $\vert n \rangle$ is the state where spin $n$ is flipped. Of all the differences between these two concepts of particles we want to stress one in particular: the relation these particles have to their respective backgrounds. The scalar field is formulated as a field on spacetime. The above Lagrangian does not include the metric. If we include the metric we obtain $$\int d^4x \sqrt{-g} \left( \frac{1}{2}g^{ab}\partial_a\phi\partial_b\phi - m^2\phi^2 + \ldots \right). $$ This way the scalar field knows about a non-trivial spacetime. Einstein told us that this is not a one-way street. The presence of a scalar field in turn changes the background. The proper equations that describe this interaction between matter and background are obtained by adding the curvature tensor to the above Lagrangian. This interaction does not change the basic fact about the scalar field that it is sitting on spacetime. It is distinct from the spacetime that it sits on. This is in contrast to the emergent particle. There is no clean separation possible between the particle and the background, i.e. the ground state in this case. The spin-wave above is an excitation of the background not an excitation on the background. We can now state what basic assumption we think needs changing. Currently we built our fundamental theories assuming the matter-on-background model. We will argue here that this assumption is wrong and that instead all matter is of the emergent type and that the excitation-of-background model applies. Our basic assumption that is wrong: : Matter does not sit on a background but is an excitation of the background. In the following we will do two things. First we will show that the assumption that matter sits on a background creates problems and how our new assumption avoids these problems. Then we will argue that the new assumption has observable consequences by showing how MOND like behavior arises naturally. The cosmological constant ========================= The basic assumption that matter sits on a background directly leads to one of the thorniest problems of theoretical physics: the cosmological constant problem. All matter is described by quantum fields which can be thought of as a collection of quantum harmonic oscillators. One feature of the spectrum of the harmonic oscillator is that it has a non-vanishing ground state energy of $\hbar\omega/2$. Because the quantum field sits on spacetime the ground state energy of all the harmonic oscillators making up the field should contribute to the curvature of space. The problem with this reasoning is that it leads to a prediction that is many orders of magnitude off. In fact if one assumes that there is some large frequency limit $\omega_\infty$ for the quantum field then the energy density coming from this ground state energy is proportional to the fourth power of this frequency: $$\epsilon \simeq \omega_\infty^4$$ If one chooses $\omega_\infty$ to be the Planck frequency then one obtains a value that is 123 orders of magnitude larger than the observed value of the cosmological constant. This has been called the worst prediction of theoretical physics and constitutes one part of the cosmological constant problem (see [@Weinberg:1989p695; @Carroll:2001p311] for more details). Let us emphasize again how crucial the basic assumption of matter-on-background is for the formulation of the cosmological constant problem. Because we have separated matter from the background we have to consider the contributions coming from the ground state energy. We have to do this even when the matter is in its ground state, i.e. when no particles are present. This is to be contrasted with the excitation-of-background model. If there are no waves present there is also no ground state energy for the excitations to consider. The cosmological constant problem can not be formulated in this model. There are two objections to this reasoning that we have to deal with. The first objection is that although the above reasoning is correct it is also not that interesting because there is no gravity in the picture yet. We will deal with this objection in the next section. The other objection concerns the ontological status of the vacuum fluctuations. IsnÕt it true that we have observed the Casimir effect? Since the Casimir effect is based on vacuum fluctuations the cosmological constant problem is not really a problem that is rooted in any fundamental assumption but in observational facts. For its formulation no theoretical foundation needs to be mentioned. We will deal with this objection now. There are two complementary views of the Casimir effect and the reality of vacuum fluctuations. In the more well known explanation of the Casimir force between two conducting plates the presence of vacuum fluctuations is assumed. The Casimir force then arises because only a discrete set of frequencies is admissible between the plates whereas no such restriction exists on the outside. The Casimir effect between two parallel plates has been observed and this has led to claims that we have indeed seen vacuum fluctuations. This claim is not quite true because there is another way to think about the Casimir effect. In this view vacuum fluctuations play no role. Instead, the Casimir effect results from forces between the currents and charges in the plates. No vacuum diagrams contribute to the total force[@Jaffe:2005p147]. If the emergent matter is described by the correct theory, say quantum electrodynamics, we will find the Casimir effect even if there are no vacuum fluctuations. We see that the cosmological constant problem can be seen as a consequence of us viewing matter as sitting on a background. If we drop this assumption we can not even formulate the cosmological constant problem. Gravity {#sec:gravity} ======= The above argument for viewing matter as an excitation of a background is only useful if we can include gravity in the picture. In [@dreyer] we have argued that this can be achieved by regarding the ground state itself as the variable quantity that is responsible for gravity. In the simplest case the vacuum is described by a scalar quantity $\theta$. If we assume that the energy of the vacuum is given by $$\label{eqn:hamiltonian} E = \frac{1}{8\pi}\int d^3x\; (\nabla \theta)^2,$$ then we can calculate the force between two objects $\mathsf{m}_i$, $i=1,2$. If we introduce the gravitational mass of an object by $$\label{eqn:gravitational} \mathsf{m} = \frac{1}{4\pi}\int_{\partial \mathsf{m}} d\sigma \cdot \nabla\theta,$$ where $\partial \mathsf{m}$ is the boundary of the object $\mathsf{m}$, then the force between them is given by $$F = \frac{\mathsf{m}_1\mathsf{m}_2}{r^2}.$$ Usually we express Newton’s law of gravitation not in terms of the gravitational masses $\mathsf{m}_i$, $i=1,2$, but in terms of the inertial masses $m_i$, $i=1,2$. In [@dreyer] we have argued that the inertial mass of an object is given by $$\label{eqn:inertial} m = \frac{2 \mathsf{m}}{3 a}\; \mathsf{m},$$ where $a$ is the radius of the object. In terms of the inertial masses $m_i$ Newton’s law then takes the usual form $$F = G \frac{m_1m_2}{r^2},$$ where $G$ has to be calculated from the fundamental theory: $$G = \left(\frac{3 a}{\mathsf{2 m}}\right)^2$$ In this picture of gravity the metric does not play a fundamental role. Gravity appears because the ground state $\theta$ depends on the matter. MOND as a consequence ===================== The picture of gravity that we have given in the last section is valid only for zero temperature. If the temperature is not zero we need to take the effects of entropy into account and instead of looking at the energy we have to look at the free energy $$F = E - TS.$$ We thus have to determine the dependence of the entropy $S$ on the temperature $T$ and the ground state $\theta$. The entropy should not depend on $\theta$ directly because every $\theta$ corresponds to the same ground state. The entropy should only be dependent on changes of $\theta$. If we are only interested in small values of $T$ we find $$S = \sigma T(\nabla\theta)^2,$$ for some constant $\sigma$. The total free energy is thus $$\begin{aligned} F & = & E - TS \\ & = & E (1 - 8 \pi \sigma T^2). \end{aligned}$$ We see that a non-zero temperature does not change the form of the force but just its strength. The new gravitational constant is given by $$G_{T} = \frac{1}{1 - \sigma T^2} \ G_{T=0}.$$ The situation changes in an interesting way if there is a large maximal length scale $L_\text{\scriptsize{max}}$ present in the problem. The contributions to the entropy of the form $(\nabla\phi)^2$ come from excitations with a wavelength of the order $$L = \vert\nabla\phi\vert^{-1}.$$ If this wavelength $L$ is larger than $L_\text{\scriptsize{max}}$ than these excitations should not exist and thus not contribute to the entropy. Instead of a simple $(\nabla\phi)^2$ term we should thus have a term of the form $$C( L_\text{\scriptsize{max}}\; \vert\nabla\theta\vert ) \cdot (\nabla\theta)^2,$$ where the function $C$ is such that it suppresses contributions from excitations with wavelengths larger than $L_\text{\scriptsize{max}}$. For wavelengths much smaller than the maximal wavelength we want to recover the usual contributions to the entropy. Thus, if $L_\text{\scriptsize{max}}\cdot\vert\nabla\theta\vert$ is much bigger than unity we want $C$ to be one: $$C(x) = 1, \text{\ \ \ for\ \ \ } x \gg 1.$$ For $x\ll 1$ we assume that the function $C$ possesses a series expansion of the form $$C(x) = \alpha x + \beta x^2 + \ldots.$$ For small values of $L_\text{\scriptsize{max}}\cdot\vert\nabla\theta\vert$ we thus find that the dependence of the entropy on $\phi$ is of the form $$T^2 \sigma \int d^4x\; \alpha L_\text{\scriptsize{max}}\vert\nabla\phi\vert^3.$$ It is here that we make contact with the Lagrangian formulation of Milgrom’s odd new dynamics (or MOND, see [@Milgrom:1983p708] and [@Famaey:2011p753] for more details). In [@Bekenstein:1984p709] Bekenstein and Milgrom have shown that a Lagrangian of the form $$\int d^3x \left( \rho\theta + \frac{1}{8\pi G}\; a_0^2\; F\left[ \frac{(\nabla\theta)^2}{a_0^2} \right]\right)$$ gives rise to MOND like dynamics if the function $F$ is chosen such that $$\mu(x) = F^\prime(x^2).$$ Here $\mu(x)$ is the function that determines the transition from the classical Newtonian regime to the MOND regime. It satisfies $$\mu(x) = \left\{ % \begin{array}{cl} 1 & x\gg 1 \\ x & x\ll 1 \end{array}\right.$$ From this it follows that the function $F$ satisfies $$F(x^2) = \left\{ % \begin{array}{cl} x^2 & x\gg 1 \\ 2/3\;x^3 & x\ll 1 \end{array}\right.$$ The behavior of the Lagrangian is then $$\label{eqn:mondlagrangian} a_0^2 F\left( \frac{\vert\nabla\theta\vert^2}{a_0^2} \right) = \left\{ % \begin{array}{cl} \vert\nabla\theta\vert^2 & \ \ \ a_0^{-1} \vert\nabla\theta\vert \gg 1 \\ \frac{2}{3 a_0}\;\vert\nabla\theta\vert^3 & \ \ \ a_0^{-1} \vert\nabla\theta\vert \ll 1 \end{array}\right.$$ This is exactly the behavior of the free energy that we have just derived if we make the identification $$\label{eqn:eqn} \frac{2}{3 a_0} = \alpha \sigma T^2 L_\text{\scriptsize{max}}.$$ There are currently two candidates for a maximal length scale $L_\text{\scriptsize{max}}$. These are the Hubble scale $$L_H = c H_0$$ and the cosmic horizon scale $$L_\Lambda = \sqrt{\frac{1}{\Lambda}}.$$ It is a remarkable fact of the universe that we live in that both of these length scales satisfy the relationship that we derived in (\[eqn:eqn\]) if we further assume that the constant $\alpha \sigma T^2$ is of order one. We have thus established a connection between the acceleration parameter $a_0$, the cosmological constant $\Lambda$, and the Hubble parameter $H_0$. In standard cosmology these coincidences remain complete mysteries. Discussion ========== Particles are either fundamental or they are emergent. If they are fundamental they are sitting on a background; if they are emergent they are excitations of a background. Rather than being a purely philosophical issue we have argued that this distinction is important and that the assumption that particles are fundamental is wrong. Assuming instead that particles are emergent leads to the resolution of theoretical problems as well as having observational consequences. We have argued that the cosmological constant problem as it is usually formulated can not even be stated if we think of particles as excitations of a background. Also, we have shown that this picture gives a straight forward way of understanding the appearance of MOND like behavior in gravity. The argument also makes clear why there are numerical relations between the MOND parameter $a_0$, the cosmological constant $\Lambda$, and the Hubble parameter $H_0$. Our derivation of MOND is inspired by recent work [@Ho:2011p703; @Ho:2010p711; @Pikhitsa:2010p769; @Klinkhamer:2012p750; @Klinkhamer:2011p756; @Li:2011p725; @Neto:2011p723; @Kiselev:2010p717; @Pazy:2012p789; @Modesto:2010p702] that uses Verlinde’s derivation of Newton’s law of gravity [@verlinde] as a starting point. Our derivation differs from these in that it does not rely on holography in any way. Our formulae for the entropy are all completely three-dimensional. Acknowledgements {#acknowledgements .unnumbered} ================ I would like to thank Seth Llloyd, Mamazim Kassankogno, and Stevi Roussel Tankio Djiokap for helpful discussion and the Foundational Questions Institute, FQXi, for financial support and for creating an environment where it is alright to play with unorthodox ideas. [WW]{} S. Weinberg. The cosmological constant problem. Reviews of Modern Physics (1989) vol. 61 (1) pp. 1-23. Carroll. The cosmological constant. Living Rev. Relativ. (2001) vol. 4 pp. 2001-1, 80 pp. (electronic). R. L. Jaffe. Casimir effect and the quantum vacuum. Phys. Rev. D (2005) vol. 72 (2) pp. 5. O. Dreyer, Internal relativity, arXiv:1203.2641. Milgrom. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. The Astrophysical Journal (1983) vol. 270 pp. 365-370. Famaey and McGaugh. Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions. Arxiv preprint arXiv:1112.3960v2 (2011). Bekenstein and Milgrom. Does the missing mass problem signal the breakdown of Newtonian gravity? The Astrophysical Journal (1984) vol. 286 pp. 7-14. Ho et al. Quantum gravity and dark matter. Gen Relativ Gravit (2011) vol. 43 (10) pp. 2567-2573. Ho et al. Cold dark matter with MOND scaling. Physics Letters B (2010) vol. 693 (5) pp. 567-570. Pikhitsa. MOND reveals the thermodynamics of gravity. Arxiv preprint arXiv:1010.0318 (2010). Klinkhamer. Entropic-Gravity Derivation of MOND. Modern Physics Letters A (2012). Klinkhamer and Kopp. Entropic gravity, minimum temperature, and modified Newtonian dynamics. Arxiv preprint arXiv:1104.2022v6 (2011). Li and Chang. Debye Entropic Force and Modified Newtonian Dynamics. Communications in Theoretical Physics (2011) vol. 55 pp. 733-736. Neto. Nonhomogeneous Cooling, Entropic Gravity and MOND Theory. Internat. J. Theoret. Phys. (2011) vol. 50 pp. 3552-3559. Kiselev and Timofeev. The holographic screen at low temperatures. arXiv (2010) Arxiv preprint arXiv:1009.1301v2 (2010). Pazy and Argaman. Quantum particle statistics on the holographic screen leads to modified Newtonian dynamics. Physical Review D (2012) vol. 85 (10) pp. 104021. Modesto and Randono. Entropic corrections to Newton’s law. arXiv (2010)Arxiv preprint arXiv:1003.1998v1 (2010). E. Verlinde, On the Origin of Gravity and the Laws of Newton, JHEP 1104:029, 2011.
--- abstract: 'We investigate the stationary characteristics of an $M/G/1$ retrial queue where the server, subject to active failures, primarily attends incoming calls and directs outgoing calls only when idle. On finding the server unavailable (busy or failed), inbound calls join the orbit and reattempt for service at exponentially-distributed time intervals. The system stability condition and probability generating functions of the number of calls in orbit and system are derived and evaluated numerically in the context of mean system size, server availability, failure frequency, and orbit waiting time.' address: - 'Department of Applied Mathematics and Computational Sciences, PSG College of Technology, Coimbatore 641-004, India' - 'School of Information and Communications, Department of Nanobio Materials and Electronics, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea' author: - Muthukrishnan Senthil Kumar - Aresh Dadlani - Kiseon Kim bibliography: - 'mybibfile.bib' title: 'Performance Analysis of an Unreliable $M/G/1$ Retrial Queue with Coupled Switching' --- Retrial queue,server breakdown,coupled switching,reliability,stationary distribution Introduction ============ Blended call centers have recently evolved as an effective and profitable communication asset in bridging companies and their customers. Such communication systems are capable of managing a mixture of both, inbound and outbound call operations that require instant service [@Bhulai2003]. Outbound calls are made by the server only when there are no inbound calls in the system. This feature, commonly also referred to as coupled switching or two-way communication, increases productivity by reducing the idle time experienced by the service agents [@Aksin2007]. Besides, incoming calls that find the server busy have an intrinsic tendency to retry for service after some random time [@Kharoufeh2006; @Artalejo2008]. As a result, stochastic behavior analysis of coupled switching in call centers under the influence of retrying customers is crucial to statistical practitioners and network service managers. There exist a number of seminal works dedicated to coupled switching in the retrial queueing literature. In [@Choi1995], the authors analyzed some expected performance measures of the $M/G/1/K$ priority retrial queue with coupled switching under the assumption that incoming and outgoing calls follow the same service distribution. Nevertheless, such an assumption limits the practicality of the model as customers may have different service needs. Although the authors of [@Artalejo2010] derive the first partial moments for a $M/G/1/1$ retrial queue model with different service time distributions using mean value analysis, it cannot be used to obtain the stationary distribution. In-depth analysis of the $M/M/1/1$ retrial queue with coupled switching and different service time distributions for single and multiple server cases have been reported in [@Artalejo2012]. Artalejo et al. [@Artalejo2013] proposed an embedded Markov chain approach to study the steady-state behavior of a couple-switched $M/G/1$ retrial queue with tailed asymptotic analysis of number of customers in the orbit. Nonetheless, in practice, the server may experience multiple failures which, to our best knowledge, has not been scrutinized in modeling systems with two-way communication. Hence, continuous-time analytical characterization of an unreliable single server retrial queue with coupled switching under steady state is imperative from the viewpoint of both, queueing as well as reliability analysis. The immediate goal of this letter is to study the impact of server failure on the performance of the $M/G/1$ queue with two-way communication having an infinite orbit and generally distributed server repair time. In particular, we obtain the system stability condition using the embedded Markov chain technique, followed by the supplementary variable approach to obtain in closed-form the probability generating functions (pgfs) for incoming calls in orbit and the system. Furthermore, we conduct numerical simulations for various performance metrics to corroborate our theoretical analysis. System Model Formulation ======================== We consider a single server retrial queue in which the primary inbound calls follow a Poisson arrival process with rate $\lambda$. If the server is idle, it makes an outgoing call in exponentially distributed time with rate $\alpha$. As in reality, the time taken to serve incoming and outgoing calls is assumed to be different. If an incoming call finds the server busy, it then enters the orbit and re-attempts to seek service after an exponentially distributed time with rate $\nu$. Otherwise, the incoming call commences service immediately. Since the server may breakdown while serving calls, without loss of generality, we assume that the lifetime of the server follows an exponential distribution with rates $\beta_1$ and $\beta_2$ during the service of inbound and outbound calls, respectively. On failure, the server is instantly sent for repair which has a generally distributed time. For the sake of consistency, we define $i\in \{1,2\}$ to differentiate between incoming and outgoing calls. Henceforth, $i=1$ refers to incoming calls, while $i=2$ indicates outgoing calls. Let $S_i(x)$ and $R_i(x)$ be the cumulative distributions of service and repair times of $i$-type calls, respectively. Similarly, let $s_i(x)$ and $r_i(x)$ denote respectively, the probability density functions of service and repair times of $i$-type calls. The Laplace transform (LT) of the service and repair times for each type of call is denoted as $\tilde{S}_i(\theta)$ and $\tilde{R}_i(\theta)$, respectively. We also define $S^o_i(x)$ and $R^o_i(x)$ as the remaining service and repair times, respectively. Moreover, let $\mu_{i,k}$ and $\gamma_{i,k}$ denote the $k^{th}$ moment of service and repair times, respectively. Additionally, the arrival flows of incoming calls, outgoing calls, service time, repair time, and intervals between successive re-attempts are all assumed to be mutually independent. Finally, let $N(t)$ be the number of incoming customer calls in orbit at time $t$, $M(t)$ be the total number of customers in the system at $t$, and $C(t)$ be the server state defined as follows: ![State transitions of the proposed system model.[]{data-label="fig1"}](./Figs/fig1.eps){width="4.0in"} $$C(t) = \begin{cases} 0, & \quad \text{if the server is } idle\\ 1, & \quad \text{if the server is } busy~with~incoming~calls\\ 2, & \quad \text{if the server is } busy~with~outgoing~calls\\ 3, & \quad \text{if the server } fails~while~serving~incoming~calls\\ 4, & \quad \text{if the server } fails~while~serving~outgoing~calls \end{cases} \label{eq1}$$ shows the state transitions of $\{(C(t),N(t));t \geq 0\}$ with state space $S\!=\!\{0,1,2,3,4\} \!\times\! Z_+$ for exponential service and repair times. Here, $n$ is the number of incoming calls in orbit and each row represents the server state as defined in (\[eq1\]). For generally distributed service and repair times, the system state at time $t$ can be expressed as the following continuous-time Markov chain: $$K(t)=\{(C(t),N(t),S^o_1(t),S^o_2(t),R^o_1(t),R^o_2(t)); t \geq 0\}. \label{eq2}$$ Based on $K(t)$, we now define the state probabilities as follows where $n,x,y \geq 0$: $$\left\{ \begin{aligned} &P_{0,n}(t) = Pr[C(t) \!=\! 0, N(t) \!=\! n]\,, \\ &P_{1,n}(x,t)dx = Pr[C(t) \!=\! 1, N(t) \!=\! n, x \!<\! S^o_1(t) \!\leq\! x \!+\! dx],\\ &P_{2,n}(x,t)dx = Pr[C(t) \!=\! 2, N(t) \!=\! n, x \!<\! S^o_2(t) \!\leq\! x \!+\! dx],\\ &P_{3,n}(x,y,t)dy = Pr[C(t) \!=\! 3, N(t) \!=\! n, S^o_1(t) \!=\! x, y \!<\! R^o_1(t) \!\leq\! y \!+\! dy],\\ &P_{4,n}(x,y,t)dy = Pr[C(t) \!=\! 4, N(t) \!=\! n, S^o_2(t) \!=\! x, y \!<\! R^o_2(t) \!\leq\! y \!+\! dy]. \end{aligned} \right. \label{eq3}$$ Here, $P_{0,n}(t)$ is the probability that the server is idle and there are $n$ calls in the orbit at time $t$. Similarly, $P_{i,n}(x,t) dx$ denotes the joint probability that the server is busy with an $i$-type call during the remaining service time $(x, x\!+\!dx)$ and there are $n$ calls in the orbit at epoch $t$. For $j \!\in\! \{3,4\}$, $P_{j,n}(x,y,t) dy$ refers to the joint probability that at time $t$ there are $n$ calls in the orbit, the remaining service time is $x$, and the failed server is fixed within the remaining repair time $(y,y\!+\!dy)$ while it was serving an inbound $(j\!=\!3)$ or an outbound $(j\!=\!4)$ call. Steady-state Distribution ========================= To identify the pgfs of orbit size and number of calls in the system, we first determine the stability condition of the system using the following theorem. The necessary and sufficient condition for system stability is given by the inequality $\lambda \mu_{1,1}(1+\beta_1 \gamma_{1,1})<1$. Let $\hat{X}_n$ be the service completion time of the $n^{th}$ call which includes possible down times (due to server failure) while providing service. For the sufficient condition, we need to prove the ergodicity of $\{L_n ; n\!\geq\! 1\}$, where $\{L_n\}$ is an irreducible and aperiodic discrete-time Markov chain of $K(t)$ in (\[eq2\]) and is defined as $L_n\!=\!N(\hat{X}_n^+)$. Using Foster’s criterion and undertaking the same approach as in [@Artalejo2013], $\{L_n\}$ is positive recurrent if $\|\eta_k\| \!<\! \infty$ and $\lim_{k \to \infty} \sup \{\eta_k\} \!<\! 0$ for all $k$, where $\eta_k\!=\!E[(L_{n\!+\!1}\!-\!L_n) / L_n \!=\! k]$. This definition of $\eta_k$ results in: $$\eta_k=\frac{kv [\lambda \mu_{1,1}(1\!+\!\beta_1\gamma_{1,1})\!-\!1]}{\lambda\!+\!kv\!+\!\alpha}\!+\!\frac{\lambda [\lambda \mu_{1,1}(1\!+\!\beta_1\gamma_{1,1})]}{\lambda\!+\!kv\!+\!\alpha}\!+\! \frac{\alpha [\lambda \mu_{2,1}(1\!+\!\beta_2 \gamma_{2,1})]}{\lambda\!+\!kv\!+\!\alpha}\,. \label{eq4}$$ Obviously, if $\lambda \mu_{1,1} (1\!+\!\beta_1 \gamma_{1,1})\!<\!1$, then for all $k$, $\eta_k \!<\! \infty$ and $\lim_{k \to \infty} \sup \{\eta_k\}\!<\!0$, which proves the sufficiency. As pointed out in [@Sennott1983], the non-ergodicity of $\{L_n\}$ can be guaranteed if Kaplan’s condition is satisfied, i.e. there exists some $k_0 \!\in\! Z_+$ such that $\eta_k \!\geq\! 0$ for $k \!\geq\!k_0$ and $\eta_k \!<\! \infty$ for all $k \!\geq\! 0$. In our case, this condition is satisfied as $r_{i,j}=0$ for $j<i-1$, where $P=[r_{i,j}]$ is the one step transition probability matrix. Therefore, $\lambda \mu_{1,1} (1+\beta_1 \gamma_{1,1})>1 $ implies the non-ergodicity of $\{L_n; n\geq 1\}$, which completes the proof. Using supplementary variable technique, we obtain the following balance equations from (\[eq3\]) as time approaches infinity: $$\left\{ \begin{aligned} &(\lambda \!+\! n \nu \!+\! \alpha) P_{0,n} = P_{1,n}(0) \!+\! P_{2,n}(0)\,, \\ &\frac{\!-d}{dx} P_{1,n}(x) = \!-(\lambda \!+\! \beta_1)P_{1,n}(x) \!+\! \lambda P_{0,n}s_1(x) \!+\! \lambda P_{1,n\!-\!1}(x) \!+\! (j \!+\! 1)\nu P_{0,n\!+\!1}s_1(x)\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \!+\! P_{3,n}(x,0)\,,\\ &\frac{\!-d}{dx} P_{2,n}(x) = \!-(\lambda \!+\! \beta_2)P_{2,n}(x) \!+\! \lambda P_{2,n\!-\!1}(x) \!+\! \alpha P_{0,n}s_2(x) \!+\! P_{4,n}(x,0)\,,\\ &\frac{\!-\partial}{\partial y} P_{3,n}(x,y) \!=\! \lambda [P_{3,n\!-\!1}(x,y) \!-\! P_{3,n}(x,y)] \!+\! \beta_1 P_{1,n}(x) r_1(y)\,,\\ &\frac{\!-\partial}{\partial y} P_{4,n}(x,y) \!=\! \lambda [P_{4,n\!-\!1}(x,y) \!-\! P_{4,n}(x,y)] \!+\! \beta_2 P_{2,n}(x) r_2(y)\,, \end{aligned} \right. \label{eq5}$$ with the following normalization condition: $$\sum\limits_{n=1}^{\infty} \big[P_{0,n}+\int_{0}^{\infty} \! W(x)dx + \int_{0}^{\infty}\!\!\int_{0}^{\infty} Z(x,y)dx dy\big] = 1 , \label{eq6}$$ where the terms $W(x)$ and $Z(x,y)$ stand for $\!P_{1,n}(x)\!+\!P_{2,n}(x)$ and $P_{3,n}(x,y)\!+\!P_{4,n}(x,y)$, respectively. Denoting LT\[$P_{i,n}(x)$\] by $\tilde{P}_{i,n}(\theta)$ and LT\[LT\[$P_{j,n}(x,y)$\]\] as $\widetilde{\tilde{P}}_{j,n}(\theta,s)$, we define the marginal generating functions as follows: $$\left\{ \begin{aligned} &P_0(z) = \sum\limits_{n=0}^{\infty} P_{0,n} z^n , \\ &\tilde{P}_i(z,\theta) = \sum\limits_{n=0}^{\infty} \tilde{P}_{i,n}(\theta) z^n\, ; \quad\textnormal{where}~ i = 1,2 ~\,,\\ &P_i(z,0) = \sum\limits_{n=0}^{\infty} P_{i,n}(0) z^n\, ; \quad\textnormal{where}~ i=1,2 ~\,,\\ &\widetilde{\tilde{P}}_j(z,\theta,s)= \sum\limits_{n=0}^{\infty} \widetilde{\tilde{P}}_{j,n}(\theta,s) z^n\, ; \quad\textnormal{where}~ j = 3,4 ~\,,\\ &\tilde{P}_{j}(z,\theta,0) = \sum\limits_{n=0}^{\infty}\tilde{P}_{j,n}(\theta,0) z^n\, ; \quad\textnormal{where}~ j = 3,4 \,. \end{aligned} \right. \label{eq7}$$ Applying LT on both sides of the equations in (\[eq5\]), multiplying $z^n$ in the resultant equations, summing over $n$, and through some algebraic manipulations, we obtain the following partial generating functions of the server state, wherein functions $\phi(z)\!=\!\exp\big(\!-\!\int_z^1 \frac{[\lambda (1-\delta_1(u)) + \alpha (1-\delta_2(u))]}{\nu [\delta_1(u)-u]} du\big)$ and for $i \!\in\! \{1,2\}$, $\delta_i(\cdot)\!=\!\tilde{S}_i(h_i(\cdot))$ and $h_i(z)\!=\!\lambda \!+\! \beta_i \!-\! \lambda z \!-\! \beta_i \tilde{R}_i(\lambda \!-\! \lambda z)$: $$\left\{ \begin{aligned} &P_{0}(z) = \frac{1 \!-\! \lambda \mu_{1,1}(1 \!+\! \beta_1 \gamma_{1,1})}{1 \!+\! \alpha \mu_{2,1}(1 \!+\! \beta_2 \gamma_{2,1})} \phi(z)\,, \\ &\tilde{P}_{1}(z,0) = \frac{[\lambda(1 \!-\! z) \!+\! \alpha(1 \!-\! \delta_2(z))][1 \!-\! \delta_1(z)]}{\big[\delta_1(z)-z\big] h_1(z)} P_0(z)\,,\\ &\tilde{P}_2(z,0) = \frac{\alpha [1 \!-\! \delta_2(z)]}{h_2(z)} P_0(z)\,,\\ &\widetilde{\tilde{P}}_3(z,0,0) = \frac{\beta_1 [1 \!-\! \tilde{R}_1(\lambda \!-\! \lambda z)]}{\lambda \!-\! \lambda z} \tilde{P}_1(z,0)\,,\\ &\widetilde{\tilde{P}}_4(z,0,0) = \frac{\beta_2 [1 \!-\! \tilde{R}_2(\lambda \!-\! \lambda z)]}{\lambda \!-\! \lambda z} \tilde{P}_2(z,0)\,.\\ \end{aligned} \right. \label{eq8}$$ The pgfs of the orbit occupancy size, $P(z)$, and the system size, $R(z)$, can be straightforwardly written in terms of the partial generating functions of (\[eq8\]) as given below: $$\begin{aligned} &P(z) \!=\! P_0(z) \!+\! \tilde{P}_1(z,0) \!+\! \tilde{P}_2(z,0) \!+\! \widetilde{\tilde{P}}_3(z,0,0) \!+\! \widetilde{\tilde{P}}_4(z,0,0)\,,\\ &R(z)\! =\! P_0(z) \!+\! z[\tilde{P}_1(z,0) \!+\! \tilde{P}_2(z,0) \!+\! \widetilde{\tilde{P}}_3(z,0,0) \!+\! \widetilde{\tilde{P}}_4(z,0,0)]. \end{aligned} \label{eq9}$$ From (\[eq8\]), the closed-form expressions for (\[eq9\]) can be easily obtained. By ignoring the non-zero probability of server failure, the following results are in complete agreement with [@Artalejo2013]: $$\begin{aligned} &P(z) = \frac{\lambda (1-z) + \alpha[1-\delta_2(z)]}{\lambda[\delta_1(z)-z]} P_0(z)\,,\\ &R(z)= \frac{(\lambda - \lambda z) \delta_1(z) + \alpha[1 - \delta_2(z)]}{\lambda[\delta_1(z) - z]} P_0(z)\,. \end{aligned} \label{eq10}$$ Performance Metrics =================== The four performance measures are defined in this section. Expected Number of Calls in the System -------------------------------------- This measure accounts for the mean number of incoming calls retrying for service, either due to server failure or it being busy, as well as those being served by the server. This is readily obtained by differentiating the equations in (\[eq10\]) and evaluating them at $z\!=\! 1$. The first equation yields the first moment of the orbit size ($E[N]$) as follows: $$\begin{aligned} E[N] =& P'(1) =\! \frac{\lambda^2[\beta_1 \mu_{1,1} \gamma_{1,2} \!+\! (1 \!+\! \beta_1 \gamma_{1,1})^2 \mu_{1,2}]}{2[1 \!-\! \rho(1 \!+\! \beta_1 \gamma_{1,1})]}\\ &\!+\! \frac{\lambda \alpha [\beta_2 \mu_{2,1} \gamma_{2,2} \!+\! (1 \!+\! \beta_2 \gamma_{2,1})^2 \mu_{2,2}]}{2[1 \!+\! \sigma(1 \!+\! \beta_2 \gamma_{2,1})]} \!+\! \frac{\lambda [\rho (1 \!+\! \beta_1 \gamma_{1,1}) \!+\! \sigma(1 \!+\! \beta_2 \gamma_{2,1})]}{\nu [1 \!-\! \rho(1 \!+\! \beta_1 \gamma_{1,1})]}\,, \end{aligned} \label{eq11}$$ where $\rho \!=\! \lambda \mu_{1,1}$ and $\sigma \!=\! \alpha \mu_{2,1}$. Similarly, the second equation in (\[eq10\]) gives the following mean system size ($E[M]$): $$\left. \begin{aligned} &E[M] \!=\! R'(1) \!=\! P'(1) \!+\! \frac{\rho (1 \!+\! \beta_1 \gamma_{1,1}) \!+\! \sigma (1 \!+\! \beta_2 \gamma_{2,1})}{1 \!+\! \sigma(1 \!+\! \beta_2 \gamma_{2,1})}\,. \end{aligned} \right. \label{eq12}$$ Server Availability ------------------- The probability that the server is operational at a given time instant $t$ is defined as its point-wise availability, $A(t)$, and its steady state availability (i.e. $\lim_{t \to \infty} A(t)\!=\!P_{a}$) is given as follows: $$\begin{aligned} P_{a} &=\! \lim_{z \!\to\! 1} \{P_0(z) \!+\! \tilde{P}_1(z,0) \!+\! \tilde{P}_2(z,0)\}\\ &=\! \frac{(1 + \sigma)[1 - \rho (1 + \beta_1 \gamma_{1,1})] + \rho [1 + \sigma(1+\beta_2 \gamma_{2,1})]}{1 + \sigma(1 + \beta_2 \gamma_{2,1})} . \end{aligned} \label{eq13}$$ Server failure frequency ------------------------ This corresponds to the probability that the server fails at time $t\!>\!0$ given that it was operating at $t\!=\!0$ [@Kumar2010]. The following closed-form results from (\[eq9\]): $$\left. \begin{aligned} P_f &= \lim_{z \!\to\! 1} \{\beta_1 \tilde{P}_1(z,0) \!+\! \beta_2 \tilde{P}_2(z,0)\} = \rho \beta_1 \!+\! \sigma \beta_2 \frac{[1 \!-\! \rho (1 \!+\! \beta_1 \gamma_{1,1})]}{[1 \!+\! \sigma (1 \!+\! \beta_2 \gamma_{2,1})]}\,. \end{aligned} \right. \label{eq14}$$ Expected Waiting Time in Orbit ------------------------------ Denoted by $W$, the steady-state delay experienced by a customer in obit depends on the total idle time of the server not serving an incoming call ($W_0$), the total service time (including the server failure time) of the server providing service to an incoming call ($W_1$), and the total service time (including the server failure time) of the server busy with an outgoing call ($W_2$). The probability of an inbound call entering the orbit ($P_w$) is thus, given as follows: $$\begin{aligned} P_w &= \lim_{z \!\to\! 1} \{\tilde{P}_1(z,0) \!+\! \tilde{P}_2(z,0)\ \!+\! \widetilde{\tilde{P}}_3(z,0,0) \!+\! \widetilde{\tilde{P}}_4(z,0,0)\}\\ &= \frac{\rho(1+\beta_1 \gamma_{1,1}) + \sigma(1 + \beta_2 \gamma_{2,1})}{1 + \sigma (1 + \beta_2 \gamma_{2,1})}\,. \end{aligned} \label{eq15}$$ Using (\[eq15\]) and the first moments of the pgfs in (\[eq8\]), we obtain the mean waiting time in the orbit to be [@Choi1995]: $$\begin{aligned} \text{E}[W] = \text{E}[W_0] + \text{E}[W_1] + \text{E}[W_2] = \text{E}[N]/\lambda\,, \end{aligned} \label{eq16}$$ where $\text{E}[W_0]= P_w/\nu$, $\text{E}[W_1]=\text{E}[N] \text{E}[B_1] + \rho(1 + \beta_1 \gamma_{1,1}) \text{E}[R_1]$, and $\text{E}[W_2]=\sigma(1 + \beta_2 \gamma_{2,1})\big(1 - \rho(1 + \beta_1 \gamma_{1,1})\big) \text{E}[R_2] / \big(1 + \sigma(1 + \beta_2 \gamma_{2,1})\big) + \sigma(1 + \beta_2 \gamma_{2,1}) \text{E}[W_0]$. Here, $\text{E}[B_i]$ and $\text{E}[R_i]$ represent the mean service time (including failure time) and the mean remaining service time (including failure time) while serving $i$-type calls, which are given as $\mu_{i,1}(1+ \beta_i \gamma_{i,1})$ and $\text{E}[B_i^2]/(2 \text{E}[B_i])$, respectively. Numerical Examples and Discussions ================================== To illustrate the impact of system parameters on the performance primitives, we present numerical examples for service and repair times with three arbitrary distributions namely, exponential with density function $c_1 e^{-c_1x}$, Erlangian of order two with density function $c_1^2 x e^{-c_1x}$ and hyperexponential given as $a c_1 e^{-c_1 x}\!+\!(1\!-\!a)c_2 e^{-c_2 x}$, where $c_1, c_2\!>\!0$ and $0\!\leq\!a\!\leq\!1$. Throughout this section, we assume $\lambda\!=\!1.2$, $\alpha\!=\!0.4$, $\nu\!=\!1$, $\mu_{2,1}\!=\!0.1$, and $\gamma_{2,1}\!=\!0.2$ to satisfy the ergodic condition of the system. We also consider an $M/G/1$ retrial queue without server failure [@Artalejo2013], i.e. $(\beta_1,\beta_2)\!=\!(0,0)$ as the baseline for comparison. \ shows the variation in mean system size as functions of the inbound arrival rate ($\lambda$), outbound rate ($\alpha$), retrial rate ($\nu$), and inbound service ($\mu_{1,1}$) and repair ($\gamma_{1,1}$) times. Increase in the number of arriving calls reduces the chances of finding the server active and idle. Consequently, these calls enter the orbit to retry for service as shown in the figure. In comparison to the failure-free scenario, we note that the system size in our model increases with the failure rate $\beta_1$ as $\lambda$ increases. A similar relationship can be observed between $E[M]$ and $\alpha$ as well. In regard to the retrial rate, $E[M]$ steeply decreases initially and gradually stabilizes to some constant value. This justifies the fact that the calls residing in orbit have higher chances of being served and thus, leaving the orbit if they retry for service more frequently. For lower values of $\beta_1$, the primary incoming calls are more probable to find the server available, resulting in a reduced orbit size. The influence of the service and repair times of primary calls is also evident in this figure. As the average time to serve incoming calls increases, $E[M]$ grows steeper. In other words, longer service times increases the number of incoming calls waiting in the orbit. Likewise, shorter the server repair time, the more active it would be thus, reducing the system size which is mainly dominated by the orbit length. ![Server availability ($P_a$) versus inbound arrival rate ($\lambda$) and server failure rate while serving such calls ($\beta_1$).[]{data-label="fig3"}](./Figs/fig3.eps){width="4.6in"} depicts the impact of $\lambda$ and $\beta_1$ on server availability. Note that all three distributions exhibit the same results for different values of $\beta_1$ and $\beta_2$. In absence of server failure, $P_a$ is always 1. However, as $\beta_1$ increases, the availability of the server to incoming calls reduces with rise in $\lambda$. For instance, at $\lambda\!=\!1.6$, $P_a$ falls by slightly less than $1\%$ as $\beta_1$ goes from 0.4 to 0.8. The figure also portrays the effect of $\beta_1$ under varying first moment values of service and repair times. Note that there is a steeper fall in $P_a$ as the service time of incoming calls increases. Thus, the probability of finding the system available is higher when $\mu_{1,1} \!<\! \gamma_{1,1}$ and decreases with increase in $\mu_{1,1}$. Similarly, describes the server failure frequency in terms of $\lambda$ and $\beta_1$. Apparent from (\[eq14\]), we see that $P_f$ monotonically increases with the number of incoming calls in our model. Additionally, at $\lambda\!=\!2$, as $\beta_1$ increases from 0.4 to 0.8, $P_f$ rises drastically by over $74\%$. The profound contribution of $(\beta_1,\beta_2)$ is also reflected in this figure. For various values of $(\mu_{1,1},\gamma_{1,1})$, $W_f$ increases constantly with $\beta_1$ and is higher when $\mu_{1,1}$ is greater than $\gamma_{1,1}$. ![Server failure frequency ($P_f$) versus inbound arrival rate ($\lambda$) and server failure rate while serving such calls ($\beta_1$).[]{data-label="fig4"}](./Figs/fig4.eps){width="4.6in"} ![Expected waiting time in orbit ($E[W]$) versus inbound arrival rate ($\lambda$) and server failure rate while serving such calls ($\beta_1$).[]{data-label="fig5"}](./Figs/fig5.eps){width="4.6in"} The mean orbit waiting time as functions of $\lambda$ and $\beta_1$ are illustrated in . With respect to the benchmark, we observe the impact of server failure while serving incoming calls on the gradual increase in $E[W]$. As $\beta_1$ increases to 2, the average orbit delay grows exponentially with increase in $\mu_{1,1}$. Conclusion ========== In this paper, we modelled call blending in call centers as an $M/G/1$ retrial queue with the possibility of server failure and studied the system behavior in steady-state. In our analysis, we derived the system stability condition and closed-form expressions for the joint distribution of the server state and the expected number of customer calls in the system. Numerical results were provided for various performance measures to validate and compare our findings with that of a system with no breakdown. Acknowledgements {#acknowledgements .unnumbered} ================ This research was a part of the project entitled ‘Domestic Products Development of Marine Survey and Ocean Exploration Equipments’, and ‘Development of an Integrated Control System of Eel Farms based on Short-range Wireless Communication’, funded by the Ministry of Oceans and Fisheries, South Korea. References {#references .unnumbered} ==========
--- abstract: 'Invariant entangled states remain unchanged under simultaneous identical unitary transformations of all their subsystems. We experimentally generate and characterize such invariant two-, four-, and six-photon polarization entangled states. This is done only with a suitable filtering procedure of multiple emissions of entangled photon pairs from a single source, without any interferometric overlaps. We get the desired states utilizing bosonic emission enhancement due to indistinguishability. The setup is very stable, and gives high interference contrasts. Thus, the process is a very likely candidate for various photonic demonstrations of quantum information protocols.' author: - 'Magnus Rådmark$^{1}$, Marcin Wieśniak$^{2}$, Marek Żukowski$^{2}$ and Mohamed Bourennane$^{1}$' title: 'Experimental filtering of two-, four-, and six-photon singlets from single PDC source' --- Entanglement is an essential tool in many quantum information tasks. Entangled states of two qubits proved to be useful in various quantum communication protocols like quantum teleportation, quantum dense coding, and quantum cryptography. They are the essence of the first versions of Bell’s theorem[@review]. However, the expansion of quantum information science has now reached a state in which many schemes are involving multiparty processes, and could require multiqubit entanglement. There is an interesting series of multiqubit states, ${\mbox{$\mid \! \Psi_{k}^{-} \, \rangle$}}$, where $k=2,4,6$ or more. They are invariant under actions consisting of identical unitary transformations of each qubit [@ZR97a]: $ U^{\otimes k}{\mbox{$\mid \! \Psi_{k}^{-} \, \rangle$}} = {\mbox{$\mid \! \Psi_{k}^{-} \, \rangle$}}, $ where $U^{\otimes k} = U\otimes...\otimes U$ denotes a tensor product of $k$ identical unitary operators $U$. The property protects the states against collective noise. The states are useful e.g. for communication of quantum information between observers who do not share a common reference frame [@BRS03]: any realignment of the receiver’s reference frame corresponds to an application of the same transformation to each of the sent qubits. The states ${\mbox{$\mid \! \Psi_{k}^{-} \, \rangle$}}$ can also be used for secure quantum multiparty cryptographic protocols, such as the multi-party secret sharing protocol [@HBB99; @GKBW07]. We generate correlations which characterize the six-photon ${\mbox{$\mid \! \Psi_{6}^{-} \, \rangle$}}$ entangled state. This is done in a six photon interference experiment. A six-photon interference was reported recently in [@LZGGZYGYP07]. To obtain graphs states the authors of ref. [@LZGGZYGYP07] used [three ]{} pulse-pumped parametric down-conversion (PDC) crystals, and interferometric overlaps, to entangle independently emitted pairs (each from a different crystal) with each other. Schemes of this kind are generalizations of those of ref. [@ZHWZ97]. However, the overlaps make the scheme fragile. In our experiment, by pulse pumping just [one crystal]{} and extracting the right order process via suitable filtering and beamsplitting (the method of [@ZZW95]), we observe simultaneously effects attributable to the multi-photon invariant entangled states ${\mbox{$\mid \! \Psi_{2}^{-} \, \rangle$}}$, ${\mbox{$\mid \! \Psi_{4}^{-} \, \rangle$}}$, and ${\mbox{$\mid \! \Psi_{6}^{-} \, \rangle$}}$. The setup has no overlaps and therefore no interferometric alignment is needed. It is strongly robust, and the output is of high fidelity with respect to the theoretical states ${\mbox{$\mid \! \Psi_{k}^{-} \, \rangle$}}$. A simple quantum optical description of two phase matched modes, of the multiphoton state that results out of a single pulse acting on a type-II PDC crystal, can be put as $$C \sum_{n=0}\frac{1}{n!}[-i\alpha(a_{0H}^{\dagger}b_{0V}^{\dagger} - a_{0V}^{\dagger}b_{0H}^{\dagger})]^n{\mbox{$\mid \! 0 \, \rangle$}}. \label{emission}$$ The symbol $a_{0H}^{\dagger}$ ($b_{0V}^{\dagger}$) denotes a creation operator for one horizontal, $H$, (vertical, $V$) photon in mode $a_{0}$ ($b_{0}$), [etc.]{} $C$ is a normalization constant, the coupling parameter $\alpha$ is a function of pump power, non-linearity and length of the crystal. This is a good approximation of the actual state, provided one collects the photons under conditions that allow full indistinguishability between separate two-photon emissions [@ZZW95]. First, second, and third order terms in the expansion in eq. (\[emission\]), correspond to an emission of two, four, and six photons, respectively, into two spatial modes. These terms can be re-interpreted as the following superpositions of photon number states: $$\begin{aligned} &{\mbox{$\mid \! 1H_{a_{0}},1V_{b_{0}} \, \rangle$}}- {\mbox{$\mid \! 1V_{a_{0}},1H_{b_{0}} \, \rangle$}}, \label{particle2}\end{aligned}$$ $$\begin{aligned} &{\mbox{$\mid \! 2H_{a_{0}},2V_{b_{0}} \, \rangle$}}- {\mbox{$\mid \! 1H_{a_{0}},1V_{a_{0}},1V_{b_{0}},1H_{b_{0}} \, \rangle$}} \nonumber \\ &+{\mbox{$\mid \! 2V_{a_{0}},2H_{b_{0}} \, \rangle$}}, \label{particle4}\end{aligned}$$ $$\begin{aligned} &{\mbox{$\mid \! 3H_{a_{0}},3V_{b_{0}} \, \rangle$}} - {\mbox{$\mid \! 2H_{a_{0}},1V_{a_{0}},2V_{b_{0}},1H_{b_{0}} \, \rangle$}}+ \nonumber \\ &+{\mbox{$\mid \! 1H_{a_{0}},2V_{a_{0}},1V_{b_{0}},2H_{b_{0}} \, \rangle$}}-{\mbox{$\mid \! 3V_{a_{0}},3H_{b_{0}} \, \rangle$}}, \label{particle3}\end{aligned}$$ where e.g. $2H_{a_{0}}$ and $3H_{a_{0}}$ denotes two and three horizontally polarized photons in mode $a_{0}$, respectively, [etc.]{} The second and third order PDC is intrinsically different than simple products of two and three entangled pairs. Due to the bosonic nature of photons, emissions of completely indistinguishable photons are more likely, than the ones giving birth to photons with orthogonal polarization. We report a [joint]{} observation, in one setup, of the correlations of the invariant two, four and six-photon polarization entangled states given by the following superpositions: $ {\mbox{$\mid \! \Psi_{2}^{-} \, \rangle$}} = \frac{1}{\sqrt{2}}({\mbox{$\mid \! HV \, \rangle$}} -{\mbox{$\mid \! VH \, \rangle$}}), $ $${\mbox{$\mid \! \Psi_{4}^{-} \, \rangle$}} = \frac{2}{\sqrt{3}}{\mbox{$\mid \! GHZ_{4}^{+} \, \rangle$}} - \frac{1}{\sqrt{3}}{\mbox{$\mid \! EPR \, \rangle$}}{\mbox{$\mid \! EPR \, \rangle$}}, \label{state4}$$ and $${\mbox{$\mid \! \Psi_{6}^{-} \, \rangle$}} = \frac{1}{\sqrt{2}}{\mbox{$\mid \! GHZ_{6}^{-} \, \rangle$}} + \frac{1}{2}({\mbox{$\mid \! \overline{W}_{3} \, \rangle$}}{\mbox{$\mid \! W_{3} \, \rangle$}} -{\mbox{$\mid \! W_{3} \, \rangle$}}{\mbox{$\mid \! \overline{W}_{3} \, \rangle$}}). \label{state6}$$ The states in the superpositions are given by: $${\mbox{$\mid \! GHZ_{4}^{+} \, \rangle$}}=\frac{1}{\sqrt{2}}({\mbox{$\mid \! HHVV \, \rangle$}}+{\mbox{$\mid \! VVHH \, \rangle$}})/\sqrt{2}$$, $${\mbox{$\mid \! GHZ_{6}^{-} \, \rangle$}}=({\mbox{$\mid \! HHHVVV \, \rangle$}}-{\mbox{$\mid \! VVVHHH \, \rangle$}}),$$ and $${\mbox{$\mid \! EPR \, \rangle$}}=\frac{1}{\sqrt{2}}({\mbox{$\mid \! HV \, \rangle$}}+ {\mbox{$\mid \! VH \, \rangle$}}).$$ Finally $${\mbox{$\mid \! W_{3} \, \rangle$}}=\frac{1}{\sqrt{3}}({\mbox{$\mid \! HHV \, \rangle$}}+{\mbox{$\mid \! HVH \, \rangle$}}+{\mbox{$\mid \! VHH \, \rangle$}}).$$ The ket ${\mbox{$\mid \! \overline{W} \, \rangle$}}$ is the spin-flipped ${\mbox{$\mid \! W \, \rangle$}}$. The states (\[state2\]-\[state6\]) are obtained out of different orders of the PDC emission (figure 1), by selecting specific double, quadruple and six-fold coincidences. In our setup we use a frequency-doubled Ti:Sapphire laser ($80$ Mhz repetition rate, $140$ fs pulse length) yielding UV pulses with a central wavelength at $390$ nm and an average power of $1300$ mW. The pump beam is focused to a $160$ $\mu$m waist in a $2$ mm thick BBO ($\beta$-barium borate) crystal. Half wave plates and two $1$ mm thick BBO crystals are used for compensation of longitudinal and transversal walk-offs. The third order emission of non-collinear type-II PDC is then coupled to single mode fibers (SMF), defining the two spatial modes at the crossings of the two frequency degenerated PDC emission cones. Leaving the fibers the down-conversion light passes narrow band ($\Delta\lambda =3$ nm) interference filters (F) and is split into six spatial modes $(a, b, c, d, e, f)$ by ordinary $50\%-50\%$ beam splitters (BS), followed by birefringent optics (to compensate phase shifts in the BS’s). Due to the short pulses, narrow band filters, and single mode fibers the down-converted photons are temporally, spectrally, and spatially indistinguishable [@ZZW95], see Fig. \[setup\]. The polarization is being kept by passive fiber polarization controllers. Polarization analysis is implemented by a half wave plate (HWP), a quarter wave plate (QWP), and a polarizing beam splitter (PBS) in each mode. The outputs of the PBS’s are lead to single photon silicon avalanche photo diodes (APD) through multi mode fibers. The APD’s electronic responses, following photo detections, are being counted by a multi-channel coincidence counter with a $3.3$ ns time window. The coincidence counter registers any coincidence event between the APD’s as well as single detection events. The states ${\mbox{$\mid \! \Psi_{k}^{-} \, \rangle$}} (k=2,4,6)$ exhibit perfect two, four, and six qubit correlations. The correlation function is defined as an expectation value of the product of local polarization “Pauli” observables. If one limits the measurement to the local observables $\cos{\theta_{l}}\sigma_{z}^{(l)} +\sin{\theta_{l}}\sigma_{x}^{(l)} $ (with eigenvectors $\sqrt{1/2}({\mbox{$\mid \! L \, \rangle$}}_{l}\pm e^{i\theta_{l}}{\mbox{$\mid \! R \, \rangle$}}_{l})$ and eigenvalues $\pm 1$), the measurements correspond to linear polarization analysis in each spatial mode ($l=a,b,c,d,e,f$). In such a case the quantum prediction for the two photon (in modes b and d) correlation function reads: $ E(\theta_{b},\theta_{d})= -\cos(\theta_{b} - \theta_{d}). $ For the four photon counts (in modes a, b, d, and e) correlation function is given by $$\begin{aligned} &E(\theta_{a},\theta_{b},\theta_{d},\theta_{e})= \frac{2}{3} \cos(\theta_{a} + \theta_{b} - \theta_{d} - \theta_{e}) \nonumber \\ &+\frac{1}{3}\cos(\theta_{a} - \theta_{b})\cos(\theta_{c} - \theta_{d}). \label{eq.correlation4}\end{aligned}$$ Finally for the six photon events one has $$\begin{aligned} &E(\theta_{a},\theta_{b},\theta_{c},\theta_{d},\theta_{e},\theta_{f})= \nonumber \\ &-\frac{1}{2} \cos(\theta_{a} + \theta_{b} + \theta_{c} - \theta_{d}- \theta_{e} - \theta_{f} ) \nonumber \\ &-\frac{1}{18}\sum{\cos(\theta_{a} \pm \theta_{b} \pm \theta_{c} \pm \theta_{d} \pm \theta_{e} \pm \theta_{f} )}, \label{eq.correlation6}\end{aligned}$$ where $\sum$ is a sum over all possible sign sequences which contain only [two positive]{} signs, with the sign sequence in the first term, proportional to $\frac{1}{2}$, excluded. Due to the invariance, the correlation functions for all measurements around any single great circle of the Bloch sphere look the same. Fig. \[correlation2\] shows three experimentally observed two-photon correlation functions, $E(\theta_{b},\theta_{l})$, where $( l= d,e,f)$. The setting $\theta_{b}$ is varied, while the other analyzer is fixed at $\theta_{l}= \theta_{m}= \theta_{n}=\pi/2$. This corresponds to diagonal/antidiagonal, $D/A$, linear polarization analysis. A sinusoidal least-square fit was made to the data. The average visibility, defined here, as the average amplitude of the three fits, is $V_2 = 0.962\%\pm 0.003\%$. Fig. \[correlation4\] shows how six experimentally observed four photon correlation functions $E(\theta_{b},\theta_{l}, \theta_{m}, \theta_{n})$ depend on $\theta_{b}$. The other analyzers were fixed at $\theta_{l}=\pi/2,$ where $(l = a,c)$, $(m = d,e)$, and $(n = e,f)$. The average value of the six visibilities is $V_4 = 0.9189\%\pm0.0049\%$. Finally, Fig. \[correlation6\] shows similar data for the experimentally observed six photon correlation function $E(\theta_{b},\theta_{a}, \theta_{c}, \theta_{d},\theta_{e},\theta_{f})$. Again $\theta_{b}$ was varied with the other five analyzers fixed at $\theta_{l}= \pi/2$ where $(l = a,c,d,e,f)$. The value on the visibility is $V=83.79\%\pm2.98\%$. In table \[t1\] we present all the experimentally obtained two-, four-, and six- photon visibilities. $k$ Modes Visibility ----- --------------- ----------------- 2 $b,d$ $0.962\pm0.004$ 2 $b,e$ $0.963\pm0.006$ 2 $b,f$ $0.962\pm0.004$ 4 $a,b,d,e$ $0.919\pm0.014$ 4 $a,b,d,f$ $0.918\pm0.011$ 4 $a,b,e,f$ $0.919\pm0.014$ 4 $a,b,e,f$ $0.919\pm0.014$ 4 $b,c,d,f$ $0.918\pm0.009$ 4 $b,c,e,f$ $0.920\pm0.012$ 6 $a,b,c,d,e,f$ $0.838\pm0.030$ : \[tab:visibility\]Visibilities of the invariant states ${\mbox{$\mid \! \Psi_k^{-} \, \rangle$}}$ (where $k = 2, 4, 6$) \[t1\] We have compared the observed visibilities with theoretical predictions, [@ARXIV]. To estimate maximal predictable visibilities one can use a less simplified description of the two-photon state emitted by SPDC event, and replace in eq. (\[emission\]) $(a_{0H}^{\dagger}b_{0V}^{\dagger} - a_{0V}^{\dagger}b_{0H}^{\dagger})$ by $$\begin{aligned} && \int dt \int d\omega_0\int d\omega_1\int d\omega_2f(\omega_1)f(\omega_2) g(\omega_0)e^{i\omega t}\times \nonumber\\ && \Delta(\omega_0-\omega_1-\omega_2) (a_{0H}^{\dagger}(\omega_1)b_{0V}^{\dagger}(\omega_2) - a_{0V}^{\dagger}(\omega_1)b_{0H}^{\dagger}(\omega_2)). \nonumber\\ &&\end{aligned}$$ This approach is rich enough to take into account the frequency phase matching conditions. The creation operators depend additionally on frequencies, and obey $[a_{0X}(\omega),a_{0X'}(\omega')] = \delta_{XX'}\delta_{XX'}(\omega- \omega')$, etc. The function $f(\omega)$ represents the shape of the filter transmission profiles, and $g(\omega_0)$ represents the frequency profile of the pump pulse. If one chooses $f(\omega) = \exp[-((\omega_f -\omega)/(2\sigma_f))^2]$ and $g(\omega) = \exp[-((\omega_p -\omega)/(2\sigma_p))^2]$ where $\sigma_p$, and $\sigma_f$ are the FWHM bandwidth of the pump, and the filters, and $\omega_f = \omega_p/2$, the following formulas for the maximal theoretical visibility as a function of the ratio $r = \sigma_f/\sigma_p$ can be reached, [@ARXIV]. For the four-photon process: $V^{temp}_4 = \sqrt{1+2r^2}/(1+r^2)$ and for the six-photon process: $V^{temp}_6 = (1+2r^2)/[(1+r^2/2)(1+3r^2/2)]$. In our experiment we used $r_{exp} = \Delta\lambda_f/(4\Delta\lambda_p) = 0.76$. This corresponds to $V_4 = 0.93$ and $V_6 = 0.90$. The actual measured values of visibility for two, four, and six photon interference are very close to the predicted ones, see table \[t1\]. Thus, the fact that our setup use only filtering and beamsplitting, has interferometric advantages. In other words the obtained four and six particles visibilities are almost as high as one get for the ratio $r_{exp}$. The high visibility has the following consequences. With just a part of our data one can use the simplest one of the “experimentally friendly” entanglement indicators introduced in [@BBLPZ08]. It guarantees that N-qubit state is entangled, if the norm of the N-particle correlation tensor is higher than 1. For the ${\mbox{$\mid \! \Psi_{2}^{-} \, \rangle$}}$ we take just $T_{xx}$, $T_{yy}$, $T_{zz}$, for ${\mbox{$\mid \! \Psi_{4}^{-} \, \rangle$}}$ again just $T_{xxxx}$, $T_{yyyy}$, $T_{zzzz}$, and for ${\mbox{$\mid \! \Psi_{6}^{-} \, \rangle$}}$ the components $T_{xxxxxx}$, $T_{yyyyyy}$ and $T_{zzzzzz}$. With our data we have obtained the $2.785\pm 0.007$, $2.517\pm0.011$, and $2.29 \pm 0.14$, respectively for each of the case. The entanglement threshold is violated by of $242$, $133$, and $9.3$ standard deviations. Additionally, according to the criteria given in [@ZB02] the state cannot be described by a local realistic model, if the sum of squares of two out the the listed components is above 1. This is again achieved by our data: for four particles we get $T_{xxxx}^2+T_{yyyy}^2=1.646\pm 0.009$ (exceeding $1$ by $74.8$ standard deviations) and for six particles we have $T_{xxxxxx}^2+T_{yyyyyy}^2=1.52\pm 0.11$ (exceeding $1$ by $4,5$ standard deviations). Thus the state can be directly (that is without still enhancing its fidelity) utilized in classical threshold beating communication complexity protocols [@BZPZ04]. In summary, we have experimentally demonstrated that a suitable filtering procedure applied to a triple emission from a [single]{} pulsed source of polarization entangled photons leads to two, four and six-photon, high visibility, interference due to entanglement, observable in a single setup. We utilize the bosonic emission enhancement occurring in the emission of three photon-pairs in PDC, thus the process is not entirely spontaneous. The six qubit state that we observed is invariant with respect to simultaneous identical (unitary) transformations of all qubits. This makes it particularly useful for multiparty quantum communication and general quantum computation tasks: it is very robust against deformations in transfer. Since we use just one source, we avoid alignment problems, and thus the setup is very stable. We would like to note that the interference contrast is high enough for our source to be used in two-, four-, and six party demonstrations of quantum reduction of communication complexity in some joint computational tasks, and for secret sharing, as well as in many other quantum informational technologies.\ This work was supported by Swedish Research Council (VR). 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--- abstract: 'We propose a universal strategy to realize a broadband control on arbitrary scatterers, through multiple coherent beams. By engineering the phases and amplitudes of incident beams, one can suppress the dominant scattering partial waves, making the obstacle lose its intrinsic responses in a broadband spectrum. The associated coherent beams generate a finite and static region, inside which the corresponding electric field intensity and Poynting vector vanish. As a solution to go beyond the sum-rule limit, our methodology is also irrespective of inherent system properties, as well as extrinsic operating wavelength, providing a non-invasive control on the wave-obstacles interaction for any kinds of shape.' author: - Jeng Yi Lee - Lujun Huang - Lei Xu - 'Andrey E. Miroshnichenko' - 'Ray-Kuang Lee' title: Broadband Control on Scattering Events with Interferometric Coherent Waves --- Making functional sub-wavelength scatterers has been attractive for a variety of applications, such as superdirective scatterers [@super1; @super2; @super3], perfect absorption objects [@absorber1; @absorber2], magnetic resonantor based devices [@magnetic1; @magnetic2; @magnetic3], Kerker effect and beyond [@kerker1; @kerker2; @kerker3], anapole [@anapole], and superscattered objects [@superscattering1; @superscattering2]. In particular, to have invisible cloaks, the concepts of transformation optics [@controlling; @conformal] and scattering cancellation method [@kerker; @alu1; @alu2] have been applied not only to electromagnetic waves, but also to acoustic [@acoustic1; @acoustic2; @acoustic3; @sc1; @sc2] and water waves [@fluid1; @fluid2], thermal diffusion science [@thermal1; @thermal2; @sc8], quantum matter waves [@quantum1; @quantum2; @quantum3; @sc6; @sc7], and elastic wave in solids [@elastic1; @elastic2; @elastic3]. However, for these methods and the consequently improved efforts [@carpet], we still suffer from the superluminal propagation [@quest1; @quest2], and limited operating bandwidth imposed by Kramers-Kronig relation [@quest3; @quest4]. To manipulate light-obstacles interaction in nanoscales, it is still desirable to have a non-invasive and efficient way to have objects working in a broadband spectrum. As pointed out by E. M. Purcell [@sum], the integration of the extinction cross section over all the spectra is related to the static electric and magnetic material parameters, leading to the sum-rule limit. Therefore, under a plane wave excitation, no scattering systems can remain stationary scattering responses in a broadband spectrum. In this Letter, we demonstrate that it is possible to turn off or amplify the target scattering partial waves with interferometric coherent waves. With a proper setting on the phases and intensities [@linear1], destructive interferometry on the dominant partial waves can be achieved, resulting in an arbitrary object invisible. At the same time, a finite and static region emerges, inside which the electric field and the corresponding Poynting vector almost completely vanish. Instead, high-order scattering partial waves are excited even the physical size of the scatterer is smaller than the size of this finite region. Moreover, the operating wavelength for such a destructive removal of excitation exists for a broadband spectrum, overcoming the fundamental sum-rule limit obtained from a single plane wave excitation. Also, there exist more than one settings for the interferometric field in the excitation, demonstrating the flexibility for experimental implementations. The robustness of our methodology on invisibility is also verified by introducing the deviations in the scatter displacements, intensities and illumination angles of incident beams, and different shapes of systems. Our results pave an alternative route to manipulate waves and obstacles in the extremely small scale. ![image](fig1.png){width="18.0cm"} Without loss of generality, we consider the illumination waves composed by a set of plane waves on a cylindrical scatterer, as illustrated in Fig. 1(a). Here, the symmetrical axis of the cylindrical axis is chosen as the $z$-axis. For a single plane wave of s-polarized electric field propagating in $x$-axis, it can be described as $E_{1}\hat{z}e^{ik_0r\cos\theta}$ with the signal wave denoted by $E_1$, the environmental wavenumber $k_0$, and the azimuthal angle $\theta$. By a combination of proper eigenstates $\nu_n(\vec{r})$, which rely on the scatterer structures, we have $E_{1}\hat{z}e^{ik_0r\cos\theta}=\hat{z}E_1\sum_{n}\phi_n\nu_n(\vec{r})$, with a complex coefficient $\phi_n$. The time dependence for each plane wave has the form $e^{-i\omega t}$. As an example for the cylindrical scatter, we adopt the Bessel and the first kind of Hankel functions obeying the Helmholtz equation for the eigenstates, i.e., $J_{n}(k_0r)e^{in\theta}$ and $H^{(1)}_{n}(k_0r)e^{in\theta}$, respectively. Then, the incident wave, denoted as [signal]{}, can be expressed as $E_{1}\hat{z}e^{ik_0r\cos\theta}=E_1\hat{z}\sum_{n=-\infty}^{\infty}i^nJ_{n}(k_0r)e^{in\theta}$, here the index, $n$, represents a series of partial waves [@book1]. The associated scattering wave generated by the scatterer has the form $\vec{E}_{sc}=E_{1}\hat{z}\sum_{n=-\infty}^{\infty}i^{n}a_{n}^{TE}H_{n}^{(1)}(k_0r)e^{in\theta}$, with the complex scattering coefficient $a_n^{TE}$. The factor $i^{n}$ indicates the excitation strength in each partial wave. For the excitation of a signal plane wave, the resulting excitation strength from this cylinder can be expressed by $\vert i^{n}\vert$, as shown in Fig. 1(b). However, as we are going to illustrate, our target is to demonstrate that the irradiation from a proper setting of signal and control waves, can turn off the initially dominant scattering events, as illustrated in Fig. 1(c), resulting in scatterers lose their functionality. To describe the total illumination and scattering waves, one has $\vec{E}_{in}=\hat{z}\sum_{n=-\infty}^{\infty}i^{n}J_{n}(k_0r)e^{in\theta}E_n^{inf}$ and $\vec{E}_{sc}=\hat{z}\sum_{n=-\infty}^{\infty}i^nE_n^{inf}e^{in\theta}H_n^{(1)}(k_0r)a_n^{TE}$, with the introduction of an interfering factor $E_n^{inf}$, which has the form $$\begin{aligned} E_n^{inf}=E_1+\sum_{m=2}^{m=s}e^{-in\Phi_m}E_m.\end{aligned}$$ Here, the first term in the right-handed side of Eq. (1) corresponds to the signal wave; while the others represent $s-1$ ($s\geq 2$) control waves whose complex wave amplitudes and incident angle are defined as $E_m$ and $\Phi_m$, respectively. The corresponding scattering and absorption powers are $P_{sc}=2/k_0\times\sqrt{\epsilon_0/\mu_0}\sum_{n=-\infty}^{\infty}\vert E_n^{inf}\vert^2\vert a_n^{TE}\vert^2$ and $P_{abs}=-2/k_0\times\sqrt{\epsilon_0/\mu_0}\sum_{n=-\infty}^{\infty}\vert E_n^{inf}\vert^2[Re(a_n^{TE})+\vert a_n^{TE}\vert^2]$, with the environmental permittivity and permeability denoted as $\epsilon_0$ and $\mu_0$, respectively. Now, suppose that our scattering system has $2N+1$ dominant partial waves (scattering channels). The only way to eliminate the scattering of these dominant partial waves is to produce the destructive interference of these target channels, i.e., $E_n^{inf}=0$ from $n=[-N, N]$. However, it can be proved straightforwardly that a total excitation of $2N+1$ irradiation waves (including the signal wave), only leads to a trivial zero solution [@linear1]. To obtain a non-trivial solution, one possibility is to expand the amount of control waves to $2N+1$ in total at least. Then, we have the following $2N+1$ equations to be satisfied: $$\begin{aligned} E_1+\sum_{m=2}^{2N+2}e^{-in\Phi_m}E_m = 0,\quad \text{for} \quad n=[-2N,2N].\end{aligned}$$ Here, in each equation there are three degrees of freedom for the extrinsic control parameters: intensity and phase of a control wave $E_i$, and the corresponding incident angle $\Phi_i$. In general, one should have a variety of solutions to satisfy the necessary condition in Eq. (2). ![image](newfig2.png){width="16.0cm"} To demonstrate our control on the scattering events, a silicon-embedded system is considered, such as silicon embedded with a high refractive index $\epsilon_1=12$ [@handbook]. Here, we tackle the first five dominant scattering channels, as shown in Fig. 2(a). These five scattering events correspond to the electric dipole ($n=0$), magnetic dipole ($n \pm 1$), and magnetic quadrupole ($n \pm 2$). The corresponding far-field scattering distribution and the intensity of the electric field are illustrated in Figs. 2(b) and 2(c) for a single wave excitation (signal only). Details on how to obtain the far-field scattering distribution are provided in Supplementary Materials. Now, in order to suppress these five dominant scattering partial waves, we construct an illumination system with another five control waves, denoted as $(E_2, E_3, E_4, E_5, E_6)$, with the corresponding incident angles $(\Phi_2,\Phi_3,\Phi_4,\Phi_5,\Phi_6)$. Then, we rewrite Eq. (1) into the following matrix presentation: $$\begin{split} \begin{bmatrix} -1\\ -1\\ -1\\ -1\\ -1\\ \end{bmatrix}= \begin{bmatrix} e^{-2i\Phi_2} & e^{-2i\Phi_3} &e^{-2i\Phi_4} &e^{-2i\Phi_5} & e^{-2i\Phi_6} \\ e^{-i\Phi_2} & e^{-i\Phi_3} &e^{-i\Phi_4} &e^{-i\Phi_5} & e^{-i\Phi_6} \\ 1 & 1 & 1 & 1 &1\\ e^{i\Phi_2} & e^{i\Phi_3} &e^{i\Phi_4} &e^{i\Phi_5} & e^{i\Phi_6} \\ e^{2i\Phi_2} & e^{2i\Phi_3} &e^{2i\Phi_4} &e^{2i\Phi_5} & e^{2i\Phi_6} \\ \end{bmatrix} \begin{bmatrix} E_2\\ E_3\\ E_4\\ E_5\\ E_6\\ \end{bmatrix}. \end{split}$$ Here, without loss of generality, we set $E_1=1$. As indicated in Fig. 2(d), we also chose the illumination angles as $[\Phi_2=\pi/9$, $\Phi_3=\pi/3$, $\Phi_4=\pi/2$, $\Phi_5=2\pi/3$, $\Phi_6=5\pi/6]$, based on which one can obtain the corresponding control wave amplitudes by solving Eq. (3), resulting in $[E_2=-2.17$, $E_3=3.28$, $E_4=-3.78$, $E_5=2.31$, $E_6=-0.64]$. In principle, one can set the incident angles arbitrarily and find out the corresponding complex amplitudes by solving Eq. (3). With these obtained results, we analyze the interfering factors for each excited scattering events, as shown in Fig. 2(e). As one can see, a complete destructive interference condition happens for the target channels $n=[-2,-1,0,1,2]$, with all the zero values. Meanwhile, non-zero interfering factors emerge on non-target scattering channels, i.e., $n= \pm 3, \pm 4$, and $\pm 5$. This result indicates that when the destructive interferometry applies to the dominant scattering channels, one can completely suppress the scattering events at the price that the originally non-dominant scattering channels are amplified. With the comparison between single and multiple excitations, Fig. 2(c) and 2(f), the intensity of electric fields, as well as the energy Poynting vectors, are totally different. With the same set of illumination configuration given in Fig. 2(d), we reveal another extreme scenario only with $n = \pm 3$rd channels supported, as shown in Fig. 2(g). Now, one can easily see that the corresponding far-field scattering pattern shown in Fig. 2(h) is significantly suppressed, i.e., at least three orders of magnitude smaller. The resulting electric field, as well as the time-averaged Poynting vectors, shown in Fig. 2(i) clearly demonstrate that the energy bypasses scatterer in the central region. Moreover, a finite and static region emerge within $x=[-0.5\lambda,0.5\lambda]$ and $y=[-0.5\lambda,0.5\lambda]$, inside which nearly all the intensity and energy Poynting vectors vanish. Here, $\lambda$ is the wavelength of illumination waves. With the comparison between Figs. 2(f) and 2(i), it is almost indistinguishable both for the field distribution and Poynting vectors, supporting the realization of invisibility. At a quick glance, as the existence of a finite and static region induced by the multiple wave excitation, one may contribute it for the reason to make the scatterer lose its functionality, as the physical size of our scatterers is smaller than the size of this zero-field region. As shown in Fig. 2(e), even though the interfering factors are completely suppressed for $n=-2, -1, 0, 1, 2$, other partial waves still survive from the wave-obstacle interaction. To highlight the size effect, we choose a bigger scatterer by changing the radius of our cylinder from $a=0.18\lambda$ to $a = 0.4\lambda$. With the same setting in Fig. 2(d), now, the resulting scattering events are enhanced for the $n= \pm 3$rd scattering channels, as shown in Fig. 2 (j). Nevertheless, with different far-field scattering pattern and field intensity (also the Poynting vector), as shown in Figs. 2(k) and 2(l), respectively, the invisibility is broken. ![ The scattering (a, c) and absorption power (b, d) spectra under the excitation of a single plane wave (signal) and multiple coherent waves (single $+$ control), depicted in Blue- and Red-colors, respectively. Here, a smaller radius of cylinder, $a=60$nm is considered in (a-b); while a larger radius, $a=170$nm is considered in (c-d).](newfig3.png){width="8.5cm"} Instead of using a structured wave to select the multipolar modes [@linear1; @linear2; @linear3; @linear4], our approach with multiple wave excitation is entirely different. Furthermore, our methodology can support broadband control through the interferometric coherent waves. If we keep all the system parameters fixed, including the illumination angles, intensities and phases of control waves, but only tune the incident wavelength. For a smaller size of the scatterer, in Fig. 3 (a-b), we set the silicon cylinder with the radius $a=60$nm and scan the incident wavelength from $500$ to $800$nm. Interestingly, compared to the plane wave excitation, depicted in Blue-color, by multiple wave excitation, both the scattering and absorption power spectra give us the zero values in this wavelength range, as depicted in Red-color. Even though it is known that for any invisible cloak illuminated by a single plane wave, Kramers-Kronig relation and sum-rule limit prevent the realization of a broadband operation. Our results demonstrate the scenario to go beyond the sum-rule limit, indicating the scatterer system working at this wavelength window with lowest-orders in the partial scattering waves. ![With the same illumination configuration in Fig. 2(d), when one places the scatterers in different shapes: (a) square, (b) triangle, (c) pentagon, and (d) hexagon, the resulting electric field distribution and Poynting vectors all remain unchanged.](appendixc.png){width="8.4cm"} For a larger size silicon cylinder, in Fig. 3(c-d), we set $a=170$nm as an example. It is known that as the size of the scatterer increases, the scattering power by the signal plane wave increases; while the corresponding absorption power decreases. Even though such a larger size system can support higher-order scattering channels, the target scattering events can remain suppressed with a multiple wave excitation. As a guideline, one solution to further suppress these higher-order scattering channels is to introduce more control waves. As for the influences on the invisibility from the mismatches in intensities and illumination angles of incident beams, as well as the scatterer displacement, one can apply Graft’s addition theorem [@addition] to have a systematic study. In Supplementary Materials, a detailed analysis is presented. Our finding reveals that the interferometric method is robust to allow these mismatching, offering flexibility toward the experimental implementation. For different kinds of shape, we also studied scatterers in the shape of a square, triangle, hexagon, and a pentagon, by Comsol[@comsol], as shown in Fig. 4. All of our outcomes can support invisibility. Even though the analysis in this work is demonstrated for the two-dimensional system, but it is readily applied to a three-dimensional scatterer or clusters. Last but not least, we note that the invisibility or enhanced scattering of the wave does not depend on individual parameters, e.g., size, structures, or materials. In summary, we have demonstrated a novel way by extrinsically imposing interferometric multiple waves to manage the excitation of partial waves. Compared to the single plane wave illumination, we reveal the possibility to support invisibility and to enhance target scattering partial waves, irrespective of internal system configuration. Unlike the known wave-obstacle interaction, which strongly relies on the material dispersion, such a multiple wave illumination provides a non-invasion way to avoid this physical constraint. It is the interferometric coherent waves, to support the existence of stationary scattering response for a broadband wavelength, beyond the sum-rule limit. 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Start with $$P_{sc}(\theta)=\frac{1}{2}Re[\vec{E}_{sc}\times\vec{H}_{sc}^{*}]\cdot\hat{r}r,$$ and apply the asymptotic analysis for the first kind of Hankel function, then, we can have $$\begin{split} \vec{E}_{sc}\sim\sqrt{\frac{2}{\pi k_{0}r}}e^{i(k_{0}r-\frac{\pi}{4})}\hat{z}\sum_{n=-\infty}^{\infty}i^{n}e^{in\theta}E_n^{inf}a_n^{TE}e^{-i\frac{n\pi}{2}},\\ (\vec{H}_{sc})_{\theta}\hat{\theta}\sim-\frac{k_{0}}{\omega\mu_0}\sqrt{\frac{2}{\pi k_{0}r}}e^{i(k_{0}r-\frac{\pi}{4})}\hat{\theta}\sum_{n=-\infty}^{\infty}i^{n}e^{in\theta}E_n^{inf}a_n^{TE}e^{-i\frac{n\pi}{2}}. \end{split}$$ Here, for magnetic fields, we only consider the $\hat{\theta}$ component, because only this term makes a contribution to radiation. Then, one can have $$P_{sc}(\theta)=\frac{1}{\pi k_{0}}\sqrt{\frac{\epsilon_0}{\mu_0}}\vert \sum_{n=-\infty}^{\infty}e^{in\theta}E_n^{inf}a_n^{TE}\vert^2.$$ ![Two coordinates denoted as $O$ and $O_{1}$, with a relative position vector, $\vec{b}$.](multiplefig1.png){width="35.00000%"} Appendix B: Graft’s addition theorem ==================================== For fields expressed in different coordinates, we can apply Graft’s addition theorem [@addition]. As shown in Fig. 5 for two coordinates, the addition theorem indicates that waves expressed in the coordinate $O$ can be transformed into the coordinate $O_1$. For our multiple wave irradiation at a different coordinate $(r_1,\theta_1)$, we have $$\begin{split} &\vec{E}_{in}(r_1,\theta_1)\\ &=\hat{z}\sum_{m=-\infty}^{\infty} i^mJ_m(k_0r)e^{im\theta}E_m^{inf},\\ &=\hat{z}\sum_{m=-\infty}^{\infty}i^m E_m^{inf}\sum_{n=-\infty}^{\infty} J_{m-n}(kb)e^{i(m-n)\beta}J_n(kr_1)e^{in\theta_1},\\ &=\hat{z}\sum_{n=-\infty}^{\infty}J_n(kr_1)e^{in\theta_1}\sum_{m=-\infty}^{\infty}i^{m}E_m^{inf}J_{m-n}(kb)e^{i(m-n)\beta},\\ &=\hat{z}\sum_{n=-\infty}^{\infty}J_n(kr_1)e^{in\theta_1}E_{n,g}^{inf}, \end{split}$$ where $(b,\beta)$ denotes the orientation of the new coordinate with respect to origin one, with $\vert\vec{b}\vert=b$ and $\beta$ be the angle with respect to x-axis. Here, we can define a general interfering factor as $$E_{n,g}^{inf}=\sum_{m=-\infty}^{\infty}i^mE_m^{inf}J_{m-n}(kb)e^{i(m-n)\beta},$$ and the corresponding multi-beam irradiation as, $$E_m^{inf}=\sum_{d=1}^{d=N}e^{-im\Phi_d}E_{d},$$ supposing there are $N$ illumination waves. Now, our interest is to find the corresponding scattering pattern when the object is placed at origin of the coordinate $O_1$. That is $$\begin{split} \vec{E}_{in}(r_1,\theta_1)&=\hat{z}\sum_nJ_n(kr_1)e^{in\theta_1}E_{n,g}^{inf},\\ \vec{E}_{sc}(r_1,\theta_1)&=\hat{z}\sum_nH^{(1)}_n(kr_1)e^{in\theta_1}E_{n,g}^{inf}a_n^{TE},\\ [\vec{H}_{sc}(r_1,\theta_1)]_{\theta_1}\hat{\theta_1}&\rightarrow\hat{\theta_1}\frac{1}{-i\omega\mu_0}\sum_n kH^{(1)'}_n(kr_1)e^{in\theta_1}E_{n,g}^{inf}a_n^{TE}.\\ \end{split}$$ At the far-field region, by applying the asymptotic analysis to the special functions, we have $$\begin{split} \vec{E}_{sc}(r_1,\theta_1)&\sim\hat{z}\sum_n\sqrt{\frac{2}{\pi k_0r_1}}e^{i(k_0r_1-\frac{\pi n}{2}-\frac{\pi}{4})}e^{in\theta_1}E_{n,g}^{inf}a_n^{TE},\\ \vec{H}_{sc}(r_1,\theta_1)&\sim-\frac{1}{\omega\mu_0}\hat{\theta_1}\sum_nk_0\sqrt{\frac{2}{\pi k_0r_1}}e^{i(k_0r_1-\frac{\pi n}{2}-\frac{\pi}{4})}e^{in\theta_1}E_{n,g}^{inf}a_n^{TE}. \end{split}$$ So the scattering power distribution when placed the antenna at this new location $O_1$ becomes $$\begin{split} P_{sc}(\theta_1)=\frac{1}{\pi k_0}\sqrt{\frac{\epsilon_0}{\mu_0}}\vert \sum_n e^{in\theta_1}e^{-i\frac{n\pi}{2}}E_{n,g}^{inf}a_n^{TE}\vert^2. \end{split}$$ ![Contour plots for the normalized scattering power, defined as $P_{sc}\frac{k_0}{2}\sqrt{\frac{\mu_0}{\epsilon_0}}$. Here, we set a derivation in the amplitude $E_i$ and incident angle $\Phi_i$ from the illumination beams, for the $i$th control wave, $i = 3, 4, 5, 6$ in (a-d), respectively. Each contour plot reveals a wide region to support invisible cloak with toleration. ](fig4.png){width="8.5cm"} Appendix C: Robustness of interferometric coherent waves ======================================================== ![image](fig5.png){width="14.5cm"} In Fig. 6, we analyze the normalized scattering powers, defined as $P_{sc}\sqrt{\mu_0/\epsilon_0}k_0/2$, with a deviation in the amplitude and incident angles of illumination waves. We find that for the control waves $E_3$, $E_4$, and $E_5$, as shown in Figs. 6(a)-6(c), the supporting invisible cloak is almost insensitive to the variations in amplitudes. Instead, the deviation in phase gives a larger change. For the control wave $E_6$, the opposite situation is observed from Fig. 6(d). Nevertheless, in terms of the variation on illumination wave, both for amplitude and angles, the advantageous of our interferometric coherent wave certainly can endure the mismatching. In addition to the derivation in the illumination waves, we also investigate the offset in displacement. Here, three different locations are arbitrarily studied for the scatterer, see the marked points in Fig. 7(a). The corresponding generalized interfering factors and far-field scattering pattern are depicted in (b) for the location at $(b=0.075\lambda,\beta=\pi/4)$, (c) $(b=0.225\lambda,\beta=3\pi/4)$, and (d) $(b=0.35\lambda,\beta=3\pi/2)$, respectively. One can see clearly that even with dislocations, the target scattering events at $n = 0, \pm 1, \pm 2$ are all still suppressed, which demonstrate the robustness of invisibility.
--- abstract: 'Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman-Von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations. A possible interpretation is that classicality requires the notorious hierarchy problem.' author: - 'Olga I. Chashchina' - Abhijit Sen - 'Zurab K. Silagadze' title: 'Planck-scale modification of classical mechanics' --- Introduction ============ Quantum mechanics is one of the most successful theories in science. At present no single experimental fact indicates its breakdown. On the contrary, we have every reason to believe that quantum mechanics encompasses every natural phenomena. Nevertheless some spell of mystery still accompanies quantum mechanics. Richard Feynman, worlds one of the best experts in quantum mechanics, expressed this feeling most eloquently [@1-1]: “We always have had (secret, secret, close the doors!) we always have had a great deal of difficulty in understanding the world view that quantum mechanics represents. At least I do, because I’m an old enough man that I haven’t got to the point that this stuff is obvious to me. Okay, I still get nervous with it. And therefore, some of the younger students … you know how it always is, every new idea, it takes a generation or two until it becomes obvious that there’s no real problem. It has not yet become obvious to me that there’s no real problem. I cannot define the real problem, therefore I suspect there’s no real problem, but I’m note sure there’s no real problem.” Superposition principle, inherent of quantum mechanics in which states of quantum systems evolve according to linear Schrödinger equation, maybe is the core reason of our uneasiness with quantum mechanics. If classical mechanics is considered as a limit of quantum mechanics then the superposition principle must hold in classical mechanics too [@1-2]. However in the classical world, as it is revealed to us by our perceptions, we never experience Schrödinger cat states (except perhaps in art, see [@1-3]) and a widespread belief is that the environment induced decoherence explains why ( see [@1-4; @1-5] and references therein. For a contrary view, however see [@1-6; @1-7]). A particularly striking example of decoherence is chaotic rotational motion of Saturn’s potato-shaped moon Hyperion. The orbit of Hyperion around Saturn is fairly predictable, but the moon tumbles unpredictably as it orbits because its rotational motion is chaotic. It was argued [@1-8; @1-9] that if Hyperion were isolated from the rest of the universe, it would evolve into a macroscopic Schrödinger cat state of undefined orientation in a time period of about 20-30 years. However this never happens. Hyperion is not isolated but constantly bombarded by photons from the rest of the universe causing its quantum state to collapse into a state of definite orientation. Gravitational interaction cannot be shielded. Therefore any object in the universe is constantly bombarded by gravitons destroying macroscopic Schrödinger cat states. It is expected, therefore, that gravity plays a prominent role in the emergence of classicality [@1-10; @1-11]. However it can be argued that the Planck scale should be viewed as a fundamental boundary of validity of the classical concept of spacetime, beyond which quantum effects cannot be neglected [@1-12]. A legitimate question then is how this expected modifications of quantum mechanics and/or gravity at the Planck scale influence the emergence of classicality. In this paper we attempt to discuss some aspects of this question in the framework of Koopman-von Neumann theory. The manuscript is organized as follows. In the second section a brief overview of the generalized uncertainty principle is given. The third section provides fundamentals of the Koopman-von Neumann formulation of classical mechanics. In the fourth section we describe modifications of classical mechanics expected then combining Koopman-von Neumann-Sudarshan perspective on classical mechanics with the generalized uncertainty principle. In the last section some concluding remarks are given. Generalized Uncertainty Principle ================================= Heisenberg’s uncertainty relation, implying the non-commutativity of the quantum mechanical observables, underlines the essential difference between classical and quantum mechanics [@2-1]. Analyzing his now famous thought experiment of measuring the position of an electron using a gamma-ray microscope, Heisenberg arrived at the conclusion that “the more precisely is the position determined, the less precisely is the momentum known, and vice versa” [@2-2]: $$\delta q\,\delta p\sim\hbar. \label{eq2-1}$$ Here $\delta q$ is the uncertainty in the determination of the position of the electron $q$, and $\delta p$ is the perturbation in its momentum $p$, canonically conjugate to $q$, induced by the measurement process. The precise meaning of “uncertainty” was not defined in Heisenberg’s paper who used heuristic arguments and some plausible measures of inaccuracies in the measurement of a physical quantity and quantified them only on a case-to-case basis as “something like the mean error” [@2-3]. After publication of [@2-2], “which gives an incisive analysis of the physics of the uncertainty principle but contains little mathematical precision” [@2-4], attempts to overcome its mathematical deficiencies were soon undertaken by Kennard [@2-5] and Weyl [@2-6]. They proved the inequality, valid for any quantum state, $$(\Delta q)^2\,(\Delta p)^2\ge\frac{\hbar^2}{4}, \label{eq2-2}$$ where $(\Delta q)^2$ and $(\Delta p)^2$ are the variances (the second moment about the mean value) of $\hat q$ and $\hat p$ defined as $(\Delta q)^2= <\nolinebreak \hat q{\,^2}>-<\hat q>^2$ and similarly for $\hat p$. As usual, the mean value of a quantum-mechanical operator $\hat A$, in the quantum state $\|\Psi>$, is defined as follows (we are considering a one-dimensional case, for simplicity) $$<\Psi\|\hat A\|\Psi\|>=\int dq\,\Psi^(q)(\hat A\Psi)(q). \label{eq2-3}$$ Taking the standard deviation $\Delta A$ (the square-root from the variance $(\Delta A)^2= <\hat A^{\,2}>-<\hat A>^2$) as a measure of indeterminacy (uncertainty) of the observable $\hat A$ in the quantum state $\|\Psi>$ seems very natural from the point of view of the classical probability theory where the standard deviation is considered as a measure of fluctuations. Indeed, soon Ditchburn established the relation $\Delta q=\delta q/\sqrt{2}$ between Heisenberg’s $\delta q$ and Weyl-Kennard’s $\Delta q$ and proved that the equality in (\[eq2-2\]) can be achieved for Gaussian probability distributions only [@2-7; @2-8]. From the mathematical point of view, we can put $<q>=0$ and $<p>=0$ in (\[eq2-2\]) without loss of generality [@2-6]. Indeed, we can achieve $<q>=0$ by suitable redefinition of the $q$-coordinate origin, and $<p>=0$ — by multiplying the wave function by $\exp{(-i<p>\,q/\hbar)}$ without changing the probability density associated with it. In this case (\[eq2-2\]) becomes a mathematical statement about (normalized) square-integrable functions $\Psi(q)$: $$-\left(\int\limits_{-\infty}^\infty q^2\,\|\Psi(q)\|^2\,dq\right) \left(\int\limits_{-\infty}^\infty \Psi^(q)\,\frac{d^2\Psi(q)}{dq^2}\,dq \right)=\left(\int\limits_{-\infty}^\infty q^2\,\|\Psi(q)\|^2\,dq\right) \left(\int\limits_{-\infty}^\infty \left\|\frac{d\Psi}{dq}\right \|^2\,dq \right)\ge\frac{1}{4}. \label{eq2-4}$$ The three-dimensional sibling of (\[eq2-4\]), known as the Heisenberg inequality, is [@2-9] $$\left(\int \mathbf{r}^2\,\|\Psi(\mathbf{r})\|^2\,d^3 r\right) \left(\int\left\|\boldsymbol{\nabla}\Psi(\mathbf{r})\right \|^2\,d^3 r\ \right)\ge\frac{9}{4}. \label{eq2-5}$$ If an electron in the hydrogen atom is localized around the origin, then (\[eq2-5\]) tells us that its momentum (and hence kinetic energy) will be large. It is tempting to use this fact to get a lower bound on the electron’s energy in the hydrogen atom and and thus prove its stability. Although it is a common practise to use such kind of reasoning for estimation of the hydrogen atom size and its ground state energy [@2-10; @2-11; @2-12; @2-13], the truth is that Heisenberg inequality is too weak to ensure the hydrogen atom stability or stability of matter in general [@2-14; @2-15]. The reason is that the first multiple in (\[eq2-5\]) can be very large even in the case when the main part of the wave function (its modulus squared) is localized around the origin, if only the remaining small part is localized very far away. However mathematically the uncertainty principle in the form of Eq.(\[eq2-5\]) is just an expression of the fact from harmonic analysis that “A nonzero function and its Fourier transform cannot both be sharply localized” [@2-4]. Heisenberg inequality (\[eq2-5\]) is just one attempt to make the above given sloppy phrase mathematically precise. But, as we have seen above for hydrogen atom, from the physics perspective the standard deviation is not always an adequate measure of localization and quantum uncertainty [@2-16; @2-17]. In such situations (in particular then considering the matter stability problem [@2-14; @2-15]) other uncertainty principles prove to be more useful. The examples include Hardy uncertainty principle [@2-15; @2-18] $$\left(\int \frac{1}{\mathbf{r}^2}\,\|\Psi(\mathbf{r})\|^2\,d^3 r\right)^{-1} \left(\int\left\|\boldsymbol{\nabla}\Psi(\mathbf{r})\right \|^2\,d^3 r\ \right)\ge\frac{1}{4}, \label{eq2-6}$$ or Sobolev inequality [@2-14] $$\left(\int \|\Psi(\mathbf{r})\|^6\,d^3 r\right)^{-1/3} \left(\int\left\|\boldsymbol{\nabla}\Psi(\mathbf{r})\right \|^2\,d^3 r\ \right)\ge\frac{3}{4}\,(4\pi^2)^{2/3}, \label{eq2-7}$$ which, in some sense, is weaker than the Hardy inequality (\[eq2-6\]) [@2-15]. From the physics side, the uncertainty principle is more than just inequalities from harmonic analysis. We can envisage at least three manifestations of uncertainty relations [@2-3; @2-19]. First of all the uncertainty relations relate intrinsic spreads of two conjugate dynamical variables in a quantum state. However Heisenberg in his seminal work speaks about unavoidable disturbance that a measurement process exerts on a pair of conjugate dynamical variables. Therefore we can understand the uncertainty relation also as an assertion about a relation between inaccuracies in measurements of conjugate dynamical variables. Namely, the relation that connects either inaccuracy of a measurement of one dynamical variable and the ensuing disturbance in the probability distribution of the conjugated variable, or inaccuracies of a pair of conjugate dynamical variables in any joint measurements of these quantities.Although conceptually distinct, these three manifestations of uncertainty relations are closely related [@2-19]. The first facet of the uncertainty principle can be formalized most easily by using second-order central moments of two conjugated quantum observables (but remember a caveat that the standard deviation is not always an adequate measure of quantum uncertainty [@2-17]). For any pair of quantum observables $\hat A$ and $\hat B$ we have three independent second-order central moments of their joint quantum distributions in a quantum state $\|\Psi>$: $$\begin{aligned} && [\Delta(A)]^2=<\Psi\|(\hat A -\bar A)^2\|\Psi>={\mkern 1.5mu\overline{\mkern-1.5muA^2\mkern-1.5mu} \mkern 1.5mu}-\bar A^{\,2},\;\; [\Delta(B)]^2=<\Psi\|(\hat B -\bar B)^2\|\Psi>={\mkern 1.5mu\overline{\mkern-1.5muB^2\mkern-1.5mu} \mkern 1.5mu}-\bar B^{\,2}, \nonumber \\ && \Delta(A,B)=\frac{1}{2}<\Psi\|(\hat A -\bar A)(\hat B -\bar B)+ (\hat B -\bar B)(\hat A -\bar A)\|\Psi>=\frac{1}{2}{\mkern 1.5mu\overline{\mkern-1.5mu(AB+BA)\mkern-1.5mu} \mkern 1.5mu}- \bar A\,\bar B, \label{eq2-8}\end{aligned}$$ where overbar denotes the mean value of the corresponding observable in the state $\|\Psi>$. We have $$\hat A\hat B=\frac{1}{2}(\hat A\hat B+\hat B\hat A)+ \frac{1}{2}(\hat A\hat B-\hat B\hat A),$$ and the first Hermitian part in the r.h.s has a real mean value, while the mean value of the second anti-Hermitian part is purely imaginary. Then the Schwarz inequality $\|<\psi\|\phi>\|^2\le\; <\psi\|\psi>\,<\phi\|\phi>$, if we take $\|\psi>=\hat A\|\Psi>$, $\|\phi>=\hat B\|\Psi>$, will give $${\mkern 1.5mu\overline{\mkern-1.5muA^2\mkern-1.5mu} \mkern 1.5mu}\,{\mkern 1.5mu\overline{\mkern-1.5muB^2\mkern-1.5mu} \mkern 1.5mu}\ge \|{\mkern 1.5mu\overline{\mkern-1.5muAB\mkern-1.5mu} \mkern 1.5mu}\|^2=\left(\frac{{\mkern 1.5mu\overline{\mkern-1.5muAB+BA\mkern-1.5mu} \mkern 1.5mu}} {2}\right)^2+\left \|\frac{{\mkern 1.5mu\overline{\mkern-1.5muAB-BA\mkern-1.5mu} \mkern 1.5mu}}{2}\right\|^2,$$ because $<\hat A\Psi\|\hat A\Psi>={\mkern 1.5mu\overline{\mkern-1.5muA^2\mkern-1.5mu} \mkern 1.5mu}$, $<\hat B\Psi\|\hat B\Psi>= {\mkern 1.5mu\overline{\mkern-1.5muB^2\mkern-1.5mu} \mkern 1.5mu}$ and for Hermitian observables $\hat A$ and $\hat B$, $<\hat A\Psi\|\hat B\Psi>=<\Psi\|\hat A\hat B\|\Psi>={\mkern 1.5mu\overline{\mkern-1.5muAB\mkern-1.5mu} \mkern 1.5mu}$. If we replace $\hat A$ and $\hat B$ by operators $\hat A-\bar A$ and $\hat B-\bar B$ in the above given reasoning, we end up with the Schrödinger uncertainty relation [@2-20] $$[\Delta(A)]^2[\Delta(B)]^2\ge [\Delta(A,B)]^2+\left \|\frac{{\mkern 1.5mu\overline{\mkern-1.5muAB-BA\mkern-1.5mu} \mkern 1.5mu}}{2} \right\|^2. \label{eq2-9}$$ Heisenberg uncertainty relation (\[eq2-2\]) is obtained from this more general uncertainty relation if we take $\hat A=\hat x$, $\hat B=\hat p$, use the canonical commutation relations $$[\hat x_i,\,\hat p_j]\equiv \hat x_i,\,\hat p_j-\hat p_j\,x_i=i\,\hbar\, \delta_{ij}, \label{eq2-10}$$ and assume that the covariance $\Delta(x,p)$ equals to zero. Although Schrödinger uncertainty relation is more general and symmetric (it remains invariant under rotations in phase space) than the Heisenberg uncertainty relation, or its generalization due to Robertson to any pair of observables with zero covariance [@2-21] $$[\Delta(A)]^2[\Delta(B)]^2\ge \left \|\frac{{\mkern 1.5mu\overline{\mkern-1.5muAB-BA\mkern-1.5mu} \mkern 1.5mu}}{2} \right\|^2, \label{eq2-11}$$ the Schrödinger uncertainty relation is strangely ignored in almost all quantum mechanics textbooks and its usefulness was appreciated only after 50 years from its discovery in connection with the description of squeezed states in quantum optics for which the covariance $\Delta(x,p)$ doesn’t equal zero [@2-8]. As we see the uncertainty relations are intimately related to canonical commutation relations (\[eq2-10\]). To our best knowledge, Gleb Wataghin was the first [@2-22; @2-23] who suggested that both the commutation relations (\[eq2-10\]) and the uncertainty principle (\[eq2-1\]) might be modified at high relative impulses in such a way that $$\delta q\,\delta p\sim\hbar\,f(p),\;\;\;\lim_{p\to 0} f(p)=1, \label{eq2-12}$$ which can lead to the existence of a lower limit of a measurable lengths. Below we give a brief outline of the modern developments of these kind of ideas. If we don’t care about the momentum, the uncertainty relation (\[eq2-2\]) does not forbid us to prepare a quantum state with the arbitrary small position uncertainty. However, as argued in 1964 by Mead [@2-24], things change if we take into account effects of gravity. The crux of the Mead’s argument is that the gravitational interaction between the electron and photon in Heisenberg microscope is a source of additional uncertainty in the electron’s position. The gravitational field of a photon was obtained in [@2-25; @2-26] by boosting the Schwarzschild space-time up to the speed of light by taking the limit $V\to c$, $m\to 0$ such that the quantity $p=mV(1-V^2/c^2)^{-1/2}$ is held constant. The resulting space-time, for the photon with momentum $p$ moving in the $z$-direction, has the metric [@2-27] $$ds^2=-2\left (du\,dv-d\zeta\,d\bar\zeta\right )-4\,\frac{p\,G}{c^3}\ln{ \left (\frac{\zeta\,\bar\zeta}{r_0^2}\right )}\,\delta(u)\,du^2, \label{eq2-13}$$ where $G$ is Newton’s constant, $u=(ct-z)/\sqrt{2}$ and $v=(ct+z)/\sqrt{2}$ are retarded and advanced null coordinates (light-cone coordinates), while the complex coordinates $\zeta=(x+i\,y)/\sqrt{2}$ and $\bar\zeta=(x-i\,y)/ \sqrt{2}$ parametrize the spatial hyperplane orthogonal to the photon’s velocity vector. The parameter $r_0$ in (\[eq2-13\]) is an arbitrary constant of the dimension of length which does not effect observable quantities [@2-27A]. The metric (\[eq2-13\]) describes an impulsive gravitational wave: the space-time remains flat everywhere except $u=0$ null hyperplane, where it develops a delta-function singularity. This gravitational shockwave moves with the photon and when it meets with the electron within the Heisenberg microscope two physical effects take place: the timelike geodesic of the electron experiences a discontinuous jump in the null coordinate $v$ and gets refracted in the transverse direction [@2-29]. There are various subtleties here. The very concept of photon with sharply defined momentum (energy), existing at $t=-\infty$, is an idealization. In reality one should take into account that the photon is produced at a finite instant of time and the corresponding light packet has a finite Fourier support [@2-28]. Besides, because (\[eq2-13\]) describes a situation when a cause (photon) and the effect (the corresponding gravitational shockwave) propagate with the same speed of light, it is not altogether clear the gravitational field is related to the photon or it arises solely in the process of emission [@2-29]. At last, to cope with the presence of ill-defined highly singular products of generalized functions in the geodesic deviation equation, precise calculation of the above mentioned physical effects of the gravitational shockwave on the test particles geodesics, requires either a suitable regularization procedure [@2-30], or making use of the Colombeau algebra of generalized functions [@2-31]. Anyway, for our purposes we need only an order of magnitude estimate of the additional uncertainty in Heisenberg microscope due to gravity. This was done in [@2-32] with the result that the additional uncertainty in electron’s position due to gravitational attraction of the photon is $$\Delta x_G\approx \frac{Gp}{c^3}\approx l_P^2\,\frac{\Delta p}{\hbar}. \label{eq2-14}$$ The second step follows from the fact that the electron momentum uncertainty $\Delta p$ must be of order of the photon momentum $p$. Here $$l_P=\sqrt{\frac{G\hbar}{c^3}} \label{eq2-15}$$ is the Planck length. If we add this new uncertainty linearly to the original Heisenberg uncertainty $\Delta x_H\approx \hbar/\Delta p$, we get the modified uncertainty principle (the so called GUP — generalized or gravitational uncertainty principle [@2-32]) $$\frac{\Delta x}{l_P}\approx \frac{\hbar}{l_P\,\Delta p}+ \frac{l_P\,\Delta p}{\hbar}. \label{eq2-16}$$ In this form the uncertainty principle is invariant under momentum inversion $\frac{\hbar}{l_P\,\Delta p}\to \frac{l_P\,\Delta p}{\hbar}$. Another remarkable property of (\[eq2-16\]) is that it predicts a minimum position uncertainty $\Delta x_{min}=2\,l_P$ at a symmetric, with respect to the above mentioned momentum inversion, point $\Delta p=\hbar/l_P$. Although the idea that a smallest length exists in nature can be traced back to Heisenberg and March [@2-33], only relatively recent attempts to reconcile quantum mechanics with general relativity in string theory produced a solid foundation for it and for the generalized uncertainty principle (see, for example, [@2-34; @2-35] and review articles [@2-36; @2-37]). As Robertson’s version (\[eq2-11\]) of the uncertainty principle shows, the generalized uncertainty principle (\[eq2-16\]) may imply a deformation of the usual Heisenberg algebra of canonical commutation relations (\[eq2-10\]). Various versions of this deformation have been proposed in the literature (see, for example, [@2-38] and references therein). As was mentioned above, Wataghin was the first to suggest modification of the canonical commutation relations at high energies. However, it was Snyder who proposed a model of noncommutative spacetime, admitting a fundamental length but nevertheless being Lorentz invariant [@2-39], non-relativistic version of which produces a concrete form of such modification [@2-40]: $$[\hat x_i,\,\hat x_j]=i\hbar\beta^\prime \hat J_{ij},\;\; [\hat p_i,\,\hat p_j]=0,\;\;[\hat x_i,\,\hat p_j]=i\hbar (\delta_{ij}+\beta^\prime \hat p_i\hat p_j), \;\; \hat J_{ij}=\hat x_i\hat p_j-\hat x_j\hat p_i, \label{eq2-17}$$ where $\beta^\prime$ is some constant, usually assumed to be of the order of $l_P^2/\hbar^2$, as (\[eq2-11\]) and (\[eq2-16\]) relations do imply. Snyder’s work was ahead of its time and its importance was not immediately recognized. Meanwhile Mead [@2-41] and Karolyhazy [@2-42] investigated uncertainties in measurements of space-time structure resulting from universally coupled gravity and concluded that it is impossible to measure distances to a precision better than Planck’s length. However very few took seriously the idea that the Planck length could ever play a fundamental role in physics [@2-37; @2-43]. The situation changed when developments in string theory revealed the very same impossibility of resolving distances smaller than Planck’s length, and these developments inspired Adler and Santiago’s 1999 paper [@2-32] that almost exactly reproduced Mead’s earlier arguments [@2-37]. Various choices of deformed commutation relations have been considered in the literature beginning from the Kempf et al. landmark paper [@2-44]. Let us mention, for example, a version that generalizes the Snyder algebra (\[eq2-17\]) [@2-45; @2-46]: $$[\hat x_i,\,\hat x_j]=-i\hbar\left [(2\beta-\beta^\prime)+\beta(2\beta+ \beta^\prime)\hat p^2\right]\,\frac{\hat J_{ij}}{1+\beta\hat p^2},\;\;\; [\hat p_i,\,\hat p_j]=0,\; \;\;[\hat x_i,\,\hat p_j]=i\hbar \left [(1+\beta \hat p^2)\delta_{ij}+ \beta^\prime \hat p_i\hat p_j\right ], \label{eq2-18}$$ where $\beta$ is a new constant of the same magnitude as $\beta^\prime$, $\hat p^2= \hat p_i \hat p^i$, and $\hat J_{ij}$ was defined in (\[eq2-17\]). A different type of modification of the canonical commutation relations was suggested by Saavedra and Utreras [@2-48]: $$[\hat x_i,\,\hat p_j]=i\hbar \left (1+\frac{l_P}{c\hbar}\,H\right) \delta_{ij}. \label{eq2-19}$$ One can say that in this case the configuration space becomes dynamical, much like the general relativity, because the commutation relations (\[eq2-19\]) depends on the system under study through the Hamiltonian $H$. As we see, commutation relations (\[eq2-17\]) and (\[eq2-18\]) imply a non-commutative spatial geometry. Mathematically this is a consequence of Jacobi identity and our tacit assumption that components of momentum operators do commute. Physical bases of this non-commutativity is a dynamical nature of space-time in general relativity: it can be argued quite generally that an unavoidable change in the space-time metric when measurement processes involve energies of the order of the Planck scale destroys the commutativity of position operators [@2-47]. There exists a vast and partly confusing literature on the modifications of quantum mechanics and quantum field theory implied by the existence of a minimal length scale (for a review and references see, for example, [@2-36; @2-37; @2-49; @2-50; @2-51]). Koopman-von Neumann mechanics ============================= It is usually assumed that classical mechanics, in contrast to quantum mechanics, is a deterministic theory with the well defined trajectories of underlying particles. However, if we realize the imperfect nature of classical measuring devices, which precludes the preparation of classical systems with precisely known initial data, it becomes clear that “the determinism of classical physics turns out to be an illusion, created by overrating mathematico-logical concepts. It is an idol, not an ideal in scientific research” [@3-1]. Therefore, one can assume that a conceptually superior appropriate statistical description of classical mechanics is then given by Liouville equation (for simplicity, we consider a one-dimensional mechanical system with canonical variables $q$ and $p$) $$i\frac{\partial \rho}{ \partial t}=\hat L \rho=i\{H,\rho\}= i\left (\frac{\partial H}{ \partial q}\;\frac{\partial \rho}{ \partial p}- \frac{\partial H}{ \partial p}\;\frac{\partial \rho}{ \partial q}\right ), \label{eq3-1}$$ which gives a time-evolution of the phase-space probability density $\rho(q,p,t)$. Here $H$ is is the Hamiltonian and $\{,\}$ denotes the Poisson bracket. However, classical and quantum mechanics are different not only by inherently probabilistic nature of the latter. Mathematical structures underlying these two disciplines are quite different. The mathematics underlying classical mechanics is a symplectic geometry of the phase space [@3-2; @3-3; @3-4], while quantum mechanics is based on the theory of Hilbert spaces [@3-5], rigged Hilbert spaces [@3-6] or on their algebraic counterpart — the theory of $C^$ algebras [@3-7]. In light of this difference in the underlying mathematical structure it is surprising that it is possible to give a Hilbert space formulation for classical mechanics too, as shown long ago in classic papers by Koopman [@3-8] and von Neumann [@3-9] (for modern presentation, see [@3-10] and references therein). [^1] This translation of classical mechanics into the language of Hilbert spaces is based on the crucial observation that, because the Liouville operator $$\hat L=i\left (\frac{\partial H}{ \partial q}\;\frac{\partial}{ \partial p}- \frac{\partial H}{ \partial p}\;\frac{\partial }{ \partial q}\right ) \label{eq3-2}$$ is linear in derivatives, the square root of the probability density $\psi(q,p,t)=\sqrt{\rho(q,p,t)}$ obeys the same Liouville equation (\[eq3-1\]): $$i\frac{\partial \psi(q,p,t)}{\partial t}=\hat L \psi(q,p,t). \label{eq3-3}$$ Moreover, if we assume that $\psi(q,p,t)$ in (\[eq3-3\]) is a complex function $\psi(q,p,t)=\sqrt{\rho(q,p,t)}\,e^{iS(q,p,t)}$, then (\[eq3-3\]) implies that the amplitude and phase evolve independently through the Liouville equations: $$i\frac{\partial \sqrt{\rho}}{\partial t}=\hat L \sqrt{\rho},\;\;\; i\frac{\partial S}{\partial t}=\hat L S, \label{eq3-4}$$ and the probability density $\rho(q,p,t)=\psi^(q,p,t)\psi(q,p,t)$ also obeys the Liouville equation (\[eq3-1\]). Therefore we can introduce a Hilbert space of square integrable complex functions $\psi(q,p,t)$, equip it with the the inner product $$<\psi\|\phi>=\int dqdp\, \psi^(p,q,t)\phi(p,q,t), \label{eq3-5}$$ and then we recover the rules that are usually associated with quantum mechanics. Namely, observables are represented by Hermitian operators and the expectation value of an observable $\hat \Lambda$ is given by $$\bar \Lambda(t)=\int dqdp\,\psi^(q,p,t) \hat \Lambda \psi(q,p,t). \label{eq3-6}$$ If $\varphi_\lambda(q,p,t)$ is an eigenstate of the observable $\hat \Lambda$, $\hat \Lambda \varphi_\lambda(q,p,t)=\lambda\,\varphi_\lambda(q,p,t)$, then the probability $P(\lambda)$ that the outcome of a measurement of $\hat \Lambda$ on a classical mechanical system with the KvN wave function $\phi(p,q,t)$ results in the eigenvalue $\lambda$ is given by the usual Born rule $$P(\lambda)=\int dqdp\,\|\varphi^_\lambda(q,p,t)\psi(q,p,t)\|^2. \label{eq3-7}$$ There are two main differences from quantum mechanics. Firstly, and most importantly in the classical theory the operators for position and momentum do commute $$[\hat q,\,\hat p]=0. \label{eq3-8}$$ In the Hilbert space formalism outlined above, these operators are realized as multiplicative operators $$\hat q \,\psi(q,p,t)=q\,\psi(q,p,t),\;\;\; \hat p\, \psi(q,p,t)=p\,\psi(q,p,t). \label{eq3-9}$$ The second important difference is that the “Hamiltonian” (Liouville operator) (\[eq3-2\]) that defines the time evolution of the KvN wave function is linear in spatial derivatives. This is quite unusual in quantum mechanics and such type of dynamical evolution was attributed to quantum systems that allow a genuine quantum chaos to emerge [@3-11]. Thanks to the imaginary unit $i$ (and that’s the reason why it was introduced), the Liouville operator $\hat L$ is Hermitian, and thus generates a unitary evolution, with respect to the inner product (\[eq3-5\]): $$\int dpdq\,\varphi^(p,q,t) \hat L \psi(p,q,t)= \int dpdq\,(\hat L \varphi)^(p,q,t)\psi(p,q,t). \label{eq3-10}$$ This can be proved through an integration by parts under reasonable assumptions about the Hamiltonian, namely the equality of mixed derivatives $\frac{\partial^2 H}{\partial q \partial p}=\frac{\partial^2 H}{\partial p \partial q}$. At that we assume that the wave functions $\varphi(q,p,t)$ and $\psi(q,p,t)$, being square integrable, vanish sufficiently fast at $q,p\to \pm\infty$. There are some mathematical subtleties here, however. Strictly speaking, not every square integrable function vanishes at infinity. The example is [@3-12] $f(x)=x^2\exp{(-x^8\sin^2{x})}$ which is square integrable but even not bounded at infinity. According to the Hellinger-Toeplitz theorem [@3-13], everywhere defined Hermitian operator is necessarily bounded. Position and momentum operators are clearly unbounded. So is the Liouville operator. Therefore, the rigorous mathematical formulation of classical mechanics in the Hilbert space KvN formalism is not as a simple task as naively can appear. However, these mathematical subtleties and difficulties are not characteristic of only KvN mechanics and is already present in ordinary quantum mechanics [@3-12]. The formalism of rigged Hilbert spaces [@3-6] can provide a possible, although a rather sophisticated solution. What the Hilbert space KvN formalism corresponds to the usual classical mechanics is most easily seen in the Heisenberg picture of time evolution. In the Schrödinger picture of evolution assumed above the operators are time-independent while the wave function evolves unitarily according to $$\psi(q,p,t)=e^{-i\hat L t}\psi(q,p,0). \label{eq3-11}$$ On the contrary, in the Heisenberg picture wave functions are assumed to be time-independent and all time dependencies of mean values of physical quantities are incorporated in the time evolution of operators according to $$\hat \Lambda(t)=e^{i\hat L t}\hat \Lambda(0)e^{-i\hat L t}. \label{eq3-12}$$ Equation of motion that follows from (\[eq3-12\]) is $$\frac{d\hat \Lambda(t)}{dt}=i[\hat L,\hat \Lambda(t)]. \label{eq3-13}$$ Namely, for multiplicative position and momentum operators we get $$\frac{d q}{dt}=i[\hat L,q]=\frac{\partial H(q,p,t)}{\partial p},\;\;\; \frac{d p}{dt}=i[\hat L,p]=-\frac{\partial H(q,p,t)}{\partial q}, \label{eq3-14}$$ which are nothing but the Hamilton’s equations. Alternatively, to show that KvN formalism corresponds to the usual Newtonian mechanics, we can apply method of characteristics in the Schröedinger picture [@3-13A]. Let us consider a curve $(q(\alpha),p(\alpha), t(\alpha)$ in the extended phase space, parametrized by a real parameter $\alpha$. Along this curve $$\frac{d\psi}{d\alpha}=\frac{\partial\psi}{\partial t}\frac{dt}{d\alpha}+ \frac{\partial\psi}{\partial q}\frac{dq}{d\alpha}+ \frac{\partial\psi}{\partial p}\frac{dp}{d\alpha}, \label{eq3-add1}$$ and if the curve is chosen in such a way that $$\frac{dt}{d\alpha}=1,\;\;\;\frac{dq}{d\alpha}=\frac{\partial H}{\partial p}, \;\;\; \frac{dp}{d\alpha}=-\frac{\partial H}{\partial q}, \label{eq3-add2}$$ we will get $$\frac{d\psi}{d\alpha}=-i\left (i\frac{\partial\psi}{\partial t}-\hat{L}\psi \right )=0, \label{eq3-add3}$$ according to the Liouville equation (\[eq3-3\]). As we see from (\[eq3-add2\]), the parameter $\alpha$ essentially coincides with time and the characteristics of the Liouville equation (\[eq3-3\]) are just classical Newtonian trajectories in the extended phase space. Moreover, the KvN wave function $\psi(q,p,t)$ remains constant along these trajectories. Thus delta-function initial date, with definite initial values of $(q_0,p_0,t_0)$, will be transported along Newtonian trajectories $(q(t),p(t),t)$, as expected for a classical point particle. In fact the Liouville operator (\[eq2-2\]) is not uniquely defined in the KvN mechanics [@3-13B]. In particular, as it is clear from (\[eq3-14\]), we can add to the Liouville operator (\[eq3-2\]) any function $F(q,p,t)$: $$\hat{L}^\prime=\hat{L}+F(q,p,t), \label{eq3-add4}$$ without changing the Hamilton’s equations (\[eq3-14\]). Of course, this gauge freedom in the choice of the Liouville operator is related to the invariance of the KvN probability density function under the phase transformations $$\psi^\prime(q,p,t)=e^{ig(q,p,t)}\psi(q,p,t). \label{eq3-add5}$$ Indeed, the new wave function $\psi^\prime(q,p,t)$ obeys the new Liouville equation $$i\frac{\partial{\psi^\prime}}{\partial t}=\hat{L}^\prime\psi^\prime, \label{eq3-add6}$$ where $$\hat{L}^\prime=e^{ig}\hat{L}e^{-ig}-\frac{\partial g}{\partial t}= \hat{L}-\frac{\partial g}{\partial t}+\{H,g\}\equiv\hat{L}+F(q,p,t). \label{eq3-add7}$$ If evolution of the KvN wave function is determined by $\hat{L}^\prime$, then along the Newtonian trajectories we will have $$F(q,p,t)=-\left(\frac{\partial g}{\partial t}+\{g,H\}\right )=- \frac{dg(q,p,t)}{dt}, \label{eq3-add8}$$ and $$\frac{d\psi^\prime}{dt}=-i\left (i\frac{\partial\psi^\prime}{\partial t}- \hat{L}\psi^\prime\right )=-iF\psi^\prime=i\frac{dg}{dt}\psi^\prime, \label{eq3-add9}$$ which implies $$\psi^\prime(q,p,t)=e^{i[g(q,p,t)-g(q_0,p_0,t_0)]}\psi^\prime(q_0,p_0,t_0). \label{eq3-add10}$$ That is, the KvN wave function no longer remains constant along Newtonian trajectories (along characteristics of the new Liouville equation (\[eq3-add6\])), but the change affects only the phase of the wave function, and such a change is irrelevant in the context of classical mechanics. An obvious difference between the KvN wave function and the true quantum wave function is the number of independent variables: KvN wave function depends typically on the phase space variables $q,p$ and time, while quantum wave function typically depends on the configuration space variables ($q$ in our case) and time. For an interesting perspective on the importance of this difference, see [@3-13B]. There were attempts to develop operator formulation of classical mechanics based on wave functions defined over configuration space, not over phase space [@3-13C; @3-13D]. In such attempts Hamilton-Jacobi equation, not the Liouville equation, is used as a starting point. Although interesting, we will not pursue such approach in the present work. A very interesting perspective on the KvN mechanics was given by Sudarshan [@3-14; @3-15]. Let us consider a quantum mechanical system with twice as many degrees of freedom as our initial classical mechanical system. Namely, besides $\hat q$ and $\hat p$ operators, let us introduce new operators $\hat Q$ and $\hat P$ so that $(\hat q,\hat P)$ and $(\hat Q, \hat p)$ form canonical pairs from the quantum mechanical point of view: $$[\hat q,\hat P]=i\hbar,\;\;\;[\hat Q,\hat p]=i\hbar. \label{eq3-15}$$ Then in the $(q,p)$-representation, where $\hat q$ and $\hat p$ operators are diagonal multiplicative operators, we will have $$\hat P=-i\hbar\,\frac{\partial}{\partial q},\;\;\;{\mathrm and}\;\;\; \hat Q=i\hbar\,\frac{\partial}{\partial p}, \label{eq3-16}$$ so that the Liouville operator (\[eq3-2\]) takes the form $$\hat L=\frac{1}{\hbar}\left (\frac{\partial H}{ \partial q}\,\hat Q+ \frac{\partial H}{ \partial p}\,\hat P\right ), \label{eq3-17}$$ and the Liouville equation (\[eq3-3\]) can be rewritten as a Schrödinger equation $$i\hbar \frac{\partial \psi}{\partial t}=\hat {\cal H} \psi, \label{eq3-18}$$ with the quantum Hamiltonian $${\cal H}=\frac{\partial H}{ \partial q}\,\hat Q+ \frac{\partial H}{ \partial p}\,\hat P. \label{eq3-19}$$ The search for a hidden variable theory for quantum mechanics is a still ongoing saga [@3-16]. Here, thanks to Sudarshan (for earlier thoughts in this direction see [@3-17]) we have an amusing situation: classical mechanics, on the contrary, is interpreted as a hidden variable quantum theory! “If we assume that not all quantum dynamical variables are actually observable, and if we set rules for distinguishing measurable from nonmeasurable operators, it is then possible to define a classical system as a special type of quantum system for which all measurable operators commute “ [@3-18]. What remains is to explain how Schrödinger cat states is avoided in KvN mechanics: the superposition principle is the basic tenet of the quantum mechanics while in the classical realm the cat is either alive or dead, any superposition of these classical states does not make sense. Of course, the fact that the amplitude and phase evolve independently, equations (\[eq3-4\]), already implies the absence of any interference effects in KvN mechanics. However, this separation of the amplitude and phase is an artifact of the $(q,p)$-representation. We can choose to work, for example, in the $(q,Q)$-representation instead [@3-10; @3-19]. In this representation $\hat q$ and $\hat Q$ are simultaneously diagonal multiplicative operators, while $\hat p$ and $\hat P$ are differential operators: $$\begin{aligned} && \hat q\,\psi(q,Q,t)=q\,\psi(q,Q,t),\hspace{11mm} \hat Q\,\psi(q,Q,t)=Q\,\psi(q,Q,t), \nonumber \\ && \hat p\,\psi(q,Q,t)=-i\hbar \frac{\partial}{\partial Q}\,\psi(q,Q,t),\;\; \hat P\,\psi(q,Q,t)=-i\hbar \frac{\partial}{\partial q}\,\psi(q,Q,t). \label{eq3-20}\end{aligned}$$ Wave functions in two representations are related by Fourier transform (the same symbol $\psi$ is used for both the function and its Fourier transform for notational simplicity): $$\psi(q,Q,t)=\frac{1}{\sqrt{2\pi}}\int dp\, e^{ipQ/\hbar}\psi(q,p,t). \label{eq3-21}$$ This follows from the following [@3-10]. If $\|q,Q>$ are the simultaneous eigenstates of the $\hat q$ and $\hat Q$ operators, while $\|q,p>$ — simultaneous eigenstates of the $\hat q$ and $\hat p$ operators: $$\begin{aligned} && \hat q\,\|q,Q>=q\,\|q,Q>,\;\;\hat Q\,\|q,Q>=Q\,\|q,Q>,\nonumber \\ && \hat q\,\|q,p>=q\,\|q,p>,\;\;\;\;\,\hat p\,\|q,p>=p\,\|q,p>, \label{eq3-22}\end{aligned}$$ then we will have $$\begin{aligned} && q<q^\prime,p^\prime\|q,Q>=<q^\prime,p^\prime\|\hat q\|q,Q>= q^\prime<q^\prime,p^\prime\|q,Q>,\nonumber \\ && Q<q^\prime,p^\prime\|q,Q>=<q^\prime,p^\prime\|\hat Q\|q,Q>= -i\hbar\frac{\partial}{\partial p^\prime}<q^\prime,p^\prime\|q,Q>, \label{eq3-23}\end{aligned}$$ which, together with the normalization condition $$<q^\prime,Q^\prime\|q,Q>=\delta(q^\prime-q)\delta(Q^\prime-Q), \label{eq3-24}$$ imply that in the $(q,p)$-representation the $\|q,Q>$ state is given by the wave function $$<q^\prime,p^\prime\|q,Q>=\frac{1}{\sqrt{2\pi}}\,\delta(q^\prime-q)e^{-ip^\prime Q/\hbar}. \label{eq3-25}$$ Because, like $\|q,p>$ states, $\|q,Q>$ states also form a complete set of orthonormal eigenstates in the KvN Hilbert space, we have $$\psi(q,Q,t)=<q,Q\|\psi(t)>=\int dq^\prime dp\, <q,Q\|q^\prime,p><q^\prime,p\|\psi(t)>, \label{eq3-26}$$ which, in light of (\[eq3-25\]), is equivalent to (\[eq3-21\]). In the $(q,Q)$-representation and for the classical Hamiltonian $H=\frac{p^2} {2m}+V(q)$, the quantum Hamiltonian (\[eq3-19\]) takes the form $${\cal H}=\frac{dV}{dq}\,Q-\frac{\hbar^2}{m}\,\frac{\partial^2}{\partial q \partial Q}. \label{eq3-27}$$ Then it follows from the Schrödinger equation (\[eq3-18\]) that the amplitude $A(q,Q)$ and the phase $\Phi(q,Q)$ of the KvN wave function $\psi(q,Q)=A\,e^{i\Phi/\hbar}$ evolve according to the equations $$\begin{aligned} && \frac{\partial A}{\partial t}+\frac{1}{m}\left(\frac{\partial A}{\partial q} \frac{\partial \Phi}{\partial Q}+\frac{\partial A}{\partial Q} \frac{\partial \Phi}{\partial q}+A\,\frac{\partial^2 \Phi}{\partial Q \partial q}\right)=0,\nonumber \\ && \frac{\partial \Phi}{\partial t}+\frac{1}{m}\left(\frac{\partial \Phi} {\partial q}\frac{\partial \Phi}{\partial Q}-\frac{\hbar^2}{A} \frac{\partial^2 A}{\partial Q\partial q}\right)+\frac{dV}{dq}\,Q=0. \label{eq3-28}\end{aligned}$$ As we see, in this representation the phase and amplitude are coupled in the equations of motion and their time evolutions become intertwined much like the ordinary quantum mechanics. This is hardly surprising because, after all, the encompassing underlying system is quantum. According to Sudarshan [@3-14; @3-15], it is the superselection principle [@3-20; @3-21] which kills the interference effects in the KvN mechanics. In classical mechanics, observables are functions of the phase space variables $q$ and $p$. Therefore, $\hat q$ and $\hat p$ commute with all classical observables and thus trigger a superselection mechanism which render the relative phase between different superselection sectors unobservable. Indeed, let $$\|\psi>=\alpha\|p,q>+\beta\|p^\prime,q^\prime>, \label{eq3-29}$$ with $\|\alpha\|^2+\|\beta\|^2=1$, be a seemingly coherent superposition of different eigenstates of $\hat q$ and $\hat p$. As we assume that $\|p,q>$ form a complete set of orthonormal states and an observable $\hat \Lambda$ commutes with $\hat q$ and $\hat p$, $\|p,q>$ is an eigenstate of $\hat \Lambda$ also and thus $<p^\prime,q^\prime\|\hat \Lambda\|p,q>=0$. Therefore, for the mean value of the observable $\hat \Lambda$ in the state $\|\psi>$ we get $$\bar \Lambda=<\psi\|\hat \Lambda\|\psi>=\|\alpha\|^2 <p,q\|\hat \Lambda\|p,q>+ \|\beta\|^2 <p^\prime,q^\prime\|\hat \Lambda\|p^\prime,q^\prime>. \label{eq3-30}$$ As we see, all interference effects are gone and the mean value is the same as if we had an incoherent mixture of the states $\|p,q>$ and $\|p^\prime,q^\prime>$ described by the diagonal density matrix $$\hat \rho=\|\alpha\|^2\|\,\|p,q><p,q\|+\|\beta\|^2\|\,\|p^\prime,q^\prime> <p^\prime,q^\prime\|. \label{eq3-31}$$ However, this use of the superselection principle in KvN mechanics differs from its conventional use in one essential aspect [@3-10; @3-14]. In quantum mechanics time evolution is governed by Hamiltonian which is by itself an observable. As a result all time evolution takes place in one superselection sector and we have genuine superselection rules that the eigenvalues of the superselecting operators cannot be changed during the time evolution. Of course, in the case of KvN mechanics this would be a catastrophe because it would imply that $q$ and $p$ cannot change during the time evolution. Fortunately, the quantum Hamiltonian (\[eq3-19\]) is not a classical observable, because it contains unobservable hidden quantum variables $\hat Q$ and $\hat P$. As a result (\[eq3-19\]) does not commute with $\hat q$ and $\hat p$ operators and thus can generate a transition from one eigenspace of these superselection operators to the other. Modification of classical mechanics =================================== Modified commutation relations alone are not enough to derive physical meaning. Many attempts were made to define the dynamics of quantum systems and their observables in the presence of a minimal length, but this research field is still far from being as logically consistent and mature as the ordinary quantum mechanics is [@2-37]. In any case, the minimal length modification of quantum mechanics entails the corresponding modification of classical mechanics, as the former is considered as the $\hbar \to 0$ limit of the latter (see, however, [@4-1]). Usually the modification of classical mechanics is obtained from the corresponding modification of quantum mechanics by replacing modified commutators with modified Poisson brackets [@2-35; @4-2]: $$\frac{1}{i\hbar}\,[\hat x_i,\,\hat p_j] \to \{x_i,\,p_j\}. \label{eq4-1}$$ At that the modified Poisson bracket of arbitrary functions $F$ and $G$ of the coordinates and momenta are defined as [@2-35] $$\{F,\,G\}=\left (\frac{\partial F}{\partial x_i}\,\frac{\partial G} {\partial p_j}-\frac{\partial F}{\partial p_i}\,\frac{\partial G} {\partial x_i}\right )\{x_i,\,p_j\}+\frac{\partial F}{\partial x_i}\, \frac{\partial G}{\partial x_j}\,\{x_i,\,x_j\}. \label{eq4-2}$$ Correspondingly, the classical equations of motion have the form $$\dot x_i=\{x_i,\,H\}=\{x_i,\,p_j\}\,\frac{\partial H}{\partial p_j}+ \,\{x_i,\,x_j\}\,\frac{\partial H}{\partial x_j},\;\; \dot p_i=\{p_i,\,H\}=-\{x_i,\,p_j\}\,\frac{\partial H}{\partial x_j}. \label{eq4-3}$$ A number of classical mechanics problem was studied within this scenario [@4-3; @4-4; @4-5; @4-6; @4-7; @4-8; @4-9; @4-10]. Koopman-von Neumann mechanics, however, provides a different and in our opinion more interesting perspective on the Planck scale deformation of classical mechanics. The main idea is the following: modification of the commutation relations, for example in the form of (\[eq2-18\]), in the encompassing (in the Sudarshan sense) quantum system will alter classical dynamics in the $(q,p)$ classical subspace. For simplicity, let us consider a one-dimensional classical harmonic oscillator with the Hamiltonian $$H=\frac{1}{2m}(p^2+m^2\omega^2\,q^2). \label{eq4-4}$$ In the Sudarshan-encompassing two-dimensional quantum system we can identify $x_1=q,\,x_2=Q,\,p_1=P,\,p_2=p$. Then Snyder commutation relations (\[eq2-17\]) take the form $$[\hat q,\,\hat P]=i\hbar (1+\beta^\prime \hat P^2),\;\; [\hat q,\,\hat p]=[\hat Q,\,\hat P]=i\hbar\beta^\prime\hat p\hat P,\;\; [\hat Q,\,\hat p]=i\hbar (1+\beta^\prime \hat p^2),\;\; [\hat q,\,\hat Q]=i\hbar\beta^\prime(\hat q\hat p-\hat Q\hat P),\;\; [\hat p,\,\hat P]=0. \label{eq4-5}$$ The first surprise is that $\hat q$ and $\hat p$ cease to be commuting. According to (\[eq2-11\]), the corresponding uncertainty relation is $$\Delta q\Delta p\ge\frac{\hbar\beta^\prime}{2}\left (\Delta(p,P)+<p><P> \right). \label{eq4-6}$$ As we see the sharply defined classical trajectories cease to exist in the $(q,p)$ phase space, much like the quantum case. The constant $\hbar\beta^\prime$, that governess the fuzziness of the “classical” $(q,p)$ phase space, is induced, we believe, by the quantum gravity/string theory effects at the Planck scale. Then $$\hbar\beta^\prime\sim \frac{l_P^2}{\hbar}=\frac{G}{c^3}, \label{eq4-7}$$ and we see that it is expected not to depend on $\hbar$! Classical trajectories will be lost even in the hypothetical world with $\hbar=0$, provided the Newton constant $G$ is not zero and the universal velocity $c$ is not infinity. However, Our troubles with classicality don’t end here. The quantum Hamiltonian (\[eq3-19\]) that corresponds to (\[eq4-4\]) has the form $${\cal H}=\frac{1}{m}\left(\hat p\hat P+m^2\omega^2 \hat q \hat Q\right ), \label{eq4-8}$$ indicating, according to (\[eq4-5\]), the following equations of motion (in the Heisenberg picture) for “classical” variables $\hat q$ and $\hat p$ : $$\frac{d\hat q}{dt}=\frac{i}{\hbar}[{\cal H},\hat q]=\frac{\hat p}{m}\left ( 1+2\beta^\prime\hat P^2\right )+ \beta^\prime m\omega^2\hat q(\hat q\hat p-\hat Q\hat P),\;\;\; \frac{d\hat p}{dt}=\frac{i}{\hbar}[{\cal H},\hat p]=-m\omega^2\hat q \left (1+\beta^\prime \hat p^2\right )- \beta^\prime m\omega^2\hat p\hat P\hat Q. \label{eq4-9}$$ As one would expect from the beginning, the equations are modified. What was probably unexpected is that the additional terms depend on the hidden variables $\hat Q$ and $\hat P$! Analogous equations hold for “hidden” variables $\hat Q$ and $\hat P$ : $$\frac{d\hat Q}{dt}=\frac{i}{\hbar}[{\cal H},\hat Q]=\left ( 1+2\beta^\prime\hat p^2\right )\frac{\hat P}{m}- \beta^\prime m\omega^2(\hat q\hat p-\hat Q\hat P)\hat Q,\;\;\; \frac{d\hat P}{dt}=\frac{i}{\hbar}[{\cal H},\hat P]=-m\omega^2 \left (1+\beta^\prime \hat P^2\right )\hat Q- \beta^\prime m\omega^2\hat q\hat p\hat P. \label{eq4-10}$$ If the modification considered emerges from the Planck scale effects, the natural scale for new phenomenological constants, like $\beta^\prime$, is $\beta^\prime\sim l_P^2/\hbar^2=1/p_P^2$, where $p_P$ is the Planck momentum. Therefore the correction terms in (\[eq4-5\]) and (\[eq4-9\]) are significant only when the momenta involved are of the order of Planck momentum. Let us suppose that this is indeed so for $(q,p)$ classical sector related momenta, while the hidden $(Q,P)$ sector related momenta for some reason remains much smaller, so that we can discard hidden variables $(Q,P)$ in (\[eq4-5\]) and (\[eq4-9\]). Then we still regain the classical sector ($\hat q$ and $\hat p$ will commute in this approximation), but the classical equations of motion will be modified (as $\hat q$ and $\hat p$ do commute, we write equations of motion for $q$ and $p$ considering them as real numbers, not operators): $$\frac{dq}{dt}=\left (1+\beta^\prime m^2\omega^2 q^2\right )\frac{p}{m},\;\;\; \frac{dp}{dt}=-m\omega^2 \left (1+\beta^\prime p^2\right ) q. \label{eq4-11}$$ Equations (\[eq4-11\]) are non-linear oscillator equations of the type introduced in [@4-11; @4-12] that model generalized one-dimensional harmonic oscillators in several important dynamical systems: $$\frac{dq}{dt}=f(q)\,p,\;\;\;\frac{dp}{dt}=-g(p)\,q, \label{eq4-12}$$ where $f(q)$ and $g(p)$ functions, with the conditions $f(0)>0,\,g(0)>0$, are assumed to be continuous with continuous first derivatives. On the other hand, (\[eq4-11\]) can be rewritten as an second order differential equation and the result is $$\ddot q -\frac{\beta^\prime m^2\omega^2 q}{1+\beta^\prime m^2\omega^2 q^2} \,\dot q^2+\omega^2(1+\beta^\prime m^2\omega^2 q^2)q=0. \label{eq4-13}$$ This equation is of the type of quadratic Liénard equation. The general quadratic Liénard equation, used in a vast range of applications, has the form $$\ddot q+f(q)\,\dot q^2+g(q)=0, \label{eq4-14}$$ where $f(q)$ and $g(q)$ are arbitrary functions that do not vanish simultaneously [@4-13; @4-13A]. The equation (\[eq4-13\]) (and in general the system (\[eq4-12\]) [@4-11]) admits a first integral which can be found as follows. From (\[eq4-11\]) we have $$\frac{dp}{dq}=-\frac{m^2\omega^2(1+\beta^\prime p^2)q} {(1+\beta^\prime m^2\omega^2 q^2)p}. \label{eq4-15}$$ The variables $q$ and $p$ in this differential equation can be separated and we get after the integration $$\ln{(1+\beta^\prime p^2)}+\ln{(1+\beta^\prime m^2\omega^2 q^2)}= \mathrm{constant},$$ which implies that $$(1+\beta^\prime p^2)(1+\beta^\prime m^2\omega^2 q^2)=1+\beta^\prime A, \label{eq4-16}$$ where $A$ is some constant. For definiteness, let us assume the following initial values $$q(0)=0,\;\;\;p(0)=p_0>0. \label{eq4-17}$$ Then $A=p_0^2$. Another interesting property of the equation (\[eq4-13\]) is that it corresponds to a Lagrangian system with a position dependent mass (in fact, any quadratic Liénard equation has such a property [@4-14]). Indeed, the Lagrangian $${\cal L}=\frac{1}{2}\mu(q)\,\dot q^2-V(q) \label{eq4-18}$$ leads to the Euler-Lagrange equation $$\ddot q+\frac{\mu^\prime}{2\mu}\,\dot q^2+\frac{V^\prime}{\mu}=0.$$ Here the prime denotes differentiation with respect to $q$. Comparing with (\[eq4-13\]), we get the identifications $$\frac{\mu^\prime}{2\mu}=-\frac{\beta^\prime m^2\omega^2 q} {1+\beta^\prime m^2\omega^2 q^2},\;\;\; V^\prime=\mu\,\omega^2 q\left (1+\beta^\prime m^2\omega^2 q^2\right). \label{eq4-19}$$ These differential equations can be integrated and we get $$\mu(q)=\frac{m}{1+\beta^\prime m^2\omega^2 q^2},\;\;\; V(q)=\frac{1}{2}m\omega^2 q^2. \label{eq4-20}$$ Comparing the conserved energy $E=\frac{p^2}{2\mu}+V$ with (\[eq4-16\]), we see that the first integral (\[eq4-16\]) represents just the energy conservation and $A=2mE$. Thanks to the first integral (\[eq4-16\]), the equation of motion (\[eq4-13\]) of the Planck scale deformed harmonic oscillator can be solved in a quadrature. Namely, from (\[eq4-16\]), assuming (\[eq4-17\]) initial conditions, we get $$p=\sqrt{\frac{p_0^2-m^2\omega^2 q^2}{1+\beta^\prime m^2\omega^2 q^2}}.$$ On the other hand, it follows from the first equation of (\[eq4-11\]) that $$p=\frac{m\dot q}{1+\beta^\prime m^2\omega^2 q^2}.$$ Combining these two expressions of $p$, we get $$m\frac{dq}{dt}=\sqrt{(p_0^2-m^2\omega^2 q^2)(1+\beta^\prime m^2\omega^2 q^2)}. \label{eq4-21}$$ Introducing a new variable $z=m\omega q/p_0$, we can integrate (\[eq4-21\]) as follows: $$\omega t=\int\limits_0^{m\omega q/p_0}\frac{dz}{\sqrt{(1-z^2)(1+ \beta^\prime p_0^2 z^2)}}=\int\limits_0^u\frac{d\theta}{\sqrt{1+ \beta^\prime p_0^2\sin^2{\theta}}}, \label{eq4-22}$$ where $\sin{u}=m\omega q/p_0$ and at the last step we have made another change of the integration variable, namely $z=\sin{\theta}$. The above integral is the incomplete elliptic integral of the first kind. Its amplitude $u$ satisfies the equation $\sin{u}=\operatorname{sn}{(\omega t,i\sqrt{\beta^\prime}p_0)}$, where $\operatorname{sn}{(\omega t,i\sqrt{\beta^\prime}p_0)}$ is the Jacobi sine elliptic function with the imaginary modulus (assuming $\beta^\prime>0$). Therefore $$q(t)=\frac{p_0}{m\omega}\,\operatorname{sn}{(\omega t,i\sqrt{\beta^\prime}p_0)}= \frac{p_0}{m\omega\sqrt{1+\beta^\prime p_0^2}}\operatorname{sd}{\left(\sqrt{1+ \beta^\prime p_0^2}\,\omega t,\,\sqrt{\frac{\beta^\prime p_0^2}{1+ \beta^\prime p_0^2}}\right )}, \label{eq4-23}$$ where at the last step we have used imaginary modulus transformation [@4-15]. Period of oscillations $T$, according to (\[eq4-22\]), is given by the relation $$\frac{\omega T}{4}=\int\limits_0^{\pi/2}\frac{d\theta}{\sqrt{1+ \beta^\prime p_0^2\sin{\theta}^2}}=\operatorname{K}(i\sqrt{\beta^\prime}), \label{eq4-24}$$ where $\operatorname{K}$ is the complete elliptic integral of the first kind. Using again the imaginary modulus transformation, we get $$T=\frac{4}{\omega\sqrt{1+\beta^\prime p_0^2}}\,\operatorname{K}{\left(\sqrt{\frac {\beta^\prime p_0^2}{1+\beta^\prime p_0^2}}\right )}\approx \frac{2\pi}{\omega} \left (1-\frac{\beta^\prime p_0^2}{4}\right ). \label{eq4-25}$$ This reduction of the period of oscillations is similar to what was found in [@4-9] within the framework of the one-dimensional Kempf modification of the commutation relations (a one-dimensional version of (\[eq2-18\])) with the standard recipe of replacing commutators by Poisson brackets when considering a classical limit. Let us also consider, as a second example, Kempf et al. modification of the commutation relations (\[eq2-18\]) with $\beta^\prime=2\beta$, so that the spatial geometry remains approximately commutative (at the first order in $\beta$). Then we will have (for our two-dimensional quantum system the square of the momentum vector is $p^2+P^2$) $$[\hat q,\hat P]=i\hbar[1+\beta(\hat p^2+3\hat P^2)],\;\; [\hat q,\,\hat p]= [\hat Q,\,\hat P]=i\hbar\,2\beta \hat p\hat P,\;\; \;\; [\hat Q,\,\hat p]=i\hbar[1+\beta(3\hat p^2+\hat P^2)],\;\; [\hat q,\,\hat Q]=0,\;\; [\hat p,\,\hat P]=0. \label{eq4-26}$$ Again $q$ and $p$ cease to be commuting. Then corresponding equations of motion are $$\frac{d\hat q}{dt}=\frac{\hat p}{m}\left [1+\beta\left(\hat p^2+ 5\hat P^2\right )\right],\;\;\;\; \frac{d\hat p}{dt}=-m\omega^2 \left \{\hat q \left[1+\beta(3\hat p^2+\hat P^2)\right ]+2\beta \,\hat p\,\hat P \hat Q \right \}, \label{eq4-27}$$ and $$\frac{d \hat Q}{dt}=\left [1+\beta\left (5\hat p^2+\hat P^2\right )\right ] \frac{\hat P}{m},\;\;\;\;\frac{d \hat P}{dt}=-m\omega^2\left\{\left[1+\beta \left (\hat p^2+3\hat P^2\right )\right ]\hat Q+2\beta\,\hat q\hat p\hat P \right \}. \label{eq4-28}$$ As in the previous example, hidden variables $P$ and $Q$ do appear in the equations of motions of the “classical” sector due to the Plank scale modification of the commutation relations. In situations when the effects of the hidden variables $P$ and $Q$ can be approximately discarded, the classical equations of motion became $$\dot q=\left (1+\beta p^2\right )\frac{p}{m},\;\;\; \dot p=-m\omega^2\left (1+3\beta p^2\right )q. \label{eq4-29}$$ This system is no longer of the type (\[eq4-12\]). Nevertheless, it gives a second order differential equation for the variable $p$ which is of the quadratic Liénard type: $$\ddot p-\frac{6\beta p}{1+3\beta p^2}\,\dot p^2+\omega^2\left (1+\beta p^2 \right )\left ( 1+3\beta p^2\right )p=0. \label{eq4-30}$$ Variable mass system (in the $p$-space) with the Lagrangian $${\cal L}=\frac{1}{2}\mu(p)\dot p^2-V(p),$$ which is equivalent to (\[eq4-30\]), is characterized by $$\mu(p)=\frac{m}{(1+3\beta p^2)^2},\;\;\;V(p)=\frac{m\omega^2}{6}\left [ p^2+\frac{2}{3\beta}\,\ln{\left(1+3\beta p^2\right )}\right ]. \label{eq4-31}$$ Of cause the integration constant in the potential $V(p)$ is irrelevant and it was chosen in such a way that when $\beta=0$ the Lagrangian ${\cal L}$ becomes the ordinary harmonic oscillator Lagrangian. The corresponding conserved “energy” $\frac{1}{2}\mu(p) \dot p^2+V(p)$ gives a first integral $$\frac{1}{2}\,\frac{m}{(1+3\beta p^2)^2}\,\dot p^2+\frac{m\omega^2}{6}\left [ p^2+\frac{2}{3\beta}\,\ln{\left(1+3\beta p^2\right )}\right ]=m^2\omega^2 E, \label{eq4-32}$$ where $E$ is some constant. Period of oscillations that follows from (\[eq4-32\]) is $$T=4\int\limits_0^{p_0}\frac{dp}{\sqrt{\frac{2}{\mu}(m^2 \omega ^2 E-V(p))}}= \frac{4}{\omega} \int\limits_0^{p_0}\frac{dp}{(1+3\beta p^2)\sqrt{2mE- \frac{1}{3}\left[p^2+\frac{2}{3\beta}\ln{(1+3\beta p^2)}\right ]}}. \label{eq4-33}$$ At the first order in $\beta$, and assuming $\dot p(0)=0,\,p(0)=p_0$ initial conditions, we have $E=\frac{p_0^2}{2m}(1-\beta p_0^2)$ and $$T\approx\frac{4}{\omega} \int\limits_0^{p_0}\frac{dp}{(1+3\beta p^2)\sqrt{ (p_0^2-p^2)\left[1-\beta(p_0^2+p^2)\right ]}}\approx \frac{4}{\omega} \int\limits_0^{p_0}\frac{dp}{\sqrt{p_0^2-p^2}}\left [1-\frac{\beta}{2} (5p^2-p_0^2)\right ]=\frac{2\pi}{\omega}\left (1-\frac{3\beta p_0^2}{4} \right ). \label{eq4-34}$$ At the final step, we have used elementary integrals $$\int\limits_0^{p_0}\frac{dp}{\sqrt{p_0^2-p^2}}=\frac{\pi}{2},\;\;\; \int\limits_0^{p_0}\sqrt{p_0^2-p^2}\,dp=\frac{\pi p_0^2}{4}.$$ Concluding remarks ================== In this note we have tried to combine Koopman-von Neumann-Sudarshan perspective on classical mechanics with the generalized uncertainty principle. We have considered two versions of the generalized commutation relations. The results were similar: classical position and momentum operators cease to be commuting and hidden variables show themselves explicitly in classical evolution equations. In situations then the effect of these hidden variables can be neglected in evolution equations, the modification of classical dynamics is similar (but not identical) to the modification obtained by using more traditional approach of replacement of commutators by Poisson brackets. We suspect that the above mentioned features are common for a large class of generalized uncertainty principle based models if they are interpreted in the Koopman-von Neumann-Sudarshan framework. Therefore, from this perspective, we can conclude that Planck scale quantum gravity effects destroy classicality. However this breakdown of classicality is controlled by a small dynamical parameter $\frac{p^2}{p_P^2}$ and can be neglected for all practical purposes thanks to the huge hierarchy between the masses of ordinary particles and the Planck mass $m_P=1.2\times 10^{19}~\mathrm{GeV}/c^2$. Usually this huge hierarchy is considered as a problem to be explained [@5-1]. As we see, for classicality it can be beneficial. For a macroscopic body the effective deformation parameter $\beta$ is approximately $N^2$ times smaller than for its elementary constituents, where $N$ is the number of constituents [@4-8]. Therefore macroscopic bodies, notwithstanding their large momenta, provide no advantage in observing Planck scale induced non-classical effects, as the small parameter controlling these non-classical effects for macroscopic bodies becomes $\frac{p^2}{N^2 p_P^2}$ It may happen that the interrelations between quantum mechanics, classical mechanics and gravity are much more tight and intimate than anticipated. The imprints left by quantum mechanics in classical mechanics are more numerous than is usually believed [@5-2; @5-3]. In fact the mathematical structure that allows quantum mechanics to emerge already exists in classical mechanics [@5-4]. Particularly surprising, maybe, is that Schrödinger-Robertston uncertainty principle has an exact counterpart in classical mechanics which can be formulated using some subtle developments in symplectic topology, namely Gromov’s non-squeezing theorem and the related notion of symplectic capacity [@5-5]. On the other hand there are unexpected and deep relations between gravity and quantum mechanics, in particular between Einstein-Rosen wormholes and quantum entanglement [@5-6; @5-7]. We believe the Koopman-von Neumann formulation of classical mechanics might be useful in investigating a twilight zone between quantum and classical mechanics. “It deserves to be better known among physicists, because it gives a new perspective on the conceptual foundations of quantum theory, and it may suggest new kinds of approximations and even new kinds of theories” [@3-13A]. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Carlo Beenakker for indicating useful references. The work of Z.K.S. is supported by the Ministry of Education and Science of the Russian Federation. [10]{} R. P. Feynman, Simulating physics with computers, Int. J. Theor. Phys. [21]{} (1982), 467-488. N. Rosen, The Relation Between Classical and Quantum Mechanics, Am. J. Phys. [32]{} (1964), 597-600. S. 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--- abstract: 'We present new parametrizations of elements of spinor and orthogonal groups of dimension 4 using Grassmann exterior algebra. Theory of spinor groups is an important tool in theoretical and mathematical physics namely in the Dirac equation for an electron.' author: - Nikolay Marchuk title: Parametrisations of elements of spinor and orthogonal groups using exterior exponents --- [^1] Steklov Mathematical Institute,\ Gubkina st.8, Moscow 119991, Russia Email: [email protected], [email protected] MSC: Primary 15A66; Secondary 15A75 An exterior algebra, invented by G. Grassmann in the year 1844 [@Grassmann], has many applications in different fields of mathematics and physics. Here we present a new application of Grassmann algebra to the theory of spinor and orthogonal groups. [Clifford algebras.]{} Let $p,q,n$ be nonnegative integer numbers and $n=p+q$. And let $\cl(p,q)$ be a real Clifford algebra [@mybook] of the signature $(p,q)$ with generators $e^1,\ldots,e^n$ such that $$e^a e^b + e^b e^a = 2\eta^{ab}e,\quad a,b=1,\ldots,n,$$ where $e$ is the identity element of Clifford algebra and $\eta^{ab}$ are elements of the diagonal matrix of dimension $n$ $$\eta=\diag(1,\ldots,1,-1,\ldots,-1)$$ with $p$ pieces of $1$ and $q$ pieces of $-1$ on the diagonal. The Clifford algebra $\cl(p,q)$ can be considered as $2^n$-dimensional vector space with basis elements $$e,e^a,e^{a_1a_2},\ldots,e^{a_1\ldots a_{n-1}},e^{12\ldots n},\quad 0\leq a_1<\ldots<a_k\leq n \label{cl:basis}$$ numbered by ordered multi-indices of lengths from $0$ to $n$. Any element of Clifford algebra $\cl(p,q)$ can be written in the form of decomposition w.r.t. the basis (\[cl:basis\]) $$U=u e +u_a e^a+\sum_{a_1<a_2}u_{a_1a_2}e^{a_1a_2}+\ldots+u_{1\ldots n}e^{1\ldots n}\label{decomp}$$ with real coefficients $u,u_a,u_{a_1a_2},\ldots,u_{1\ldots n}$. Elements of the form $$U=\sum_{a_1<\ldots<a_k}u_{a_1\ldots a_k}e^{a_1\ldots a_k}$$ are called [elements of rank $k$]{}. Denote by $\cl_k(p,q)$ the subspace of rank $k$ elements. We have $$\cl(p,q)=\cl_0(p,q)\oplus\ldots\oplus\cl_n(p,q).$$ An element $U\in\cl(p,q)$ is called [even (odd)]{} if this element is a sum of elements of even (odd) ranks. Hence $$\begin{aligned} \cl_\Even(p,q)&=&\cl_0(p,q)\oplus\cl_2(p,q)\oplus\ldots,\\ \cl_\Odd(p,q)&=&\cl_1(p,q)\oplus\cl_3(p,q)\oplus\ldots,\\ \cl(p,q)&=&\cl_\Even(p,q)\oplus\cl_\Odd(p,q).\end{aligned}$$ [Reverse operation.]{} Let us define a linear [reverse]{} operation $\sim : \cl(p,q)\to\cl(p,q)$ with the aid of the following rules: $$e^\sim= e,\quad (e^a)^\sim=e^a,\quad (U V)^\sim=V^\sim U^\sim,\quad\forall U,V\in\cl(p,q).$$ In particular, $$(e^{a_1}\ldots e^{a_k})^\sim=e^{a_k}\ldots e^{a_1}.$$ [Spinor groups.]{} Let $n=p+q\leq 5$. Consider the following set of even elements of the Clifford algebra $\cl(p,q)$: $$\Spin_+(p,q)=\{S\in\cl_\Even(p,q) : S^\sim S=e\}.$$ This set is closed w.r.t. Clifford product and contains the identity element $e$. Elements of this set are invertible. Therefore $\Spin_+(p,q)$ can be considered as a group (Lie group) w.r.t. the Clifford product. This group is called [spinor group]{}. The set of second rank elements $\cl_2(p,q)$ with the commutator $[A,B]=AB-BA$ is the Lie algebra of the Lie group $\Spin_+(p,q)$ [@Lounesto]. Let us define [an exponent]{} of Clifford algebra elements $\exp : \cl(p,q)\to\cl(p,q)$ by the formula $$\exp\,A=e+A+\frac{1}{2!}A^2+\frac{1}{3!}A^3+\ldots.$$ [Spinor groups in cases $p+q=4$.]{} The sign $\simeq$ denotes group isomorphisms. It is known [@Lounesto] that - $\Spin_+(4,0)\simeq\Spin_+(0,4)\simeq\Spin(4)$. - $\Spin_+(1,3)\simeq\Spin_+(3,1)$. - $\forall S\in\Spin(4)$ there exists $B\in\cl_2(4,0)$ such that $S=\exp\,B$. - $\forall S\in\Spin_+(1,3)$ there exists $B\in\cl_2(1,3)$ such that $S=\exp\,B$, or $S=-\exp\,B$. Note that the set of exponents of rank 2 elements do not form a group. - There exist elements $S\in\Spin_+(2,2)$ such that these elements can’t be represented in the form $\pm\exp\,B$, where $B\in\cl_2(2,2)$. [Exterior (Grassmann) multiplication of Clifford algebra elements.]{} Let us define the associative and distributive operation of exterior multiplication of Clifford algebra elements (denoted by $\wedge$) $$e^{a_1}\wedge e^{a_2}\wedge\ldots\wedge e^{a_k}=e^{[a_1}e^{a_2}\ldots e^{a_k]},$$ where square brackets denote the operation of alternation of indices. The Clifford algebra $\cl(p,q)$, considered with the exterior product, can be identified with the Grassmann algebra of dimension $n$. [Exterior exponent.]{} Consider the exterior exponent $\extexp : \cl(p,q) \to \cl(p,q)$ $$\extexp(B)=e+B+\frac{1}{2!} B\wedge B+\frac{1}{3!} B\wedge B\wedge B +\ldots, \label{extexp}$$ where $B\in\cl(p,q)$. We interested in exterior exponent of second rank elements. In this case in the right hand part of (\[extexp\]) there are finite number of nonzero summands. In particular, for the case $n=4$ there are only three summands $$\extexp(B)=e+B+\frac{1}{2} B\wedge B,$$ where $B\in\cl_2(p,q)$, $p+q=4$. We see that $$(\extexp(B))^\sim=\extexp(-B)=e-B+\frac{1}{2} B\wedge B.$$ It is not hard to prove that $$(\extexp(B))^\sim \extexp(B)=\lambda e,$$ where $\lambda=\lambda(B)$ is a scalar that depends on coefficients of the element $B$. [Main theorem.]{} If $U\in\cl(p,q)$ is written in the form (\[decomp\]), then we denote $\Tr\,U=u$. Also denote $\ell=e^1e^2e^3e^4$ and $\epsilon=\Tr(\ell^2)$. Let $n=p+q=4$. Any element $S$ of the group $\Spin_+(p,q)$ can be represented in one of two following forms: - If $\Tr\,S\neq 0$, then there exists $B\in\cl_2(p,q)$ such that $\lambda=\lambda(B)>0$ and $$S=\pm\frac{1}{\sqrt{\lambda}}\extexp\,B.\label{exex}$$ The sign (plus or minus) at the right hand part is equal to the sign of the number $\Tr\,S$. - If $\Tr\,S=0$, then there exists $B\in\cl_2(p,q)$ such that $B\wedge B=0$, $\epsilon(1+\beta)\geq 0$, where $\beta=\Tr(B^2)$ and $$S=B\pm\ell\sqrt{\epsilon(1+\beta)}.\label{S:Badj}$$ The sign (plus or minus) at the right hand part is equal to the sign of the number $\Tr(\ell^{-1}S)$. . Let $S\in\Spin_+(p,q)$ be such that $\Tr\,S=\alpha\neq 0$. We white $S$ and $S^\sim$ in the form $$S=\alpha e+U+F,\quad S^\sim=\alpha e-U+F,$$ where $U\in\cl_2(p,q)$, $F\in\cl_4(p,q)$. For $n=4$ it follows that $$U^2=U\wedge U+\gamma e,\quad \alpha^2 e-F^2-\gamma^2 e\in\cl_0(p,q),\quad 2\alpha F-U\wedge U\in\cl_4(p,q).$$ The identity $$S^\sim S=(\alpha^2 e-F^2-\gamma^2 e)+(2\alpha F-U\wedge U)=e$$ gives us $$2\alpha F-U\wedge U=0\ \Rightarrow\ F=\frac{1}{2\alpha}U\wedge U.$$ That means $$S=\alpha e+U+\frac{1}{2\alpha}U\wedge U=\alpha\ \extexp(\frac{1}{\alpha}U).$$ Denoting $$B=\frac{1}{\alpha}U,\quad \alpha=\pm\frac{1}{\sqrt{\lambda}},$$ we get $$S=\pm\frac{1}{\sqrt{\lambda}}\extexp\,B,$$ where $$\lambda e=\extexp(B)\ \extexp(-B).\label{lam:B}$$ It is easy to prove that if an element $S\in\Spin_+(p,q)$ is such that $\Tr\,S=0$, then $S$ can be represented in the form (\[S:Badj\]). This completes the proof. If $S\in\Spin_+(p,q)$ has the form (\[exex\]), then we say that $S$ is given with the aid of semi-polynomial parametrisation. That means, coefficients of $S$ in the basis (\[cl:basis\]) are polynomials of second degree of coefficients $b_{ij}$ multiplied on one and the same factor $1/\sqrt{\lambda}$, where $\lambda=\lambda(B)$ is the polynomial of second degree of coefficients $b_{ij}$. If $S\in\Spin_+(p,q)$ has the form (\[S:Badj\]), then we say that $S$ is given with the aid of adjoint semi-polynomial parametrisation. Let us write down explicit form of elements $S\in\Spin_+(p,q)$, $p+q=4$ using real coefficients of $b_{ij}\in B\in\cl_2(p,q)$ $$B= b_{12}\,e^{12} + b_{13}\,e^{13} + b_{14}\,e^{14} + b_{23}\,e^{23} + b_{24}\,e^{24} + b_{34}\,e^{34}.$$ We have $$\begin{aligned} \extexp(B) &=& e + b_{12}\,e^{12} + b_{13}\,e^{13} + b_{14}\,e^{14} + b_{23}\,e^{23} +\\ && b_{24}\,e^{24} + b_{34}\,e^{34} + (b_{14}\,b_{23} - b_{13}\,b_{24} + b_{12}\,b_{34})\,e^{1234}\end{aligned}$$ Expressions for the scalar $\lambda=\lambda(B)$ that satisfy (\[lam:B\]) and for $\epsilon(1+\beta)$ are depend on a signature $(p,q)$. For $(p,q)=(0,4),(4,0)$ $$\begin{aligned} \lambda &=& 1 + b_{12}{}^2 + b_{13}{}^2 + b_{14}{}^2 + b_{23}{}^2+ b_{14}{}^2\,b_{23}{}^2 - 2\,b_{13}\,b_{14}\,b_{23}\,b_{24} + b_{24}{}^2 +\\ && b_{13}{}^2\,b_{24}{}^2 + 2\,b_{12}\,b_{14}\,b_{23}\,b_{34} - 2\,b_{12}\,b_{13}\,b_{24}\,b_{34} + b_{34}{}^2 + b_{12}{}^2\,b_{34}{}^2,\\ \epsilon(1+\beta) &=& 1 - b_{12}{}^2 - b_{13}{}^2 - b_{14}{}^2 - b_{23}{}^2 - b_{24}{}^2 - b_{34}{}^2.\end{aligned}$$ For $(p,q)=(1,3)$ $$\begin{aligned} \lambda &=& 1 - b_{12}{}^2 - b_{13}{}^2 - b_{14}{}^2 + b_{23}{}^2 - b_{14}{}^2\,b_{23}{}^2 + 2\,b_{13}\,b_{14}\,b_{23}\,b_{24} + b_{24}{}^2 -\\&& b_{13}{}^2\,b_{24}{}^2 - 2\,b_{12}\,b_{14}\,b_{23}\,b_{34} + 2\,b_{12}\,b_{13}\,b_{24}\,b_{34} + b_{34}{}^2 - b_{12}{}^2\,b_{34}{}^2,\\ \epsilon(1+\beta) &=& -1 - b_{12}{}^2 - b_{13}{}^2 - b_{14}{}^2 + b_{23}{}^2 + b_{24}{}^2 + b_{34}{}^2.\end{aligned}$$ For $(p,q)=(2,2)$ $$\begin{aligned} \lambda &=& 1 + b_{12}{}^2 - b_{13}{}^2 - b_{14}{}^2 - b_{23}{}^2 + b_{14}{}^2\,b_{23}{}^2 - 2\,b_{13}\,b_{14}\,b_{23}\,b_{24} - b_{24}{}^2 +\\&& b_{13}{}^2\,b_{24}{}^2 + 2\,b_{12}\,b_{14}\,b_{23}\,b_{34} - 2\,b_{12}\,b_{13}\,b_{24}\,b_{34} + b_{34}{}^2 + b_{12}{}^2\,b_{34}{}^2,\\ \epsilon(1+\beta) &=& 1 - b_{12}{}^2 + b_{13}{}^2 + b_{14}{}^2 + b_{23}{}^2 + b_{24}{}^2 - b_{34}{}^2.\end{aligned}$$ For $(p,q)=(3,1)$ $$\begin{aligned} \lambda &=& 1 + b_{12}{}^2 + b_{13}{}^2 - b_{14}{}^2 + b_{23}{}^2 - b_{14}{}^2\,b_{23}{}^2 + 2\,b_{13}\,b_{14}\,b_{23}\,b_{24} - b_{24}{}^2 -\\&& b_{13}{}^2\,b_{24}{}^2 - 2\,b_{12}\,b_{14}\,b_{23}\,b_{34} + 2\,b_{12}\,b_{13}\,b_{24}\,b_{34} - b_{34}{}^2 - b_{12}{}^2\,b_{34}{}^2,\\ \epsilon(1+\beta) &=& -1 + b_{12}{}^2 + b_{13}{}^2 - b_{14}{}^2 + b_{23}{}^2 - b_{24}{}^2 - b_{34}{}^2.\end{aligned}$$ With the aid of these formulas we get the general form(\[exex\]),(\[S:Badj\]) of elements $S\in\Spin_+(p,q)$. The proof of the following known theorem is straightforward. Let $n=p+q=2,3$. Any element $S\in\Spin_+(p,q)$ can be represented in the following form $$S=\pm e\sqrt{1+\beta}+B,\label{S:adj}$$ where $B\in\cl_2(p,q)$ is such that $\beta=\Tr(B^2)\geq -1$. The sign (plus or minus) at the right hand part of (\[S:adj\]) is equal to the sign of the number $\Tr\,S$. [Orthogonal groups.]{} Consider the Lie groups of special orthogonal matrices of dimension $n=p+q$ $$\SO(p,q)=\{P\in GL(n,\R) : P^T\eta P=\eta,\ det\,P=1\},$$ where $P^T$ is the transposed matrix. For $pq\neq 0$ the groups $\SO(p,q)$ have two disconnected components [@Diedonne]. The component of the group $\SO(p,q)$ that contains the identity matrix is a subgroup $\SO_+(p,q)$. In cases $pq=0$ we have $\SO(0,n)=SO(n,0)=SO(n)$ and this group has only one component, i.e. $\SO_+(n)=SO(n)$. For $p+q=4$ the following propositions are valid [@Lounesto]: - $\SO_+(4,0)\simeq\SO_+(0,4)\simeq\SO(4)$. - $\SO_+(1,3)\simeq\SO_+(3,1)$. - $\forall P\in\SO(4)$ there exists an anti-Hermitian matrix of fourth order ($A^T=-A$) such that $P=\exp\,A$. - $\forall P\in\SO_+(1,3)$ there exists a matrix $A$ of fourth order such that $\eta A^T\eta=-A$, ($\eta=diag(1,-1,-1,-1)$) and $P=\exp\,A$. - There exist matrices from $\SO_+(2,2)$ that can’t be represented in the form $\pm\exp\,A$, where $\eta A^T\eta=-A$, ($\eta=diag(1,1,-1,-1)$). [Connection between spinor and orthogonal groups.]{} It is known [@Lounesto] that the spinor group $\Spin_+(p,q)$ double cover the orthogonal group $\SO_+(p,q)$. This connection can be expressed by the formula $$S^\sim e^a S=p^a_b e^b. \label{spin:ort}$$ In this formula a pair of elements $\pm S$ of spinor group are connected with the matrix $P=\\|p^a_b\\|$ from the orthogonal group. Consider in more details the case $(p,q)=(1,3)$, which is important for physics. Let us take expressions for $S\in\Spin_+(1,3)$ from Theorem 1 and, using (\[spin:ort\]), calculate corresponding elements of the matrix $P=\\|p^a_b\\|$. Then we get the following formulas. If $S$ has the form (\[exex\]) and $\lambda=\lambda(B)>0$, then elements of the matrix $T=\lambda\,P$ have the form $$\begin{aligned} t^1_1 &=& 1 + b_{12}{}^2 + b_{13}{}^2 + b_{14}{}^2 + b_{23}{}^2 + b_{14}{}^2\,b_{23}{}^2 - 2\,b_{13}\,b_{14}\,b_{23}\,b_{24} +\\&& b_{24}{}^2 + b_{13}{}^2\,b_{24}{}^2 + 2\,b_{12}\,b_{14}\,b_{23}\,b_{34} - 2\,b_{12}\,b_{13}\,b_{24}\,b_{34} + b_{34}{}^2 + b_{12}{}^2\,b_{34}{}^2,\\ t^1_2 &=& 2\,b_{12} + 2\,b_{13}\,b_{23} + 2\,b_{14}\,b_{24} + 2\,b_{14}\,b_{23}\,b_{34} - 2\,b_{13}\,b_{24}\,b_{34} + 2\,b_{12}\,b_{34}{}^2,\\ t^1_3 &=& 2\,b_{13} - 2\,b_{12}\,b_{23} - 2\,b_{14}\,b_{23}\, b_{24} + 2\,b_{13}\,b_{24}{}^2 + 2\,b_{14}\,b_{34} - 2\,b_{12}\,b_{24}\,b_{34},\\ t^1_4 &=& 2\,b_{14} + 2\,b_{14}\,b_{23}{}^2 - 2\,b_{12}\,b_{24} - 2\,b_{13}\,b_{23}\,b_{24} - 2\,b_{13}\,b_{34} + 2\,b_{12}\,b_{23}\,b_{34},\\ t^2_1 &=& 2\,b_{12} - 2\,b_{13}\,b_{23} - 2\,b_{14}\,b_{24} + 2\,b_{14}\,b_{23}\,b_{34} - 2\,b_{13}\,b_{24}\,b_{34} + 2\,b_{12}\,b_{34}{}^2,\\ t^2_2 &=& 1 + b_{12}{}^2 - b_{13}{}^2 - b_{14}{}^2 - b_{23}{}^2 + b_{14}{}^2\,b_{23}{}^2 - 2\,b_{13}\,b_{14}\,b_{23}\,b_{24} -\\&& b_{24}{}^2 + b_{13}{}^2\,b_{24}{}^2 + 2\,b_{12}\,b_{14}\,b_{23}\,b_{34} - 2\,b_{12}\,b_{13}\,b_{24}\,b_{34} + b_{34}{}^2 + b_{12}{}^2\,b_{34}{}^2,\\ t^2_3 &=& 2\,b_{12}\,b_{13} - 2\,b_{23} + 2\,b_{14}{}^2\,b_{23} - 2\,b_{13}\,b_{14}\,b_{24} + 2\,b_{12}\,b_{14}\,b_{34} - 2\,b_{24}\,b_{34},\\ t^2_4 &=& 2\,b_{12}\,b_{14} - 2\,b_{13}\,b_{14}\,b_{23} - 2\,b_{24} + 2\,b_{13}{}^2\,b_{24} - 2\,b_{12}\,b_{13}\,b_{34} + 2\,b_{23}\,b_{34},\\ t^3_1 &=& 2\,b_{13} + 2\,b_{12}\,b_{23} - 2\,b_{14}\,b_{23}\,b_{24} + 2\,b_{13}\,b_{24}{}^2 - 2\,b_{14}\,b_{34} - 2\,b_{12}\,b_{24}\,b_{34},\\ t^3_2 &=& 2\,b_{12}\,b_{13} + 2\,b_{23} - 2\,b_{14}{}^2\,b_{23} + 2\,b_{13}\,b_{14}\,b_{24} - 2\,b_{12}\,b_{14}\,b_{34} - 2\,b_{24}\,b_{34},\\ t^3_3 &=& 1 - b_{12}{}^2 + b_{13}{}^2 - b_{14}{}^2 - b_{23}{}^2 + b_{14}{}^2\,b_{23}{}^2 - 2\,b_{13}\,b_{14}\,b_{23}\,b_{24} +\\&& b_{24}{}^2 + b_{13}{}^2\,b_{24}{}^2 + 2\,b_{12}\,b_{14}\,b_{23}\,b_{34} - 2\,b_{12}\,b_{13}\,b_{24}\,b_{34} - b_{34}{}^2 + b_{12}{}^2\,b_{34}{}^2,\\ t^3_4 &=& 2\,b_{13}\,b_{14} + 2\,b_{12}\,b_{14}\,b_{23} - 2\,b_{12}\,b_{13}\,b_{24} - 2\,b_{23}\,b_{24} - 2\,b_{34} + 2\,b_{12}{}^2\,b_{34},\\ t^4_1 &=& 2\,b_{14} + 2\,b_{14}\,b_{23}{}^2 + 2\,b_{12}\,b_{24} - 2\,b_{13}\,b_{23}\,b_{24} + 2\,b_{13}\,b_{34} + 2\,b_{12}\,b_{23}\,b_{34},\\ t^4_2 &=& 2\,b_{12}\,b_{14} + 2\,b_{13}\,b_{14}\,b_{23} + 2\,b_{24} - 2\,b_{13}{}^2\,b_{24} + 2\,b_{12}\,b_{13}\,b_{34} + 2\,b_{23}\,b_{34},\\ t^4_3 &=& 2\,b_{13}\,b_{14} - 2\,b_{12}\,b_{14}\,b_{23} + 2\,b_{12}\,b_{13}\,b_{24} - 2\,b_{23}\,b_{24} + 2\,b_{34} - 2\,b_{12}{}^2\,b_{34},\\ t^4_4 &=& 1 - b_{12}{}^2 - b_{13}{}^2 + b_{14}{}^2 + b_{23}{}^2 + b_{14}{}^2\,b_{23}{}^2 - 2\,b_{13}\,b_{14}\,b_{23}\,b_{24} -\\&& b_{24}{}^2 + b_{13}{}^2\,b_{24}{}^2 + 2\,b_{12}\,b_{14}\,b_{23}\,b_{34} - 2\,b_{12}\,b_{13}\,b_{24}\,b_{34} - b_{34}{}^2 + b_{12}{}^2\,b_{34}{}^2.\end{aligned}$$ If $S$ has the form (\[S:Badj\]), $\epsilon(1+\beta)\geq 0$ and $B\wedge B=0$, then elements of the matrix $P$ have the form $$\begin{aligned} p^1_1 &=& -1 + 2\,b_{23}{}^2 + 2\,b_{24}{}^2 + 2\,b_{34}{}^2,\\ p^1_2 &=& 2\,b_{13}\,b_{23} + 2\,b_{14}\,b_{24} + 2\,b_{34}\,\sqrt{\rho},\\ p^1_3 &=& -2\,b_{12}\,b_{23} + 2\,b_{14}\,b_{34} - 2\,b_{24}\,\sqrt{\rho},\\ p^1_4 &=& -2\,b_{12}\,b_{24} - 2\,b_{13}\,b_{34} + 2\,b_{23}\,\sqrt{\rho},\\ p^2_1 &=& -2\,b_{13}\,b_{23} - 2\,b_{14}\,b_{24} + 2\,b_{34}\,\sqrt{\rho},\\ p^2_2 &=& -1 - 2\,b_{13}{}^2 - 2\,b_{14}{}^2 + 2\,b_{34}{}^2,\\ p^2_3 &=& 2\,b_{12}\,b_{13} - 2\,b_{24}\,b_{34} + 2\,b_{14}\,\sqrt{\rho},\\ p^2_4 &=& 2\,b_{12}\,b_{14} + 2\,b_{23}\,b_{34} - 2\,b_{13}\,\sqrt{\rho},\\ p^3_1 &=& 2\,b_{12}\,b_{23} - 2\,b_{14}\,b_{34} - 2\,b_{24}\,\sqrt{\rho},\\ p^3_2 &=& 2\,b_{12}\,b_{13} - 2\,b_{24}\,b_{34} - 2\,b_{14}\,\sqrt{\rho},\\ p^3_3 &=& -1 - 2\,b_{12}{}^2 - 2\,b_{14}{}^2 + 2\,b_{24}{}^2,\\ p^3_4 &=& 2\,b_{13}\,b_{14} - 2\,b_{23}\,b_{24} + 2\,b_{12}\,\sqrt{\rho},\\ p^4_1 &=& 2\,b_{12}\,b_{24} + 2\,b_{13}\,b_{34} + 2\,b_{23}\,\sqrt{\rho},\\ p^4_2 &=& 2\,b_{12}\,b_{14} + 2\,b_{23}\,b_{34} + 2\,b_{13}\,\sqrt{\rho},\\ p^4_3 &=& 2\,b_{13}\,b_{14} - 2\,b_{23}\,b_{24} - 2\,b_{12}\,\sqrt{\rho},\\ p^4_4 &=& -1 - 2\,b_{12}{}^2 - 2\,b_{13}{}^2 + 2\,b_{23}{}^2,\end{aligned}$$ where $$\rho =\epsilon(1+\beta)= -1 - b_{12}{}^2 - b_{13}{}^2 - b_{14}{}^2 + b_{23}{}^2 + b_{24}{}^2 + b_{34}{}^2\geq 0.$$ [99]{} Grassmann G., Die lineare Ausdehnungslehre. Leipzig: Wiegand (1844). English translation, 1995, by Lloyd Kannenberg, A new branch of mathematics. Chicago: Open Court. Marchuk N.G., Field theory equations and Clifford algebras, (in Russian), RCD (2009). Lounesto P., [Clifford Algebras and Spinors]{}, Cambridge Univ. Press (1997, 2001). Dieudonne J., La geometrie des groupes classiques, Springer-Verlag (1971). [^1]: This work was partially supported by the grant of the President of the Russian Federation (project NSh-3224.2008.1) and by Division of mathematics of RAS (project “Modern problems in theoretical mathematics”).
--- abstract: 'We have simulated the optical properties of micro-fabricated Fresnel zone plates (FZPs) as an alternative to spatial light modulators (SLMs) for producing non-trivial light potentials to trap atoms within a lensless Fresnel arrangement. We show that binary (1-bit) FZPs with wavelength ($1\,\mu$m) spatial resolution consistently outperform kinoforms of spatial and phase resolution comparable to commercial SLMs in root mean square error comparisons, with FZP kinoforms demonstrating increasing improvement for complex target intensity distributions. Moreover, as sub-wavelength resolution microfabrication is possible, FZPs provide an exciting possibility for the creation of static cold-atom trapping potentials useful to atomtronics, interferometry, and the study of fundamental physics.' address: 'Dept. of Physics, SUPA, University of Strathclyde, Glasgow, G4 0NG, UK' author: - 'V A Henderson, P F Griffin, E Riis and A S Arnold$^{1}$' title: Comparative simulations of Fresnel holography methods for atomic waveguides --- Introduction ============ Atom interferometry is a powerful tool for precise measurements and metrological technologies. It can be used for a wide range of applications, from the determination of fundamental constants and cosmological phenomena [@Burrage2015; @Gupta2002] to navigation applications such as accelerometers and gyroscopes [@Barrett2014; @Cronin2009; @Gustavson1997]. Developments in laser cooling, trapping and atom manipulation have allowed a wide range of atom interferometers to be developed [@Burrage2015; @Gupta2002; @Barrett2014; @Cronin2009; @Gustavson1997; @Pritchard2012; @Zawadzki2010; @Arnold2006], and for the exploration of light based atom traps [@Nshii2013; @Schonbrun2008; @McGilligan2015; @Bruce2011; @Trypogeorgos2013; @Gaunt2012; @Pasienski2008]. Optical traps can offer a method of production for much more complex micrometer scale traps such as atomtronic optical circuits [@Seaman2007]. Toroidal trapping of cold atoms for use as atom circuits has many applications beyond interferometry [@Halkyard2010; @Marti2015], such as the study of persistent currents in superfluids [@Ramanathan2011; @Wright2013; @Murray2013], and low-dimensional atomic systems [@lowd; @lowd2]. However, trapping ultra-cold atoms requires a very smooth trap, as the presence of very small perturbations in a potential can result in heating of a cold atom cloud or fragmentation of a trapped Bose-Einstein condensate [@Fortagh2002]. Within previous demonstrations of all-optical ring trapped BECs, the azimuthal variation of the ring minimum was far below the chemical potential of the BEC, with these rings produced through a variety of methods such as painted potentials [@Ryu2013] or combinations of confining light sheets with shaped light, for instance, Laguerre-Gaussian beams [@Ramanathan2011; @Wright2013; @Murray2013; @Beattie2013], co-axial focused beams [@Marti2015], or conical refraction based beams [@Turpin2015]. To successfully produce trapping potentials for BEC, we must aim to match or surpass the above limit on azimuthal variation, thus aiming to produce traps of $\mu$K depth with a roughness of below 1%. There are many methods which can be used to produce tailored optical potentials, ranging from acousto-optic beam deflection techniques [@Trypogeorgos2013; @Henderson2009] to holographic phase manipulation using a phase adjustable spatial light modulator (SLM) [@Bruce2011; @Gaunt2012; @Pasienski2008; @Bruce2015] or digital micromirror device (DMD) [@Ha2015; @Muldoon2012]. To date, the holographic method has proved to be very adaptable, paving the way for the production of novel optical lattices for quantum simulation [@Nogrette2014], dark spontaneous-force optical traps [@Radwell2013] and exotic Laguerre-Gauss modes [@Amico2005; @Franke-Arnold2007; @Arnold2012]. Despite these successes, SLM holography for atom trapping still remains an imperfect and computationally intensive technique, notwithstanding significant improvement in the iterative algorithms used [@Pasienski2008; @Gaunt2012; @Harte2014]. This is due to a combination of system aberrations, low spatial resolution, dead space between pixels, and the difficulty of creating an algorithm that converges on a solution suitable for atom trapping (i.e. smooth and without background light which could cause low loading rates or tunnelling out of the trap [@Gaunt2012]) without lowering light usage efficiency. ![a) Spherical light wave phasefronts (separated in phase by steps of $\pi$) emanating from a focused light beam form a distinctive Fresnel phase pattern when intersecting a plane. b), c) Binary transmission holograms with equivalent phase characteristics are made from refractive index $n$ material, with half-wavelength steps in optical depth $(n-1)d$. Higher bit-depths of phase resolution enable hologram blazing.\[fzp\]](Fig0.pdf){width="\textwidth"} Fresnel Zone Plates (FZPs) work by spatially modulating either the amplitude or phase of a light beam, resulting in interference of the optical field after propagation; by design of the modulated region one can in principle then produce an arbitrary optical pattern, or trapping potential for atomtronics. The prototypical FZP is one that acts as a lens, resulting in a focused spot in the selected focal plane ($z = f$). While the operation of such an FZP is standard in the teaching literature of diffraction, we find it intuitive to briefly consider the FZP required to generate a single focus, shown diagrammatically in figure \[fzp\] a). We make use of the time/direction symmetry of linear optics by starting from the desired result and finding the full electric field pattern at a defined plane. Our goal is now to create an optical element, the FZP, that matches an input beam, for example an idealised plane wave, to the field pattern that we produced in the plane. The FZP can then be considered the hologram generated by a plane wave and the backward-propagating field from the focus. For a binary FZP, we obtain a two–level map of the phases of the electric field in the plane of the FZP required to generate the desired focus. In the next section we discuss in detail the theory and numerical methods to implement this. This type of plate (Fig. \[fzp\]) consists of alternating Fresnel zones forming concentric rings that alternate between the chosen binary states at radii, $$r_j = \sqrt{j\lambda f + \frac{j^2 \lambda^2}{4}}~, \label{eq:fres_r}$$ where $j$ can take any integer value, and $\lambda$ is the wavelength of the incident light. Successive rings can be blocked, allowing only those that constructively interfere at the target plane to propagate. Alternatively, a phase shift of $\pi$ can be added to otherwise ‘destructive’ zones, increasing the useful power at the focal plane. Figures \[fzp\] b), c) demonstrate an envisaged transmissive binary FZP etched into a substrate, with consecutive zones that would be completely out of phase experiencing an increased optical path length. A similar approach can be used to make straight waveguides with a linear symmetric FZP pattern, or to create arbitrary FZP-like patterns by recording the phase of a near-field diffraction pattern. In this work we will calculate and simulate phase plate patterns, or kinoforms, for single focii, rings, and beamsplitters, as shown in figure \[fig:targets\]. These target intensity distribution have been chosen due to their applicability to cold-atom trapping and atomtronics. The single focus allows both the calculation and propagation methods to be evaluated and compared to the simplest FZP model, whereas the ring allows for comparison of this method to existing toroidal traps which are the simplest nontrivial closed-loop circuits. In order to extend the simulations to consider complex elements for atom optics we finally consider a beam splitter, as such an element is essential as a building block to create a circuit type interferometer. ![\[fig:targets\]The target intensity distributions used to simulate a range of potentials useful to atomtronics and interferometry; a), b), and c) show a focused spot, a ring and a beam-splitter, respectively. These simulation distributions are formed of Gaussians with $1/e^2$ widths of 2 $\mu$m (or 5 $\mu$m for the focus) and ring radii of $200~\mu$m, however, for visibility, the distributions shown above have a larger width and are cropped to show only the $600~\mu$m$\times 600~\mu$m area around the non-zero intensity.](Targets.pdf){width="80.00000%"} We anticipate that microfabricated FZPs will overcome many of the limitations posed by the use of SLMs in atom trapping experiments. The higher spatial resolution and sharper edges between pixels presents the ability to reach higher spatial frequency and thus produce a wider range of more accurate holograms. Additionally, due to their size and transmissive operation, we expect that FZPs should be placed inside a vacuum chamber (as with the grating MOTs shown in [@Nshii2013; @McGilligan2015]), thus immediately addressing the major system aberration of propagation through a vacuum chamber window. Further information will be discussed pertaining to the nature of FZPs in future sections. Simulation methods ================== ![\[fig:scheme\] Schematic of the kinoform, or phase plate pattern, design process used. The target (T) electric field distribution is propagated backwards a distance $f$ using Fourier techniques and maximum spatial resolution (4096$\times$4096). The electric field is spatially averaged over a variable size (larger) grid of pixels, then separated into phase ($\phi_{\rm -f}$) and amplitude ($I_{\rm -f}$) components, with the phase rounded to 1-8 bit resolution. The kinoform is then illuminated to create an image.](Creation.pdf){width="50.00000%"} The phase patterns required to produce the optical traps shown in figure \[fig:targets\] are calculated using a Fourier–optics method of modelling the propagation of an initial electric field $\mathcal{E}^{(0)} = E(x', y', z=0)$ to a distance $z$. This uses the angular spectrum of the field, ($\mathcal{A}^{(0)}$), and the Helmholtz propagator, $\mathcal{H}$, such that, $$\mathcal{E}^{(z)} = \mathcal{F}^{-1}\left [\mathcal{H}(z) \mathcal{A}^{(0)} \right ] = \mathcal{F}^{-1}\left [e^{ik_z z} \mathcal{F}\left [ \mathcal{E}^{(0)} \right ] \right ]~, \label{eq:fourier_prop}$$ where the z-component of the wave vector is $k_z = \sqrt{k^2 - k_x^2 - k_y^2}$ for an electric field with wave vector $k = 2 \pi / \lambda$ [@McDonald2015; @Novotny2012]. We use this method, following the details in [@McDonald2015], and references therein, to complete the design algorithm shown in figure \[fig:scheme\]; firstly, a target intensity is calculated and then propagated backwards, using equation \[eq:fourier\_prop\], by the focal length. The phase of the resulting electric field in this plane is rounded to the desired bit depth, as discussed later in the text. This routine acts to calculate the required kinoform, and the performance of the result is tested numerically by simulating a desired input beam (either a plane wave or a Gaussian beam with defined width) that is then propagated forward by the focal length. Our method of simulation means that the pixel sizes of the kinoform and simulation (the electric field) are independent. Although we set the input beam and target plane to have flat phase fronts, we allow for phase freedom in the resultant distribution. As we are not utilising a feedback algorithm, our method intrinsically avoids the presence of optical vortices, which can be confirmed through observations of simulation results. We consider the case in which the kinoform acts as a transmissive element and the incident light only illuminates the patterned area. It should also be noted that no optimisation is used to improve the kinoform. This full Helmholtz propagation method is computationally efficient and accurate, reducing the possibility of fringing artifacts in comparison to the paraxial approximation utilised in many hologram calculations as also highlighted in Ref. [@Gaunt2012]. To evaluate the success of each kinoform, we calculate the root mean squared (RMS) error for the normalised two dimensional final and target intensities, $$\epsilon = \sqrt{\frac{1}{N} \sum \left ( \tilde{I}- \tilde{T}\right )^2}~, \label{eq:rmserror}$$ where $N$ is the number of pixels (in the simulation), $\tilde{I}$ is the final intensity, and $\tilde{T}$ is the target intensity distribution, both intensity distributions are normalised by the mean of the pixels in $T$ that are brighter than $50\%$ of the maximum value [@Bruce2015]. The target distributions we have chosen to simulate are shown in figure \[fig:targets\]: a) a focus with Gaussian waist (e$^{-2}$ radius) $w_0=5~\mu$m; b) a ring of radius $r=200~\mu$m and radial Gaussian waist $w_r=2~\mu$m; c) a beam splitter formed from straight segments and radii as given in b), again with waist $w_b=2~\mu$m. Laser parameters of 2 mW (30 mW) power at a wavelength of 1064 nm were used for the focus (ring and beam splitter) simulations as these parameters give trap depths of a few $\mu$K. Moreover, trap frequencies are 2 kHz in the direction of tightest confinement, which is higher than existing ring shaped dipole potentials [@Marti2015; @Wright2013; @Murray2013; @Eckel2014; @Jendrzejewski2014; @Ryu2013; @Beattie2013] and permits access to lower dimensional regimes. The ring radius is larger than these previous demonstrations to increase its applicability to interferometry application where sensitivity scales with the area enclosed. Within the simulations, we run the calculations for a wide range of kinoform pixel sizes and phase resolution (or bit depth), allowing the comparison of binary FZP-type kinoforms with simulated pixel size of 1 $\mu$m to 8-bit SLM type kinoforms with simulated pixel sizes of 12 $\mu$m or more. The 12 $\mu$m pixel size corresponds to the state of the art for SLMs, which have an effective area of approximately 2 cm$^2$, whereas FZPs can be manufactured with pixel size as small as 10 nm and with large total areas of up to 25 cm$^2$ [@Nshii2013]. Despite these evident spatial advantages for FZPs, one must remember that SLMs typically operate with 8-bit precision and are updatable, whereas FZPs, by their very nature are static, with only two levels of phase control. Both technologies are already being utilised for trapping, in the form of optical tweezers [@Schonbrun2008; @Thalhammer2011]. Throughout the simulation process, the electric field propagation is calculated to a resolution of a wavelength with a simulation area of 4.38$\times$4.38 mm$^2$ ($2^{12}\,\lambda\times2^{12}\,\lambda$), limited solely by the reverse propagation technique and computation memory requirements. For illumination by Gaussian beams, the choice of the input beam e$^{-2}$ radius, $w(z)$, is determined by the desired focal length and the Gaussian width, $w_0$ of the desired features by $w(z)=w_0\sqrt{1+(z/z_{\rm R})^2}$, with Rayleigh length $z_{\rm R}=\pi\,w_0^2/\lambda$. We do note that these computation limitations mean that the active area is smaller than, if comparable to, typical SLM active areas. Results & Discussion ==================== Maps of RMS error, calculated in the simulations using equation \[eq:rmserror\], are shown in figures \[fig:RMSErrorPlane\] and \[fig:RMSErrorGaus\]. For all three target patterns and illumination beams (except the plane focus), there is a clear increase in RMS error with increasing pixel size and decreasing bit depth. The simulations also show that a two level FZP consistently has an RMS error lower than that of a kinoform comparable to an SLM. In addition, we can note that, at low pixel size, increasing the bit depth from 2 to 4 level phase resolution significantly reduces the RMS error, thus improvements in microfabrication techniques would significantly increase the accuracy of the FZP kinoforms by allowing for non-binary phase. Examples of the calculated kinoforms for FZPs illuminated by a Gaussian and producing a ring and beam splitter are shown in figure \[fig:FZPexample\]. ![\[fig:RMSErrorPlane\] Plot of RMS error for kinoforms of varying spatial and phase resolution, illuminated by plane waves. The target intensity distributions, labelled T and with a scale bar, are to the right of the corresponding RMS error plot. The obtained intensity distributions for the lowest RMS error, typical FZP, and typical SLM are labeled by the triangle, 5-point star and 7-point star respectively.](fig4.jpg){width=".95\textwidth"} ![\[fig:RMSErrorGaus\] Plot of RMS error for kinoforms of varying spatial and phase resolution, illuminated by Gaussian beams of optimised widths. The obtained intensity distributions for the lowest RMS error, typical FZP, and typical SLM are labelled by the triangle, 5-point star and 7-point star respectively, shown logarithmically. Line graphs of intensity versus radial position is shown below the full intensity plots. For the focus and ring, the area around the (symmetrical) brightest region is shown at an appropriate scale. The equivalent for the beam splitter shows the intensity distribution along the vertical line of symmetry, with the peak offset from the distribution centre indicating the position of split.](fig5.jpg){width=".85\textwidth"} ![\[fig:FZPexample\] Fresnel Zone Plates calculated for producing a ring and a beam splitter using Gaussian beam illumination (as highlighted by the 5-point star in figure \[fig:RMSErrorGaus\]). The inset shows the central section of the kinoform, magnified to allow the zone plate features to be easily seen. Note that the outer regions of the zone plates appear grey due to pixel dithering where the Fresnel zones would be smaller than a pixel. The pure black area denotes the masked area, where the plate is non-transmissive or light is blocked. The off-centre appearance of rings in the ring kinoform are artefacts of the finite simulation pixel size.](fig6a.pdf "fig:"){width=".4\textwidth"} ![\[fig:FZPexample\] Fresnel Zone Plates calculated for producing a ring and a beam splitter using Gaussian beam illumination (as highlighted by the 5-point star in figure \[fig:RMSErrorGaus\]). The inset shows the central section of the kinoform, magnified to allow the zone plate features to be easily seen. Note that the outer regions of the zone plates appear grey due to pixel dithering where the Fresnel zones would be smaller than a pixel. The pure black area denotes the masked area, where the plate is non-transmissive or light is blocked. The off-centre appearance of rings in the ring kinoform are artefacts of the finite simulation pixel size.](fig6b.pdf "fig:"){width=".4\textwidth"} The RMS error map, shown in figure \[fig:RMSErrorPlane\], for a focus kinoform illuminated by a plane wave clearly shows an unexpected increase in RMS error at high phase (high bit depth) and spatial resolution (small pixel size). In this area of higher RMS error, we observe that the optical power is concentrated in a tighter focus than the target 5 $\mu$m e$^{-2}$ radius. This can be understood because each pixel of the kinoform is illuminated equally, unlike Gaussian optics where a concomitant Gaussian illumination of the optical element is required. We can explore the consequences of this by considering three of the contributors to the RMS error: phase resolution error, spatial resolution error, and illumination error. In our algorithm all spatial intensity information from back-propagation is lost and replaced by the intensity information of the illumination beam, whereas the phase information loss is only limited by the pixel size and phase resolution. At large pixel size and low phase resolution, these sources of error dominate over the intensity error, but at high resolutions, the lost intensity information becomes dominant. It is a standard result in Gaussian optics that a smaller focus diverges more rapidly than a larger focus, meaning that the tighter the focus desired, the larger a kinoform or lens should be used, such that the numerical aperture can be increased. Conversely, this means that the size of the illuminated area of the kinoform, rather than the phase across it, affects the size of the focus produced. So, for the plane wave case, the illumination is more similar to that required for a smaller focus than 5 $\mu$m. We do not see this in the Gaussian illumination simulations, figure \[fig:RMSErrorGaus\], due to the Gaussian weighting of the intensity at the kinoform. As one can see from the RMS errors shown in figures \[fig:RMSErrorPlane\] and \[fig:RMSErrorGaus\], accuracy of intensity reproduction is reduced with pattern complexity and for distributions with less obvious symmetries: reproduction of the beam splitter is much less accurate than for either the focus and the ring. Both the ring and the focus have been masked to form circularly symmetric kinoforms, meaning that artefacts caused by the square shape of the active area are reduced, however, the reduced symmetry of the beam splitter makes this process more complex. The masking makes pixels outside of a desired area completely dark, thus creating an ‘active area’ of illuminated pixels and excluding pixels which cause abberations. In the beam-splitter case, we were able to use the symmetry properties of a straight waveguide Fresnel zone plate to shape the active area appropriately, thus blocking light incident more than a certain distance from centre of the intensity lines. This technique greatly improved accuracy, but requires further fine-tuning to allow the approach to be applied to an arbitrary intensity pattern. However, we note that if the appropriate spatial distribution of the incident field, with a flat phase front, can be produced at the kinoform, then the errors would rapidly tend to zero, as for the single focus in the upper plots of figure \[fig:RMSErrorGaus\]. Indeed, producing such a large scale pattern is well suited to the coarser resolution of an SLM, suggesting that SLMs and FZPs can be used together synergistically. In all the error maps, particularly for the Gaussian illumination, we see non-monotonic variations in the errors between consecutive pixel sizes. This is due to aliasing between the three length scales involved in the kinoform design calculations: the length scale of phase change, the simulation pixel size ($\lambda$), and the kinoform pixel size. Due to the involvement of three length scales we were not able to reduce this roughness with suitable choice of any of these values. The roughness in RMS error is less pronounced for plane wave illumination as the overall RMS error is higher and so this aliasing is less prominent. We can also note that the discontinuous nature of the example beam splitter has also increased the error in its production, this led to us using a target that reached the edges of the simulation area to avoid such issues. In a useful intensity distribution for atomtronics, one would want to produce a target intensity with no discontinuities (i.e. a closed-loop circuit), such as a ring with a beam splitter at either end for use in interferometry; hence, the discontinuity based artifacts and errors are not critical to the success of these simulations. ![\[fig:Gausringprop\] Propagation through the focus for a ring hologram generated using an FZP (left) and the best kinoform (right). The average intensity of points within 0.5$\mu$m of the ring radius is shown in the cross-section plots on the right of each image.](fig7a.pdf "fig:"){width="48.00000%"} ![\[fig:Gausringprop\] Propagation through the focus for a ring hologram generated using an FZP (left) and the best kinoform (right). The average intensity of points within 0.5$\mu$m of the ring radius is shown in the cross-section plots on the right of each image.](fig7b.pdf "fig:"){width="48.00000%"} In order to demonstrate the applications of the hologram method of optical trap generation (particularly the potential for three dimensional trapping), we have demonstrated propagation through the focus of the ring distribution in figure \[fig:Gausringprop\]. This is shown both for the best kinoform and for an FZP, with the average intensity of the ring minimum at each distance shown as a scatter plot alongside the full intensity distributions. Both cases demonstrate a full-width-half-maximum (FWHM) in the propagation direction of 20 $\mu$m, similar to that expected for a focussed Gaussian beam. Azimuthal plots of intensity are shown as line graphs in figure \[fig:RMSErrorGaus\], allowing for intensity noise to be seen. We can note that the intensity distribution in the case of the focus and ring are too narrow to show any noise due to the pixel size of the simulation, however, we can see significant noise along the vertical waveguide section of the beam splitter. The beam splitter is noise is largely due to beating between the vertical and horizontal sections of the waveguides and could be minimised with more careful target distribution design. In the simulations of RMS errors in Figures \[fig:RMSErrorPlane\], \[fig:RMSErrorGaus\] we adopted a compromise position whereby we compared both target and image distributions across the whole grid size. This means that even the background wings (i.e. non-target zone) of the intensity distribution - which could affect the atomtronic circuit loading efficiency - contribute to the error. However, for a given application one may be mainly interested in a subset of the image and target, e.g. the pixel region where the top 50% of the target intensity distribution. This region is where the coldest atoms would be trapped and in this case it makes sense to modify equation \[eq:rmserror\] to only consider pixels in this zone. Moreover, one should then adapt $\tilde{I}$ the final intensity, and $\tilde{T}$ the target intensity distribution, so that the intensity distributions are independently normalised by their maximum value over the pixels in $T$ which are brighter than $50\%$ of the maximum value. This gives a more realistic estimate of the in-situ trap roughness, which can be seen in figure \[fig:targetweighted\]. The lowest RMS error, typical FZP, and typical SLM have corresponding errors of $0.0\%,$ $3.7\,\%$ and $32.7\%,$ respectively for a ring shaped target. In this situation, rather than the plane wave/Gaussian illumination considered in Figs. \[fig:RMSErrorPlane\] and \[fig:RMSErrorGaus\], the hologram is illuminated by its ideal spatial intensity distribution – a realistic assumption we elucidate on in our conclusions. ![\[fig:targetweighted\] For the ring-shaped target, image (a) is a demonstration of how the RMS error is modified if one considers only the grid points in which the target is within $50\%$ of the maximum intensity. The target is normalised to its maximum value within this pixel range, and the image is scaled by a constant which minimises the RMS error. Note the much higher overall error, as the large background content of the image can give a false impression of pattern smoothness. The lowest RMS error, typical FZP, and typical SLM are labeled by a triangle, 5-point star and 7-point star, with corresponding errors of $0.0\%,$ $3.7\,\%$ and $32.7\%$, respectively. In image (b), for benchmarking, we consider a complex target ‘OR’ gate ($150\times 280\, \mu$m$^2$) similar to that used in Refs. [@Gaunt2012; @Pasienski2008]. Note that in this case the lowest RMS error, typical FZP, and typical SLM are labeled by a triangle, 5-point star and 7-point star, with corresponding errors of $0.0\%,$ $17.4\,\%$ and $19.0\%$, respectively. Such values appear high, however it is important to consider the small target size, and that there is no additional hologram optimisation. The phase profile across the target is flat in all cases, with no observable vortices.](fig8_ring1024_3.pdf "fig:"){width="48.00000%"} ![\[fig:targetweighted\] For the ring-shaped target, image (a) is a demonstration of how the RMS error is modified if one considers only the grid points in which the target is within $50\%$ of the maximum intensity. The target is normalised to its maximum value within this pixel range, and the image is scaled by a constant which minimises the RMS error. Note the much higher overall error, as the large background content of the image can give a false impression of pattern smoothness. The lowest RMS error, typical FZP, and typical SLM are labeled by a triangle, 5-point star and 7-point star, with corresponding errors of $0.0\%,$ $3.7\,\%$ and $32.7\%$, respectively. In image (b), for benchmarking, we consider a complex target ‘OR’ gate ($150\times 280\, \mu$m$^2$) similar to that used in Refs. [@Gaunt2012; @Pasienski2008]. Note that in this case the lowest RMS error, typical FZP, and typical SLM are labeled by a triangle, 5-point star and 7-point star, with corresponding errors of $0.0\%,$ $17.4\,\%$ and $19.0\%$, respectively. Such values appear high, however it is important to consider the small target size, and that there is no additional hologram optimisation. The phase profile across the target is flat in all cases, with no observable vortices.](fig8_or1024_3.pdf "fig:"){width="48.00000%"} Outlook and Conclusion ====================== By calculating and simulating kinoforms for focii, rings and beam splitters, we have shown that under most circumstances spatial resolution is much more critical than the bit-depth of the hologram. Specifically, we demonstrated that, in the lensless Fresnel regime, FZPs with wavelength spatial resolution consistently show improved root mean square error over kinoforms of spatial and phase resolution comparable to commercial SLMs which are typically 8—bit, with 12 $\mu$m pixels. FZP kinoforms become increasingly superior for complex target intensity distributions, indicating their suitability for use to produce static atomtronic circuits for trapping ultracold atoms. This is accompanied by the illustration of 3D trapping capabilities through propagation of a ring shaped potential through its focus. By extension of a FZP from a binary kinoform to a 4 level kinoform, the fidelity of intensity distributions can be greatly increased, showing the potential of these kinoforms to improve with increasing micro-fabrication capabilities [@Harvey2014]. Despite the success of these simulations, they are limited to wavelength resolution due to the Helmholtz propagation method used and to a size of 4.38$\times$4.38 mm$^2$ by the memory requirements of the simulation. The calculation process explicitly does not include an algorithm for iterative error correction of the kinoform meaning that both FZP and SLM RMS errors may find improvements with the use of algorithms similar to those used in [@Bruce2015; @Gaunt2012]. Future work will extend this method of kinoform calculation to include an optimisation algorithm, whilst the manufacture of potentially useful FZPs will allow for predictions to be tested experimentally. There is also great potential for combining the strengths of different techniques: a laser incident on a DMD pattern could be re-imaged onto the FZP in order to provide flat-phase front spatially-tailored intensity illumination (assumed in Fig. \[fig:targetweighted\]), with RMS errors substantially reduced beyond those from plane wave/Gaussian illumination. Whilst only red-detuned (bright) dipole potentials were considered in this paper, extension to blue-detuned (dark) traps should be straightforward, however such patterns rely more heavily on destructive interference which is likely to impinge on the smoothness of the final patterns. Additionally, FZPs should lend themselves to future extension work involving multi-wavelength hologram production following a similar approach to that shown in [@Bowman2015]. For the dataset associated with this paper please see [@doi]. 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--- abstract: 'A new type of superstring in four dimension is proposed which has the central charge 26. The Neveu Schwarz and the Ramond vacua are both tachyonic. The self energy of the scalar tachyon cancels from the contribution of the fermionic loop of the Ramond sector. The NS tachyonic vacuum is used to construct a massless graviton. Coulombic vector excitations of zero mass, referred as a ‘Newtonian’ graviton are also shown to exist. The propagators are explicitly evaluated. Following the method of Weinberg, we deduce the Einstein’s field equation of general relativity.' author: - 'B. B. Deo and L. Maharana[^1]' title: 'Derivation of Einstein’s Equation from a New Type of four Dimensional Superstring' --- The string theory has come to prominence due to appearance of the graviton in the mass spectrum of Nambu-Goto string. The researches on this line has been to develop gravity from a ten dimensional superstring by writing the group invariant but nonrenormalisable supergravity Lagrangian and studying the phenomenology arising therefrom at low energies. Recently one of us [@deo] has proposed a new type of superstring having central charge 26 and two tachyons whose self energies cancel. The salient features are given below for completeness. There is an effective dimensional reduction by adding 11 fermions which are the Lorentz vectors in SO(3,1) bosonic sector, $\psi^{\mu,j}$, j=1,..,6 ; $\phi^{\mu,k}$, k=7,..,11 in the world sheet ($\tau, \sigma$), to the 4 bosonic co-ordinates $X^{\mu}(\sigma,\tau)$. The action is [@deo] $$S= -\frac{1}{2\pi}\int d^2\sigma\left [ \partial_{\alpha}X^{\mu}\partial^{\alpha}X_{\mu} -i \bar{\psi}^{\mu,j}\rho^{\alpha}\partial_{\alpha} \psi_{\mu,j} + i \bar{\phi}^{\mu,k}\rho^{\alpha}\partial_{\alpha} \phi_{\mu,k}\right ].\label{a}\\$$ The upper indices (j,k) refer to rows and the lower indices to columns. Here $\psi^{\mu,j }=\psi^{(+)\mu,j} +\psi^{(-)\mu,j}$ but\ $\phi^{\mu,k} = \phi^{(+)\mu,k} - \phi^{(-)\mu,k}$ where $+,-$ refer to the positive and the negative parts of the Majorana massless fermions. There are in all 13 metric ghosts, two bosonic and eleven fermionic. In the Lorentz metric, the fermionic sector light cone action is\ $$S_{l.c.}=\frac{i}{2\pi}\;\int d^2\sigma \sum_{\mu =0,3}\left( \bar{\psi}^{\mu j}\rho^ {\alpha}\partial_{\alpha}\psi_{\mu j} - \bar{\phi}^{\mu k}\rho^ {\alpha}\partial_{\alpha}\phi_{\mu k}\right)$$\ contributes eleven to the central charge like the super conformal Faddeev-Popov ghosts. They will be eliminated from the Fock space by the susidiary conditions. $\rho$-matrices are given in [@deo] and [@gr] along with other objects. The action (\[a\]) is invariant under the transformations $$\begin{aligned} \delta X^{\mu} =\bar{\epsilon}\;(\;e^j\;\psi^{\mu}_j - e^k\;\phi^{\mu}_k),\\ \delta\psi^{\mu,j}= - i\;e^j\;\rho^{\alpha}\;\partial_{\alpha}X^{\mu}\;\epsilon,\\ \delta\phi^{\mu,k}= ie^k\;\rho^{\alpha}\partial_{\alpha}X^{\mu}\;\epsilon,\end{aligned}$$ with $\epsilon$ as the constant anticommutating spinor. ${e}^j$ and ${e}^k$ are the two unit component c-number row vectors with the property that ${e}^j{e}_{j'}=\delta^j_{j'}$ and ${e}^k{e}_{k'}=\delta^k_{k'}$ and it follows that ${e}^j{e}_{j}$=6 and ${e}^k{e}_{k}$=5. The commutator of two supersymmetric transformation is a translation $$[ \delta_1 ,\delta_2 ] X^{\mu} = a^{\alpha}\partial_{\alpha}X^{\mu},$$ where $a^{\alpha}=2i\bar{\epsilon}_1\rho^{\alpha}\epsilon_2$. Similar are the results for the spinors $\psi^{\mu,j}= e^j\Psi^{\mu}$ and $\phi^{\mu,k}= e^k\Psi^{\mu}$ where $$\Psi^{\mu}=e^j\;\psi^{\mu}_j - e^k\;\phi^{\mu}_k.$$ It is easy to verify that $\delta X^{\mu} =\bar{\epsilon}\Psi^{\mu}$, $\delta\Psi^{\mu}=-i\rho^{\alpha} \partial_{\alpha}X^{\mu}\epsilon $ and $[ \delta_1 ,\delta_2 ] \Psi^{\mu} = a^{\alpha}\partial_{\alpha}\Psi^{\mu}$. Thus $\Psi^{\mu}$ is the supersymmetric partner of $X^{\mu}$. Introducing another supersymmetric pair, the zweibein $e_{\alpha}(\sigma,\tau)$ and the ‘gravitino’ $\chi_{\alpha}=\nabla_{\alpha}\;\epsilon$, the local 2d supersymmetric action is [@gr] $$S= -\frac{1}{2\pi}\int d^2\sigma ~e~\left [ h^{\alpha\beta}\partial_{\alpha}X^{\mu } \partial_{\beta}X_{\mu } -i\bar \Psi^{\mu}\rho^{\alpha}\nabla_{\alpha} \Psi_{\mu}+ 2\bar{\chi}_{\alpha}\rho^{\beta}\rho^{\alpha}\Psi^{\mu} \partial_{\beta}X_{\mu}+\frac{1}{2} \bar{\Psi }^{\mu}\Psi_{\mu}\bar{\chi}_{\beta} \rho^{\beta}\rho^{\alpha}\chi_{\alpha}. \right ] \label{s1}.$$ The variation of this equation with respect to $\chi^{\alpha}$ and $e^{\alpha}_b$ leads to the equations for current and energy momentum tensors. In the gauge where the gravitino $\chi_{\alpha}$ wii be zero, and $e^a_{\alpha} = \delta^a_{\alpha}$, $$J_{\alpha}= \frac{\pi}{2e}\frac{\delta S}{\delta \bar\chi^{\alpha}}=\frac{1}{2}\rho^{\beta} \rho_{\alpha}\bar{\Psi}^{\mu} \partial_{\beta}X_{\mu} - \frac{1}{2}\bar\Psi^{\mu}\Psi_{\mu}\rho_{\alpha}\rho^{\beta} \chi_{\beta} = 0,$$ $$T_{\alpha\beta}=\partial_{\alpha}X^{\mu } \partial_{\beta}X_{\mu }- \frac{i}{2}\bar{\Psi}^{\mu}\rho_{(\alpha}\partial_{\beta )} \Psi_{\mu}-(trace)=0.$$ In the light cone basis $\tilde {\psi}=(\psi^+,\psi^-)$ and $\tilde {\phi}=(\phi^+,\phi^-)$, the above equations translate to $$J_{\pm}=\partial_{\pm}X_{\mu}\Psi^{\mu}_{\pm}=0\label{J},$$ and $$T_{\pm\pm}= \partial_{\pm}X^{\mu}\;\partial_{\pm}X_{\mu}+\frac{i}{ 2}\psi^{\mu,j }_{\pm}\;\partial_{\pm} \psi_{\pm\mu,j}- \frac{i}{2}\phi_{\pm}^{\mu k}\;\partial_{\pm}\phi_{\pm\mu,k}=0\label{t}$$ where $\partial_{\pm}=\frac{1}{2}(\partial_{\tau} \pm \partial_{\sigma})$. The component constraints are $$\begin{aligned} \partial_{\pm}X_{\mu}\;\psi_{\pm}^{\mu,j} = \partial_{\pm}X_{\mu}\; e^j\;\Psi^{\mu}_{\pm }=0,~~~~~~~~j=1,2...6.\\ \partial_{\pm}X_{\mu}\;\phi_{\pm}^{\mu,k} = \partial_{\pm}X_{\mu}\;e^k\;\Psi^{\mu} _{\pm }=0,~~~~~~~~k=7,...11.\end{aligned}$$ These eleven constraints are enough to eliminate the extra eleven Lorentz metric fermionic ghosts from physical spectrum. However, there will be one current constraint, equation (\[J\]), from which the eleven follow. So there will be eleven subsidiary current generators combinable to one and eleven pairs of $(\beta^j,~\gamma^j)$, $(\beta^k,~\gamma^k)$ ghosts combinable to one pair $(\beta,~\gamma)$ for the construction of nilpotent BRST charge. We write the action $S_F^{l.c.}$ in terms of light cone superpartner $\Psi^{\pm}$. The Hilbert space is supplemented by a space where particles obey bose statistics and the light cone fermions are Grassmanians $\theta, \bar{\theta}$. In particular, $\bar{\Psi}^+(\bar{\theta})=\frac{1}{2}\bar{\theta} \bar{\gamma}$ and $\rho^{\alpha} \bar{\Psi}^-(\theta)=\frac{1}{2}\theta\beta^{\alpha}$. Integrating over $\theta\bar\theta$, we obtain $S_F^{l.c.}=-\frac{1}{2\pi}\int d^2\sigma \bar{\gamma}\partial_{\alpha}\beta^{\alpha}$. Standard text book results follow. For the two groups of fermions, the group invariant constraints are $$\partial_{\pm}X_{\mu}\;e^k\;\phi^{\mu}_{\pm k} = \partial_{\pm}X_{\mu}\; e^j\;\psi^{\mu}_{\pm j}=0.\label{5}$$ In the covariant formulation, the total number of physical degrees of freedom is the total number forty four minus the total number of constraints. Due to the above four constraints ( \[5\]), there are 40 physical space time fermionic modes in the theory.\ SO(3,1) has Dirac spinorial representation denoted by $\theta_{j,\delta}$ and $\theta_{k,\delta} ,where~~\delta=1,2,3,4$. So we can construct a genuine space time spinor with four components $$\theta_{\delta}=\sum^6_{j=1}e^j\theta_{j\delta} - \sum^{11}_{k=7}e^k\theta_{k\delta}.$$ The Green Schwarz action [@gr] exhibiting local four dimensional N=1 supersymmetry is $$S=\frac{1}{2\pi}\int d^2\sigma \left ( \sqrt{g}g^{\alpha\beta}\Pi_{\alpha}\cdot\Pi_{\beta} +2i\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\bar{\theta}\Gamma_{\mu} \partial_{\beta}\theta \right ),$$ where $\Gamma_{\mu}$ are the Dirac gamma matrices and $$\Pi^{\mu}_{\alpha}=\partial_{\alpha}X^{\mu}- i\bar{\theta}\Gamma^{\mu}\partial_ {\alpha}\theta.$$ It is difficult to quantise this action, so we fall back on the Neveu-Schwarz [@ne] and the Ramond [@ra] formulations with the G.S.O [@gl] projection. The GSO operator is to project out the odd number of fermionic modes from the Hilbert space and is defined as $$G= \frac{1}{2}\left ( 1 +(-)^F\right ).$$ where F is the fermion number. The forty fermionic modes can be placed in five identical groups, each group containing eight of them. The total partition function is that of a group of eight, raised to the power of five. It has been shown by Seiberg and Witten [@se] that the partition function of eight fermions with Neveu-Schwarz and Ramond boundary conditions, vanish due to the famous Jacobi equality among the $\Theta$-functions. Thus the total product partition functions of the string states vanish. This is also the condition for a local supersymmetry.\ The commutators and anticommutators between fields follow from the action of equation(\[a\]). The fields can be quantised in the usual way [@gr]. Let the bosonic quanta be denoted by $\alpha^{\mu}_m$, the ($b^{\mu}_{r,j},\; b^{\prime\mu}_{r,k}$) with r half integral be the quanta of ($\psi^{\mu}_j $, $\phi^{\mu}_j$) satisfying NS boundary condition and ($d^{\mu}_{m,j}$ , $\; d^{\prime\mu}_{m,k}$) with m integral, satisfying Ramond boundary condition. The nonvanishing commutation and anticommutation relations are $$[\alpha_m^{\mu}, \alpha_n^{\nu}]=m\delta_{m,-n}g^{\mu\nu},$$ $$\begin{aligned} \{ b^{\mu ,j}_r , b^{\nu,j'}_s\}&=&g^{\mu\nu}\delta^{j,j'}\delta_{r,-s} ,\nonumber\\ \{ b^{'\mu ,k}_r , b^{'\nu,k'}_s\}&=& -g^{\mu\nu}\delta^{k,k'}\delta_{r,-s},\\ \{ d^{\mu ,j}_m , d^{\nu,j'}_n \}&=&g^{\mu\nu}\delta^{j,j'}\delta_{m,-n}, \nonumber\\ \{ d^{'\mu ,k}_m, d^{'\nu,k'}_n \}&=&-g^{\mu\nu}\delta^{k,k'}\delta_{m,-n}.\end{aligned}$$ The phase of the creation operator is such that for r, m $ >$ 0, $b_{-r}^{\prime} =-b_r^{\prime\dag}$ and $d^{\prime}_{-m}=-d_m^{\dag}$ It is necessary to identify the relations $b_r^{\mu,j}~=~e^jB_r^{\mu},~~ b_r^{\prime\mu,k} = e^k B_r^{\mu}$, such that $B_r^{\mu}=e^j b_{r,j}^{\mu} - e^k b_{r,k}^{\prime\mu}$. Similarly for the d, $d^{\prime}$; $D_m^{\mu}= e^j d_{m,j}^{\mu}- e^k d_{m,k}^{\prime\mu}$. The Virasoro generators $$\begin{aligned} L_m&=&\frac{1}{\pi}\int_{-\pi}^{\pi}d\sigma e^{im\sigma}T_{++}\nonumber\\ &= &\frac{1}{2}\sum^{\infty}_{-\infty}:\alpha_{-n}\cdot\alpha_{m+n}: +\frac{1}{2} \sum_{r\in z+\frac{1}{2}}(r+\frac{1}{2}m): (b_{-r} \cdot b_{m+r} - b_{-r}' \cdot b_{m+r}'):~~~~~~~~~NS \nonumber\\ &=&\frac{1}{2}\sum^{\infty}_{-\infty}:\alpha_{-n}\cdot\alpha_{m+n}: +\frac{1}{2} \sum^{\infty}_{n=-\infty}(n+\frac{1}{2}m): (d_{-n} \cdot d_{m+n} - d_{-n}' \cdot d_{m+n}'):~, ~~~~~~~R\\ G_r &=&\frac{\sqrt{2}}{\pi}\int_{-\pi}^{\pi}d\sigma e^{ir\sigma}J_{+}= \sum_{n=-\infty}^{\infty}\alpha_{-n}\cdot B_{n+r}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ NS\\ F_m &=&\sum_{-\infty}^{\infty} \alpha_{-n}\cdot D_{n+m}.~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ R\end{aligned}$$ satisfy the super Virasoro algebra [@v] $$\begin{aligned} \left [L_m , L_n\right ] & = &(m-n)L_{m+n} +\frac{C}{12}(m^3-m)\delta_{m,-n},\\ \left [L_m , G_r\right ] & = &(\frac{1}{2}m-r)G_{m+r}, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~NS\\ \{G_r , G_s\} & =& 2L_{s+r} +\frac{C}{3}(r^2-\frac{1}{4})\delta_{r,-s},\\ \left [L_m , F_n\right ] & = & (\frac{1}{2}m-n)F_{m+n},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~R\\ \{F_m, F_n\} & = & 2L_{m+n} +\frac{C}{3}(m^2-1)\delta_{m,-n},\;\;\;\; m\neq 0.\end{aligned}$$ Since, from equation (\[t\]), $<T_{\pm\pm}(z)T_{\pm\pm}(\omega)>~=~\frac{26}{2}(z-\omega)^{-4}$ + ......, the central charge C=26. It is worth while to note that $<T_{\pm\pm}^{l.c.}(z)T_{\pm\pm}^{l.c.}(\omega)>~=~\frac{11}{2}(z-\omega)^{-4}$ with central charge 11. The terms, containing the central charge C=26, are the anomaly terms due to the normal ordering [@gr]. As is well known, they are cancelled by the contribution from the conformal ghosts (b,c). This is also known that all anomalies will cancel if the normal ordering constant of $L_o$ is equal to one. We define the physical states as satisfying $$\begin{aligned} (L_o-1)\|\phi>&=&0,~~~L_m\|\phi>=0,~~~G_r\|\phi>=0~~~ for~~~~ r,m>0,\;\;\;\; :NS\;\;\;Bosonic\\ L_m\|\psi>&=&F_m\|\psi>=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; for\;\;\;\;\; m>0,\;\;\;\;\;\;\;\; ~~:R\;\;\;\;Fermionic\\ (L_o-1)\|\psi>_{\alpha}&=&(F_o^2-1)\|\psi>_{\alpha}=0.\end{aligned}$$ So we have $$(F_o +1)\|\psi_+>_{\alpha}=0\;\;\;\; and\;\;\; (F_o-1)\|\psi_->_{\alpha}=0$$ These conditions shall make the string model ghost free. It can be seen in a simple way. Applying $L_o$ condition the state $\alpha_{-1}^{\mu}\|0,k>$ is massless and the $L_1$ constraint gives the Lorentz condition $k^{\mu}\|0,k>=0$ implying a transverse photon and Gupta Bleuler impose that $\alpha_{-1}^0\|\phi>=0$. Applying $L_2,\;L_3\; ....$, constraints, one obtains $\alpha_{m}^0\|\phi>=0$. Further, since $[\alpha_{-1}^0 , G_{r+1}]\|\phi>=0$,$\;\; B_r^0\|\phi>=0$ and $b^0_{rj}\|\phi>=e_{j}B^0_r\|\phi>=0$; $b^{\prime 0}_{rk}\|\phi>= e_{k}B^0_r\|\phi>=0$. All the time components are eliminated from Fock space. From fourier transforms and defination, the eleven subsidiary physical state conditions are $$\begin{aligned} G^j_r\|\phi>&=&~~e^j G_r\|\phi>=0,~~~G_r^k\|\phi>=e^k G_r\|\phi> = 0,~~~ for~~~~ r~~>~~0\label{gjr},\;\;\;\; NS\;\;\;\\ F^j_m\|\phi>&=&~~e^j F_m\|\phi>=0,~~~F_m^k\|\phi>=e^k F_m\|\phi> = 0,~~~ for~~~~ m~~>~~0.\label{fjm}\;\;\;\; R\;\;\;\end{aligned}$$ We now proceed to write the nilpotent BRST charge. The part which comes from the usual conformal Lie algebra technique is $$(Q_1)^{NS,R} = \sum (L_{-m}c_m)^{NS,R} -\frac{1}{2}\sum (m-n) : c_{-m}c_{-n}b_{m+n}:~ - ~a~ c_0 ;~~~~~Q^2_1=0 ~~~for ~a=1.$$ The eleven pairs of commuting ghost quanta ($\beta_r,\gamma_r$) of ghost fields ($\beta(\tau),\gamma(\tau)$), satisfying $d\gamma = d\beta = 0$, needed for subsidiary current generator operator conditions (\[gjr\]) and (\[fjm\]), are related as $$\begin{aligned} \beta^j_r& =&~~e^j \beta_r,~~~\beta_r^{'k} = e^k \beta_r, ~~~\beta_r = e^j \beta_{r,j}- e^k \beta'_{r,k}\label{b},\\ \gamma^j_r &=&~~e^j \gamma_r,~~~\gamma_r^{'k} = e^k \gamma'_r, ~~~\gamma_r = e^j \gamma_{r,j}- e^k \gamma'_{r,k}.~~~~(NS)\end{aligned}$$ There are identical ones for the Ramond sector with half integral ‘r’ replaced by integral ‘m’ . The light cone ghost quanta satisfy $\{ b_r^{+,j}, b_s^{-,l}\}=-\delta_{r,-s}\delta^{j,l}$ and $\{ b_r^{'+,k}, b_s^{'-,m}\}=-\delta_{r,-s}\delta^{k,m}$, where as $[\gamma_s^j, \beta_r^{j'}]=\delta_{r,-s}\delta^{j,j'}$ and $[ \gamma_s^k, \beta_r^{k'}]=\delta_{r,-s}\delta^{k,k'}$ and it follows that $[ \gamma_s, \beta_r ]=\delta_{r,-s}$. All that we want to know is the conformal dimensions, $\gamma$ with $-\frac{1}{2}$, and $\beta$ with $\frac{3}{2}$, $$[~L_m, \gamma_r^{(j,k)}~]=-(\frac{3}{2}m +n)\gamma^{(j,k)}_{r+m},~~~~~~~ [~L_m, \beta_r^{(j,k)}~]=(\frac{1}{2}m -n)\beta^{(j,k)}_{r+m}.$$ Using the Graded Lie algebra, we get the additional BRST charge. $$\begin{aligned} Q_{NS}'&=&\sum G_{-r}\gamma_r -\sum\gamma_{-r}\gamma_{-s}b_{r+s},\nonumber\\ Q_{R}'&=&\sum F_{-m}\gamma_m -\sum\gamma_{-m}\gamma_{-n}b_{n+m}.\end{aligned}$$ It is to be noted that the products $$G_{-r}\gamma_{-r}=G^j_{-r}\gamma_{-r,j}- G^k_{-r}\gamma_{-r,k},$$ so that all the eleven pairs of ghosts are present in the charge. As constructed, the BRST charge $$Q_{BRST}=Q_1 + Q',$$ is such that $Q_{BRST}^2=0$ in both NS and R sector [@deo]. In proving $\{Q',Q'\} + 2\{Q_1,Q'\}=0$, we have used the fourier transforms, wave equations and integration by parts to show that $\sum\sum r^2\gamma_r\gamma_s\delta_{r,-s}=\sum_r\sum_s\gamma_r\gamma_s\delta_{r,-s}=0$. The theory is unitary and ghost free. The mass spectrum is $$\begin{aligned} NS:~~~~~~~\alpha^{\prime}M^2 &=& -1,-\frac{1}{2},0,\frac{1}{2},1,\frac{3}{2}....,\nonumber\\ R:~~~~~~~~\alpha^{\prime}M^2 &=& -1, 0, 1, 2, 3....~.\end{aligned}$$ The GSO projection eliminates the half integral masses. The scalar bosonic vacuum energy $<0\|(L^{NS}_o -1)^{-1}\|0>$ is cancelled by the fermionic energy $-<0\|(F_o -1)^{-1}(F_o +1)^{-1}\|0>$, the negative sign arising due to the normal ordering of the fermions. In both the sectors, we have the Regge trajectories $\alpha^{\prime}M^2= -1, 0, 1, 2, 3....$ .\ Thus, satisfying ourselves that we have an anomaly free, ghost free and harmless but useful tachyons, we attempt to tackle the problem of gravitational field theory from the above string theory. Let us construct tensors like $b^{\mu}_{-\frac{1}{2} i}b^{\nu}_{-\frac{1}{2} j}$. For simplicity, we drop the suffix $-\frac{1}{2}$. Consider the string state $${\bf a}^{\dag\mu\nu}(p)=\sum_{i,j} C^{ij} (b^{\mu}_i b^{\nu}_j + b^{\nu}_ib^{\mu}_j -2\eta^{\mu\nu}b^{\lambda}_i b^{\lambda}_j)\|0,p>.$$ This is symmetric and traceless. Further $L_o$ will be taken as the free Hamiltonian $H_o$ in the interaction representation. Operating on this state, one gets $L_0{\bf a}^{\dag\mu\nu}(p)=0$. So this is massless. Further $p_{\mu}{\bf a}^{\dag\mu\nu} = p_{\nu}{\bf a}^{\dag\mu\nu}=0$ if $C^{ij}=C^{ji}$. If $C^{ij}$ is symmetric, a little algebra shows that $$G_{\frac{1}{2}}{\bf a}^{\dag\mu\nu}(p)= [ G_{\frac{1}{2}}, {\bf a}^{\dag\mu\nu}(p) ]=0.$$ Thus ${\bf a}^{\dag\mu\nu}(p)$ satisfy all the physical state conditions due to Virasoro algebraic relations. This is the graviton. The commutator is $$[{\bf a}^{\mu\nu}(p), {\bf a}^{\dag\lambda\sigma}(q)] = 2\pi \|C\|^2 \delta^4(p-q).\label{c}$$ To switch over to the quantum field theory, we define the gravitational field by space time fourier transform of ${\bf a}^{\mu\nu}(p)$ $${\bf a}^{\mu\nu}(x)=\frac{1}{(2\pi)^3}\int \frac{d^3p}{\sqrt{2p^o}}\left [ {\bf a}^{\mu\nu}(p)\;e^{ipx} +{\bf a}^{\dag\mu\nu}(p)\;e^{-ipx}\right ],$$ with the commutator $$\;\;[\;\;{\bf a}^{\mu\nu}(x), {\bf a}^{\lambda\sigma}(y)\;\;]\;\; = \frac{1}{(2\pi)^3}\int\frac{d^3p}{2p^o} [\;\; e^{ip\cdot (x-y)} - e^{-ip\cdot(x- y)}\;\;]\;\;f^{\mu\nu,\lambda\sigma},\label{d}$$ where $$f^{\mu\nu,\lambda\sigma}=g^{\mu\lambda}\; g^{\nu\sigma} + g^{\nu\lambda}\; g^{\mu\sigma} - g^{\mu\nu}\; g^{\lambda\sigma}.$$ The Feynman propagator is $$\Delta^{\mu\nu,\lambda\sigma}(x -y)=<0\|\;T({\bf a}^{\mu\nu}(x)\;\; {\bf a}^{\lambda\sigma}(y)\;)\|0>\;=\;\frac{1}{(2\pi)^4}\int d^4p\;e^{i\;p\cdot(x-y)}\; \Delta^{\mu\nu,\lambda\sigma}(p),$$ where $$\Delta^{\mu\nu,\lambda\sigma}(p) = \frac{1}{2}\;\;f^{\mu\nu,\lambda\sigma}\frac{1}{p^2 - i \epsilon}.$$ This is the propagator of the graviton in the interaction representation.\ As already noted in this superstring theory there is also a massless vector following from the $L_0$ condition acting on $\alpha^{\mu}_{-1}\|0,p>$ with Lorentz relation $p^{\mu}\|0,p>$=0, due to the $L_1$ condition. With the help of a time like vector $n^{\mu}$ , we can construct another traceless second rank ‘Newtonian’(N) tensor $${\bf a}^{\dag\mu\nu}_{N,1} =\left (n^{\mu}\alpha^{\nu}_{-1} + n^{\nu}\alpha^{\mu}_{-1} - g^{\mu\nu} (n\cdot\alpha_{-1}\right )\|0,p>_{NS}.$$ There are several points which are conflicting. While forming the commutator like equation(\[d\]) ,we will arrive at a term with $g^{\mu\nu}g^{\lambda\sigma}$ which is already there in the graviton propagator. So there will be over counting. Secondly, $n\cdot\alpha_{-1}\; \sim \;\alpha_{-1}^o$ and is excluded in the Fock space, but is necessary for vanishing of the trace.\ We now proceed to construct the propagators, remembering that this is local and there is no pole at $\|{\bf p}\|^2={p_o}^2$ except in the $g^{\mu\nu}g^{\lambda\sigma}$ term $$\Delta^{\mu\nu,\lambda\sigma}_{N,1}=\frac{1}{2}\left (\;\frac{f^{\mu\nu\lambda\sigma}_N} {{\|\bf p\|}^2} - \frac{ g^{\mu\nu}\;n^{\lambda}\;n^{\sigma} + g^{\lambda\sigma}\;n^{\mu}\;n^{\nu}}{{\|\bf p\|}^2}- \frac{ g^{\mu\nu}g^{\lambda\sigma} }{ p^2 }\right ),$$ where $$f^{\mu\nu\lambda\sigma}_N = g^{\mu\lambda}\;n^{\nu\sigma} + g^{\mu\sigma}\; n^{\nu\lambda} + g^{\nu\lambda}\;n^{\mu\sigma}+ g^{\mu\lambda}\;g^{\nu\sigma} - g^{\mu\nu}\;n^{\lambda}n^{\sigma}- g^{\lambda\sigma}\;n^{\mu}n^{\nu}.$$ We have still the traceless tensor $\Pi^{\mu\nu}(p)$ [@we] and the creation operator $L_{-1}\|o,{\bf p}>$. Since\ $<o,p\| L_1\;L_{-1} \|o,p>$=2, we construct a third traceless tensor, $${\bf a}^{\dag\mu\nu}_{N,2}=\frac{1}{\sqrt{2}}\Pi^{\mu\nu}\;L_{-1}\;\|o,p>_R.$$ The propagator is simply $$\Delta^{\mu\nu,\lambda\sigma}_{N,2} = \frac{1}{2(p^2-i\epsilon)}\Pi^{\mu\nu} \Pi^{\lambda\sigma}.$$ The gradient terms do not contribute when contracted with conserved energy momentum tensor. Ignoring the gradient terms $$\Pi^{\mu\nu}\rightarrow g^{\mu\nu} +\frac{p^2}{\|{\bf p}\|^2}n^{\mu}\;n^{\nu},$$ leading to $$\Delta^{\mu\nu,\lambda\sigma}_{N,2}=\frac{1}{2(p^2-i\epsilon)}\left ( g^{\mu\nu}g^{\lambda\sigma} + (g^{\mu\nu}n^{\lambda}n^{\sigma}+n^{\mu}n^{\nu}g^{\lambda\sigma})\frac{p^2}{\|{\bf p}\|^2} +n^{\mu}n^{\nu}n^{\lambda}n^{\sigma}\frac{(p^2)^2}{\|{\bf p}\|^4}\right ).$$ The total ‘Newtonian’ graviton propagator is $$\Delta^{\mu\nu\lambda\sigma}_{N} =\frac{1}{2}\frac{ f^{\mu\nu\lambda\sigma} } { \|{\bf p}\|^2} +\frac{1}{2}n^{\mu}n^{\nu}n^{\lambda}n^{\sigma}\frac{(p^2)}{\|{\bf p}\|^4}.$$ In all $$\Delta^{\mu\nu\lambda\sigma}_{F} = \Delta^{\mu\nu\lambda\sigma}+\Delta^{\mu\nu\lambda\sigma}_{N}.$$ In space time, the fourier transformed propagator is $$\begin{aligned} \Delta^{\mu\nu\lambda\sigma}_{N}(x-y) & =&\frac{1}{(2\pi)^4}\int d^4p\;e^{ip\cdot(x-y)}\Delta^{\mu\nu\lambda\sigma}_{N}(p) \nonumber\\ &=&\frac{1}{2}\; \left [ \left ( f^{\mu\nu\lambda\sigma}_N + n^{\mu}n^{\nu}n^{\lambda}n^{\sigma}\right )\delta(x^o-y^o)\;D({\bf x}-{\bf y}) + n^{\mu}n^{\nu}n^{\lambda}n^{\sigma}\ddot{\delta}(x^o-y^o)E({\bf x}-{\bf y})\;\right ],\end{aligned}$$ where $$E({\bf x})=\frac{1}{(2\pi)^3}\int d^3q e^{i{\bf q.x}}\frac{1}{\|{\bf q}\|^4} =E(o)-\frac{\|{\bf x}\|}{4\pi},$$ and $$D({\bf x})=\frac{1}{(2\pi)^4}\int d^3q e^{i{\bf q.x}}\frac{1}{\|{\bf q}\|^2} =\frac{1}{4\pi \|{\bf x}\|}.$$ Thus we have obtained the quantum field theoretic result of Weinberg [@we] from this new string theory. $\Delta^{\mu\nu\lambda\sigma}_{N}(x-y)$ is highly divergent and to cancel this we must add to the free Hamiltonian $H_o$, an interaction Hamiltonian $H_N'(t)$ as specified below $$\begin{aligned} H_N'(t)&=&\frac{1}{2} \int\int d^3x\;d^3y \;\; (\;\; 2\theta^{\mu}_o({\bf x},t) \theta_{\mu o}({\bf y},t) - \frac{1}{2} \theta^{\mu}_{\mu}({\bf x},t) \theta_{o o}({\bf y},t)\nonumber\\ &-& \frac{1}{2} \theta_{o o}({\bf x},t) \theta_{\mu}^{\mu}({\bf y},t) - \frac{1}{2} \theta_{o o}({\bf x},t) \theta_{o o}({\bf y},t) \;\;) D({\bf x}-{\bf y}) +\frac{1}{2} \int \int d^3x\;d^3y\; \theta_{o o}({\bf x},t) \ddot{\theta}_{o o}({\bf y},t) E({\bf x}-{\bf y}).\end{aligned}$$ As Weinberg puts it, ‘this term ugly as it seems, is precisely what is needed to generate Einstein Field Equations when we pass to Heisenberg representation’. $\theta_{\mu\nu}$ is the symmetric energy momentum tensor with vanishing divergence and includes the self energy of the gravitational field, the matter field and the Pauli term. In the Heisenberg representation, the spatial components are defined as $$a_H^{ij}(x) - \frac{1}{3}\delta^{ij}\delta^{kl}a_H^{kl}(x)= U(x^o)a^{ij}(x) U^{-1}(x^o),$$ where $$U(x^o) = e^{iH_o t}\;\;\; and\;\;\; \partial_{\mu}\partial^{\mu} \; a^{ij}(x)=0,\;\;\;\;\;\; \partial_i a^{ij}(x) =0.$$ The only nonvanishing commutator is $$\left [a^{ij}(x),\; \dot{a}^{kl}(y)\right ] = i\;D^{ij,kl}({\bf x-y}).$$ To evaluate the D’s from the derived propagation function , we introduce a four vector $\hat{p}^{\mu}=(1, \hat{p})$ and note that $$\hat{p}^{\mu} = \frac{p^{\mu} + n^{\mu}(\|{\bf p}\| - p_o)}{\|{\bf p}\|}.$$ The terms containing $p^{\mu}$ when contracted with conserved currents vanish. So we have the effective equality $$\frac{n^{\mu}n^{\nu}}{\|{\bf p}\|^2} = \frac{1}{p^2} \left [ \hat{q}^{\mu}n^{\nu}+ \hat{q}^{\nu}n^{\mu}-\hat{q}^{\mu}\hat{q}^{\nu} \right ] +\;\;\; gradient \;\;\; terms.$$ With the poles at $p^2$, the Green’s functions are easily calculated. After some algebra, retaining the spatial parts, we get $$\begin{aligned} D^{ij,kl}({\bf x-y})& =&\frac{1}{2}\;\;[ \delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk}-\delta^{ij} \delta^{kl}\;\;]\delta^{(3)}({\bf x-y})\nonumber\\ &+&\frac{1}{2} [ \partial^i\partial^k\delta^{jl} + \partial^i\partial^l\delta^{jk}+ \partial^l\partial^k\delta^{il}- \partial^i\partial^j\delta^{kl}- \partial^k \partial^l\delta^{ij}] D({\bf x-y})\nonumber\\ &+&\frac{1}{2}\partial^i\partial^j\partial^k\partial^l\;E({\bf x-y}).\end{aligned}$$ The solutions to the Heisenberg field equations are easily worked out and are given by Weinberg [@we]. With $\theta_{H,kl}$ as the energy momentum tensor in the Heisenberg representation $$\partial^{\mu}\partial_{\mu}\left [\;\; a_H^{ij}({\bf x},t)-\frac{1}{3}\delta^{ij}\delta_{kl}a_H^{kl} ({\bf x},t)\;\;\right ] =-\int d^3y\;D^{ij,kl}({\bf x}-{\bf y})\theta_{H,kl}({\bf y},t).$$ With Weinberg we also invent the time derivatives from the direct contact term $$\begin{aligned} H'_N(t)&=&\frac{1}{2}\int\int \;\;d^3xd^3y\;\;(\;\; [\;\; 2\;\theta^i_o({\bf x},t) \theta_{io}({\bf y},t) -\frac{1}{2}\theta^i_i({\bf x},t)\theta_{oo}({\bf y},t)\nonumber\\ &-&\frac{1}{2}\theta_{oo}({\bf x},t)\theta_{i}^i({\bf y},t)- \frac{1}{2}\theta_{oo}({\bf x},t)\theta_{oo}({\bf y},t)\;\; ]D({\bf x-y})\nonumber\\ &+&\theta_{oo}({\bf x},t)\ddot{\theta}_{oo}({\bf y},t)E({\bf x-y})\;\;).\end{aligned}$$ and define $$\begin{aligned} &a_H^{io}({\bf x},t)&=\int d^3y\;\theta_H^{io}({\bf y},t)D({\bf x-y});\;\;\;\; \nabla^2a_H^{io}(x)= -\theta_H^{io}(x),\nonumber\\ &a_{iH}^{i}({\bf x},t)&=\frac{3}{2}\int d^3y\;\theta_H^{oo}({\bf y},t)D({\bf x-y});\;\;\;\; \nabla^2 a_{iH}^{o}(x)=-\frac{3}{2}\theta_H^{oo}(x),\nonumber\\ &a_H^{oo}({\bf x},t)&=\frac{1}{2}\int d^3y\; [ \theta_H^{oo}({\bf y},t)+ \theta^{i}_{Hi}({\bf y},t) ] D({\bf x-y}) -\frac{1}{2}\int d^3y\ddot{\theta}^{oo}({\bf y},t)E({\bf x-y}),\nonumber\\ &\nabla^2a^{oo}_H(x)& =-\frac{1}{2}\theta^{i}_{Hi}(x)-\frac{1}{2} \theta^{oo}_{H}(x) +\frac{1}{3}\ddot{a}_{Hi}^i(x) .\end{aligned}$$ Using tracelessness condition and current conservation condition $$\begin{aligned} &\partial_i a_H^{ij}(x)& = \frac{1}{2}\partial^j\theta_{Hi}^i(x),\nonumber\\ &\partial_i a_{Ho}^{i}(x)& = -\frac{2}{3}\partial_o\theta_{Hi}^i(x).\end{aligned}$$ and after some algebra we arrive at the result obtained by Weinberg $$R^{\mu\nu}_H(x)= -\theta^{\mu\nu}(x) +\frac{1}{2} g^{\mu\nu}\theta^{\lambda}_{\lambda}(x),$$ where $$R^{\mu\nu}_H(x) =\partial^{\mu}\partial_{\mu} a_H^{\mu\nu}(x)-\partial^{\mu}\partial_{\lambda}a_H^{\lambda\nu}(x) -\partial^{\nu}\partial_{\lambda}a_H^{\lambda\mu}(x)+ \partial^{\mu}\partial^{\nu}a_{H\lambda}^{\lambda}(x).$$ This equation can be put in the Einsteinian form $$R^{\mu\nu}_H(x)-\frac{1}{2}g^{\mu\nu} R_{H\lambda}^{\lambda} = -\theta^{\mu\nu}_H(x).$$ Thus the presence of the tachyons in superstring theory has provided the massless states which have led to the construction of graviton and Newtonian graviton and finally enabled us to deduce the Einstein’s field equations following Weinberg. This was first deduced by S.N Gupta [@gu].We have made a direct contact from spin 2 quanta string amplitude to the field of the graviton with the help of the tachyonic vacuum. To our knowledge, this is the first direct derivation of the Einstein’s field equation from the superstring theory following Weinberg [@we]. Since superstring theory is renormalisable,we hope that our research will help in probing further into the subtelities of Quantum Gravity. [99]{} B. B. Deo, [A new type of Superstring in four dimension]{} hep-th/0211223 and Phys. Lett. [B]{} to be published. M. B. Green, J. H. Schwarz and E. Witten,[Superstring Theory]{} Vol-I,\ Cambridge University Press, Cambridge, England(1987); A. Neveu and J. H. Schwarz, Nucl. Phys. [B31]{}(1971)86; P. Ramond, Phys. Rev. [D3]{}(1971)2415; F. Gliozzi, J .Sherck and D. Olive, Phys. Lett.[65B]{}(1976)282; N. Seiberg and E. Witten, Nucl. Phys.[B276]{}(1986)27; M.A. Virasoro, Phys. Rev. [D1]{}(1970)2933 S. Weinberg, Phys.Rev. [B138]{}(1965)988; S.N.Gupta, Proc. Phys. Soc. [A65]{}(1952)608; Phys. Rev. [90]{}(1954)1683 [^1]: E-Mail: [email protected]
--- abstract: 'In this paper, we perform a detail analysis on leptogenesis and dark matter form low scale seesaw. In the framework of $\nu$2HDM, we further introduce one scalar singlet $\phi$ and one Dirac fermion singlet $\chi$, which are charged under a $Z_2$ symmetry. Assuming the coupling of $\chi$ is extremely small, it serves as a FIMP dark matter. The heavy right hand neutrinos $N$ provide a common origin for tiny neutrino mass (via seesaw mechanism), leptogenesis (via $N\to \ell_L \Phi_\nu^,\bar{\ell}_L \Phi_\nu$) and dark matter (via $N\to \chi\phi$). With hierarchical right hand neutrino masses, the explicit calculation shows that success thermal leptogenesis is viable even for TeV scale $N_1$ with $0.4~{{\rm GeV}}\lesssim v_\nu\lesssim1$ GeV and lightest neutrino mass $m_1\lesssim 10^{-11}$ eV. In such scenario, light FIMP dark matter in the keV to MeV range is naturally expected. The common parameter space for neutrino mass, natural leptogenesis and FIMP DM is also obtained in this paper.' author: - Ang Liu - 'Zhi-Long Han' - Yi Jin - 'Fa-Xin Yang' title: Leptogenesis and Dark Matter from Low Scale Seesaw --- Introduction ============ Besides the success of standard model (SM), there are still several open questions. In particular, tiny neutrino mass, baryon asymmetry of the Universe (BAU) and dark matter (DM) are the three outstanding evidences that require physics beyond SM. The discovery of neutrino oscillations [@Fukuda:1998mi; @Ahmad:2002jz] indicate that the mass of neutrinos are at sub-eV scale, which is at least six order of magnitudes smaller than charged leptons. Known as type-I seesaw mechanism [@Minkowski:1977sc; @Mohapatra:1979ia], this extensively considered way to naturally incorporate neutrino masses is via introducing three right hand neutrinos $N$ together with high scale Majorana masses of $N$, $$-\mathcal{L}_Y\supset y\overline{L} \widetilde{\Phi} N +\frac{1}{2}\overline{N ^c}m_{N } N + {{\rm h.c.}},$$ where $\Phi$ is the SM Higgs doublet. After spontaneous electroweak symmetry breaking, neutrinos achieve masses as $$m_\nu = - \frac{v^2}{2} y~ m_{N}^{-1} y^T.$$ Typically, $m_\nu\sim\mathcal{O}(0.1)$ eV is obtained with $y\sim\mathcal{O}(1)$ and $m_N\sim\mathcal{O}(10^{14})$ GeV. Meanwhile, the heavy neutrino can also account for BAU via leptogenesis [@Fukugita:1986hr]. For canonical thermal leptogenesis with hierarchal right hand neutrinos, an upper limit on the CP asymmetry exists, thus a lower limit on mass of lightest right hand neutrino $M_1$ should be satisfied [@Davidson:2002qv], $$M_1\gtrsim 5\times10^8 ~{{\rm GeV}}\left(\frac{v}{246~{{\rm GeV}}}\right)^2.$$ Therefore, both tiny neutrino mass and leptogenesis favor high scale $N$ in type-I seesaw. However for such high scale $N$, a naturalness problem might arise [@Vissani:1997ys]. By requiring radiative corrections to the $m_\Phi^2 \Phi^\dag\Phi$ term no larger than $1~{{\rm TeV}}^2$, it is found that[@Clarke:2015gwa] $$\label{eq:M1} M_1\lesssim3\times10^7 ~{{\rm GeV}}\left(\frac{v}{246~{{\rm GeV}}}\right)^{2/3},$$ should be satisfied. Clearly, naturalness is incompatible with leptogenesis. One viable pathway to overcome this is lowering the leptogenesis scale by imposing resonant leptogenesis [@Pilaftsis:2003gt], ARS mechanism via neutrino oscillation [@Akhmedov:1998qx; @Asaka:2005pn], or from Higgs decays [@Hambye:2016sby; @Hambye:2017elz]. All the success of these scenarios depend on the degenerate mass of right hand neutrinos [@Baumholzer:2018sfb], which seems is another sense of unnatural. An alternative scenario with hierarchal right hand neutrinos is employing intrinsic low scale neutrino mass model, e.g., $\nu$2HDM [@Chao:2012pt; @Clarke:2015hta] or Scotogenic model [@Ma:2006fn; @Kashiwase:2012xd; @Kashiwase:2013uy; @Racker:2013lua; @Hugle:2018qbw; @Borah:2018uci; @Borah:2018rca; @Mahanta:2019sfo]. In this paper, we consider the $\nu$2HDM [@Ma:2000cc]. Based on previous brief discussion in Ref. [@Haba:2011ra; @Haba:2011yc; @Clarke:2015hta], we perform a detailed analysis on leptogenesis, especially focus on dealing with the corresponding Boltzmann equations to obtain the viable parameter space. On the other hand, dark matter accounts for more than five times the proportion of visible baryonic matter in our current cosmic material field. In principle, one can regard the lightest right hand neutrino $N_1$ at keV scale as sterile neutrino DM [@Dodelson:1993je; @Adhikari:2016bei; @Adulpravitchai:2015mna; @Han:2018pek]. However, various constraints leave a quite small viable parameter space [@Boyarsky:2018tvu]. Meanwhile, leptogenesis with two hierarchal right hand neutrinos is actually still at high scale [@Hugle:2018qbw; @Antusch:2011nz; @Mahanta:2019gfe]. In this paper, we further introduce a dark sector with one scalar singlet $\phi$ and one Dirac fermion singlet $\chi$, which are charged under a $Z_2$ symmetry [@Chianese:2019epo]. The stability of DM $\chi$ is protected by the $Z_2$ symmetry, therefore the tight X-ray limits can be avoided [@Boyarsky:2018tvu]. In light of the null results from DM direct detection [@Aprile:2018dbl] and indirect detection [@Ackermann:2015zua], we consider $\chi$ as a FIMP DM [@Bernal:2017kxu]. The structure of the paper is as follows. In Sec. \[SEC:TM\], we briefly introduce our model. Leptogenesis with hierarchal right hand neutrinos is discussed in Sec. \[SEC:LG\]. The relic abundance of FIMP DM $\chi$ and constraint from free streaming length are considered in Sec. \[SEC:DM\]. Viable parameter space for leptogenesis and DM is obtained by a random scan in Sec. \[SEC:CA\]. We conclude our work in Sec. \[SEC:CL\]. The Model {#SEC:TM} ========= The original TeV-scale $\nu$2HDM for neutrino mass was proposed in Ref. [@Ma:2000cc]. The model is extended by one neutrinophilic scalar doublet $\Phi_\nu$ with same quantum numbers as SM Higgs doublet $\Phi$ and three right hand heavy neutrino $N$. To forbid the direct type-I seesaw interaction $\overline{L}\tilde{\Phi}N$, a global $U(1)_L$ symmetry should be employed, under which $L_\Phi=0$, $L_{\Phi_\nu}=-1$ and $L_{N}=0$. Therefore, $\Phi_\nu$ will specifically couple to $N$, and $\Phi$ couple to quarks and charge leptons as in SM. For the dark sector, one scalar singlet $\phi$ and one Dirac fermion singlet $\chi$ are further introduced, which are charged under a $Z_2$ symmetry. Provided $m_\chi<m_\phi$, then $\chi$ serves as DM candidate. The scalar doublets could be denoted as $$\begin{aligned} \Phi=\left( \begin{array}{c} \phi^+\\ \frac{v+\phi^{0,r}+i \phi^{0,i}}{\sqrt{2}} \end{array}\right),~ \Phi_\nu=\left( \begin{array}{c} \phi^+_\nu\\ \frac{v_{\nu}+\phi^{0,r}_{\nu}+i \phi^{0,i}_{\nu}}{\sqrt{2}} \end{array}\right).\end{aligned}$$ The corresponding Higgs potential is then $$\begin{aligned} V & = & m_{\Phi}^2 \Phi^\dag \Phi + m_{\Phi_\nu}^2 \Phi^\dag_\nu \Phi_\nu +m_\phi^2 \phi^\dag\phi+\frac{\lambda_1}{2} (\Phi^\dag \Phi)^2+\frac{\lambda_2}{2} (\Phi^\dag_\nu \Phi_\nu)^2\\ \nonumber & & +\lambda_3 (\Phi^\dag \Phi)(\Phi^\dag_\nu \Phi_\nu)+\lambda_4(\Phi^\dag \Phi_\nu)(\Phi^\dag_\nu \Phi) - (\mu^2 \Phi^\dag\Phi_\nu +{{\rm h.c.}})\\ && +\frac{\lambda_5}{2} (\phi^\dag\phi)^2 + \lambda_6 (\phi^\dag \phi)(\Phi^\dag\Phi)+\lambda_7 (\phi^\dag \phi) (\Phi_\nu^\dag\Phi_\nu),\end{aligned}$$ where the $U(1)_L$ symmetry is broken explicitly but softly by the $\mu^2$ term. For the unbroken $Z_2$ symmetry, $\langle\phi\rangle=0$ should be satisfied. Meanwhile, VEVs of Higgs doublets in terms of parameters of the Higgs potential can be found by deriving the minimization condition $$\begin{aligned} v\left[m_{\Phi}^2+\frac{\lambda_1}{2} v^2 +\frac{\lambda_3+\lambda_4}{2}v_\nu^2\right]-\mu^2 v_\nu=0 \\ v_\nu\left[m_{\Phi_\nu}^2+\frac{\lambda_2}{2} v_\nu^2 +\frac{\lambda_3+\lambda_4}{2}v^2\right]-\mu^2 v=0.\end{aligned}$$ Taking the parameter set $$m_{\Phi}^2<0, m_{\Phi_\nu}^2>0, \|\mu^2\|\ll m_{\Phi_\nu}^2,$$ we can obtain the relations of VEVs as $$v\simeq \sqrt{\frac{-2 m_{\Phi}^2}{\lambda_1}}, v_\nu \simeq \frac{\mu^2 v}{m_{\Phi_\nu}^2+(\lambda_3+\lambda_4)v^2/2}.$$ Typically, $v_\nu\sim1$ GeV is obtained with $\mu\sim10~{{\rm GeV}}$ and $m_{\Phi_\nu}\sim100$ GeV. Since $\mu^2$ term is the only source of $U(1)_L$ breaking, radiative corrections to $\mu^2$ are proportional to $\mu^2$ itself and are only logarithmically sensitive to the cutoff [@Davidson:2009ha]. Thus, the VEV hierarchy $v_\nu\ll v$ is stable against radiative corrections [@Morozumi:2011zu; @Haba:2011fn]. After SSB, the physical Higgs bosons are given by [@Guo:2017ybk] $$\begin{aligned} H^+=\phi^+_\nu\cos\beta-\phi^+\sin\beta&,~& A=\phi^{0,i}_\nu\cos\beta-\phi^{0,i}\sin\beta, \\ H=\phi^{0,r}_\nu\cos\alpha-\phi^{0,r}\sin\alpha&,~& h=\phi^{0,r}\cos\alpha+\phi^{0,r}\sin\alpha,\end{aligned}$$ where the mixing angles $\beta$ and $\alpha$ are determined by $$\label{mix} \tan\beta=\frac{v_\nu}{v},~\tan2\alpha\simeq2\frac{v_\nu}{v} \frac{-\mu^2+(\lambda_3+\lambda_4)vv_\nu}{-\mu^2+\lambda_1 vv_\nu}.$$ Neglecting terms of $\mathcal{O}(v_\nu^2)$ and $\mathcal{O}(\mu^2)$, masses of the physical Higgs bosons are $$\begin{aligned} m_{H^+}^2\simeq m_{\Phi_\nu}^2\!+\frac{1}{2}\lambda_3v^2,~m_A^2\simeq m_H^2\simeq m_{H^+}^2\!+\frac{1}{2}\lambda_4v^2,~ m_h^2 \simeq \lambda_1 v^2.\end{aligned}$$ Since the mixing angles are suppressed by the small value of $v_\nu$, $h$ is almost identically to the $125$ GeV SM Higgs boson [@Aad:2012tfa; @Chatrchyan:2012xdj]. A degenerate mass spectrum of $\Phi_\nu$ as $m_{H^+}\!=\!m_{H}\!=\!m_A\!=\!m_{\Phi_\nu}$ is adopted in our following discussion for simplicity, which is certainly allowed by various constraints [@Machado:2015sha]. Due to the unbroken $Z_2$ symmetry, the dark scalar singlet $\phi$ do not mix with the Higgs doublets. The new Yukawa interaction and mass terms are $$\label{yuk} -\mathcal{L}_Y\supset y\overline{L} \widetilde{\Phi}_\nu N +\lambda \bar{\chi}\phi N +\frac{1}{2}\overline{N ^c}m_{N } N + m_\chi \bar{\chi} \chi+ {{\rm h.c.}},$$ where $\widetilde{\Phi}_\nu=i\sigma_2 \Phi_\nu^$. Similar to the canonical Type-I seesaw [@Minkowski:1977sc], the mass matrix for light neutrinos can be derived from Eq. (\[yuk\]) as: $$\label{eq:mv} m_\nu = - \frac{v_\nu^2}{2} y~ m_{N}^{-1} y^T = U_{\text{PMNS}}\, \hat{m}_\nu U^T_{\text{PMNS}},$$ where $\hat{m}_\nu=\mbox{diag}(m_1,m_2,m_3)$ is the diagonalized neutrino mass matrix, and $U_{\text{PMNS}}$ is the PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix: $$\begin{aligned} U_{\text{PMNS}}\! =\! \left( \begin{array}{ccc} c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{i\delta}\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{-i\delta} & c_{12}c_{23}-s_{12}s_{23}s_{13} e^{-i\delta} & s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{-i\delta} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{-i\delta} & c_{23}c_{13} \end{array} \right)\!\times\! \text{diag}(e^{i \varphi_1/2},1,e^{i\varphi_2/2})\end{aligned}$$ Here, we use abbreviations $c_{ij}=\cos\theta_{ij}$ and $s_{ij}=\sin\theta_{ij}$, $\delta$ is the Dirac phase and $\varphi_1,\varphi_2$ are the two Majorana phases. Due to smallness of $v_\nu$, TeV scale $m_{N}$ could be viable to realise $0.1$ eV scale light neutrino masses. Using the Casas-Ibarra parametrization [@Casas:2001sr; @Ibarra:2003up], the Yukawa matrix $y$ can be expressed in terms of neutrino oscillation parameters $$\label{eq:CI} y=\frac{\sqrt{2}}{v_\nu}U_{\text{PMNS}}\hat{m}_\nu^{1/2} R (\hat{m}_{N})^{1/2},$$ where $R$ is an orthogonal matrix in general and $\hat{m}_N=\mbox{diag}(M_1,M_2,M_3)$ is the diagonalized heavy neutrino mass matrix. In this work, we parameterize matrix $R$ as $$\begin{aligned} R = \left( \begin{array}{ccc} \cos\omega_{12} & -\sin\omega_{12} & 0 \\ \sin\omega_{12} & \cos\omega_{12} & 0 \\ 0 & 0& 1 \end{array} \right) \left( \begin{array}{ccc} \cos\omega_{13} & 0& -\sin\omega_{13} \\ 0 & 1 & 0 \\ \sin\omega_{13} & 0& \cos\omega_{13} \end{array} \right) \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos\omega_{23} & -\sin\omega_{23} \\ 0 & \sin\omega_{23}& \cos\omega_{23} \end{array} \right),\end{aligned}$$ where $\omega_{12,13,23}$ are arbitrary complex angles. Leptogenesis {#SEC:LG} ============ Now we consider the leptogenesis in this model. The lepton asymmetry is generated by the out-of-equilibrium CP-violating decays of right hand neutrino $N\to \ell_L \Phi_\nu^,\bar{\ell}_L \Phi_\nu$. Neglecting the flavor effect [@Nardi:2006fx], the CP asymmetry is given by $$\epsilon_i=\frac{1}{8\pi (y^\dag y)_{ii}} \sum_{j\neq i} \text{Im}[(y^\dag y)^2_{ij}] G\left(\frac{M_j^2}{M_i^2},\frac{m_{\Phi_\nu}^2}{M_i^2}\right),$$ where the function $G(x,y)$ is defined as [@Mahanta:2019gfe] $$G(x,y)=\sqrt{x}\left[\frac{(1-y)^2}{1-x}+1+\frac{1-2y+x}{(1-y^2)^2} \ln \left(\frac{x-y^2}{1-2y+x}\right)\right].$$ Using the parametrization of Yukawa coupling $y$ in Eq. (\[eq:CI\]), it is easy to verify $$\label{eq:yy} y^\dag y = \frac{2}{v_\nu^2} \hat{m}_{N}^{1/2} R^\dag \hat{m}_\nu R \hat{m}_{N}^{1/2}.$$ Hence, the matrix $y^\dag y$ does not depend on the PMNS matrix, which means that the complex matrix $R$ is actually the source of CP asymmetry $\epsilon_i$. The asymmetry is dominantly generated by the decay of $N_1$. Further considering the hierarchal mass spectrum $m_{\Phi_\nu}^2\ll M_1^2\ll M_{2,3}^2$, the asymmetry $\epsilon_1$ is simplified to $$\epsilon_1\simeq -\frac{3}{16\pi(y^\dag y)_{11}}\sum_{j=2,3} \text{Im}[(y^\dag y)^2_{1j}] \frac{M_1}{M_j}$$ Similar to the Davidson-Ibarra bound [@Davidson:2002qv], an upper limit on $\epsilon_1$ can be derived $$\label{eq:eps1} \|\epsilon_1\|\lesssim \frac{3}{16\pi}\frac{M_1 m_3}{v_\nu^2}.$$ Comparing with the bound in type-I seesaw, the asymmetry could be enhanced due to the smallness of VEV $v_\nu$. Therefore, low scale leptogenesis seems to be viable in the $\nu$2HDM [@Haba:2011ra; @Clarke:2015hta]. Meanwhile, the washout effect is quantified by the decay parameter $$K = \frac{\Gamma_1}{H(z=1)},$$ where $\Gamma_1$ is the decay width of $N_1$, $H$ is the Hubble parameter and $z\equiv M_1/T$ with $T$ being the temperature of the thermal bath. The decay width is given by $$\Gamma_1=\frac{M_1}{8\pi}(y^\dag y)_{11}\left(1-\frac{m_{\Phi_\nu}^2}{M_1^2}\right)^2,$$ and the Hubble parameter is $$H=\sqrt{\frac{8\pi^3 g_}{90}}\frac{T^2}{M_{pl}}=H(z=1)\frac{1}{z^2},$$ with $g_$ the effective number of relativistic degrees of freedom and $M_{pl}=1.2\times10^{19}$ GeV. Using Eq. (\[eq:yy\]), one can verify $$K\simeq 897 \left(\frac{v}{v_\nu}\right)^2 \frac{(\hat{m}_\nu^R)_{11}}{{{\rm eV}}},$$ where $\hat{m}_\nu^R\equiv R^\dag \hat{m}_\nu R$, and thus $$\label{eq:mvR} (\hat{m}_\nu^R)_{11}=m_1 \|\cos\omega_{12}\|^2\|\cos\omega_{13}\|^2+m_2 \|\sin\omega_{12}\|^2\|\cos\omega_{13}\|^2+m_3\|\sin\omega_{13}\|^2.$$ It is obvious that the decay parameter $K$ does not depend on $\omega_{23}$, and it is also enhanced by smallness of $v_\nu$. Since $(\hat{m}_\nu^R)_{11}$ is typically of the order of $m_3\sim0.1$ eV, the decay parameter $K\simeq 5.4\times10^6$ when $v_\nu=1$ GeV. So even with maximum asymmetry $\epsilon_1^\text{max}\sim-6.0\times10^{-7}$ for $M_1=10^5$ GeV obtained from Eq. , a rough estimation of final baryon asymmetry gives $Y_{\Delta B}\sim-10^{-3} \epsilon_1^\text{max}/K\sim1.1\times10^{-16}$ for strong washout [@Davidson:2008bu], which is far below current observed value $Y_{\Delta B}^\text{obs}=(8.72\pm0.04)\times10^{-11}$ [@Aghanim:2018eyx]. Hence, only obtaining an enhanced CP asymmetry $\epsilon_1$ is not enough, one has to deal with the washout effect more carefully. ![Decay parameter $K$ as a function of $m_1$ with $v_\nu=10$ GeV. Because $R$ must be a complex matrix, we have set $\omega_{ij R}=\omega_{ij I}$.[]{data-label="FIG:K-m1"}](K-m1-1.pdf "fig:"){width="0.45\linewidth"} ![Decay parameter $K$ as a function of $m_1$ with $v_\nu=10$ GeV. Because $R$ must be a complex matrix, we have set $\omega_{ij R}=\omega_{ij I}$.[]{data-label="FIG:K-m1"}](K-m1-2.pdf "fig:"){width="0.45\linewidth"} One promising pathway is to reduce the decay parameter $K$. For instance, if weak washout condition $K\lesssim1$ is realised, then $Y_{\Delta B}\sim-10^{-3} \epsilon_1^\text{max}\sim 6.0\times10^{-10}>Y_{\Delta B}^\text{obs}$. Thus, correct baryon asymmetry can be obtained by slightly tunning $\epsilon_1$. As pointed out in Ref. [@Hugle:2018qbw], small value of $K$ can be realised by choosing small $\omega_{12,13}$. In Fig. \[FIG:K-m1\], we illustrate the dependence of $K$ on lightest neutrino mass $m_1$ with $v_\nu=10$ GeV. The left panel shows the special case $\omega_{13}=0$, where Eq. is simplified to $(\hat{m}_\nu^R)_{11}=m_1 \|\cos\omega_{12}\|^2+m_2\|\sin\omega_{12}\|^2\geq \sqrt{\Delta m_{21}^2} \|\sin\omega_{12}\|^2$. It is clear that the weak washout condition $K<1$ favors $\|\omega_{12}\|\lesssim10^{-2}$ and $m_1\lesssim10^{-6}$ eV. The right panel shows the special case $\omega_{12}=0$. Similar results are observed with left panel. On the other hand, the $\Delta L=2$ washout processes become more significant for small $v_\nu$ [@Haba:2011ra; @Clarke:2015hta]. Notably, for low scale seesaw, the narrow width condition $\Gamma_1/M_1\ll1$ is satisfied. Therefore, the evolution of lepton asymmetry and DM abundance actually decouple from each other [@Falkowski:2011xh; @Falkowski:2017uya]. The evolution of abundance $Y_{N_1}$ and lepton asymmetry $Y_{\Delta L}$ are described by the Boltzmann equations $$\begin{aligned} \frac{dY_{N_1}}{dz} &=& - D (Y_{N_1}-Y_{N_1}^{eq}),\\ \frac{dY_{\Delta L}}{dz} &=&- \epsilon_1 D (Y_{N_1}-Y_{N_1}^{eq}) - W Y_{\Delta L}.\end{aligned}$$ The decay term is given by $$D = K z \frac{\mathcal{K}_1(z)}{\mathcal{K}_2(z)}.$$ For the washout term, two contributions are considered, i.e., $W=W_{ID}+W_{\Delta L=2}$, where the inverse decay term is $$W_{ID}= \frac{1}{4} K z^3 \mathcal{K}_1(z),$$ and the $\Delta L=2$ scattering term at low temperature is approximately [@Buchmuller:2004nz] $$W_{\Delta L=2}\simeq \frac{0.186}{z^2} \left(\frac{246~{{\rm GeV}}}{v_\nu}\right)^4 \left(\frac{M_1}{10^{10}~{{\rm GeV}}}\right) \left(\frac{\bar{m}}{{{\rm eV}}}\right)^2.$$ Here, $\bar{m}$ is the absolute neutrino mass scale, which is calculated as $$\bar{m}^2=m_1^2+m_2^2+m_3^2 =3m_1^2+ \Delta m^2_{21} + \delta m^2_{31},$$ for normal hierarchy. According to latest global fit, we use the best fit values, i.e., $\Delta m^2_{21}=7.39\times10^{-5}~{{\rm eV}}^2$ and $\delta m^2_{31}=2.525\times10^{-3}~{{\rm eV}}^2$ [@Esteban:2018azc]. For tiny lightest neutrino mass $m_1\ll10^{-2}$ eV, we actually have $\bar{m}\simeq\sqrt{\delta m_{31}^2}\sim0.05$ eV. Notably, the $\Delta L=2$ scattering term would be greatly enhanced when $v_\nu\ll v$, so this term is much more important than in vanilla leptogenesis. Then, the sphaleron processes convert the lepton asymmetry into baryon asymmetry as [@Harvey:1990qw] $$Y_{\Delta B}= \frac{28}{79} Y_{\Delta(B-L)}=-\frac{28}{51}Y_{\Delta L}.$$ ![The washout effect of $\Delta L=2$ processes. The cyan lines are the observed value $Y_{\Delta B}^\text{obs}=8.72\times10^{-11}$.[]{data-label="FIG:YB-Z"}](YB-Z-1.pdf "fig:"){width="0.45\linewidth"} ![The washout effect of $\Delta L=2$ processes. The cyan lines are the observed value $Y_{\Delta B}^\text{obs}=8.72\times10^{-11}$.[]{data-label="FIG:YB-Z"}](YB-Z-2.pdf "fig:"){width="0.45\linewidth"}\ ![The washout effect of $\Delta L=2$ processes. The cyan lines are the observed value $Y_{\Delta B}^\text{obs}=8.72\times10^{-11}$.[]{data-label="FIG:YB-Z"}](YB-Z-3.pdf "fig:"){width="0.45\linewidth"} ![The washout effect of $\Delta L=2$ processes. The cyan lines are the observed value $Y_{\Delta B}^\text{obs}=8.72\times10^{-11}$.[]{data-label="FIG:YB-Z"}](YB-Z-4.pdf "fig:"){width="0.45\linewidth"} Fig. \[FIG:YB-Z\] shows the washout effect of $\Delta L=2$ processes. In Fig. \[FIG:YB-Z\] (a), weak washout scenario is considered by fixing $K=10^{-2},\|\epsilon_1\|=10^{-6}, M_1=10^6~{{\rm GeV}}$ while varying $v_\nu=10,1,0.1~{{\rm GeV}}$. It shows that for $v_\nu=10~{{\rm GeV}}$, the $\Delta L=2$ effect is not obvious, but for $v_\nu=1~{{\rm GeV}}$, the final baryon asymmetry $Y_{\Delta B}$ is diluted by over three orders of magnitude. While for $v_\nu=0.1~{{\rm GeV}}$, the $\Delta L=2$ effect is so strong that the final baryon asymmetry is negligible. The strong washout scenario with $K=10^2, \|\epsilon_1\|=10^{-4}, M_1=10^6~{{\rm GeV}}$ and varying $v_\nu=10,1,0.1~{{\rm GeV}}$ is illustrated in Fig. \[FIG:YB-Z\] (b), where the final baryon asymmetry $Y_{\Delta B}$ for $v_\nu=1~{{\rm GeV}}$ is decreased by about six orders comparing with the case for $v_\nu=10~{{\rm GeV}}$. Therefore, the $\Delta L=2$ washout effects set a lower bound on $v_\nu$, i.e., $v_\nu\gtrsim0.3~{{\rm GeV}}$ as suggested by Ref. [@Clarke:2015hta]. Furthermore, since the $\Delta L=2$ washout term is also proportional to $M_1$, the larger $M_1$ is, the more obvious the washout effect is. The corresponding results are depicted in Fig. \[FIG:YB-Z\] (c) for the weak washout and Fig. \[FIG:YB-Z\] (d) for the strong washout. In this way, for certain value of $v_\nu$, an upper bound on $M_1$ can be obtained. For instance, when $v_\nu=1~{{\rm GeV}}$, then $M_1\lesssim10^5~{{\rm GeV}}$ should be satisfied [@Haba:2011ra]. Dark Matter {#SEC:DM} =========== In our extension of the $\nu$2HDM, the right-handed heavy neutrinos $N$ also couple with fermion singlet $\chi$ and scalar singlet $\phi$ via the Yukawa interaction. The complex Yukawa coupling coefficient $\lambda$ can lead to CP violation in $N$ decays, and eventually producing asymmetric DM $\chi$ [@Falkowski:2011xh]. Instead, we consider another interesting scenario, i.e., the FIMP case with the real coupling $\lambda \ll1$ [@Falkowski:2017uya]. In this way, the interaction of DM $\chi$ is so weak that it never reach thermalization. Its relic abundance is determined by the freeze-in mechanism [@Hall:2009bx], which is obtained by solving the following Boltzmann equation $$\begin{aligned} \label{eq:BE} \frac{dY_{\chi}}{dz} &=& D ~Y_{N_1} \text{BR}_\chi,\end{aligned}$$ where $\text{BR}_\chi$ is the branching ratio of $N_1\to \chi \phi$. Due to the FIMP nature of $\chi$, the hierarchal condition $\text{BR}_\chi\ll\text{BR}_\ell\simeq1$ is easily satisfied. The out of equilibrium condition for $N_1\to\chi\phi$ decay is $\Gamma_\chi/H(z=1)\simeq \text{BR}_\chi \Gamma_1/H(z=1)=\text{BR}_\chi K<1$. In following studies, we mainly take $\text{BR}_\chi<10^{-2}$ and $K\lesssim10$, thus the out of equilibrium condition is always satisfied. ![ Evolution of dark matter abundance with parameter $z=M_1/T$. We fix $K=10$ in the left panel and BR$_\chi=10^{-3}$ in the right panel. The dashed horizontal lines correspond to the estimated results with Eq. . DM mass $m_\chi$ is obtained by setting $\Omega_\chi h^2=0.12$ with the numerical results of $Y_\chi(\infty)$.[]{data-label="FIG:YX-Z"}](YX-Z-1.pdf "fig:"){width="45.00000%"} ![ Evolution of dark matter abundance with parameter $z=M_1/T$. We fix $K=10$ in the left panel and BR$_\chi=10^{-3}$ in the right panel. The dashed horizontal lines correspond to the estimated results with Eq. . DM mass $m_\chi$ is obtained by setting $\Omega_\chi h^2=0.12$ with the numerical results of $Y_\chi(\infty)$.[]{data-label="FIG:YX-Z"}](YX-Z-2.pdf "fig:"){width="45.00000%"} According to the above Boltzmann equation, we can estimate the asymptotic abundances of $\chi$ as [@Falkowski:2017uya] $$\label{eq:yx} Y_{\chi}(\infty)\simeq Y_{N_1}(0)\text{BR}_{\chi}\left(1 +\frac{15\pi\zeta(5)}{16\zeta(3)}K\right).$$ Then, the corresponding relic abundance is $$\label{eq:ra} \Omega_{\chi}{h^2}=\frac{m_\chi s_0 Y_\chi(\infty)}{\rho_c}h^2\simeq 0.12\times\left(\frac{m_{\chi}}{{{\rm keV}}}\right) \left(\frac{\text{BR}_{\chi}}{10^{-3}}\right)\left(0.009+\frac{K}{44}\right),$$ where $s_0=2891.2~\text{cm}^{-3}$, $\rho_c=1.05371\times10^{-5}h^2 ~{{\rm GeV}}~\text{cm}^{-3}$ [@Tanabashi:2018oca]. Typically, the observed relic abundance can be obtained with $m_\chi\sim4$ keV, $\text{BR}_\chi\sim10^{-3}$ and $K\sim10$. The evolution of DM abundances are shown in Fig. (\[FIG:YX-Z\]). It is clear that when the temperature goes down to $z=m_\chi/T\sim5$, the abundances $Y_\chi$ freeze in and keep at a constant. The left panel of Fig. (\[FIG:YX-Z\]) indicates that $m_\chi$ is inverse proportional to $\text{BR}_\chi$ when the decay parameter $K$ is a constant. For instance, sub-MeV scale light DM is obtained when $\text{BR}_\chi>10^{-6}$ with $K=10$. Right panel of Fig. (\[FIG:YX-Z\]) shows the impact of decay parameter $K$. Affected by the constant term before $K$ in Eq. , we can only conclude that the smaller the $K$ is, the larger the $m_\chi$ is. Besides, we also find that the discrepancy between the numerical and analytical results of $Y_\chi(\infty)$ increases when $K$ decreases. Therefore, we adopt the numerical result of $Y_\chi(\infty)$ for a more precise calculation in the following discussion. ![Influence of free streaming on DM mass. The red area ($r_{FS}>0.1$ Mpc), white area ($0.1~\text{Mpc}>r_{FS}>0.01$ Mpc) and blue area ($r_{FS}<0.01$ Mpc) correspond to hot, warm and cold DM scenario [@Merle:2013wta], respectively.[]{data-label="FIG:rFS"}](rFS.pdf){width="45.00000%"} The dominant constraint on FIMP DM $\chi$ comes from its free streaming length, which describes the average distance a particle travels without a collision [@Falkowski:2017uya] $$r_{FS}=\int_{a_{rh}}^{a_{eq}}\frac{\langle v\rangle}{a^2H}da \approx \frac{a_{NR}}{H_0\sqrt{\Omega_R}} \left(0.62+\ln\left(\frac{a_{eq}}{a_{NR}}\right)\right),$$ where $\langle v\rangle$ is the averaged velocity of DM $\chi$, $a_{eq}$ and ${a_{rh}}$ represent scale factors in equilibrium and reheating, respectively. We use the results $H_0=67.3~\text{km}~ \text{s}^{-1}\text{Mpc}^{-1},\Omega_R=9.3\times10^{-5}$ and $a_{eq}=2.9\times10^{-4}$ obtained from Ref. [@Ade:2015xua]. The non-relativistic scale factor for FIMP DM is $$a_{NR}=\frac{T_0}{2m_{\chi}}\left(\frac{g_{,0}}{g_{,rh}}\right)^{\frac{1}{3}} K^{-\frac{1}{2}}.$$ Taking $g_{,0}=3.91$, $g_{,rh}=106.75$ and $T_0=2.35\times 10^{-4}$ eV, finally we can get $$r_{FS} \simeq2.8\times 10^{-2}\left(\frac{{{\rm keV}}}{m_{\chi}}\right) \left(\frac{50}{K}\right)^{\frac{1}{2}}\times \left(1+0.09\ln\left[\left(\frac{m_{\chi}}{{{\rm keV}}}\right) \left(\frac{K}{50}\right)^{\frac{1}{2}}\right]\right) \text{Mpc}.$$ The most stringent bound on $r_{FS} $ comes from small structure formation $r_{FS}<0.1$ Mpc [@Berlin:2017ftj]. The relationship between the mass of $\chi$ and its free streaming length is depicted in Fig. (\[FIG:rFS\]). Basically speaking, warm DM is obtained for $m_\chi\sim10$ keV while $K\in[0.01,100]$. Meanwhile, $\chi$ becomes cold DM when $\chi$ is sufficient heavy and/or the decay parameter $K$ is large enough. Combined Analysis {#SEC:CA} ================= ![Viable parameter space for DM. The red, orange and blue points correspond to hot, warm and cold DM, respectively.[]{data-label="FIG:DM"}](DM-1.pdf "fig:"){width="0.45\linewidth"} ![Viable parameter space for DM. The red, orange and blue points correspond to hot, warm and cold DM, respectively.[]{data-label="FIG:DM"}](DM-2.pdf "fig:"){width="0.45\linewidth"}\ ![Viable parameter space for DM. The red, orange and blue points correspond to hot, warm and cold DM, respectively.[]{data-label="FIG:DM"}](DM-3.pdf "fig:"){width="0.45\linewidth"} ![Viable parameter space for DM. The red, orange and blue points correspond to hot, warm and cold DM, respectively.[]{data-label="FIG:DM"}](DM-4.pdf "fig:"){width="0.45\linewidth"} After studying some benchmark points, it would be better to figure out the viable parameter space for success leptogenesis and DM. We then perform a random scan over the following parameter space: $$\begin{aligned} m_1\in[10^{-12},10^{-2}]~{{\rm eV}}, ~M_1\in[10^3,10^8]~{{\rm GeV}}, v_\nu\in[10^{-2},10^2]~{{\rm GeV}},\\\nonumber \text{Re}(\omega_{12,13,23})\in[10^{-10},1],~\text{Im}(\omega_{12,13,23})\in[10^{-10},1],~ \text{BR}_\chi\in[10^{-6},10^{-2}].\end{aligned}$$ During the scan, we have fixed $M_2/M_1=M_3/M_2=10$. The final obtained baryon asymmetry $Y_{\Delta B}$ is required to be within $3\sigma$ range of the observed value, i.e., $Y_{\Delta B}\in[8.60,8.84]\times10^{-11}$. The results are shown in Fig. \[FIG:DM\] and Fig. \[FIG:LG\] for DM and leptogenesis, respectively. ![Same as Fig \[FIG:DM\], but for leptogenesis.[]{data-label="FIG:LG"}](LG-1.pdf "fig:"){width="0.45\linewidth"} ![Same as Fig \[FIG:DM\], but for leptogenesis.[]{data-label="FIG:LG"}](LG-2.pdf "fig:"){width="0.45\linewidth"}\ ![Same as Fig \[FIG:DM\], but for leptogenesis.[]{data-label="FIG:LG"}](LG-3.pdf "fig:"){width="0.45\linewidth"} ![Same as Fig \[FIG:DM\], but for leptogenesis.[]{data-label="FIG:LG"}](LG-4.pdf "fig:"){width="0.45\linewidth"} Let’s consider the DM results in Fig. \[FIG:DM\] first. According to the dominant constraint from free streaming length $r_{FS}$, we can divide the viable samples into three scenarios in Fig. \[FIG:DM\] (a). Of course, the hot DM scenario is not favored by small structure formation. For warm DM, $m_\chi\in[0.3,2\times10^3]$ keV is possible. Meanwhile for cold DM, $m_\chi\in[10,2\times10^5]$ keV is allowed. And $r_{FS}$ is down to about $10^{-5}$ Mpc when $m_\chi\sim10^5$ keV. From Fig. \[FIG:DM\] (b), we aware that the hot DM samples correspond to those with small DM mass $m_\chi$ and very weak washout effect $K\lesssim10^{-2}$. Fig. \[FIG:DM\] (c) shows the samples in the $m_\chi-M_1$ plane. Three kinds of DM are all possible for certain value of $M_1$. By the way, it is interesting to obtain an upper limit on $m_\chi$ when $M_1\lesssim10^6$ GeV. This indicates that for TeV scale leptogenesis, FIMP DM should be keV to sub-MeV. The result for $\text{BR}_\chi$ is shown in Fig. \[FIG:DM\] (d), which tells us that warm DM requires $\text{BR}_\chi\gtrsim10^{-4}$ and cold DM requires $\text{BR}_\chi\lesssim10^{-3}$, respectively. Then we consider the leptogenesis results in Fig. \[FIG:LG\]. The generalised Davidson-Ibarra bound is clearly seen in Fig. \[FIG:LG\] (a). The (warm and cold DM) allowed samples show that the mass of $N_1$ for success leptogenesis could be down to about 3 TeV. The viable region in the $v_\nu-M_1$ plane is shown in Fig. \[FIG:LG\] (b), which is consistent with the theoretical bounds discussed in Ref. [@Clarke:2015hta]. For completeness, the naturalness bound in Eq. is also shown. Therefore, natural leptogenesis is viable for $3\times10^3~{{\rm GeV}}\lesssim M_1\lesssim7\times10^6$ GeV with $0.4~{{\rm GeV}}\lesssim v_\nu\lesssim30~{{\rm GeV}}$. The result for decay parameter $K$ is given in Fig. \[FIG:LG\] (c), which shows that $K\lesssim10$ should be satisfied when $M_1\lesssim10^8$ GeV. Actually for $M_1\lesssim10^5$ GeV, all the samples are within weak washout region. An upper bound on lightest neutrino mass $m_1$ is clearly seen in Fig. \[FIG:LG\] (d). Success leptogenesis in the $\nu$2HDM requires $m_1$ must be extremely tiny, i.e., $m_1\lesssim10^{-11}$ eV for $M_1\sim10^4$ GeV. Before ending this section, we give a brief discussion on the collider signature. According to the results of leptogenesis in Fig. \[FIG:LG\], not too small $v_\nu$ is favored. In such scenario, the branching ratios of neutrinophilic scalars are quite different from the scenario with small $v_\nu$ [@Guo:2017ybk; @Haba:2011nb; @Wang:2016vfj; @Huitu:2017vye], but are similar with type-I 2HDM [@Branco:2011iw]. Currently, if $m_{\Phi_\nu}$ is smaller than $m_t$, the most stringent constraint comes from $t\to b H^\pm (H^\pm\to \tau^\pm\nu)$ [@Sirunyan:2019hkq], which could exclude the region $v_\nu\gtrsim18$ GeV [@Sanyal:2019xcp]. Meanwhile, if $m_Z+m_h\lesssim m_{\phi_\nu}\lesssim2m_t$, the channel $A\to Zh(h\to b\bar{b})$ could exclude the region $v_\nu\gtrsim 24$ GeV [@Aaboud:2017cxo]. For heavier additional scalars with $m_{\Phi_\nu}>2 m_t$, the signature $A/H\to t\bar{t}$ is only able to probe the region $v_\nu\gtrsim 174$ GeV [@Aaboud:2017hnm; @Chen:2019pkq]. Therefore, the experimental bounds on neutrinophilic scalars can be easily escaped provided $m_{\Phi_\nu}$ is large enough. At HL-LHC, the signature $A\to Zh(h\to b\bar{b})$ would reach $v_\nu\sim10$ GeV [@Chen:2019pkq]. Then the observation of this signature will indicate $M_1\sim10^6$ GeV and $m_1\lesssim10^{-7}$ eV. Conclusion {#SEC:CL} ========== In this paper, we propose an extended $\nu$2HDM to interpret the neutrino mass, leptogenesis and dark matter simultaneously. This model contains one neutrinophilic scalar doublet $\Phi_\nu$, three right hand heavy neutrino $N$, which account for low scale neutrino mass generation similar to type-I seesaw. Leptogenesis is generated due to the CP-violating decays of right hand neutrino $N\to \ell_L \Phi_\nu^,\bar{\ell}_L \Phi_\nu$. The dark sector contains one scalar singlet $\phi$ and one Dirac fermion singlet $\chi$, which are charged under a $Z_2$ symmetry. Provided $m_\chi<m_\phi$ and $\lambda\ll1$, $\chi$ is a FIMP DM candidate within this paper. The relic abundance of $\chi$ is produced by $N\to \chi \phi$. Therefore, we have a common origin, i.e., the heavy right hand neutrino $N$, for tiny neutrino mass, baryon asymmetry and dark matter. In the frame work of $\nu$2HDM, the asymmetry $\epsilon_1$ and decay parameter $K$ are both enhanced by the smallness of $v_\nu$. By explicit calculation, we show that the decay parameter $K$ can be suppressed under certain circumstance. The importance of $\Delta L=2$ washout process is also illustrated. As for FIMP DM, the relic abundance mainly depends on the branching ratio $\text{BR}_\chi$ and decay parameter $K$, and $m_\chi$ is typically at the order of keV to MeV scale. Meanwhile the free streaming length sets stringent bound. The viable parameter space for success leptogenesis and DM is obtained by solving the corresponding Boltzmann equations. 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--- abstract: 'We provide an algorithm for unfolding the surface of any orthogonal polyhedron that falls into a particular shape class we call Manhattan Towers, to a nonoverlapping planar orthogonal polygon. The algorithm cuts along edges of a $4 \times 5 \times 1$ refinement of the vertex grid.' address: - 'Dept. of Computer Science, Villanova University, Villanova, PA 19085, USA.' - 'Dept. of Computer Science, Siena College, Loudonville, NY 12211, USA.' - 'Dept. of Computer Science, Smith College, Northampton, MA 01063, USA.' author: - Mirela Damian - Robin Flatland - 'Joseph O’Rourke' bibliography: - 'MT.bib' title: Unfolding Manhattan Towers --- and and [ @lab\[\#1\]\#2[[\[\#2 —[\#1]{}\]]{}]{} @nolab\#1[[\[\#1\]]{}]{} ]{} Unfolding, orthogonal, genus-zero, polyhedra. Introduction {#sec:Introduction} ============ It is a long standing open problem to decide whether the surface of every convex polyhedron can be edge unfolded: cut along edges and unfolded flat to one piece without overlap [@do-sfucg-05]. It is known that some nonconvex polyhedra have no edge unfolding; a simple example is a small box sitting on top of a larger box. However, no example is known of a nonconvex polyhedron that cannot be unfolded with unrestricted cuts, i.e., cuts that may cross the interior of faces. The difficulty of these questions led to the exploration of orthogonal polyhedra, those whose faces meet at right angles. Progress has been made in two directions: firstly, by restricting the shapes to subclasses of orthogonal polyhedra, such as the “orthostacks” and “orthotubes” studied in [@bddloorw-uscop-98]; and secondly, by generalizing the cuts beyond edges but with some restrictions. In particular, a grid unfolding partitions the surface of the polyhedron by coordinate planes through every vertex, and then restricts cuts to the resulting grid. The box-on-box example mentioned earlier can be easily grid unfolded. Recent work on grid unfolding of orthostacks is reported in [@dm-geuoo-04] and [@dil-gvuo-04]. Because on the one hand no example is known of an orthogonal polyhedron that cannot be grid unfolded, and on the other hand no algorithm is known for grid unfolding other than very specialized shapes, the suggestion was made in [@do-op02-04] to seek $k_1 \times k_2 \times k_3$ refined grid unfoldings, where every face of the vertex grid is further refined into a grid of edges. Positive integers $k_1$, $k_2$ and $k_3$ are associated with the amount of refinement in the $x$, $y$ and $z$ directions, respectively; e.g., $z$ perpendicular faces are refined into a $k_1 \times k_2$ grid, and similarly $x$ ($y$) perpendicular faces are refined into a $k_2 \times k_3$ ($k_1 \times k_3$) grid. It is this line we pursue in this paper, on a class of shapes not previously considered. We define “Manhattan Tower (MT) polyhedra” to be the natural generalization of “Manhattan Skyline polygons.” Although we do not know of an unrefined grid unfolding for this class of shapes, we prove (Theorem \[theo:5x5\]) that there is a $4 \times 5 \times 1$ grid unfolding. Our algorithm peels off a spiral strip that winds first forward and then interleaves backward around vertical slices of the polyhedron, recursing as attached slices are encountered. Definitions =========== Let $Z_k$ be the plane $\{ z = k \}$, for $k \ge 0$. Define $\P$ to be a Manhattan Tower (MT) if it is an orthogonal polyhedron such that: 1. $\P$ lies in the halfspace $z \ge 0$, and its intersection with $Z_0$ is a simply connected orthogonal polygon; 2. For $k < j$, $\P \cap Z_k \supseteq \P \cap Z_{j}$: the cross-section at higher levels is nested in that for lower levels. Manhattan Towers are terrains in that they meet each vertical (parallel to $z$) line in a single segment or not at all; thus they are monotone with respect to $z$. Fig. \[fig:MTex\]a shows an example. Manhattan Towers may not be monotone with respect to $x$ or $y$, and indeed $\P \cap Z_k$ will in general have several connected components (cf. Fig. \[fig:Z\]c), and may have holes (cf. Fig. \[fig:Z\]b), for $k > 0$. ![Manhattan Tower $\P$.[]{data-label="fig:MTex"}](Figures/MTex0.abs.eps){width="0.6\linewidth"} ![ Cross-sections of Manhattan Tower $P$ from Fig. \[fig:MTex\]: (a) $Z_0 \cap P$ is a simple orthogonal polygon; (b) $Z_2 \cap P$ is an orthogonal polygon with one hole; (c) $Z_5 \cap P$ has two disjoint components.[]{data-label="fig:Z"}](Figures/Z0Z2Z5.eps){width="0.90\linewidth"} As an $xy$-plane sweeps from $Z_0$ upwards, the cross-section of $\P$ changes at finitely many locations. Thus a Manhattan Tower $\P$ may be viewed as consisting of nested layers, with each layer the extrusion of a set of orthogonal polygons. The [base]{} of $\P$ is its bottom layer, which is bounded below by $Z_0$ and above by the $xy$-plane passing through the first vertex with $z>0$. Note that, unlike higher layers, the base is simply connected, since it is an extrusion of $\P \cap Z_0$. We use the following notation to describe the six types of faces, depending on the direction in which the outward normal points: [front]{}: $-y$; [back]{}: $+y$; [left]{}: $-x$; [right]{}: $+x$; [bottom]{}: $-z$; [top]{}: $+z$. An $x$-$edge$ is an edge that is parallel to the $x$-axis; $y$-$edges$ and $z$-$edges$ are defined similarly. Clockwise (cw) and counterclockwise (ccw) directions are defined with respect to the viewpoint from $y = -\infty$. Later we will rotate the coordinate axes in recursive calls, with all terms tied to the axes altering appropriately. Recursion Tree ============== We start with the partition $\Pi$ of the base layer induced by the $xz$-planes passing through every vertex of $\P$. (The restriction of the partition to planes orthogonal to $y$ will facilitate processing in the $\pm y$ directions below.) Such a partition consists of rectangular boxes only (see Fig. \[fig:basepartition\]a). The dual graph of $\Pi$ has a node for each box and an edge between each pair of nodes corresponding to adjacent boxes. Since the base is simply connected, the dual graph of $\Pi$ is a tree $T$ (Fig. \[fig:basepartition\]b), which we refer to as the [recursion tree]{}. The root of $T$ is a node corresponding to a box (the root box) whose front face has a minimum $y$-coordinate (with ties arbitrarily broken). ---------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------- ![ (a) Partition $\Pi$ of $\P$’s base; (b) Recursion tree $T$.[]{data-label="fig:basepartition"}](Figures/part2d.abs.eps "fig:"){width="0.55\linewidth"} ![ (a) Partition $\Pi$ of $\P$’s base; (b) Recursion tree $T$.[]{data-label="fig:basepartition"}](Figures/tree.abs.eps "fig:"){width="0.15\linewidth"} (a) (b) ---------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------- It turns out that nearly all unfolding issues are present in unfolding single-layer MTs, due to the nested-layer structure of MTs. In Sec. \[sec:TMT\] we describe an algorithm for unfolding single-layer MTs. The algorithm is then extended to handle multiple-layer MTs in Sec. \[sec:MT\]. $(4 \times 5 \times 1)-$Refined Manhattan Towers ================================================ Fig. \[fig:basepartition2\] illustrates the refinement process, using the base from Fig. \[fig:basepartition\]a as an example. ---------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------- ![ (a) Gridded MT base; (b) $(4 \times 5 \times 1)$-refined MT base.[]{data-label="fig:basepartition2"}](Figures/grid.eps "fig:"){width="0.5\linewidth"} ![ (a) Gridded MT base; (b) $(4 \times 5 \times 1)$-refined MT base.[]{data-label="fig:basepartition2"}](Figures/refined.eps "fig:"){width="0.5\linewidth"} (a) (b) ---------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------- The gridded base (Fig. \[fig:basepartition2\]a) contains additional surface edges induced by $yz$-coordinate planes through each vertex. A $4 \times 5 \times 1$ refinement of the gridded base further partitions each horizontal grid rectangle into a $4 \times 5$ grid. In addition to gridedges of the gridded base, the $(4 \times 5 \times 1)$-refined base (Fig. \[fig:basepartition2\]b) contains all surface edges induced by coordinate planes passing through each gridpoint in the refinement. In the following we show that every $(4 \times 5 \times 1)$-refined MT can be edge-unfolded. Single-Layer MTs {#sec:TMT} ================ A single-layer Manhattan Tower consists of a single layer, the [base]{} layer. We describe the unfolding algorithm recursively, starting with the base case in which the layer is a single rectangular box. Single Box Unfolding {#sec:basecase} -------------------- Let $r$ be a $(4 \times 5 \times 1)$-gridded rectangular box and let $T$, $R$, $B$, $L$, $K$ and $F$ be the top, right, bottom, left, back and front faces of $r$, respectively. Let $s$ and $t$ be two gridpoints either adjacent on the same $x$-edge of $r$ (as in Fig. \[fig:basecase.a\]a), or one on the top $x$-edge and one on the bottom $x$-edge of the front face of $r$ (as in Fig \[fig:basecase.b\]a). Let $y_s$ and $y_t$ be the ($y$ parallel) gridedges incident to $s$ and $t$. The unfolding of $r$ starts at $y_s$ and ends at $y_t$. More precisely, this means the following. Let $\xi_{2d}$ ($\xi_{3d}$) denote the planar (three-dimensional) embedding of the cut surface piece. Then $\xi_{2d}$ has $y_s$ on its far left and $y_t$ on its far right (as in Figs. \[fig:basecase.a\]c and \[fig:basecase.b\]c). ![Single box unfolding: $s$ adjacent to $t$ (a) Front view of box $r$ and mirror view of right ($R$), bottom ($B$) and back ($K$) faces, marked with unfolding cuts (b) Faces of $r$ flattened out (front face not shown) (c) Spiral unfolding of $r$; labels identify faces containing the unfolded pieces.[]{data-label="fig:basecase.a"}](Figures/basecase.a.eps){width="0.95\linewidth"} ![Single box unfolding: $s$ and $t$ on opposite front edges (a) Front view of box $r$ and mirror view of right ($R$), bottom ($B$) and back ($K$) faces, marked with unfolding cuts (b) Faces of $r$ flattened out (front face not shown) (c) Spiral unfolding of $r$; labels identify faces containing the unfolded pieces.[]{data-label="fig:basecase.b"}](Figures/basecase.b.eps){width="0.95\linewidth"} The main unfolding idea is to cut the top, right, bottom and left faces so that they unfold into a staircase-like strip and attach front and back faces to it vertically without overlap. We collectively refer to the top, right, bottom and left faces as support faces (intuitively, they support the front and back faces). Roughly stated, $\xi_{3d}$ starts at $y_s$, spirals cw around the support faces toward the back face, crosses the back face, then spirals ccw around the support faces back to $y_t$. This idea is illustrated in Figs. \[fig:basecase.a\] and \[fig:basecase.b\]. In the following we provide the details for the case when $s$ and $t$ are adjacent on the top front edge of $r$ (Fig. \[fig:basecase.a\]). The case when $s$ lies on a bottom front edge and $t$ lies on a top front edge of $r$ is similar and is illustrated in Fig. \[fig:basecase.b\]; the case when $s$ is on the top and $t$ is on the bottom is identical, when viewed through an $xy$-mirror. As illustrated in Fig. 5, let $w$ be the $x$-extent and let $h$ be the $y$-extent of $r$. We implicitly define the unfolding cuts by describing the surface pieces encountered in a walk along $\xi_{3d}$ on the surface of $r$ (delineated by unfolding cuts). Starting at $y_s$, walk cw along a rectangular strip of $y$-extent equal to $2h/5$ (two gridfaces-wide) that spirals around the support faces from $y_s$ to $y_t$. This spiral strip lies adjacent to the front face of $r$; we will refer to it as the front spiral of $\xi_{3d}$. At $y_t$, take a left turn and continue along a rectangular strip (orthogonal to the front spiral and right-aligned at $t$) of $y$-extent equal to $2h/5$ (two gridfaces-wide) and $x$-extent equal to $w/4$ (one gridface-long). At the end of this strip, take a right turn and continue along a rectangular strip of $y$-extent equal to $h/5$, until the right face $R$ is met; at this point, the strip thickens to a $y$-extent equal to $2h/5$ (two gridfaces-wide), so that it touches the back face $K$ of the box. The strip touching $K$ consumes the entire length of the right face $R$, plus an additional $w/4$ (one gridface) amount onto the adjacent bottom face $B$. At the end of this bottom strip, take a left turn and continue along a $w/4$-wide strip across back face $K$ and up onto the top face $T$. The piece of $\xi_{3d}$ traversed so far is called the forward spiral; the remaining piece is called the backward spiral, conveying the fact that from this point on $\xi_{3d}$ spirals ccw around the support faces back to $y_t$. The piece of the backward spiral adjacent to the back face is the back spiral of $r$. The planar piece $\xi_{2d}$ (obtained by laying $\xi_{3d}$ out in the plane) has the staircase-like shape illustrated in Fig \[fig:basecase.a\]c. Conceptually, the front face $F$ and the back face $K$ are not part of the unfolding described so far; however, they can be flipped up and attached vertically to $\xi_{2d}$ without overlap (see the striped faces in Fig. \[fig:basecase.a\]c), a point to which we return in Sec. \[sec:front.back\]. Recursion Structure {#sec:Recursion} ------------------- In general, a box $r$ has children (adjacent boxes) attached along its front and/or back face. Call a child attached on the front a [front child]{} and a child attached on the back a [back child]{}. In unfolding $r$, we unwind the support (top, bottom, left, right) faces into a staircase-like strip just as described for the single box. But when the front/back spiral runs alongside the front/back face of $r$ and encounters an adjacent child, the unfolding of $r$ is temporarily suspended, the child is recursively unfolded, then the unfolding of $r$ resumes where it left off. At any time in the recursive algorithm there is a forward direction, corresponding to the initial spiraling from front to back (the lighter strip in Figs. \[fig:basecase.a\] and \[fig:basecase.b\]), and an opposing backward direction corresponding to the subsequent reverse spiraling from back to front (the darker strip in Figs. \[fig:basecase.a\] and \[fig:basecase.b\]). When the recursion processes a front child, the sense of forward/backward is reversed: we view the coordinate system rotated so that the $+y$ axis is aligned with the forward direction of the child’s spiral, with all terms tied to the axes altering appropriately. In particular, this means that the start and end unfolding points $s'$, $t'$ of a front child $r'$ lie on the front face of $r'$, as defined in the rotated system. For example, in Fig. \[fig:forward.recursion\], boxes $a$, $b$, $c$, $d$ are processed from front to back. But recursion on $e$, a front child of $d$, reverses the sense of forward, which continues through $e$, $f$, and $g$. We can view the coordinate system rotated so that $+y$ is aligned with the arrows shown. Thus $f$ is a back child of $e$, $g$ is a back child of $f$, and $k$ a front child of $g$. Again the sense of forward is reversed for the processing of $k$. ![Arrows indicate which direction is forward in the recursive processing.[]{data-label="fig:forward.recursion"}](Figures/forward.recursion.eps){width="0.6\linewidth"} Suturing Techniques {#sec:suturing} ------------------- We employ two methods to “suture” a child’s unfolding to its parent’s unfolding. The first method, [same-direction suture]{}, is used to suture all front children to their parent. If there are no back children, then a strip from the back face of the parent ($K_0$ in Figs \[fig:basecase.a\] and \[fig:basecase.b\]) is used to reverse the direction of the spiral to complete the parent’s unfolding, as described in Sec. \[sec:basecase\] for the single box. However, if the parent has one or more back children, these children cover parts or perhaps all of the back face of the parent, and the back face strip may not be available for the reversal. So instead we use a second suturing method, [reverse-direction suture]{}, for one of the back children. This suture uses the child’s unfolding to reverse the direction of the parent’s spiral, and does not require a back-face strip. We choose [exactly one]{} back child for reverse-direction suturing. Although any such child would serve, for definiteness we select the rightmost child. Our suturing rules are as follows: 1. For every front child, use same-direction suturing. 2. For the rightmost back child, use reverse-direction suturing. 3. For remaining back children, use same-direction suturing. ### Same-direction suture {#sec:same.direction} We first note that a front child $r'$ never entirely covers the front face of its parent box $r$, because the parent of $r$ is also adjacent to the front face of $r$. This is evident in Fig. \[fig:forward.recursion\], where $e$ cannot cover the front face of $d$ because $d$’s parent, $c$, is also adjacent along that side. Similarly, $k$ cannot cover the “front” face of $g$ (where here the sense of front is reversed with the forward direction of processing) because $g$’s parent $f$ is also adjacent along that side. The same-direction suture may only be applied in such a situation of non-complete coverage of the shared front face, for it uses a thin (one gridface-wide) vertical strip off that face. ![Same-direction suture. (a) Front view of faces root box $r$ and front child $r'$, with mirror bottom, left and back views. (b) Result $\xi_{2d}$ of recursive unfolding.[]{data-label="fig:same.suture"}](Figures/same.suture.abs.eps){width="0.74\linewidth"} This suture begins at the point where the parent’s spiral meets an adjacent child as it runs alongside its front or back face. To be more specific, consider the case when $r'$ is a front child of $r$, and the parent’s front spiral meets $r'$ as it runs along the top of $r$. This situation is illustrated in Fig. \[fig:same.suture\]. The same-direction suture begins by cutting a vertical strip $I$ off the front face of parent $r$, which includes all vertical gridfaces that lie alongside child $r'$ (see Fig. \[fig:same.suture\]a), then it takes a bite $J$ one gridface-thick and three gridfaces-long (in the $x$-direction) off the bottom face of the parent. This marks the gridedge $y_{s'}$ on $r'$ where the child’s spiral unfolding starts. The child’s spiral unfolding ends at top gridedge $y_{t'}$ of the same $x$-coordinate as $y_{s'}$. When the child’s unfolding is complete, the spiral unfolding of the parent resumes at the $y$-gridedge it left off (see the cut labeled $\gamma$ in Fig. \[fig:same.suture\]). The other cases are similar: if $r'$ is a back child of $r$, $I$ occurs on the back face of $r$; and if the parent’s front spiral meets $r'$ as it runs along the bottom of $r$,[^1] $J$ occurs on the top face of $r$ (see child $r_4$ and parent $r_2$ in Fig. \[fig:3hshape\]). It is this last case that requires a $5$ refinement in the $y$ direction: the front spiral must be two gridfaces-thick so that cutting $J$ off it does not disconnect it. As the name suggests, this suturing technique preserves the unwinding direction (cw or ccw) of the parent’s spiral. In Fig. \[fig:same.suture\], notice that the parent’s spiral unfolds in cw direction on top face $T$ before the suture begins. The parent’s cw unfolding is suspended at $y$-gridedge marked $\gamma$, and after the child is unfolded, the parent’s spiral resumes its unfolding in cw direction at $\gamma$. The unfolded surface $\xi_{2d}$ is shown in Fig. \[fig:same.suture\]b. ### Reverse-direction suture {#sec:rev.direction} This suture begins after the parent’s spiral completes its first cycle around the support (top, right, bottom, left) faces, as illustrated in Fig. \[fig:rev.suture\] for parent $r$ and back child $r'$. As in the single box case (Sec. \[sec:basecase\]), after a forward move in the $+y$-direction, the spiral starts a second cycle around the support faces. However, unlike in the single box case, the spiral stops as soon as it reaches a $y$-gridedge of the same $x$-coordinate as the rightmost gridpoint $u$ that the parent shares with a back child. At that point, the parent’s spiral continues with a gridface-thick strip $S$ in the $+y$-direction, right-aligned at $y_u$. Let $s'$ be the left corner of $S$ on the boundary shared by $r$ and $r'$. The unfolding of $r'$ begins at gridedge $y_{s'}$ and ends at gridedge $y_{t'}$ immediately to the left of $y_{s'}$ on top of $r'$. When the child’s unfolding is complete, the unfolding of the parent resumes at the gridedge it left off, with the spiral unwinding in reverse direction. ![Reverse-direction suture. (a) Front view of faces of root box $r$ and back child $r'$, with mirror bottom, left and back views. (b) Result $\xi_{2d}$ of recursive unfolding.[]{data-label="fig:rev.suture"}](Figures/rev.suture.abs.eps){width="0.86\linewidth"} As the name suggests, this suturing technique reverses the unwinding direction (cw or ccw) of the parent’s spiral. In Fig. \[fig:rev.suture\], notice that the parent’s spiral unfolds in cw direction on top face $T$ before the suture begins. After the child is unfolded, the parent’s spiral resumes its unfolding in ccw direction at $y_{s'}$. The result $\xi_{2d}$ of this unfolding is shown in Fig. \[fig:rev.suture\]b. Attaching Front and Back Faces {#sec:front.back} ------------------------------ The spiral strip $\xi_{3d}$ covers all of the top, bottom, right, and left faces of the base. It also covers the gridface-thick strips of a front/back face used by the same-direction sutures ($I$ in Fig. \[fig:same.suture\]) and the gridface-thick strips of back faces used to reverse the spiral direction in the base cases ($K_0$ in Figs \[fig:basecase.a\] and \[fig:basecase.b\]). The staircase structure of $\xi_{2d}$ (shown formally in Theorem \[thm:correctedness\]) guarantees that no overlap occurs. We now show that remaining exposed front and back pieces that are not part of $\xi_{3d}$ can be attached orthogonally to $\xi_{2d}$ without overlap. Consider the set of top gridedges shared by top faces with front/back faces. These gridedges occur on the horizontal boundaries of $\xi_{2d}$ as a collection of one or more contiguous segments. We partition the front/back faces by imagining these top gridedges illuminate downward lightrays on front/back faces. Then all front and back pieces are illuminated, and these pieces are attached to corresponding illuminating gridedges (see Figs. \[fig:basecase.a\]c, \[fig:basecase.b\]c, \[fig:same.suture\]b and \[fig:rev.suture\]b). Although no interior points overlap in the unfolding, we allow [edge overlap]{}, which corresponds to the physical model of cutting out the unfolded piece from a sheet of paper. For example, in Fig. \[fig:rev.suture\]b a left gridedge of $F'$ overlaps a gridedge of $\xi_{2d}$. It is not difficult to avoid edge overlap (e.g. by making the portion of the strip causing the edge overlap narrower to separate it from $F'$), but doing so requires increasing the degree of refinement. The next section summarizes the entire unfolding process for single-layer MTs. Unfolding Algorithm for Single-Layer MTs {#sec:base.algorithm} ---------------------------------------- Consider an arbitrary base partitioned into rectangular boxes with $xz$-planes $Y_0, Y_1, \ldots$ through each vertex. Select a root box $r$ adjacent to $Y_0$ (breaking ties arbitrarily) and set the forward unwinding direction $d$ to be cw. Let $y_s$ and $y_t$ be top $y$-gridedges of $r$, as described in Sec. \[sec:basecase\] for the single-box case. Our recursive unfolding starts at root box $r$ and proceeds as follows. [Algorithm UNFOLD($r, y_s, y_t$)]{} This algorithm can be easily implemented to run in $O(n^2)$ time on a polyhedron $\P$ with $n$ vertices. Fig. \[fig:3hshape\] illustrates the recursive unfolding algorithm on a $3$-legged $H$-shaped base. The unfolding starts at gridedge $y_{s_1}$ of root box $r_1$ and ends at gridedge $y_{t_1}$. (Only the endpoints $s_1$ and $t_1$ of these two gridedges are marked in Fig \[fig:3hshape\].) The spiral strip encounters the boxes in the order $r_1, r_2, r_3, r_4, r_5, r_6$ and $r_7$, which corresponds to the ordering of the recursive calls. For each $i$, $y_{s_i}$ and $y_{t_i}$ are gridedges of $r_i$ where the unfolding of $r_i$ starts and ends.The algorithm uses reverse-direction suture to attach back child $r_2$ to parent $r_1$; same-direction suture to attach front child $r_3$, and then $r_4$, to parent $r_2$; reverse-direction suture to attach back child $r_5$ to parent $r_2$; and same-direction suture to attach back child $r_6$, and then $r_7$, to parent $r_2$. Note that a refinement of 5 in the $y$ direction is necessary on top of box $r_2$ for this unfolding. ![Unfolding a 3-legged $H$-shaped base.[]{data-label="fig:3hshape"}](Figures/3hshape.eps){width="\linewidth"} The [UNFOLD]{}($r,y_s,y_t$) algorithm unfolds all boxes in the recursion tree rooted at $r$ into a staircase-like strip $\xi_{2d}$ completely contained between the vertical lines passing through $y_s$ and $y_t$. \[thm:correctedness\] The proof is by induction on the height $k$ of the recursion tree rooted at $r$. The base case is $k = 0$ and corresponds to single node trees. This is the case illustrated in Figs. \[fig:basecase.a\] and \[fig:basecase.b\], which satisfy the claim of the theorem. The inductive hypothesis is that the theorem is true for any recursion tree of height $k-1$ or less. To prove the inductive step, consider a recursion tree $T$ of height $k$ rooted at $r$. The staircase strip $\xi_{2d}(r)$ of $r$ alone, ignoring all children, fits between the vertical lines passing through $y_s$ and $y_t$ (cf. Figs. \[fig:basecase.a\]c and \[fig:basecase.b\]c). Assume, w.l.o.g., that $r$ unfolds cw. There are two possible placements of $s$ and $t$ on $r$: (i) $s$ and $t$ are on [opposite]{} top/bottom edges of the front face of $r$ (Fig. \[fig:basecase.b\]a), as placed by a same-direction suture, or (ii) $s$ and $t$ are on a [same]{} top/bottom edge of $r$ (Fig. \[fig:basecase.a\]a), as placed by a reverse-direction suture. In either case, $s$ and $t$ are placed in such a way that no children exist along the path extending cw from $t$ to $s$ on $r$. This means that all front children of $r$ are encountered during the unwinding of $r$’s front spiral from $s$ to $t$ on $r$. That all back children are encountered during the unwinding of $r$’s back spiral is clear: starting at the rightmost back child, the back spiral makes a complete cycle around the back face. Consider now an arbitrary child $r'$ of $r$ in $T$ and let $T'$ be the subtree rooted at $r'$. As noted above, $r'$ will be encountered during the unfolding of $r$. Let $y_{s'}$ and $y_{t'}$ be the gridedges on $r'$ where the unfolding of $r'$ starts and ends. The inductive hypothesis applied on $T'$ tells us that the strip $\xi_{2d}(r')$ corresponding to $T'$ fits between the vertical lines passing through $y_{s'}$ and $y_{t'}$. Fig. \[fig:same.suture\]b illustrates the same-direction suture: when $\xi_{2d}(r')$ is sutured to $\xi_{2d}(r)$, the strip $\xi_{2d}(r)$ expands horizontally and remains contained between the vertical lines passing through $y_s$ and $y_t$. The reverse-unfolding suture has a similar behavior (illustrated in Fig. \[fig:rev.suture\]b), thus completing this proof. Multiple-Layer MTs {#sec:MT} ================== Few changes are necessary to make the single-layer unfolding algorithm from Sec. \[sec:base.algorithm\] handle multiple-layer Manhattan Towers. In fact, the view of the cuts used to form $\xi_{3d}$ from $z = \pm \infty$ in the multi-layer case is identical to that in the single-layer unfolding. All the differences lie in vertical ($z$-parallel) strips used to adjust for differing tower heights. When there are multiple-layers, the basic unit to unfold is a vertical [slab]{} $S(r)$ consisting of a box $r$ in the partition $\Pi$ of the base layer and all the towers that rest on top of $r$ (see Fig. \[fig:basecase2\]). ![Front view of single slab $S(r)$, with mirror bottom, left and back views.[]{data-label="fig:basecase2"}](Figures/basecase2.eps){width="0.78\linewidth"} A slab is a Manhattan Skyline polygon parallel to the $xz$-plane extruded in the $y$ direction: the projection of the top faces of the slab on the $xy$-plane forms a partition of the (unique) bottom face (face $B$ in Fig. \[fig:basecase2\]). It is here that we make essential use of the assumptions that $\P \cap Z_0$ is a simply connected orthogonal polygon, and the cross-sections at higher levels are nested in those for lower levels. The unfolding of a slab $S(r)$ is similar to the unfolding of a single box: 1. Select an arbitrary top face $T$ of the slab. 2. Select start and end gridedges $y_s$ and $y_t$ on $T$ as in the single box case. 3. Unfold $S(r)$ using the procedure described in Sec. \[sec:basecase\] for $r$. The only difference is that a slab may have multiple left/right/top faces, causing the spiral $\xi_{3d}$ to cycle up and down over the towers of $S(r)$, as illustrated in Fig. \[fig:basecase2\]. As a result, $\xi_{2d}$ lengthens horizontally, but still maintaining its staircase structure. As in the case of a single box, $\xi_{3d}$ covers all of the top, right, bottom and left faces. The remaining front and back pieces are attached to $\xi_{2d}$ using the illumination scheme described in Sec. \[sec:front.back\]. ![Unfolding multiple-layer MTs. (a) Spiral $\xi_{3d}$; bottom and back mirror views are as shown in Fig. \[fig:3hshape\] (b) $\xi_{2d}$, strips $J_1$ and $J_2$ attached above; transitions between towers are striped; piece labels correspond to MT boxes to which they belong.[]{data-label="fig:3towers"}](Figures/3towers.eps){width="0.98\linewidth"} In general, a multiple-layer MT $\P$ consists of many slabs; in this case, we use the recursion tree for the base of $\P$ to unfold $\P$ recursively (in this sense, single-layer and multiple-layer MTs have identical recursion structures). The recursive unfolding algorithm is similar to the algorithm described in Sec. \[sec:base.algorithm\] for single-layer MTs, with some minor modifications to accommodate the existence of towers. In the following we describe these modifications with the help of the MT example from Fig. \[fig:3towers\], whose base is the $3$-legged $H$-shape single-layer MT from Fig. \[fig:3hshape\]. Let $S(r')$ be the slab corresponding to a child $r'$ of $r$. When the unfolding strip for $S(r)$ first encounters a top/bottom face $f$ of $S(r')$ (when viewed from $z = +\infty$), the unfolding of $S(r)$ is suspended in favor of $S(r')$. Next we discuss the two suturing techniques used to glue the unfolding of $S(r')$ to the unfolding of $S(r)$. [Same-direction suture.]{} In this case, the bottom/top face opposite to $f$ is used to accommodate the start unfolding gridedge $y_{s'}$ for $S(r')$; the end unfolding gridedge $y_{t'}$ is selected as in the single-layer case. Consider first the case when $r'$ is a front child of $r$. If $S(r')$ is encountered while $\xi_{3d}$ runs along the top of $S(r)$, the suture is identical to the single-layer case: a vertical strip across the front of $S(r)$ is used to reach the bottom of $S(r')$ (see strip $I_2$ in Fig. \[fig:3towers\], reaching front child $S(r_3)$). If $S(r')$ is encountered while $\xi_{3d}$ runs along the bottom of $S(r)$, the suture is similar to the single-layer case, with two simple modifications: 1. After using a vertical strip to reach the top of $S(r)$, a small “bite” is taken out of the top of $S(r)$ to reach the top of $S(r')$ in the single-layer case. In the multiple-layer case, it may be necessary to extend such a bite up/down a $z$-face in order to reach the point of the same $x$-coordinate as $y_{s'}$. This is the case of slab $S(r_4)$ in Fig. \[fig:3towers\]: strip $I_3$ is used to get from the bottom of $S(r_2)$ to the top of $S(r_2)$, after which the “bite” labeled $L$ extends up a right face of $S(r_2)$ to reach the $x$-coordinate of $y_{s_4}$. 2. Unlike the single-layer case, a top bite used in the same-direction suture is not necessarily adjacent to child $S(r')$. In this case, a second $z$-strip (such as $I_4$ in Fig. \[fig:3towers\]) is used to reach the top of $S(r')$. The case in which $r'$ is a back child of $r$ is similar and is illustrated in Fig. \[fig:3towers\]: strips $I_7$ and $I_{9}$ (visible in Fig. \[fig:3towers\]b, but not in \[fig:3towers\]a) are used to make the transition from $S(r_2)$ to $S(r_6)$ and $S(r_7)$ respectively, and strips $I_8$ and $I_{10}$ are used to return to $S(r_2)$. [Reverse-direction suture.]{} As in the same-direction suture case, a vertical strip may be needed to make transitions between the top of a parent $S(r)$ and the top of a child $S(r')$ that uses reverse-direction suture. This is the case for $S(r_3)$ in Fig. \[fig:3towers\], where the vertical strip $I_6$ ($I_7$) is used to move from (to) $S(r_2)$ to (from) $S(r_5)$. The result of these alterations is that $\xi_{2d}$ may lengthen vertically, but it remains monotone in the horizontal direction. One final modification is necessary due to the difference in height between towers that belong to a same slab (see for instance towers $T_a$ and $T_b$ of $S(r_2)$ in Fig. \[fig:3towers\]a). In such cases it is possible that the spiral $\xi_{3d}$ does not completely cover the left/right faces of the slab. We resolve this problem by thickening $\xi_{3d}$ in the $y$-direction to cover the uncovered pieces. To be more precise, consider the vertical strip marked $J_1$ in Fig. \[fig:3towers\] (in the mirror view of right face $R$). The reason $J_1$ remains uncovered is because in unfolding $S(r_3)$, the unfolding of $S(r_2)$ suspends at the top $y$-gridedge of $J_1$ and resumes at the bottom $y$-gridedge of $J_1$. Similarly, $\xi_{3d}$ skips over the strip marked $J_2$ in Fig. \[fig:3towers\]: when the back spiral of $S(r_2)$ meets $S(r_6)$, the unfolding of $S(r_2)$ suspends at the top $y$-gridedge of $J_2$ and resumes at the bottom $y$-gridedge of $J_2$. We resolve the problem of uncovered strips as follows. First, note that every uncovered strip is on a left/right face (never a back/front face) of a slab. This means that each left/right piece of $\xi_{3d}$ adjacent to an uncovered strip can be thickened until it completely covers it. This results in vertically thicker pieces in the planar embedding $\xi_{2d}$ of $\xi_{3d}$. Because $\xi_{2d}$ is monotonic in the horizontal direction, thickening it vertically cannot result in overlap. It also cannot interfere with the hanging of the front/back faces, since front/back faces attach along horizontal ($x$-parallel) sections of $\xi_{3d}$, whereas the thickened strips occur along otherwise unused vertical ($z$-parallel) sections of $\xi_{3d}$. Thus we have the following result. \[theo:5x5\] Every Manhattan Tower polyhedron can be edge-unfolded with a $4 \times 5 \times 1$ refinement of each face of the vertex grid. Conclusion ========== We have established that every $(4 \times 5 \times 1)$-refined Manhattan Tower polyhedron may be edge-unfolded. This is the second nontrivial class of objects known to have a refined grid-unfolding, besides orthostacks. This is the first unfolding algorithm for orthogonal polyhedra that uses recursion, something we believe will be useful in developing algorithms to unfold more general shapes that can branch in many directions. The algorithm works on some orthogonal polyhedra that are not Manhattan Towers, and we are working on widening its range of applicability. #### Acknowledgements. {#acknowledgements. .unnumbered} We thank the anonymous referees for their careful reading and insightful comments. [^1]: This only happens if $r'$ is a front child of $r$.
--- abstract: 'Orbit codes are a family of codes applicable for communications on a random linear network coding channel. The paper focuses on the classification of these codes. We start by classifying the conjugacy classes of cyclic subgroups of the general linear group. As a result, we are able to focus the study of cyclic orbit codes to a restricted family of them.' author: - bibliography: - '../../huge.bib' - '../../to\_update\_publications.bib' title: On conjugacy classes of subgroups of the general linear group and cyclic orbit codes --- Introduction {#introduction .unnumbered} ============ The interest on constructions of codes for random linear network coding arises with the paper [@ko08]. This paper introduces the notion of a code as a subset of $\mathcal{P}({\mathcal{V}})$, that is the set of all subspaces of a vector space over a finite field ${\mathbb{F}}_q$. This set is equipped with a metric, suitable for the model of communication introduced, called subspace distance, defined as follows: for every ${\mathcal{U}}_1,{\mathcal{U}}_2\in \mathcal{P}({\mathcal{V}})$, $$d({\mathcal{U}}_1,{\mathcal{U}}_2)=\dim({\mathcal{U}}_1)+\dim({\mathcal{U}}_2)-2\dim({\mathcal{U}}_1\cap {\mathcal{U}}_2).$$ The set of all subspaces of dimension $k$ is called the Grassmannian and denoted by ${\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$. Some effort has been done in the direction of constructing codes for random linear network coding in the last few years. Some results can be found in [@ko08; @ma08p; @ko08p; @et09; @sk10; @tr10p]. In order to introduce orbit codes, we first recall the notion of the right action of the group $GL_n({\mathbb{F}}_q)$ of the invertible matrices on the Grassmannian. Let ${\mathcal{U}}\in {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$ and $U\in {\mathbb{F}}_q^{k\times n}$ a matrix such that ${\mathcal{U}}:={\mathrm{rowsp}}(U)$. We define the following operation $${\mathcal{U}}A:={\mathrm{rowsp}}(UA).$$ As a consequence we obtain the following right action of $GL_n({\mathbb{F}}_q)$ on ${\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$ $$\begin{aligned} {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}\times GL_n({\mathbb{F}}_q) & \rightarrow & {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}\\ ({\mathcal{U}},A)& \mapsto & {\mathcal{U}}A. \end{aligned}$$ The action just defined on ${\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$ is independent of the choice of the representation matrix $U\in {\mathbb{F}}_q^{k\times n}$ it is distance preserving. For more information the reader is referred to [@tr10p]. Orbit codes are a certain class of constant dimension codes. Let ${\mathcal{U}}\in {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$ and ${\mathfrak{S}}<GL_n({\mathbb{F}}_q)$ a subgroup. Then $${\mathcal{C}}= \{{\mathcal{U}}A\mid A\in {\mathfrak{S}}\}$$ is called orbit code. An orbit code is called cyclic if there exists a subgroup defining it that is cyclic. In [@tr10p] the authors show that orbit codes satisfy properties that are similar to the ones of linear codes for classical coding theory. Moreover, some already known constructions, such as the ones contained in [@ko08] and [@ma08p], are actually orbit codes. This paper focuses on the classification of orbit codes. In order to do so, we are going to give a classification of the conjugacy classes of subgroups of $GL_n({\mathbb{F}}_q)$. The paper is structured as follows. The first section is dedicated to the classification of subgroups of $GL_n({\mathbb{F}}_q)$. More in detail, we are able to characterize the properties of a unique representative for the conjugacy classes of cyclic subgroups of $GL_n({\mathbb{F}}_q)$. The result is contained in Theorem \[t:rep\_grp\]. With some examples we also show that the classification as it is cannot be extended to arbitrary subgroups. In the second section we apply these results to cyclic orbit codes. The main result is that we can focus on the study of cyclic orbit codes defined by a cyclic group generated by a matrix in rational canonical form. Moreover we study the construction of codes in this case and relate them to completely reducible cyclic orbit codes. At last we give some conclusions. Characterization of cyclic subgroups of $GL_n({\mathbb{F}}_q)$ {#s:1} ============================================================== In this section we investigate the cyclic subgroups of $GL_n({\mathbb{F}}_q)$. The goal is to characterize them in a way that is suitable for the construction of orbit codes. More specifically we are interested in answering the question about when two cyclic groups are conjugate to each other. Consider $GL_n({\mathbb{F}}_q)$ and the following equivalence relation on it: Given $A,B\in GL_n({\mathbb{F}}_q)$ then $$A\sim_c B \quad \iff \quad \exists L\in GL_n({\mathbb{F}}_q): \ A=L^{-1}BL.$$ A natural choice of representatives of the classes of $ GL_n({\mathbb{F}}_q)/\sim_c$ is given by the rational canonical form. Rational canonical forms are based on companion matrices, whose definition is as follows. Let $p=\sum_{i=0}^sp_ix^i\in {\mathbb{F}}_q[x]$ be a monic polynomial. Its companion matrix is the matrix $$M_p:={\left(\begin{matrix}0&1&0 &\cdots &0\\0 & 0 & 1 & &0\\\vdots & & & \ddots &\vdots\\ 0&0&0& & 1 \\ -p_0& -p_1 & -p_2 & \cdots & -p_{s-1}\end{matrix} \right)}\in {\mathbb{F}}_q^{s\times s}.$$ The following theorem states the existence and uniqueness of a rational canonical form. Let $A\in GL_n({\mathbb{F}}_q)$. Then there exists a matrix $L\in GL_n({\mathbb{F}}_q)$ such that $$\begin{aligned} \label{e:rcf} \nonumber L^{-1}AL = {\mathrm{diag}}(&M_{p_1^{e_{11}}},\dots, M_{p_1^{e_{1r_1}}},\\ &\dots,M_{p_m^{e_{m1}}}, \dots, M_{p_m^{e_{mr_m}}}) \end{aligned}$$ is a block diagonal matrix where $p_i\in {\mathbb{F}}_q[x]$ are irreducible polynomials, $e_{ij}\in {\mathbb{N}}$ are such that $e_{i1}\geq \dots \geq e_{ir_i}$, $\chi_A=\prod_{i,j}p_i^{e_{ij}}$ and $\mu_A=\prod_i p_i^{e_{i1}}$ represent respectively the characteristic and the minimal polynomials of $A$ and $M_{p_i^{e_{ij}}}$ denotes the companion matrix of the polynomial $p^{e_{ij}}$. Moreover, the matrix is unique for any choice of $A\in GL_n({\mathbb{F}}_q)$. \[d:rcf\] Let $A\in GL_n({\mathbb{F}}_q)$. The matrix is called rational canonical form of $A$ and the polynomials $p_1^{e_{11}},\dots,p_1^{e_{1r_1}},\dots,p_m^{e_{m1}},\dots,p_m^{e_{mr_m}} \in {\mathbb{F}}_q[x]$ are its elementary divisors. The following lemma motivates why rational canonical forms are a good choice of representatives for the classes of $GL_n({\mathbb{F}}_q)/\sim_c$. \[l:conj\_mat\] Let $A,B\in GL_n({\mathbb{F}}_q)$. Then the following statements are equivalent: 1. $A\sim_c B$, and 2. $A$ and $B$ have the same rational canonical form. This lemma is well-known and is a direct consequence of the uniqueness of the rational canonical form. Now we want to extend the previous characterization to subgroups of $GL_n({\mathbb{F}}_q)$. Consider the set of all subgroups of $GL_n({\mathbb{F}}_q)$ $$\mathbf{G}:=\{{\mathfrak{S}}\mid {\mathfrak{S}}< GL_n({\mathbb{F}}_q)\}$$ and the following equivalence relation on it. Given ${\mathfrak{S}}_1,{\mathfrak{S}}_2\in \mathbf{G}$ then $${\mathfrak{S}}_1\sim_c {\mathfrak{S}}_2 \quad \iff \quad \exists L\in GL_n({\mathbb{F}}_q): \ {\mathfrak{S}}_1=L^{-1}{\mathfrak{S}}_2L.$$ The following theorem extends the arguments of Lemma \[l:conj\_mat\] to the case of cyclic subgroups. \[t:conj\_grp\] Let $A,B\in GL_n({\mathbb{F}}_q)$ and ${\mathfrak{S}}_A=\langle A\rangle, {\mathfrak{S}}_B=\langle B\rangle< GL_n({\mathbb{F}}_q)$ be the two cyclic groups generated by them. Then, ${\mathfrak{S}}_A\sim_c {\mathfrak{S}}_B$ if and only if $\|{\mathfrak{S}}_A\|=\|{\mathfrak{S}}_B\|$ and there exists an $i\in {\mathbb{N}}$ with $\gcd(i,\|{\mathfrak{S}}_B\|)=1$ such that $A\sim_cB^i$. : Since ${\mathfrak{S}}_A\sim_c {\mathfrak{S}}_B$, it follows that there exists an $L\in GL_n({\mathbb{F}}_q)$ such that ${\mathfrak{S}}_A=L^{-1}{\mathfrak{S}}_BL$, implying that the two groups have the same order. Moreover, it follows that the group homomorphism $$\begin{aligned} \varphi:{\mathfrak{S}}_A & \rightarrow & GL_n({\mathbb{F}}_q) \\ A^i & \mapsto & LA^iL^{-1} \end{aligned}$$ is an isomorphism if restricted to the image of $\varphi$. As a consequence, the generator $A$ of ${\mathfrak{S}}_A$ is mapped to a generator of $L{\mathfrak{S}}_AL^{-1}={\mathfrak{S}}_B$, i.e., an element of $\{B^i\mid \gcd(i,\|{\mathfrak{S}}_B\|)=1\}$. Then, there exists an $i\in {\mathbb{N}}$ with $\gcd(i,\|{\mathfrak{S}}_B\|)=1$ such that $LAL^{-1}=B^i$, which implies that $A\sim_c B^i$. : From the hypothesis we know that $\langle B^i \rangle ={\mathfrak{S}}_B$ and that there exists $L\in GL_n({\mathbb{F}}_q)$ such that $A=L^{-1}B^iL$. The statement follows as a consequence. We introduce the following definition. Let $p\in {\mathbb{F}}_q[x]$ be a nonzero polynomial. If $p(0)\neq 0$, then the least integer $e\in {\mathbb{N}}$ such that $p$ divides $x^e-1$ is called the order of $p$. The definition is generalizable to any $p\in {\mathbb{F}}_q[x]$ but it is not interesting for the purpose of this paper since we will only consider irreducible polynomials. In order to give unique representatives for the classes of cyclic groups contained in $\mathbf{G}/\sim_c$ we need the following lemma. \[l:elem\_div\] Let $A\in GL_n({\mathbb{F}}_q)$, $p_{A,1}^{e_{A,1}},\dots,p_{A,m}^{e_{A,m}} \in {\mathbb{F}}_q[x]$ its elementary divisors, where $p_{A,j}$ for $j\in \{1,\dots,m\}$ are not necessarily distinct, and ${\mathfrak{S}}_A< GL_n({\mathbb{F}}_q)$ the cyclic group generated by $A$. Then, for every $i\in {\mathbb{N}}$ with $\gcd(i,\|{\mathfrak{S}}_A\|)=1$, the elementary divisors of $A^i$ are exactly $m$ many. If we denote them by $p_{A^i,1}^{e_{A^i,1}},\dots, p_{A^i,m}^{e_{A^i,m}}\in {\mathbb{F}}_q[x]$, then, up to reordering, the order of $p_{A,j}$ is the same as the one of $p_{A^i,j}$ and $e_{A,j}=e_{A^i,j}$ for $j=1,\dots,m$. First we prove the case where the elementary divisor is unique. At the end of the proof we will give the main remark that implies the generalized statement. Let $p_A^{e_A}\in {\mathbb{F}}_q[x]$ be the elementary divisor of a matrix $A\in GL_n({\mathbb{F}}_q)$ and $k:=n/e_A$. Let ${\mathbb{F}}_{q^k}:={\mathbb{F}}_q[x]/(p_A)$ be the splitting field of the polynomial $p_A$ and $\mu\in {\mathbb{F}}_{q^k}$ a primitive element of it. There exists a $j\in {\mathbb{N}}$ such that $p_A=\prod_{u=0}^{k-1}(x-\mu^{jq^u})$. Since $p_A^{e_A}$ is the unique elementary divisor of the matrix $A$, it corresponds to the characteristic and the minimal polynomial of $A$. As a consequence we obtain that the Jordan normal form of $A$ over ${\mathbb{F}}_{q^k}$ is $$J_A={\mathrm{diag}}\left(J_{A,\mu^j}^{e_a},\dots,J_{A,\mu^{jq^{k-1}}}^{e_a}\right)$$ where $J_{A,\mu^{jq^u}}^{e_a}\in GL_{e_A}({\mathbb{F}}_{q^k})$ is a unique Jordan block with diagonal entries $\mu^{jq^u}$ for $u=0,\dots,k-1$. By the Jordan normal form of $A$ it follows that for every $i\in {\mathbb{N}}$ the characteristic polynomial of $A^i$ is $p_{A^i}=(\prod_{u=0}^{k-1}x-\mu^{ijq^u})^{e_A}$. Let us now focus on the $i$’s such that $\gcd(i,\|{\mathfrak{S}}_A\|)=1$. $A^i$ is then a generator of ${\mathfrak{S}}_A$, i.e., $p_{A^i}\in {\mathbb{F}}_q[x]$ is a monic irreducible polynomial whose order is the same as the one of $p_A$. In order to conclude that $p_{A^i}^{e_A}$ is the elementary divisor of $A^i$ we consider its rational canonical form. Assume that the elementary divisors of $A^i$ were more than one. Without loss of generality we can consider them to be two, i.e., $p_{A^i}^{e_{A,1}}$ and $p_{A^i}^{e_{A,2}}$. This means that its rational canonical form is $\mathrm{RCF}(A^i)= {\mathrm{diag}}(M_{p_{A^i}^{e_{A,1}}},M_{p_{A^i}^{e_{A,2}}})$ where we use the operator $\mathrm{RCF}$ as an abbreviation for rational canonical form and $e_A=e_{A,1}+e_{A,2}$. For any $j\in {\mathbb{N}}$ we obtain that the matrix $\mathrm{RCF}((\mathrm{RCF}(A^i))^j)$ is a block diagonal matrix with at least two blocks. Let $j\in {\mathbb{N}}$ such that $ij\equiv 1 \pmod{\|{\mathfrak{S}}_A\|}$ and $L\in GL_n({\mathbb{F}}_q)$ be a matrix such that $\mathrm{RCF}(A^i)=L^{-1}A^iL$, then $$(\mathrm{RCF}(A^i))^j=(L^{-1}A^iL)^j=L^{-1}AL\sim_c A$$ implying that $$\mathrm{RCF}(A)=\mathrm{RCF}((\mathrm{RCF}(A^i))^j)$$ This leads to a contradiction since $\mathrm{RCF}(A)=M_{p_A^{e_A}}$ has only one block. We conclude that $p_{A^i}^{e_A}$ is the elementary divisor of $A^i$. The only difference in the case where $m>1$ consists in the choice of the splitting field. Given $p_{A,1}^{e_{A,1}},\dots,p_{A,m}^{e_{A,m}} \in {\mathbb{F}}_q[x]$ the elementary divisors of $A$ and $p_{A,l_1},\dots p_{A,l_r}$ with $l_1,\dots l_r \in \{1,\dots,m\}$ the maximal choice distinct polynomials from the elementary divisors, the splitting field on which the proof is based is ${\mathbb{F}}_q[x]/(\prod_{t=1}^r p_{A,l_t})$. We are now ready to characterize cyclic subgroups of $GL_n({\mathbb{F}}_q)$ via the equivalence relation $\sim_c$ based only on their elementary divisors. \[t:rep\_grp\] Let $A,B\in GL_n({\mathbb{F}}_q)$ and ${\mathfrak{S}}_A,{\mathfrak{S}}_B\in \mathbf{G}$ the cyclic subgroups generated by them. Then, ${\mathfrak{S}}_A\sim_c {\mathfrak{S}}_B$ if and only if the following conditions hold: 1. $A$ and $B$ have the same number of elementary divisors, and 2. if $p_{A,1}^{e_{A,1}},\dots,p_{A,m}^{e_{A,m}} \in {\mathbb{F}}_q[x]$ and $p_{B,1}^{e_{B,1}},\dots,p_{B,m}^{e_{B,m}} \in {\mathbb{F}}_q[x]$ are the elementary divisors of respectively $A$ and $B$, then, up to a reordering argument, the orders of $p_{A,j}$ and $p_{B,j}$ are the same and $e_{A,j}=e_{B,j}$ for $j=1,\dots,m$. : By Theorem \[t:conj\_grp\], there exists a power $i\in {\mathbb{N}}$ with $\gcd(i,\|{\mathfrak{S}}_A\|)=1$ such that $A\sim_c B^i$, i.e., they have the same elementary divisors. The statement follows with Lemma \[l:elem\_div\]. : Let $p_{B,l_1},\dots p_{B,l_r}\in {\mathbb{F}}_q[x]$ with $l_1,\dots l_r \in \{1,\dots,m\}$ be the maximal choice of pairwise coprime polynomials from the elementary divisors of $B$, ${\mathbb{F}}$ the splitting field of $\prod_{t=1}^rp_{B,l_t}$ and $\mu\in {\mathbb{F}}$ a primitive element of it. Consider the notation $k_j:= \deg p_{B,l_j}$ for $j=1,\dots, r$. Then, there exist $i_{B,1},\dots,i_{B,r}\in {\mathbb{N}}$ such that $p_{B,l_j}=\prod_{u=0}^{k_j-1}(x-\mu^{i_{B,j}q^u})$ for $j=1,\dots,r$. The same holds for the matrix $A$, i.e., there exist $i_{A,1},\dots,i_{A,r}\in {\mathbb{N}}$ such that $p_{A,l_j}=\prod_{u=0}^{k_j-1}(x-\mu^{i_{A,j}q^u})$ for $j=1,\dots,r$. By the condition on the orders, there exists a unique $i\in {\mathbb{N}}$ such that $i_{A,j}\equiv i\cdot i_{B,j} \pmod{{\mathrm{ord}}(p_{B,l_j})}$ for $j=1,\dots,r$. It follows that the elementary divisors of $B^i$ and the ones of $A$ are the same, i.e., $A\sim_c B^i$. The theorem states that we can uniquely represent the classes of cyclic subgroups in $\mathbf{G}/\sim_c$ by considering the cyclic subgroups generated by a rational canonical form based on the choice of a sequence of polynomials of the type $p_1^{e_1},\dots,p_m^{e_m}\in {\mathbb{F}}_q[x]$ where the polynomials $p_1,\dots,p_m$ are irreducible and $\sum_{j=1}^me_j\cdot \deg(p_j)=n$. Moreover, what matters in the choice of the polynomials $p_j$’s is only their degrees and orders. Trivially, the following holds for the cardinality of a cyclic group. Let ${\mathfrak{S}}_A=\langle A \rangle < GL_n({\mathbb{F}}_q)$. Then the order of ${\mathfrak{S}}_A$ is the least common multiple of the orders of the elementary divisors $p_1^{e_1},\dots,p_m^{e_m}\in {\mathbb{F}}_q[x]$ of the matrix $A$. To conclude the section we are going to give an example explaining why a straight forward generalization of Theorem \[t:rep\_grp\] to any subgroup of $GL_n({\mathbb{F}}_q)$ does not work. [ ]{} 1. Consider the following matrix over ${\mathbb{F}}_2$: $$A={\left(\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0\end{matrix} \right)}.$$ Although the elementary divisor of $A$ and the one of its transpose $A^t$ is the same, the groups ${\mathfrak{S}}_A=\langle A\rangle = \langle A,A\rangle$ and $GL_3({\mathbb{F}}_2)=\langle A,A^t\rangle$ are not conjugate. 2. Let ${\mathbb{F}}_{4}={\mathbb{F}}_2[x]/(x^2+x+1)$ and $\mu\in {\mathbb{F}}_4$ a primitive element. Consider the following matrices over ${\mathbb{F}}_4$: $$\begin{aligned} A={\left(\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0\end{matrix} \right)}, \ &B_1={\left(\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 1\end{matrix} \right)},\\ \mbox{ and } &B_2={\left(\begin{matrix}\mu+1 & 1 & \mu \\ \mu & \mu & \mu+1 \\ 0 & 1 & 0\end{matrix} \right)}.\end{aligned}$$ Although $B_1\sim_c B_2$, i.e., they have the same unique elementary divisor, it holds that $\|\langle A,B_1\rangle\|\neq \|\langle A,B_2\rangle\|$, meaning that the two groups are not conjugate. Conjugate groups and cyclic orbit codes ======================================= We now apply the results from the previous section to the characterization of cyclic codes. Let ${\mathfrak{S}}_1,{\mathfrak{S}}_2<GL_n({\mathbb{F}}_q)$ and ${\mathcal{C}}_1:=\{{\mathcal{U}}_1A\mid A\in {\mathfrak{S}}_1\},{\mathcal{C}}_2:=\{{\mathcal{U}}_2A\mid A\in {\mathfrak{S}}_2\}\subseteq {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$ be two orbit codes. We say that ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are conjugate or simply ${\mathcal{C}}_1\sim_c {\mathcal{C}}_2$ if there exists a matrix $L\in GL_n({\mathbb{F}}_q)$ such that $${\mathcal{U}}_2={\mathcal{U}}_1L \mbox{ and } {\mathfrak{S}}_2=L^{-1}{\mathfrak{S}}_1L,$$ i.e., ${\mathcal{C}}_2=\{{\mathcal{U}}_1AL\mid A\in {\mathfrak{S}}_1\}=\{{\mathcal{U}}_1L(L^{-1}AL)\mid A\in {\mathfrak{S}}_1\}$. In order to further study properties of orbit codes, we need to introduce the notion of distance distribution for orbit codes. Due to [@tr10p], we are able to adapt the definition of weight enumerator from classical coding theory to orbit codes. But first we recall some facts from [@tr10p]. Let ${\mathcal{U}}\in {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$. Then the stabilizer group of ${\mathcal{U}}$ is defined as $$Stab({\mathcal{U}}):=\{A\in GL_n({\mathbb{F}}_q)\mid {\mathcal{U}}A={\mathcal{U}}\}<GL_n({\mathbb{F}}_q).$$ The following proposition is important in order to define the distance distribution. Let ${\mathcal{C}}=\{{\mathcal{U}}A\mid A\in {\mathfrak{S}}<GL_n({\mathbb{F}}_q)\}$ be an orbit code. Then it holds that $$\|{\mathcal{C}}\|=\frac{\|{\mathfrak{S}}\|}{\|{\mathfrak{S}}\cap Stab({\mathcal{U}})\|}$$ and $$d({\mathcal{C}})=\min_{A\in {\mathfrak{S}}\setminus Stab({\mathcal{U}})}d({\mathcal{U}},{\mathcal{U}}A).$$ Let ${\mathcal{C}}=\{{\mathcal{U}}A\mid A \in {\mathfrak{S}}< GL_n({\mathbb{F}}_q)\}\subseteq {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$ be an orbit code. The distance distribution of ${\mathcal{C}}$ is the tuple $(D_0,\dots,D_k)\in {\mathbb{N}}^{k+1}$ such that $$D_i:=\frac{\|\{A\in {\mathfrak{S}}\mid d({\mathcal{U}},{\mathcal{U}}A)=2i\}\|}{\|{\mathfrak{S}}\cap Stab({\mathcal{U}})\|}.$$ As a consequence we obtain that $D_0=1$ and $\sum_{i=0}^kD_i=\|{\mathcal{C}}\|$. We are able to state the following theorem that characterizes conjugate orbit codes and that is a generalization of Theorem 9 from [@tr11]. \[t:conj\_orb\] The binary relation $\sim_c$ on orbit codes is an equivalence relation. Moreover, let ${\mathcal{C}}_1,{\mathcal{C}}_2$ be two orbit codes such that ${\mathcal{C}}_1\sim_c {\mathcal{C}}_2$, then $\|{\mathcal{C}}_1\|=\|{\mathcal{C}}_2\|$ and they have the same distance distribution. The fact that $\sim_c$ is an equivalence relation on orbit codes is a consequence of Theorem \[t:conj\_grp\]. Let ${\mathcal{C}}_1:=\{{\mathcal{U}}A\mid A\in {\mathfrak{S}}<GL_n({\mathbb{F}}_q)\}$ and $L\in GL_n({\mathbb{F}}_q)$ such that ${\mathcal{C}}_2=\{{\mathcal{U}}AL\mid A\in {\mathfrak{S}}\}$. The same cardinality is consequence of the fact that given $A,B\in {\mathfrak{S}}$ then $${\mathcal{U}}AL={\mathcal{U}}BL \iff {\mathcal{U}}A={\mathcal{U}}B.$$ The same distance distribution follows from the distance preserving property of the $GL_n({\mathbb{F}}_q)$ action on ${\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$, i.e., $d({\mathcal{U}}L,{\mathcal{U}}AL)=d({\mathcal{U}},{\mathcal{U}}A)$. The importance of this last theorem is that two conjugate orbit codes are not distinguishable from the point of view of cardinality and distance distribution. Theorem \[t:rep\_grp\] translates as follows in the language of orbit codes. Every cyclic orbit code is conjugate to a cyclic orbit code defined by a cyclic group generated by a matrix in rational canonical form. This fact gives us the opportunity to consider only cyclic orbit codes out of matrices in rational canonical form for the study of codes with good parameters. We are now interested in these orbits codes. \[t:codes\] Let $M:={\mathrm{diag}}(M_{p_1^{e_1}},\dots,M_{p_t^{e_t}})\in GL_n({\mathbb{F}}_q)$ a matrix such that $p_i\in {\mathbb{F}}_q[x]$ are monic irreducible polynomials and $d_i:=\deg(p_i^{e_i})$ for $i=1,\dots,t$. Let ${\mathcal{U}}={\mathrm{rowsp}}(U_1,\dots,U_t)\in {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$ with $U_i\in {\mathbb{F}}_q^{k\times d_i}$ and where $(U_1,\dots,U_t)$ is in row reduced echelon form. For any $i\in \{1,\dots,t\}$, let $\bar{U}_i$ be a submatrix of $U_i$ as depicted in Figure \[f:rre\]. If ${\mathcal{C}}:=\{{\mathcal{U}}M^i\mid i\in {\mathbb{N}}\}$ and ${\mathcal{C}}_i:=\{{\mathrm{rowsp}}(\bar{U}_i)M_{p_i^{e_i}}^j\mid j\in {\mathbb{N}}\}$, then $$\begin{aligned} \label{e:modular_dmin} d({\mathcal{C}})\!\geq\! 2k\!-\!2\!\sum_{i=1}^t \!\max_{j\in {\mathbb{N}}} \dim\!\left(\!{\mathrm{rowsp}}\left(\bar{U}_i\right) \!\cap\! {\mathrm{rowsp}}\left(\!\bar{U}_iM_{p_i^{e_i}}^j\!\right)\!\right)\!,\!\!\! \end{aligned}$$ and $\|{\mathcal{C}}\|:=\mathrm{lcm}(\|{\mathcal{C}}_i\|,\dots,\|{\mathcal{C}}_t\|)$. Consider the following projections $$\begin{array}{rccc} \pi_i:&{\mathbb{F}}_q^n & \longrightarrow &{\mathbb{F}}_q^{d_i}\\ &(v_1,\dots,v_n) & \longmapsto & (v_{l_{i-1}+1},\dots,v_{l_i}) \end{array}$$ where $l_i=\sum_{j=1}^id_i$ for $i=1,\dots,t$. Since $(U_1,\dots,U_t)$ has full rank and is in row reduced echelon form, the matrices $\bar{U}_i$ have full rank. Let $\bar{{\mathcal{U}}}_i\subset {\mathbb{F}}_q^n$ be the space spanned by the rows of $(U_1,\dots,U_t)$ indexed by the rows corresponding to $\bar{U}_i$. Since $\bar{U}_i$ has full rank it follows that $\pi_i\vert_{\bar{{\mathcal{U}}}_i}$ is injective for $i=1,\dots,t$. As a consequence we obtain that for any $i=1,\dots,t$, if we define $m_i\in {\mathbb{N}}$ such that $$\dim \left(\bar{{\mathcal{U}}}_i\cap \bar{{\mathcal{U}}}_iM_{p_i^{e_i}}^{m_i}\right)\geq \dim \left(\bar{{\mathcal{U}}}_i\cap \bar{{\mathcal{U}}}_iM_{p_i^{e_i}}^j\right), \quad \forall j\in {\mathbb{N}}$$ and ${\mathcal{V}}_i:= \bar{{\mathcal{U}}}_i\cap \bar{{\mathcal{U}}}_iM_{p_i^{e_i}}^{m_i}$, then $$\pi_i({\mathcal{V}}_i)\subseteq {\mathrm{rowsp}}(\bar{U}_i)\cap{\mathrm{rowsp}}(\bar{U}_iM_{p_i^{e_i}}^{m_i}).$$ It follows that $$\dim({\mathcal{V}}_i)\leq \max_{j\in {\mathbb{N}}}\dim({\mathrm{rowsp}}(\bar{U}_i)\cap{\mathrm{rowsp}}(\bar{U}_iM_{p_i^{e_i}}^{j})).$$ Since ${\mathcal{U}}=\oplus_{i=1}^t\bar{{\mathcal{U}}}_i$ we conclude that $$\begin{aligned} d({\mathcal{C}})&=2k-2\max_{j\in {\mathbb{N}}}\dim ({\mathcal{U}}\cap{\mathcal{U}}M^j)\\ &\geq 2k-2\sum_{i=1}^t \max_{j\in {\mathbb{N}}} \dim\left({\mathrm{rowsp}}\left(\bar{U}_i\right) \cap {\mathrm{rowsp}}\left(\bar{U}_iM_{p_i^{e_i}}^j\right)\right) \end{aligned}$$ The cardinality of ${\mathcal{C}}$ is a direct consequence of the fact that $${\mathrm{diag}}(M_{p_1^{e_1}},\dots,M_{p_t^{e_t}})^i= {\mathrm{diag}}(M_{p_1^{e_1}}^i,\dots,M_{p_t^{e_t}}^i)$$ and of the minimality of the least common multiple. It is possible to find examples for which the lower bound given by is attained. The following lemmas depict these examples. Let $M:={\mathrm{diag}}(M_{p_1^{e_1}},\dots,M_{p_t^{e_t}})\in GL_n({\mathbb{F}}_q)$ a matrix such that $p_i\in {\mathbb{F}}_q[x]$ are monic irreducible polynomials and $d_i:=\deg(p_i^{e_i})$ for $i=1,\dots,t$. Let $k\leq d_i$ for $i=1,\dots,t$ and ${\mathcal{U}}:={\mathrm{rowsp}}(U_1,\dots,U_t)\in {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$ where $U_i\in {\mathbb{F}}_q^{k\times d_i}$ are matrices having full rank for $i=1,\dots,t$. If we define ${\mathcal{C}}:=\{{\mathcal{U}}M^i\mid i\in {\mathbb{N}}\}$ and ${\mathcal{C}}_i:=\{{\mathrm{rowsp}}(U_i)M_{p_i^{e_i}}^j\mid j\in {\mathbb{N}}\}$ and it holds $\gcd(\|{\mathcal{C}}_i\|,\|{\mathcal{C}}_j\|)=1$ for all $i\neq j$, then $$\begin{aligned} d({\mathcal{C}})= \min_{i\in \{1,\dots, t\}}d({\mathcal{C}}_i). \end{aligned}$$ We only need to show that there exists a codeword of ${\mathcal{C}}$ that satisfies this minimum. Up to a permutation of $\{1,\dots,t\}$ we can consider that the code ${\mathcal{C}}_1$ is satisfying the minimum distance. Let $g_1\in {\mathbb{N}}$ be such that $d({\mathrm{rowsp}}(U_1),{\mathrm{rowsp}}(U_1)M_{p_1^{e_1}}^{g_1})=d({\mathcal{C}}_1)$. Since the cardinalities of the codes ${\mathcal{C}}_i$ are pairwise coprime, it follows that there exists $g\in {\mathbb{N}}$ such that $$g \equiv g_1 \pmod{\|{\mathcal{C}}_1\|}\quad \mbox{and}\quad g\equiv 0 \pmod{\|{\mathcal{C}}_j\|}$$ for $j=2,\dots,m$. We obtain that $$\begin{aligned} d({\mathcal{U}},{\mathcal{U}}M^g)&=&d({\mathcal{U}},{\mathcal{U}}{\mathrm{diag}}(M_{p_1^{e_1}}^{g_1},I,\dots,I))\\ &=&d({\mathrm{rowsp}}(U_1),{\mathrm{rowsp}}(U_1)M_{p_1^{e_1}}^{g_1})=d({\mathcal{C}}_1) \end{aligned}$$ Let $M:={\mathrm{diag}}(M_{p_1^{e_1}},\dots,M_{p_t^{e_t}})\in GL_n({\mathbb{F}}_q)$ such that $p_i\in {\mathbb{F}}_q[x]$ are monic irreducible polynomials and $d_i:=\deg(p_i^{e_i})$ for $i=1,\dots,t$. Let $k_i\leq d_i$, $\bar{U}_i\in {\mathbb{F}}_q^{k_i\times d_i}$ be matrices with full rank and ${\mathcal{U}}:={\mathrm{diag}}(\bar{U}_1,\dots,\bar{U}_t)\in {\mathcal{G}_{{\mathbb{F}}_q}(k,n)}$. If we define ${\mathcal{C}}:=\{{\mathcal{U}}M^i\mid i\in {\mathbb{N}}\}$ and ${\mathcal{C}}_i:=\{{\mathrm{rowsp}}(\bar{U}_iM_{p_i^{e_i}})^j\mid j\in {\mathbb{N}}\}$ and it holds $\gcd(\|{\mathcal{C}}_i\|,\|{\mathcal{C}}_j\|)=1$ for all $i\neq j$, then $$\begin{aligned} d({\mathcal{C}})\!= \!2k-2\!\sum_{i=1}^t \max_{j\in {\mathbb{N}}} \dim\!\left({\mathrm{rowsp}}\left(\bar{U}_i\right) \cap {\mathrm{rowsp}}\left(\!\bar{U}_iM_{p_i^{e_i}}^j\!\right)\right). \end{aligned}$$ Also here we show a codeword of ${\mathcal{C}}$ which satisfies the relation. Let $g_1,\dots,g_t\in {\mathbb{N}}$ be such that $\dim({\mathrm{rowsp}}(\bar{U}_j)\cap {\mathrm{rowsp}}(\bar{U}_jM_{p_j^{e_j}}^{g_j})$ is maximal for $j=1,\dots,m$. Since the cardinalities of the codes are pairwise coprime, it follows that there exists a $g\in{\mathbb{N}}$ such that $$g\equiv g_j \pmod{\|{\mathcal{C}}_j\|}$$ for any $j=1,\dots,t$. Then, $$\begin{aligned} d_{\min}({\mathcal{C}})&= d({\mathcal{U}},{\mathcal{U}}{\mathrm{diag}}(M_{p_1^{e_1}},\dots,M_{p_m^{e_m}})^g)\\ &=d({\mathcal{U}},{\mathcal{U}}{\mathrm{diag}}(M_{p_1^{e_1}}^{g_1},\dots,M_{p_m^{e_m}}^{g_m}))\\ &\phantom{=2k-2\sum_{j=1}^m\dim({\mathrm{rowsp}}(\bar{U}_j)\cap{\mathrm{rowsp}}(\bar{U}_jM_{p_j^{e_j}}^{g_j})).} \end{aligned}$$ $$\begin{aligned} \phantom{d_{\min}({\mathcal{C}})}&=2k-2\sum_{j=1}^m\dim({\mathrm{rowsp}}(\bar{U}_j)\cap{\mathrm{rowsp}}(\bar{U}_jM_{p_j^{e_j}}^{g_j})). \end{aligned}$$ A matrix $M\in GL_n({\mathbb{F}}_q)$ is called completely reducible if its elementary divisors are all irreducible, i.e., from Definition \[d:rcf\] if $e_{i,j}=1$ for all $i,j$. One can use the theory of irreducible cyclic orbit codes from [@tr11] to compute the minimum distances of the block component codes in the extension field representation and hence with Theorem \[t:codes\] a lower bound for the minimum distance of the whole code. Conclusions {#conclusions .unnumbered} =========== Due to the characterization of conjugacy classes of cyclic subgroups of $GL_n({\mathbb{F}}_q)$, we were able to conclude that every cyclic orbit code is conjugated to a cyclic orbit code defined by the cyclic group generated by a matrix in rational canonical form. The research of orbit codes with good parameters can then be restricted to this subclass of cyclic orbit codes. The following step in this research direction is to completely classify orbit codes. In order to do so we have to find a characterization of the conjugacy classes of subgroups of $GL_n({\mathbb{F}}_q)$ that possibly coincides with the one presented in Section \[s:1\] if restricted to cyclic subgroups of $GL_n({\mathbb{F}}_q)$.
--- author: - 'C. Schreiber' - 'D. Elbaz' - 'M. Pannella' - 'L. Ciesla' - 'T. Wang' - 'M. Franco' bibliography: - '../bbib/full.bib' date: 'Received 4th of July 2017; accepted 24th of October 2017' title: \| Dust temperature and mid-to-total infrared color\ distributions for star-forming galaxies at $0<z<4$[^1] [^2] --- Introduction \[SEC:introduction\] ================================= Properly accounting for the amount of stellar light absorbed by dust has proven to be a key ingredient to study star formation in galaxies. The most obvious breakthrough linked to deep infrared (IR) surveys was probably the exciting new outlook they provided on the cosmic history of star formation [e.g., @smail1997; @hughes1998; @barger1998; @blain1999; @elbaz1999; @flores1999; @lagache1999; @gispert2000; @franceschini2001; @elbaz2002; @papovich2004; @lefloch2005; @elbaz2007; @daddi2009-a; @magnelli2009; @gruppioni2010; @elbaz2011; @rodighiero2011; @magdis2012; @madau2014 and references therein]. In the meantime, the emission of dust in distant galaxies has also been used to study the dust itself, which turned out to be a valuable tool to learn more about non-stellar baryonic matter, present in the interstellar medium (ISM) either in the form of dust grains or atomic and molecular gas [e.g., @chapman2003; @hwang2010-a; @elbaz2011; @magdis2012; @berta2013; @scoville2014; @santini2014; @bethermin2015-a; @genzel2015; @tacconi2017]. In the Local Universe, the large amount of infrared data acquired in the Milky Way and nearby galaxies has given birth to detailed models aiming to provide a description of the dust content from first principles [e.g., @zubko2004; @draine2007; @galliano2011; @jones2013 to only name a few of the most recent ones]. These models typically contain three main components [see, e.g., @desert1990]: big grains (BGs, $>0.01\,{\mu{\rm m}}$), very small grains (VSGs, $<0.01\,{\mu{\rm m}}$), and complex molecules (polycyclic aromatic hydrocarbon, or PAH). The most prominent one is the emission of big grains, which are at thermal equilibrium with the ambient interstellar radiation. These grains radiate like gray bodies with a typical temperature of ${T_{\rm dust}}\sim 20$–$40{{\rm K}}$, and therefore emit the bulk of their energy in the far-infrared (FIR) around the rest-frame $100\,{\mu{\rm m}}$. Smaller grains have a too small cross-section to be at equilibrium with the ambient radiation, and are instead only transiently heated to temperatures $\sim1000\,{{\rm K}}$. This produces continuum emission in the mid-infrared (MIR). Lastly, PAHs are large carbonated molecules which cool down through numerous rotational and vibrational modes, and thus produce a group of bright and broad emission lines between $\lambda=3.3$ and $12.3\,{\mu{\rm m}}$ [@leger1984; @allamandola1985]. To reproduce a set of observations in the IR, one can vary the total mass of dust encompassing all ISM components (${M_{\rm dust}}$), the distribution of energy they receive from their surrounding medium ($U$), coming mostly from stellar light, and the properties of each species of grains and molecules, including their size distribution, their chemical state and composition (neutral vs. ionized PAHs, silicate vs. carbonated grains). Given these parameters, models can output the expected infrared spectrum and interpret the observed data. However, most of these parameters are degenerate or unconstrained and the number of degrees of freedom is too large, hence assumptions have to be made when applying such models to photometric data. Typical approaches (e.g., @draine2007, hereafter or @dacunha2008) assume fixed grain distributions (motivated by observations from clouds of the Milky Way), and consider simplified geometries of dusty regions (e.g., birth clouds, diffuse ISM, hot torus around a super-massive black hole). This still allows much flexibility in the output spectrum, and can describe observations accurately (e.g., @dacunha2015 [@gobat2017]). But even then, properly constraining the fit parameters (in particular the dust temperature) requires exquisite IR spectral energy distributions (SEDs) with good wavelength sampling, at a level of quality that can currently only be reached either in the Local Universe or at high-redshifts for the most extreme starbursts [e.g., @hwang2010-a; @magdis2010-c; @riechers2013], strongly lensed galaxies [@sklias2014 e.g.,], or on stacked samples [e.g., @magnelli2014; @bethermin2015-a]. For the typical higher redshift galaxy, the available IR SED is limited to one or two photometric points [e.g., @elbaz2011], and even simpler approaches are often preferred. A number of empirical libraries have been constructed to this end, each composed of a reduced number of template SEDs. These templates are typically associated to different values of a single parameter, for example the $8$-to-$1000\,{\mu{\rm m}}$ luminosity (${L_{\rm IR}}$) [@chary2001 hereafter ], a FIR color [@dale2002 hearafter ], or the average intensity of the interstellar radiation field, ${\left<U\right>}$ [@magdis2012; @bethermin2015-a]. In all cases the number of free parameters is reduced to two: the normalization of the template (which can be linked either to the mass or luminosity of the dust) and its “shape” (essentially its average dust temperature, which defines the wavelength at which the template peaks). Despite their extreme simplicity, these models are sufficient to reproduce the observed IR features of the vast majority of distant galaxies, illustrating the fact that the dust SED of a galaxy taken as a whole is close to universal [see, e.g., @elbaz2010; @elbaz2011]. This universality echoes another key observation of the last decade: the main sequence of star-forming galaxies [@noeske2007-a; @elbaz2007]. This tight correlation between the star formation rate (${{\rm SFR}}$) and the stellar mass (${M_\ast}$) has been observed across a broad range of redshifts up to $z\sim6$ [e.g., @daddi2007-a; @pannella2009-a; @rodighiero2011; @whitaker2012-a; @bouwens2012; @whitaker2014; @salmon2015; @pannella2015; @schreiber2015; @schreiber2017-a]. Since its discovery, the main sequence has been used to put upper limits on the variability of the star formation histories, showing that galaxies form their stars mostly through a unique and secular way, as opposed to random bursts [see, e.g., the discussion in @noeske2007-a]. Through observations of their molecular gas content, galaxies belonging to the main sequence have also been shown to form their stars with a roughly constant efficiency (${{\rm SFE}}\equiv{{\rm SFR}}/{M_{\rm gas}}$) (e.g., @daddi2010-a [@tacconi2013; @genzel2015]). The same conclusion can be drawn from the dust emission of these galaxies and the measurement of their average dust temperature (@magdis2012 [@bethermin2015-a], but see however @schreiber2016). Indeed, ${T_{\rm dust}}$ is a proxy for ${L_{\rm IR}}/{M_{\rm dust}}$, which itself can be linked directly to ${{\rm SFE}}/Z$ [@magdis2012], where $Z$ is the gas-phase metallicity. The universality of the dust SED therefore also suggests that star formation in galaxies is the product of a universal mechanism, which still remains to be fully understood (see, e.g., @dekel2013 or @tacchella2015). Departures from this “universal” SED do exist however. Galaxy-to-galaxy variations of ${T_{\rm dust}}$ have been observed, with a first correlation identified with ${L_{\rm IR}}$ [e.g., @soifer1987; @soifer1989; @dunne2000; @chapman2003; @chapin2009; @symeonidis2009; @amblard2010; @hwang2010-a]. It was later argued that this correlation is not fundamental, but in fact consequential of two effects: on the one hand a global increase of the temperature with redshift [e.g., @magdis2012; @magnelli2014; @bethermin2015-a], and on the other hand an additional increase of temperature for galaxies that are offset from the main sequence [@elbaz2011; @magnelli2014; @bethermin2015-a], suggesting these galaxies form stars more efficiently than the average. Quantifying changes of the dust temperature can thus provide crucial information about the star formation efficiency in galaxies, and it is therefore an important ingredient in any library. In addition, significant galaxy-to-galaxy variations have been observed in the MIR around the rest-frame $3$ to $12\,{\mu{\rm m}}$. As written above, the dust emission in this wavelength domain is mostly produced by small grains and PAHs. The PAH emission lines are so bright that they typically contribute about $80\%$ of the observed broadband MIR fluxes [e.g., @helou2000; @huang2007], however their strength is strongly reduced in starbursts and active galactic nuclei (AGNs, e.g., [@armus2007]), in which hot dust takes over. Therefore, the observed diversity in the MIR can be expected to come mostly from a diversity of PAH properties, at least for galaxies without strong AGNs (e.g., @fritz2006). The interplay between the overall strength of PAHs and physical conditions inside the host galaxy is not yet fully understood. Two main trends are known at present: on the one hand an anti-correlation with metallicity [e.g., @madden2006; @wu2006; @ohalloran2006; @smith2007; @draine2007-a; @galliano2008; @ciesla2014; @remy-ruyer2015], and on the other hand a correlation with ${L_{\rm IR}}$ [e.g., @pope2008-a; @elbaz2011; @nordon2012]. Although this latter correlation suffers from a significant scatter, it implies that PAH features or the $8\,{\mu{\rm m}}$ luminosity can be used as a rough tracer of star formation rate [@pope2008-a; @shipley2016]. An interesting property of PAHs is that they are set aglow mostly in photo-dissociation regions, at the interface between the ionized and molecular interstellar medium [e.g., @tielens1993], whereas the FIR dust continuum is emitted from the whole volume of the dust clouds. Therefore, by relating the dust continuum to the PAH emission one can probe the geometry of star-forming regions, and in particular the filling factor of regions. Using this approach and combining [[Spitzer]{}]{}and [[Herschel]{}]{}data, [@elbaz2011] have used the ${{\rm IR8}}={L_{\rm IR}}/{L_8}$ ratio as a tracer of compactness in distant galaxies: at fixed ${L_{\rm IR}}$, a lower ${L_8}$ indicates a higher filling factor of regions, hence a higher compactness. main sequence galaxies have a constant ${{\rm IR8}}\sim4$, while, as for the dust temperature, ${{\rm IR8}}$ increases as a function of the distance to the main sequence [see also @nordon2012; @rujopakarn2013; @murata2014]. These trends confirm that galaxies above the main sequence form their stars in a different way, with a higher efficiency and in more compact volumes. While the study of the physical origin of the MIR emission is obviously of interest on its own, it is also important to the extra-galactic community for practical observational reasons. Since the PAH emission is strong and found at low infrared wavelengths, the wavelength domain around the rest-frame $8\,{\mu{\rm m}}$ is easier to observe than than the FIR continuum. It is particularly the case for galaxies at $z\sim2$, where the rest-frame $8\,{\mu{\rm m}}$ shifts into the very deep [[Spitzer]{}]{}MIPS $24\,{\mu{\rm m}}$ band, and allows the detection of galaxies significantly fainter than the detection limit of other infrared observatories like [[Herschel]{}]{}. However, these galaxies have by construction a very poorly constrained infrared SED, and extrapolating the total ${L_{\rm IR}}$ from the $8\,{\mu{\rm m}}$ alone is challenging [see @daddi2007-a; @elbaz2011; @magdis2011-a; @rujopakarn2013; @shivaei2017]. Doing so requires an accurate understanding of the ${{\rm IR8}}$ ratio. Another important practical interest for the rest-frame $8\,{\mu{\rm m}}$ is that it will be easily accessed by the [James Webb Space Telescope]{} ([[JWST]{}]{}) in the near future, for both local and distant galaxies. Once this satellite is launched, there will be a need for a properly calibrated library to exploit these data together with ancillary [[Herschel]{}]{}and [[Spitzer]{}]{}observations, and in particular to cope with their absence for the faintest objects. Our goal in this paper is the following. We introduce in [section \[SEC:irsed\]]{} a new SED library in which both ${T_{\rm dust}}$ and ${{\rm IR8}}$ are free parameters. This library provides an increased level of detail compared to standard libraries (e.g., , ), but still keeps the number of adjustable parameters low. In [section \[SEC:irsed\_stack\]]{} we determine the redshift evolution of both ${T_{\rm dust}}$ and ${{\rm IR8}}$ using the MIR-to-FIR stacks introduced in [@schreiber2015], to which we add stacks of $16\,{\mu{\rm m}}$ and ALMA $870\,{\mu{\rm m}}$ to better constrain the PAH features and the dust temperature at high redshifts. We then apply this model to individual [[Herschel]{}]{}detections in [section \[SEC:irsed\_indiv\]]{} to constrain the scatter on the model parameters, and also to quantify their enhancements for those galaxies that are offset from the main sequence. Using these results, we derive in [section \[SEC:irsed\_recipe\]]{} a set of recipes for optimal SED fitting in the IR, in particular when a single photometric band is available. Finally, we quantify the accuracy of such measurements using mock galaxy catalogs in [section \[SEC:irsed\_mono\]]{}, and provide in [section \[SEC:conv\]]{} conversion factors to determine dust masses and infrared luminosities from ALMA fluxes and [[JWST]{}]{}MIRI luminosities. These are valid for $0<z<4$, and are extrapolated to $z=8$ for ALMA. In the following, we assume a $\Lambda$CDM cosmology with $H_0 = 70\ {\rm km}\,{\rm s}^{-1} {\rm Mpc}^{-1}$, $\Omega_{\rm M} = 0.3$, $\Omega_\Lambda = 0.7$ and a [@salpeter1955] initial mass function (IMF), to derive both star formation rates and stellar masses. All magnitudes are quoted in the AB system, such that $M_{\rm AB} = 23.9 - 2.5\log_{10}(S_{\!\nu}\ [{\mu{\rm Jy}}])$. Sample and observations ======================= We based this analysis on the sample and data described in [@schreiber2015] , which covers redshifts from $z=0.3$ to $z=4$. We complemented this sample with $z=0$ galaxies from the [[Herschel]{}]{}Reference Survey (HRS; @boselli2010), and $z=2$ to $4$ galaxies in the Extended Chandra Deep Field South (ECDFS) observed by ALMA as part of the ALESS program [@hodge2013]. In this section, we make a brief summary of these observations. CANDELS ------- The catalogs we used in this work are based on the CANDELS [@grogin2011; @koekemoer2011] Hubble Space Telescope ([[HST]{}]{}) WFC3 $H$ band images in the fields covered by deep [[Herschel]{}]{}PACS and SPIRE observations, namely GOODS–South [@guo2013-a], UDS [@galametz2013] and COSMOS [@nayyeri2017]. For the GOODS–North fied the CANDELS catalog was not yet finalized, and we used instead the [$K_{\rm s}$]{}-selected catalog of [@pannella2015]. Each of these fields is about $150\,{\rm arcsec}^2$ and they are evenly distributed on the sky to mitigate cosmic variance. We also used a catalog of the COSMOS $2\,\sq\degr$ field [@muzzin2013-a], which has overall shallower data but covers a much larger area; this field provides important statistics for the rarest and brightest objects. The ancillary photometry varies from one field to another, being a combination of both space- and ground-based imaging from various facilities. The UV to near-IR wavelength coverage typically goes from the $U$ band up the [[Spitzer]{}]{}IRAC $8\,{\mu{\rm m}}$, including at least the [[HST]{}]{}bands F606W, F814W, F125W, and F160W in CANDELS, and a deep $K$ (or $K_{\rm s}$) band. All these images are among the deepest available views of the sky. These catalogs therefore cover most of the important galaxy spectral features across a wide range of redshifts, even for intrinsically faint objects. We augmented these catalogs with mid-IR photometry from [[Spitzer]{}]{}MIPS and far-IR photometry from [[Herschel]{}]{}PACS and SPIRE taken as part of the GOODS–[[Herschel]{}]{}[@elbaz2011], CANDELS–[[Herschel]{}]{}programs (PI: M. Dickinson), PEP [@lutz2011] and HerMES [@oliver2010]. Photometric redshifts and stellar masses were computed following [@pannella2015] using EAzY [@brammer2008]; for COSMOS $2\,\sq\degr$ we used the redshifts from [@muzzin2013-a] which were computed the same way. For all catalogs, stellar masses were then computed using FAST [@kriek2009] by fixing the redshift to the best-fit photo-$z$ and fitting the observed photometry up to the IRAC $4.5\,{\mu{\rm m}}$ band[^3] using the [@bruzual2003] stellar population synthesis model, assuming a [@salpeter1955] IMF, a [@calzetti2000] extinction law and a delayed exponentially-declining star formation history. Galaxies with an uncertain photometric redshift (redshift `odds` less than $0.8$) or bad SED fitting (reduced $\chi^2$ larger than $10$) were excluded from our sample. The resulting sample is the one we used for stacking the [[Herschel]{}]{}images in . In this previous work, we estimated the evolution of the stellar mass completeness (at the $90\%$ level) of these catalogs at all redshifts, and found that all the stacked samples with significant signal were complete in mass. For example, at $z=1$ the completeness is as low as $5\times10^8\,{{\rm M}_\odot}$. We estimated ${{\rm SFR}}$s by summing the emerging UV light and the dust obscured component observed in the mid- to far-IR, following [@daddi2007-a] and [@kennicutt1998-a] to convert the observed luminosities into star formation rates: $${{\rm SFR}}= 2.17\times10^{-10}\,{L_{\rm UV}}\,[{L_\odot}] + 1.72\times10^{-10}\,{L_{\rm IR}}\,[{L_\odot}]\,.$$ UV luminosities ($1500\,\text{\AA}$) were computed from the best-fit photo-$z$ template from EAzY, while IR luminosities were computed from the best-fit dust SED obtained with our new library (see [section \[SEC:irsed\_recipe\]]{}). We applied this method to both the stacked samples and to individual galaxies with mid- or far-IR detections. For individual galaxies, we did not attempt to measure the ${{\rm SFR}}$s of the rest of the sample without IR data since estimates based on the UV light alone are less reliable [@goldader2002; @buat2005; @elbaz2007; @rodighiero2011; @rodighiero2014]. When working with individual galaxies (as opposed to stacking), we therefore only considered the sub-sample of IR-detected galaxies, which implicitly corresponds to a ${{\rm SFR}}$ threshold at each redshift [see @elbaz2011]. The resulting selection biases are discussed in [section \[SEC:selection\_effects\]]{}. Lastly, the rest-frame $U$, $V$ and $J$ magnitudes were computed for each galaxy using EAzY, by integrating the best-fit galaxy template from the photo-$z$ estimation. These colors were used, following [@williams2009], to separate galaxies that are “quiescent” from those that are “star-forming”. We used the same selection criteria as those described in , that is, a galaxy was deemed star-forming if its colors satisfy $$UVJ_{\rm SF} = \left\{\begin{array}{rcl} U - V &<& 1.3\,\text{, or} \\ V - J &>& 1.6\,\text{, or} \\ U - V &<& 0.88\times(V - J) + 0.49\,. \end{array}\right.\label{EQ:uvj}$$ otherwise the galaxy was considered as quiescent as was thus excluded from the present study. As shown in , only a very small fraction of the IR-detected galaxies are classified as [$UVJ$]{}quiescent, therefore this selection is only important for stacking. ALESS ----- To improve the statistics on the dust temperature of individual galaxies at $z>1$, we complemented this sample with the $99$ galaxies observed by ALMA in the ALESS program [@hodge2013]. ALESS is a targeted program aiming to deblend the $870\,{\mu{\rm m}}$ emission of sources detected in the single-dish LABOCA image of the ECDFS, which covers about $0.3\,\deg^2$ centered on the CANDELS GOODS–South field. The high resolution of the ALMA imaging allows a precise localization of the sub-millimeter source and avoids flux boosting from blended neighbors. We used the photometric redshifts and stellar masses determined in [@dacunha2015] using Magphys [@dacunha2008]. We used the $24\,{\mu{\rm m}}$ and PACS photometry from PEP, the SPIRE photometry as measured in [@swinbank2014] using the ALMA detections to improve the deblending, and the ALMA photometry from [@hodge2013]. The galaxies were then treated the same way as those from CANDELS, and used only in [section \[SEC:irsed\_indiv\]]{}. HRS --- To complement our sample toward the local Universe, we used the [[Herschel]{}]{}Reference Survey (HRS; @boselli2010). This is a volume-limited survey targeting a few hundred galaxies in and out of the Virgo cluster, obtaining in particular [[Herschel]{}]{}PACS and SPIRE photometry for a full sampling of their dust SEDs. All the galaxies in the HRS also have UV-to-NIR coverage to determine stellar masses and colors. Of this sample, we only considered the [$UVJ$]{}star-forming galaxies which do not belong to the Virgo cluster, to avoid systematic effects caused by this peculiar environment, for a total of 131 galaxies with a minimum stellar mass of $10^{9}\,{{\rm M}_\odot}$. This same sample was studied in [@schreiber2016]; further informations can be found there. We used the photometry from [@ciesla2012] and modeled the HRS galaxies using CIGALE [@noll2009; @roehlly2014], which fits simultaneously the stellar and dust emission. Using CIGALE proved necessary because the contribution of the stellar continuum to the $8\,{\mu{\rm m}}$ luminosity (or more generally to the PAH domain in the mid-IR) can be non-negligible at $z=0$, owing to the overall lower star-formation activity. We performed the fits using same SED libraries as for the CANDELS sample, that is, the dust SEDs introduced in this paper and the @bruzual2003 templates with a delayed SFH. Our dust SEDs are made available to all CIGALE users in the official package. A new far infrared template library \[SEC:irsed\] ================================================= Description of the model ------------------------ ![Cartoon picture illustrating the two effective parameters impacting the shape of our FIR SEDs. The total SED, normalized to unit ${L_{\rm IR}}$, is shown with a black solid line, while the dust continuum and PAH components are shown with solid orange and blue lines, respectively. We also show how the shape of the SED varies with dust temperature ${T_{\rm dust}}$ by displaying several templates of different ${T_{\rm dust}}$ in orange lines of varying intensities. The orange and blue arrows illustrate how the SED is modified by increasing ${T_{\rm dust}}$ and ${{\rm IR8}}$, respectively.[]{data-label="FIG:irseds"}](sed_cartoon.eps){width="9cm"} Since it was published, the library has been used routinely to derive infrared luminosities, and therefore star formation rates, for large samples of galaxies at various redshifts. In , we found that, in spite of the relatively small number of different SEDs it contains, it is able to fit well our stacked FIR photometry (rest-frame $30$-to-$500\,{\mu{\rm m}}$) at all $z=0.5$ to $z=4$. However, once properly adjusted to the observed FIR data, the behavior of these SEDs, calibrated on local starbursts, may not adequately describe the observed MIR photometry. Indeed, the library enforces a unique relation between ${T_{\rm dust}}$, ${{\rm IR8}}$ and ${L_{\rm IR}}$ which was calibrated from Local Universe galaxies. The relevance of this assumption for distant galaxies was unknown at the time, and in fact it was shown to break when applied to $24\,{\mu{\rm m}}$-detected $z=2$ galaxies [@papovich2007; @daddi2007-a]. It was later understood that this was caused by an evolving ${L_{\rm IR}}$–${{\rm IR8}}$ relation, and another more universal calibration was proposed where the ${{\rm IR8}}$ varies as a function of the distance to the main sequence [@elbaz2011; @nordon2012; @rujopakarn2013] rather than with the absolute luminosity. Similar conclusions have been drawn for the dust temperature [@elbaz2011; @magnelli2014]. To build a more up to date and versatile library, we started by making each of these observables independent of one another, that, we created a set of templates that allow us to vary ${T_{\rm dust}}$, ${{\rm IR8}}$ and ${L_{\rm IR}}$ simultaneously. To do so, we made the arbitrary choice of separating the IR emission into two components: on the one hand, the dust continuum of varying ${T_{\rm dust}}$ created by big and small grains, and on the other hand the MIR features emitted by PAH molecules (see [Fig. \[FIG:irseds\]]{}): $$S_{\nu} = {M_{\rm dust}}^{\rm cont}\,\bar{S}_{\nu}^{\rm cont} + {M_{\rm dust}}^{\rm PAH}\,\bar{S}_{\nu}^{\rm PAH}\,. \label{EQ:sed_comp}$$ ${M_{\rm dust}}^{\rm PAH}$ and ${M_{\rm dust}}^{\rm cont}$ are defined as the mass of dust grain found in the form of PAH molecules and silicate+carbonated grains, respectively, while $\bar{S}_{\nu}^{\rm PAH}$ and $\bar{S}_{\nu}^{\rm cont}$ are the spectra of each grain population normalized to unit dust mass (these are described in the next sections). This decomposition implies that PAHs are almost exclusively responsible for the observed diversity in ${{\rm IR8}}$. Indeed, decomposing the MIR emission into multiple components (PAHs, very small grains, and AGN torus) is a degenerate problem when only broadband photometry is used, and a choice needs to be made. Since our objective is mostly to study galaxies and not AGNs, we neglect the presence of AGNs torus emission, and assume a fixed fraction of small vs. big grains (see below). It is certainly possible to use our library in combination with an AGN template if sufficient MIR data is available, but this goes out of the scope of this work. To create our templates, we used the amorphous carbon dust model of [@galliano2011] (hereafter ). This model can output the mid- to far-IR spectrum emitted by a dust cloud of mass $1\,{{\rm M}_\odot}$ under the influence of a uniform radiation field of integrated intensity $U$ (taken here in units of the @mathis1983 interstellar radiation field in the solar neighborhood, $U_\sun$). In the following, we call each spectrum generated by this model an “elementary” template. The next two sections describe how our library is built from these templates, as well as the underlying assumptions. ### Dust continuum In the model, the dust continuum elementary templates are produced by a combination of silicate and amorphous carbon grains of varying sizes, split into “big” (thermalized) and “small” (transiently heated) grains. The size distributions of these grains were taken from [@zubko2004] and were assumed to be universal. Here we assumed the Milky Way (MW) mass-fraction of carbonated vs. silicate grains as derived by Zubko et al. for big grains, but fixed the mass-fraction of small silicate grains to zero [as in @compiegne2011] instead of $11\%$. While this is compatible with observational constraints [@li2001], our motivation for this choice was purely empirical: reducing the emission of small grains in the mid-IR increased the range of ${{\rm IR8}}$ values that our model can reach. The only remaining free parameter is the radiation intensity $U$. This parameter controls the energy of the radiation field to which each grain is exposed. For grains big enough to be thermalized, an increase of this energy implies an enhanced grain temperature (this is quantified later in [section \[SEC:relations\]]{}), and affects the shape of their FIR spectrum. For smaller grains which are not thermalized, only the overall normalization of the spectrum is modified. ### PAH emission To produce the associated PAH emission, we assumed that these molecules are subject to the same $U$ as the other dust grains, although in this case this choice has very little consequence since, as for small grains, the PAH molecules are not thermalized and therefore the effect of increasing $U$ is essentially only to increase the normalization of the PAH SED at fixed dust mass. This will affect the absolute values of the PAH masses, in which we have limited interest here. The only free parameter in the model regarding the PAH composition is the fraction of neutral vs. ionized molecules, which we chose here to be the MW value of $50\%$ [@zubko2004]. This parameter mostly influences the relative strengths of the $8$ vs. $12\,{\mu{\rm m}}$ PAH features, and we found that the MW value provided indeed a good match to the stacked [[Spitzer]{}]{}IRS spectra of $z=2$ ULIRGs (see [section \[SEC:irs\_spec\]]{}), as well as to our stacked $S_{24}/S_{\!16}$ broadband flux ratio at $z=1$ (see [section \[SEC:irsed\_stack\]]{}). ### Radiation field distribution The elementary templates introduced above are not well suited to describe an entire galaxy, since the hypothesis of a uniform $U$ usually does not hold in such kind of extended systems. Instead, a “composite” template must be built by adding together the emission of different dusty regions, heated by different radiation intensities. As in [@dale2001], assumed that the distribution of $U$ in a composite system follows a power law in ${\ensuremath{d}}{M_{\rm dust}}/ {\ensuremath{d}}U \sim U^{-\alpha}$, where ${\ensuremath{d}}{M_{\rm dust}}$ is the mass of dust associated to the elementary region illuminated with an intensity $[U, U+{\ensuremath{d}}U]$. This distribution is then integrated from $U=U_{\rm min}$ to $U=U_{\rm max}$ to form the final template. Contrary to , they did not assume the presence of an additional component linked to photo-dissociation regions since it was shown not to provide significant improvement to the fits (as demonstrated in the Appendix of ). The parameter $\alpha$ is the slope of the mass distribution of $U$ within a galaxy. This parameter affects the composite spectrum in a non-trivial way: large and small values of $\alpha$ will accumulate most of the dust mass close to $U_{\rm min}$ and $U_{\rm max}$, respectively ($\alpha=1$ and $2$ give a uniform weighting in mass and luminosity, respectively). To avoid this complexity, we fixed $\alpha=2.6$ for all our templates. This value was chosen to reproduce the width of the SEDs between $15$ and $500\,{\mu{\rm m}}$, since these templates are known to provide a good description of the FIR emission of distant galaxies [@elbaz2010]. We also checked a posteriori that this value provided a satisfactory fit to our stacked FIR photometry (see [section \[SEC:irsed\_stack\]]{}). With our adopted $U$ distribution, the final SED is relatively insensitive to the precise choice of $U_{\rm max}$, provided $U_{\rm max} \gg U_{\rm min}$ [@draine2007-a]. Hence the only remaining parameter that allows us to tune the SED shape is $U_{\rm min}$ or, equivalently, the mass-weighted intensity ${\left<U\right>}$. In particular, we note in [section \[SEC:relations\]]{} that ${\left<U\right>}$ is related to the average dust temperature through a simple power law, which is one of the parameter our library aims to describe. Therefore, we generated a logarithmic grid of $U_{\min}$ ranging from $0.1$ to $5\,000\,U_\sun$ with $250$ samples, and took $U_{\rm max} = 10^6\,U_\sun$ [@draine2007-a]. This allows our library to describe dust temperatures ranging from about $15$ to $100\,{{\rm K}}$ with a roughly constant step of $0.3\,{{\rm K}}$. The resulting SEDs can be obtained on-line (a link is provided on the first page). ### Amorphous carbon or graphite? \[SEC:amorphous\] Compared to more standard dust models (e.g., ), the one we used here assumes that carbonated grains are found exclusively in the form of amorphous carbon grains, rather than graphites. While this has no visible impact on the shape of the generated spectra, it systematically lowers the value of the measured dust masses by a factor of about $2.0$ compared to graphite dust, owing to the different emissivities of these grain species. This was in fact the motivation for using amorphous carbon in : lowering the measured dust masses eases the tension between the observed dust-to-gas ratio and stellar abundances in the Large Magellanic Cloud (LMC). This conclusion was not only reached in the LMC, which has a particularly low metallicity, but also in more normal galaxies including the MW [e.g., @compiegne2011; @jones2013; @fanciullo2015; @planckcollaboration2016]. As stressed in , purely amorphous carbon is just one possibility to achieve higher emissivities. We therefore do not give much credit to carbon dust being truly amorphous, but since this type of grains does describe the content of the ISM in a more consistent way, we chose to favor it instead of graphite. The impact of this choice on gas-to-dust ratios is discussed in [section \[SEC:gas\]]{}. Basic usage of the library and useful relations \[SEC:relations\] ----------------------------------------------------------------- In this section we provide a set of simple relations to relate the internal parameters of the library, namely ${\left<U\right>}$, ${M_{\rm dust}}^{\rm cont}$, and ${M_{\rm dust}}^{\rm PAH}$, to more commonly used observables such as the total infared luminosity, the dust mass, and the ${{\rm IR8}}$. We then provide instructions for the most basic usage of this template library. ### Dust temperature The dust temperature of our model SEDs was computed by applying Wien’s law to each elementary template for the dust continuum: $${T_{\rm dust}}[{{\rm K}}] = 2.897\times10^{3}/(\lambda_{\rm max} [{\mu{\rm m}}])\,,$$ determining $\lambda_{\rm max}$ as the wavelength corresponding to the peak of $\lambda^\beta L_{\nu}$ (the term $\lambda^\beta$ takes into account the effective emissivity of the templates, $\beta \simeq 1.5$). We then weighted each value by the dust mass associated to the corresponding template (${\ensuremath{d}}{M_{\rm dust}}$), therefore producing a mass-weighted average. We found that the following relation links together ${T_{\rm dust}}$ and the radiation field intensity: $$\frac{{\left<U\right>}}{U_\sun} = \left(\frac{{T_{\rm dust}}}{18.2{{\rm K}}}\right)^{5.57}\,. \label{EQ:tdust_umean}$$ As stated earlier, our library covers ${T_{\rm dust}}$ values ranging from $15$ to $100\,{{\rm K}}$. We also applied Wien’s law (as above) to the peak of the final dust template, to obtain a light-weighted average ${T_{\rm dust}}^{\rm light}$. In practice, we find the difference between the two to be simply a constant factor, with $${T_{\rm dust}}= 0.91\times{T_{\rm dust}}^{\rm light}\,. \label{EQ:tdust_weight}$$ ${T_{\rm dust}}^{\rm light}$ is less stable because the summed dust template is broader, making it harder to accurately locate the position of the peak. Its physical meaning is also less clear, since our templates do not have a single temperature, but it has the advantage of being less model-dependent and is essentially the temperature one would measure by using a modified blackbody model with a single temperature and an emissivity of $\beta=1.5$. We therefore provide tabulated values for both temperatures in the library, and in the following, unless otherwise stated, we will refer to the dust temperature as the mass-weighted value. Likewise, using [Eq. \[EQ:tdust\_umean\]]{} we mapped a dust temperature to each value of ${\left<U\right>}$ and will refer to the two quantity interchangably. ### Total infrared and $8\,{\mu{\rm m}}$ luminosities Each template and component in our library, both for the continuum and PAH emission, is associated with a value of ${L_{\rm IR}}$ ($8$-to-$1000\,{\mu{\rm m}}$) and ${L_8}$ (integrated with the response of the [[Spitzer]{}]{}IRAC channel 4, i.e., $6.4$-to-$9.3\,{\mu{\rm m}}$). Both luminosities were computed as the integral of $L_\lambda\,{\ensuremath{d}}\lambda$ within their respective wavelength interval, and are given in units of total solar luminosity (${L_\odot}=3.839\times10^{26}\,{\rm W}$). The luminosities are proportional to the mass of dust, and depend linearly on ${\left<U\right>}$: $$\begin{aligned} {L_{\rm IR}}&= {L_{\rm IR}}^{\rm cont} + {L_{\rm IR}}^{\rm PAH}, \\ {L_{\rm IR}}^{\rm cont} [{L_\odot}] &= 191 \, ({M_{\rm dust}}^{\rm cont}/{{\rm M}_\odot}) \, ({\left<U\right>}\!/U_\sun), \label{EQ:lir_mdust} \\ {L_{\rm IR}}^{\rm PAH} [{L_\odot}] &= 325 \, ({M_{\rm dust}}^{\rm PAH}/{{\rm M}_\odot}) \, ({\left<U\right>}\!/U_\sun), \\ {L_8}&= {L_8}^{\rm cont} + {L_8}^{\rm PAH}, \\ {L_8}^{\rm cont} [{L_\odot}] &= 7.05 \, ({M_{\rm dust}}^{\rm cont}/{{\rm M}_\odot}) \, ({\left<U\right>}\!/U_\sun), \\ {L_8}^{\rm PAH} [{L_\odot}] &= 755 \, ({M_{\rm dust}}^{\rm PAH}/{{\rm M}_\odot}) \, ({\left<U\right>}\!/U_\sun).\end{aligned}$$ Defining the mass fraction of PAHs as ${f_{\rm PAH}}\equiv {M_{\rm dust}}^{\rm PAH}/{M_{\rm dust}}$, we have $$\begin{aligned} {{\rm IR8}}\equiv \frac{{L_{\rm IR}}}{{L_8}} &= \frac{{L_{\rm IR}}^{\rm cont}\,(1-{f_{\rm PAH}}) + {L_{\rm IR}}^{\rm PAH}\,{f_{\rm PAH}}}{{L_8}^{\rm cont}\,(1-{f_{\rm PAH}}) + {L_8}^{\rm PAH}\,{f_{\rm PAH}}}\,, \nonumber \\ \frac{1}{{f_{\rm PAH}}} &= 1 - \frac{{L_{\rm IR}}^{\rm PAH} - {L_8}^{\rm PAH}\,{{\rm IR8}}}{{L_{\rm IR}}^{\rm cont} - {L_8}^{\rm cont}\,{{\rm IR8}}}\,. \label{EQ:fpah_ir8_th}\end{aligned}$$ Using the above approximate equations, this becomes: $$\begin{aligned} {{\rm IR8}}= \frac{191 + 134\,{f_{\rm PAH}}}{7.05 + 748\,{f_{\rm PAH}}}\, \quad \text{and} \quad {f_{\rm PAH}}= \frac{191 - 7.05\,{{\rm IR8}}}{-134 + 748\,{{\rm IR8}}}\,. \label{EQ:fpah_ir8}\end{aligned}$$ By varying the relative contribution of PAHs to the total dust mass, our library can reach ${{\rm IR8}}$ values in the range $0.48$ to $27.7$, which covers the vast majority of the observed parameter space [@elbaz2011]. ### Basic usage Our library is composed of multiple templates, each corresponding to a given dust temperature ${T_{\rm dust}}$. As illustrated in [Fig. \[FIG:irseds\]]{}, each template is composed of two components: dust continuum on the one hand, and PAH emission on the other hand. The amplitude of each component is internally dictated by the corresponding mass of dust grains, ${M_{\rm dust}}^{\rm cont}$ and ${M_{\rm dust}}^{\rm PAH}$, respectively ([Eq. \[EQ:sed\_comp\]]{}), and both can be freely adjusted to match the observed data, effectively varying the dust mass (or ${L_{\rm IR}}$) and the ${{\rm IR8}}$. To perform the fit, both PAH and continuum components must be redshifted to the assumed redshift of the source, and the expected flux in each observed passband is computed by integrating the redshifted template multiplied with the corresponding filter response curve. At this stage, the expected fluxes can be fit to the observed ones through a linear solver, varying simultaneously ${M_{\rm dust}}^{\rm cont}$ and ${M_{\rm dust}}^{\rm PAH}$. By performing such a fit for each value of ${T_{\rm dust}}$ in the library and picking the smallest $\chi^2$, one can find the optimal model (in the $\chi^2$ sense) corresponding to the provided photometry. This is the simplest way to use the library, and it will work in most cases if enough photometry is available and if the redshift is precisely known. In other cases, a more careful approach should be used, and we describe it later in [section \[SEC:irsed\_recipe\]]{}. The immediate products of this fit are the dust masses, ${M_{\rm dust}}^{\rm cont}$ and ${M_{\rm dust}}^{\rm PAH}$ (expressed in solar mass), and the dust temperature ${T_{\rm dust}}$. The total dust mass can be obtained by summing the two components, ${M_{\rm dust}}= {M_{\rm dust}}^{\rm cont} + {M_{\rm dust}}^{\rm PAH}$, while ${L_{\rm IR}}$ and ${L_8}$ are tabulated (per unit solar mass of dust) for each ${T_{\rm dust}}$ value in the library, or can be estimated using the above relations. In the next section, we check the accuracy of our PAH templates by fitting stacked MIR spectra from the [[Spitzer]{}]{}IRS spectrometer. Comparison against stacked MIR spectroscopy \[SEC:irs\_spec\] ------------------------------------------------------------- ![Comparison of the our templates (black solid line) against stacked [[Spitzer]{}]{}IRS spectra of $z=1$ LIRGs (blue, top) and $z=2$ ULIRGs (red, bottom) from [@fadda2010]. The relative residuals of the fits are shown above the plot for each sample; the region of perfect agreement is shown with a dashed line, surrounded by a $\pm20\%$ confidence interval. We also show the best fit using other models: (gray), (green), and [@magdis2012] (purple). In all cases the fit only uses observations at $\lambda > 5\,{\mu{\rm m}}$, since shorter wavelength can be contaminated by the stellar continuum, as illustrated with the hashed region in the residual plots.[]{data-label="FIG:irs_spec"}](irs_spec.eps){width="50.00000%"} During the cryogenic phase of the [[Spitzer]{}]{}mission, the IRS spectrometer could observe in the MIR from $5.3$ to $38\,{\mu{\rm m}}$ at low ($R=90$) and medium ($R=600$) spectral resolutions. It has therefore provided valuable measurements of the PAH emission, both in local and distant galaxies. In particular, [@fadda2010] have observed a sample of $z=1$ LIRGs and $z=2$ ULIRGs in the GOODS–South field (plus a few in the wider ECDFS). The galaxies in these two samples have been selected based on their redshift and [[Spitzer]{}]{}MIPS $24\,{\mu{\rm m}}$ flux ($0.2$–$0.5\,{{\rm mJy}}$), therefore their measured properties cannot be straightforwardly compared to the mass-complete stacks that we will analyze in [section \[SEC:irsed\_stack\]]{}. Because of the $24\,{\mu{\rm m}}$ flux limit, the sample of Fadda et al. is biased toward starburst galaxies located above the main sequence. With this caveat in mind, this dataset can still be used as a consistency check for our SED library. Indeed, although these samples may be biased, our library must be able to at least broadly reproduce their PAH spectra. We therefore applied the fitting method described in the previous section to the stacked spectra of both $z=1$ and $z=2$ samples. Since the dust temperature cannot be constrained from IRS spectroscopy alone, we fixed it to its redshift-average value of ${T_{\rm dust}}=28$ and $33\,{{\rm K}}$, respectively (see [section \[SEC:irsed\_stack\]]{}), although this choice had no impact on the quality of the fit. We also did not attempt to fit rest-frame wavelengths $\lambda<5\,{\mu{\rm m}}$ to avoid contamination from stellar continuum. The result is shown in [Fig. \[FIG:irs\_spec\]]{}, together with fits from a number of other libraries from the literature. It can be seen from this figure that we are able to fit these spectra with good accuracy (typically better than $20\%$) and obtain a significantly improved match compared to the or libraries. The [@magdis2012] library (which uses the models) yields a similarly good fit as ours, with only subtle differences. This should not come as a surprise, since the physical properties of the PAHs in both the Magdis et al. library and ours are adapted from . Observed dust temperatures and IR8 \[SEC:calib\] ================================================ Average values from stacked photometry \[SEC:irsed\_stack\] ----------------------------------------------------------- ![image](seds_z.eps){width="80.00000%"} ### Description of the stacked data We now proceed to apply our new library to model the stacked [[Spitzer]{}]{}and [[Herschel]{}]{}photometry in the CANDELS fields. These stacks were presented in detail in , and we briefly recall here the essential information. We selected all galaxies in the GOODS–North, GOODS–South, UDS and COSMOS CANDELS fields that are brighter than $H=26$ and [$UVJ$]{}star-forming ([Eq. \[EQ:uvj\]]{}). These galaxies were then binned according to their redshift and stellar mass, for a total of $24$ bins (six redshift bins from $z=0.3$ to $5$, and four mass bins from ${M_\ast}=3\times10^9$ to $3\times10^{11}\,{{\rm M}_\odot}$). The number of stacked galaxies in each bin is given in Figure 4 of . The smallest number of stacked galaxy were $53$ and $28$, in the highest mass bins at $z\sim 0.5$ and $z\sim 4$, respectively. Otherwise, all the bins had at least $100$ galaxies (median of $628$). In the bins where we estimated our stellar-mass completeness to be above $90\%$, we stacked the [[Spitzer]{}]{}and [[Herschel]{}]{}images at the positions of all the galaxies in the bin. The fluxes in the resulting stacked images were measured through point spread function (PSF) fitting with a free background. The measured fluxes were corrected for clustering [a posteriori]{} using an empirical recipe calibrated on simulated images, and the method is described fully in the appendix of . Briefly, we simulated the [[Herschel]{}]{}maps using the positions of the real galaxies of CANDELS and the prescriptions described in [@schreiber2017-b] to predict (statistically) their ${L_{\rm IR}}$ and FIR fluxes. The resulting images have the same statistical properties (pixel distribution and number counts) as the real images. We then applied the same stacking method and compared the measured fluxes to the true flux averages in the stacked sample. We found that the flux boosting caused by clustering is roughly constant in a given band, hence we assumed a fixed correction in each [[Herschel]{}]{}band throughout, at all redshifts and masses (see also @bethermin2015-a). The largest correction was a reduction of the fluxes by $25\%$ for the SPIRE $500\,{\mu{\rm m}}$ band. The uncertainty on each flux measurement was finally computed by taking the maximum value of two independent estimates: first by doing bootstrapping, that is, repeatedly removing half of the sample from the stack, and second from the RMS of the residual image after subtraction of the best-fit PSF model. In , we only stacked the [[Spitzer]{}]{}MIPS $24\,{\mu{\rm m}}$ band as well as the [[Herschel]{}]{}bands redward of $100\,{\mu{\rm m}}$, which are the only bands covered in all the four CANDELS fields. For the present work, we extended these stacks to also include the [[Spitzer]{}]{}IRS $16\,{\mu{\rm m}}$ imaging [acquired in the GOODS fields only; @teplitz2011] as well as the [[Herschel]{}]{}PACS $70\,{\mu{\rm m}}$ (acquired in the GOODS–South field only). Because these images only cover some of the CANDELS fields, the stacked fluxes can be affected by field-to-field variations. To correct for this effect, we first computed the $S_{16}/S_{24}$ flux ratio observed when stacking only the two GOODS fields. We then multiplied the stacked $24\,{\mu{\rm m}}$ flux obtained with the four fields by this ratio to estimate the corresponding $16\,{\mu{\rm m}}$ flux. The same procedure is used for the $70\,{\mu{\rm m}}$ flux, based on the $S_{70}/S_{100}$ flux ratio observed in GOODS–South. Lastly, to better constrain the dust temperature for our $z=4$ bin we added the average ALMA $890\,{\mu{\rm m}}$ of our galaxies as measured in [@schreiber2017-a] (in all fields but GOODS–North). ### Fitting of the stacked SEDs Using the fitting method described in [section \[SEC:relations\]]{}, we used our new library to model the stacked fluxes from (avoiding again $\lambda<5\,{\mu{\rm m}}$). We used the mean redshift of the stacked sample to shift the SED in the observed frame. Uncertainties on the redshifts are of the order of $5\%$ and are thus much smaller than the bins (which have a constant width of $25\%$), we therefore ignored them, but we did broaden the templates by the width of the redshift bins. To derive accurate error estimates on the fitting parameters, we bootstrapped the stacked sample and applied the fitting procedure on each bootstrapped SED. This allows a better treatment of correlated noise (flux fluctuations affecting multiple bands simultaneously due, e.g., to contamination from neighboring sources on the map). The resulting models are compared to the measured photometry in [Fig. \[FIG:irstacks\]]{}. The first thing to note is the excellent agreement between our templates and the photometry. With only three free parameters, no clear tension is observed, reinforcing the idea of a universal SED. Compared to our previous fits with the library, we found very similar values of ${L_{\rm IR}}$. The most extreme difference arose in the lowest redshift bin ($0.3 < z < 0.7$) where we obtained values that are systematically $0.1\,{{\rm dex}}$ lower with the new library. This difference is caused by a peculiar feature of the previously adopted best-fit template. This particular SED (ID $40$) shows an enhanced flux around the rest-frame $30\,{\mu{\rm m}}$ compared to our library. Without any data to constrain this feature, we cannot say whether it is real or not, although we tend to favor the result of our new SED library which has a consistent shape at all ${T_{\rm dust}}$. ### Best fit parameters and systematic bias corrections \[SEC:stack\_correct\] One major issue when interpreting stacked photometry is that the stacked SED is the flux-weighted average in each band, which is not necessarily equal to the SED of the average galaxy of the sample. This is particularly true if the brightest galaxies have a different SED than the average galaxy, and indeed such situation is expected in our case: starburst galaxies, which have the highest ${{\rm SFR}}$s in a given bin of mass, have an increased temperature and ${{\rm IR8}}$. Therefore our stacked SED will be biased towards higher temperatures and higher ${{\rm IR8}}$ compared to their true average, and these biases need to be accounted for before going further. To quantify these biases, we used the Empirical Galaxy Generator (EGG; @schreiber2017-b). Using a set of empirical prescriptions, this tool can generate mock galaxy catalogs matching exactly the observed stellar mass functions at $0<z<6$ and the galaxy main sequence with its scatter and starburst population (). We also implemented the relations derived in the present paper for ${T_{\rm dust}}$ and ${{\rm IR8}}$ and their dependence on ${R_{\rm SB}}$, namely [Eqs. \[EQ:tdust\_ms\]]{} to \[EQ:ir8\_sb\], adding the observed residual scatter (see [sections \[SEC:indiv\_tdust\_result\]]{} and \[SEC:indiv\_ir8\_result\]) as a random Gaussian perturbation to match the observed distribution of both quantities. Thanks to these prescriptions, the tool can produce realistic mock catalogs of galaxies with a full infrared SED, and we showed in [@schreiber2017-b] that this empirical model reproduces faithfully the observed number counts in all FIR bands. Generating a large mock catalog of about a million galaxies with ${M_\ast}> 3\times 10^{9}\,{{\rm M}_\odot}$ at $0.3<z<5$, we simulated our stacked SEDs by computing the average flux in each band for each of the stellar mass and redshift bins displayed in [Fig. \[FIG:irstacks\]]{}. The size of the mock catalog was chosen so that each stacked bin contained at least $1\,000$ galaxies. No noise was added to the stacked fluxes, since we only looked for systematic biases. We then applied exactly the same fitting method to these synthetic stacks as used for the real stacks, and compared the resulting fit parameters to their true average. Since the recipes for ${T_{\rm dust}}$ and ${{\rm IR8}}$ used in EGG depend to some extent on the bias correction we are now discussing, we proceeded iteratively: we derived a first estimate of [Eqs. \[EQ:tdust\_ms\]]{} to \[EQ:ir8\_sb\] without applying any bias correction, implemented these relations in EGG, and determined a first value of the corrections. We then applied these corrections to the observed values, re-evaluated the relations, updated EGG and the mock catalog, then determined the final corrections. The obtained values were not significantly different from the first estimates, so we only did two iterations of this procedure. We found that the “raw” stacked values (before correction) were on average larger than their true average by $1.5\pm0.3\,{{\rm K}}$ for ${T_{\rm dust}}$, and by a factor of $10\pm1\%$ for ${{\rm IR8}}$. These are relatively small corrections which do not impact our conclusions, but we applied them nevertheless to our best fit values. ![image](tdust_pah.eps){width="\textwidth"} [ccccccc]{}\ $z$ & $\log_{10}{M_\ast}$ & ${L_{\rm IR}}$ & ${M_{\rm dust}}$ & ${T_{\rm dust}}$ & ${f_{\rm PAH}}$ & ${{\rm IR8}}$\ & $\log_{10}{{\rm M}_\odot}$ & $10^{10}{L_\odot}$ & $10^7{{\rm M}_\odot}$ & ${{\rm K}}$ & % &\ \ 0.3 – 0.7 & 9.5 – 10.0 & $1.49^{+0.08}_{-0.09}$ & $0.136^{+0.034}_{-0.025}$ & $23.3^{+1.0}_{-1.1}$ & $3.08^{+0.40}_{-0.46}$ & $6.4^{+0.8}_{-0.6}$\ & 10.0 – 10.5 & $4.61^{+0.20}_{-0.20}$ & $0.180^{+0.019}_{-0.017}$ & $27.9^{+0.5}_{-0.4}$ & $4.66^{+0.36}_{-0.28}$ & $4.66^{+0.22}_{-0.28}$\ & 10.5 – 11.0 & $12.9^{+0.9}_{-0.9}$ & $0.57^{+0.10}_{-0.07}$ & $27.2^{+1.2}_{-1.2}$ & $3.8^{+0.5}_{-0.5}$ & $5.5^{+0.5}_{-0.4}$\ & 11.0 – 11.5 & $14.9^{+1.5}_{-1.5}$ & $1.43^{+0.38}_{-0.26}$ & $23.0^{+1.0}_{-1.3}$ & $6.4^{+0.9}_{-0.9}$ & $3.6^{+0.5}_{-0.5}$\ 0.7 – 1.2 & 9.5 – 10.0 & $2.24^{+0.13}_{-0.18}$ & $0.130^{+0.064}_{-0.035}$ & $25.7^{+1.8}_{-1.7}$ & $1.76^{+0.35}_{-0.24}$ & $9.4^{+0.9}_{-1.0}$\ & 10.0 – 10.5 & $8.48^{+0.23}_{-0.35}$ & $0.354^{+0.061}_{-0.031}$ & $27.4^{+0.8}_{-0.9}$ & $4.73^{+0.26}_{-0.25}$ & $4.61^{+0.16}_{-0.17}$\ & 10.5 – 11.0 & $21.2^{+0.8}_{-0.7}$ & $0.88^{+0.08}_{-0.08}$ & $27.4^{+0.6}_{-0.4}$ & $5.47^{+0.33}_{-0.26}$ & $4.11^{+0.15}_{-0.19}$\ & 11.0 – 11.5 & $35.2^{+1.8}_{-2.3}$ & $2.58^{+0.31}_{-0.31}$ & $24.3^{+0.5}_{-0.7}$ & $5.36^{+0.33}_{-0.31}$ & $4.19^{+0.22}_{-0.23}$\ 1.2 – 1.8 & 9.5 – 10.0 & $2.91^{+0.22}_{-0.14}$ & $0.063^{+0.010}_{-0.016}$ & $31.1^{+1.5}_{-0.3}$ & $1.40^{+0.23}_{-0.27}$ & $10.7^{+1.1}_{-0.8}$\ & 10.0 – 10.5 & $13.0^{+0.6}_{-0.5}$ & $0.40^{+0.10}_{-0.06}$ & $29.2^{+0.3}_{-1.7}$ & $3.36^{+0.20}_{-0.21}$ & $5.98^{+0.30}_{-0.26}$\ & 10.5 – 11.0 & $36.3^{+1.3}_{-1.5}$ & $0.85^{+0.07}_{-0.07}$ & $30.6^{+0.3}_{-0.5}$ & $3.75^{+0.19}_{-0.21}$ & $5.50^{+0.22}_{-0.21}$\ & 11.0 – 11.5 & $84^{+4}_{-4}$ & $2.35^{+0.37}_{-0.24}$ & $29.8^{+0.7}_{-0.6}$ & $3.72^{+0.38}_{-0.38}$ & $5.5^{+0.5}_{-0.4}$\ 1.8 – 2.5 & 10.0 – 10.5 & $16.3^{+0.7}_{-0.8}$ & $0.131^{+0.069}_{-0.024}$ & $36.2^{+0.8}_{-1.8}$ & $2.13^{+0.13}_{-0.19}$ & $8.10^{+0.45}_{-0.32}$\ & 10.5 – 11.0 & $65.4^{+2.3}_{-2.8}$ & $0.76^{+0.16}_{-0.13}$ & $34.0^{+0.9}_{-1.1}$ & $2.75^{+0.12}_{-0.14}$ & $6.83^{+0.24}_{-0.26}$\ & 11.0 – 11.5 & $144^{+6}_{-7}$ & $2.04^{+0.28}_{-0.28}$ & $33.1^{+0.9}_{-0.5}$ & $2.44^{+0.15}_{-0.20}$ & $7.46^{+0.36}_{-0.32}$\ 2.5 – 3.5 & 10.0 – 10.5 & $26.5^{+2.2}_{-2.0}$ & $0.213^{+0.069}_{-0.033}$ & $36.2^{+1.2}_{-1.3}$ & $1.52^{+0.30}_{-0.21}$ & $10.0^{+0.9}_{-1.0}$\ & 10.5 – 11.0 & $102^{+7}_{-10}$ & $0.58^{+0.14}_{-0.13}$ & $38.6^{+1.4}_{-1.9}$ & $2.98^{+0.36}_{-0.37}$ & $6.4^{+0.6}_{-0.5}$\ & 11.0 – 11.5 & $257^{+17}_{-16}$ & $2.9^{+0.5}_{-0.5}$ & $34.3^{+0.9}_{-0.6}$ & $2.96^{+0.25}_{-0.25}$ & $6.49^{+0.46}_{-0.37}$\ 3.5 – 5.0 & 11.0 – 11.5 & $357^{+37}_{-61}$ & $1.41^{+0.26}_{-0.18}$ & $41.8^{+1.6}_{-1.6}$ & — & —\ In [Fig. \[FIG:tdust\_fpah\_stack\]]{} (top left and top right) we show the best-fit values we obtain for ${T_{\rm dust}}$ and ${{\rm IR8}}$ in all bins of mass and redshifts where they could be measured. These are also tabulated in [Table \[TAB:stack\_best\_fits\]]{}. To complement our stacks toward the local Universe, we also compute the average ${T_{\rm dust}}$ and ${{\rm IR8}}$ for the [$UVJ$]{}star-forming galaxies of the HRS. The evolution with redshift and mass of ${T_{\rm dust}}$ and ${{\rm IR8}}$ are quantified in the following sections. ### Evolution of the dust temperature In most cases, varying the stellar mass has no influence on the dust temperature. The most massive galaxies (${M_\ast}> 10^{11}\,{{\rm M}_\odot}$) tend to have colder temperatures by $2.3\,{{\rm K}}$ on average (see also @matsuki2017 who report a similar trend at $z=0$), but this is only significant at $z<1$, in the domains where the main sequence departs from a linear relation [e.g., @whitaker2015; @schreiber2015]. This reduced temperature was already interpreted in our previous work as a sign that massive star-forming galaxies at low redshifts are in the process of shutting down star-formation, with a slowly declining efficiency [@schreiber2016]. Since this is a minor effect compared to the overall redshift evolution, and since it only affects the few most massive galaxies, we do not discuss it further here. Averaging the dust temperatures of the mass bins unaffected by the above effect, we recovered a previously reported trend for the dust temperature to increase with redshift [e.g., @magdis2012; @magnelli2013; @bethermin2015-a]. Contrary to what [@magdis2012] claimed, we observed a continuous rise of the temperature up to $z=4$, confirming the results of [@bethermin2015-a]. Fitting the evolution of ${T_{\rm dust}}$ with redshift as an empirical power law, we obtained $${T_{\rm dust}}^{\rm MS} [{{\rm K}}] = (32.9 \pm 2.4) + (4.60 \pm 0.35) \times (z - 2)\,.\label{EQ:tdust_ms}$$ This relation is displayed in [Fig. \[FIG:tdust\_fpah\_stack\]]{} (bottom left), where we compare it to results from the literature. In particular [@magnelli2014] found ${T_{\rm dust}}= 26.5 \times (1+z)^{0.18}$. The normalization of this relation is higher than the one we report here, but the evolution with redshift is milder. This higher normalization is linked to the fact that [@magnelli2014] measured ${T_{\rm dust}}$ by fitting modified blackbodies, making their dust temperatures light-weighted. Correcting for this difference using [Eq. \[EQ:tdust\_weight\]]{} (as was done in [Fig. \[FIG:tdust\_fpah\_stack\]]{}), their $z=0$ value is fully consistent with ours, but their measurement at $z=2$ falls short of ours by $5{{\rm K}}$. The same is true for the stacks of [@magdis2012]. This could be caused by the absence of clustering correction in these two studies, since it affects preferentially the long wavelength [[Herschel]{}]{}bands which are crucial to determine the temperature. [@magnelli2014] does exclude the bands for which they predict the flux is doubled by the effect of clustering, however this threshold is extreme and will leave substantial clustering signal in their photometry. To further check for systematic issues in our stacked fluxes, we applied our model to the stacked SEDs of [@bethermin2015-a], which were obtained for a different sample, in a single mass bin, and include longer wavelength data from LABOCA $870\,{\mu{\rm m}}$ and AzTEC $1.1\,{{\rm mm}}$ at all redshifts. The correction for clustering is also performed with another method. Despite these differences, the evolution of ${T_{\rm dust}}$ in these data is in excellent agreement with the above relation, suggesting that our stacked fluxes are robustly measured. We also compare the ${\left<U\right>}$ value as reported by [@bethermin2015-a] for reference; the model used in that work assumes a different functional form for the $U$ distribution, so the comparison is limited in scope, but we nevertheless recover a similar slope for the redshift dependence of the temperature. We therefore confirm that the dust temperature increases continuously from about $25\,{{\rm K}}$ at $z=0$ up to more than $40\,{{\rm K}}$ at $z=4$, with little to no dependence on stellar mass (hence ${L_{\rm IR}}$) at fixed redshift. This is important to take into account when only limited FIR photometry is available and ${T_{\rm dust}}$ needs to be assumed to extrapolate the total IR luminosity, particularly for sub-millimeter samples which do not probe the emission blueward of the peak of the dust emission (see [section \[SEC:irsed\_mono\]]{}). ### Evolution of the IR8 [@elbaz2011] proposed that a unique value of ${{\rm IR8}}=4.9$ holds for all main sequence galaxies, however it could be seen already from their stacked data (see their Fig. 7) that the average ${{\rm IR8}}$ is closer to $8$ at $z=2$ (see also @reddy2012). After applying the correction described in [section \[SEC:stack\_correct\]]{}, we found ${{\rm IR8}}\sim 7$ at $z=2$, mildly but significantly larger than the value first reported in [@elbaz2011] at $z=1$. On the other hand, we found the $z=0$ galaxies from the HRS have a marginally smaller ${{\rm IR8}}$ of about $3.5$, very similar to our $z=1$ value of $4$. This implies that the average value has evolved continuously between $z=2$ and $z=1$ only, and our stacks at $z=3$ suggest that this evolution stops at higher redshifts. We thus fit the evolution of the average ${{\rm IR8}}$ as a broken linear relation and obtained $${{\rm IR8}}^{\rm MS} = 4.08\pm0.29 + (3.29\pm0.24) \times \left\{\begin{array}{lcl} 0 & \text{if} & z < 1\,, \\ (z-1) & \text{if} & 1 \leq z \leq 2\,, \\ 1 & \text{if} & z > 2\,. \label{EQ:ir8_ms} \end{array}\right.$$ This relation is displayed in [Fig. \[FIG:tdust\_fpah\_stack\]]{} (bottom right) and compared to the values obtained by @elbaz2011 [Fig. 7] and [@magdis2012], which are in rough agreement. The ${{\rm IR8}}$ value of [@magdis2012] at $z=0$ is significantly higher than ours, probably because their local sample was flux-limited [@dacunha2010-a], hence is biased toward starbursts which have a systematically higher ${{\rm IR8}}$ (see [section \[SEC:selection\_effects\]]{}). Interestingly, we found that low mass galaxies (${M_\ast}< 10^{10}\,{{\rm M}_\odot}$) have systematically higher ${{\rm IR8}}$ values, implying that their PAH emission is reduced. [@shivaei2017] observed a similar trend in $z\sim2$ galaxies with spectroscopic redshifts. Shivaei et al. also observe that this trend of reduced PAH emission can be linked to decreasing metallicity, as observed in the local Universe [e.g., @galliano2003; @ciesla2014]. The physical origin of this trend is still debated. One plausible explanation is that a metal-poor interstellar medium blocks less efficiently the UV radiation of young stars, and makes it harder for PAH molecules to survive [e.g., @galliano2003]. Other scenarios have been put forward, suggesting either that low metallicity objects are simply too young to host enough carbon grains to form PAH complexes [@galliano2008], or that this is instead caused by a different filling factor of molecular clouds in metal poor environments [@sandstrom2012]. Metallicity, in turn, is positively correlated with the stellar mass through the mass-metallicity relation [@lequeux1979; @tremonti2004], and this relation has been found to evolve with time, so that galaxies were more metal-poor in the past [e.g., @erb2006]. If metallicity was the main driver of PAH emission, one would expect to find the strongest PAH features (and the lowest ${{\rm IR8}}$) within massive low-redshift galaxies, which is indeed what we found. To test this hypothesis quantitatively, we used the Fundamental Metallicity Relation (FMR; @mannucci2010) to estimate the average metallicity of our stacked galaxies from their stellar masses and SFRs. The resulting relation between metallicity and ${{\rm IR8}}$ is shown in [Fig. \[FIG:fpah\_metal\]]{}. We found a clear anti-correlation between the two quantities, quantitatively matching the trend observed in the local Universe by [@galliano2008] and confirming the results of [@shivaei2017]. The best fitting power law is $${{\rm IR8}}= (3.5 \pm 0.3)\times(Z/Z_\sun)^{-0.99\pm0.15}\,,$$ where $Z$ is the metallicity (which can be substituted for the oxygen abundance $O/H$ under the assumption that $O/H$ scales linearly with $Z$). The galaxies of the HRS, for which we have individual metallicity measurements [@boselli2010], also follow a similar trend. Unfortunately, the power of the FMR in predicting metallicities is limited [particularly at high redshifts, e.g., @bethermin2015-a], and measuring metallicities from emission lines for individual galaxies is both expensive and prone to systematics [@kewley2008]. Excluding the masses below $10^{10}\,{{\rm M}_\odot}$, the model of [Eq. \[EQ:ir8\_ms\]]{} with a simple dependence on redshift provides a fit of equal quality to the data. Since [Eq. \[EQ:ir8\_ms\]]{} is more easily applicable to large samples, we chose to adopt it as the fiducial model and caution that the relations derived below for ${{\rm IR8}}$ only apply to galaxies more massive than $10^{10}\,{{\rm M}_\odot}$. ![Relation between the ${{\rm IR8}}$ observed in stacked [[Spitzer]{}]{}and [[Herschel]{}]{}photometry, and the gas-phase metallicity. The metallicity, expressed in terms of oxygen abundance $12+\log_{10}(O/H)$ (the solar value is $8.73$, @asplund2009), was estimated using the Fundamental Metallicity Relation [FMR, @mannucci2010]. Data points are colored with redshift as indicated in the legend. We also show the galaxies from the HRS as gray circles, using their measured metallicities. The black line shows the best fitting power law to the stacked data. The $z=0$ relation obtained by [@galliano2008] is shown for reference with a dotted gray line, converting their measured PAH mass fractions to ${{\rm IR8}}$ using [Eq. \[EQ:fpah\_ir8\]]{}.[]{data-label="FIG:fpah_metal"}](ir8_metal.eps){width="48.00000%"} Values for individual galaxies \[SEC:irsed\_indiv\] --------------------------------------------------- In this section we describe the scatter on both ${T_{\rm dust}}$ and ${{\rm IR8}}$ about the average “main sequence” values we obtained in [section \[SEC:irsed\_stack\]]{}. We also describe how these quantities are modified for starburst galaxies, that is, those galaxies that have an excess ${{\rm SFR}}$ at a given stellar mass compared to the main sequence. To quantify this latter excess, we used the “starburstiness” [@elbaz2011] which is defined as ${R_{\rm SB}}\equiv {{\rm SFR}}/{{\rm SFR}_{\rm MS}}$; galaxies with ${R_{\rm SB}}=1$ are on the main sequence, and those with ${R_{\rm SB}}>1$ are located above the sequence. ### Fitting individual galaxies ![image](indiv_mstar_z.eps){width="90.00000%"} We used our library to fit the FIR-detected galaxies in the CANDELS fields and COSMOS $2\,\sq\degr$. To ensure reliable fits, we selected galaxies with a good enough wavelength coverage and robust photometry. In particular we only used the [[Herschel]{}]{}photometry for clean sources [@elbaz2011] to avoid biases toward low ${T_{\rm dust}}$ because of blending, and excluded all sources for which the matching of the $24\,{\mu{\rm m}}$ emission to a $H$ or [$K_{\rm s}$]{}-band counterpart was ambiguous [as defined in @schreiber2016]. We also excluded fits of poor quality by rejecting galaxies with $\chi^2$ larger than $10$ (less than $10\%$ of our sample), indicative of uncaught issues in the photometry and counterpart identification. For the measurement of ${T_{\rm dust}}$, following [@hwang2010-a] we required at least one photometric measurement with significance greater than $5\sigma$ on either side of the peak of the dust continuum. This produced a sample of $438$ galaxies, which are shown on the left panel of [Fig. \[FIG:indiv\_sample\]]{}. For the measurement of ${{\rm IR8}}$, we selected only the galaxies at $0.5<z<1.5$ with at least a $3\sigma$ detection in [[Spitzer]{}]{}IRS $16\,{\mu{\rm m}}$ and the galaxies at $1.5<z<2.5$ with at least a $3\sigma$ detection in [[Spitzer]{}]{}MIPS $24\,{\mu{\rm m}}$ to ensure the rest-frame $8\,{\mu{\rm m}}$ emission is well constrained. Of these, we only kept the galaxies for which ${L_{\rm IR}}$ could be independently measured using longer wavelength photometry at better than $5\sigma$. This produced a sample of $1068$ galaxies, shown on the right panel of [Fig. \[FIG:indiv\_sample\]]{}, and $264$ of these are in COSMOS $2\,\sq\degr$. ### Completeness and selection biases \[SEC:selection\_effects\] To quantify the selection biases introduced by the numerous criteria listed above, we created a new mock catalog with EGG (see [section \[SEC:stack\_correct\]]{}) over $10\deg^2$ and down to a [[Spitzer]{}]{}IRAC $3.6\,{\mu{\rm m}}$ magnitude of $25$ (slightly deeper than the typical $5\sigma$ depth in CANDELS). We then perturbed the generated fluxes within the uncertainties typical for CANDELS, and mimicked the [[Spitzer]{}]{}and [[Herschel]{}]{}flux extraction procedure described in [@elbaz2011]: we only kept the $16$ and $24\,{\mu{\rm m}}$ fluxes for galaxies with a $3.6\,{\mu{\rm m}}$ flux larger than $0.5\,{\mu{\rm m}}$ [@magnelli2011], we only kept the $70$ and $100\,{\mu{\rm m}}$ fluxes for galaxies with a $24\,{\mu{\rm m}}$ flux larger than $21\,{\mu{\rm Jy}}$ ($3\sigma)$, we only kept the $160$ to $500\,{\mu{\rm m}}$ fluxes for galaxies with a $24\,{\mu{\rm m}}$ flux larger than $35\,{\mu{\rm Jy}}$ ($5\sigma$), and finally we only kept the $350$ and $500\,{\mu{\rm m}}$ fluxes for galaxies with a $250\,{\mu{\rm m}}$ flux larger than $3.8\,{{\rm mJy}}$ ($2\sigma$). Since most of our sample has no ALMA data, we did not include ALMA photometry in the mock catalog. We then ran the same fitting procedure as for the real galaxies, and applied the same selection criteria to produce the two samples with robust ${T_{\rm dust}}$ and ${{\rm IR8}}$. Normalizing the number of objects in the mock catalog over an area equal to that of CANDELS (and accounting for the loss of effective area caused by the rejection of the non-clean [[Herschel]{}]{}sources), the mock catalog predicts $396$ galaxies with a robust ${T_{\rm dust}}$ measurement, and $1101$ galaxies with a robust ${{\rm IR8}}$. These numbers are in excellent agreement with the observed ones, and confirm the validity of the mock catalog. We display in [Fig. \[FIG:mz\_comp\]]{} the completeness of various samples: selecting galaxies with an ${L_{\rm IR}}$ measured at better than $3\sigma$, selecting galaxies with a robust ${T_{\rm dust}}$, and selecting galaxies with a robust ${{\rm IR8}}$. It is clear from this plot that one can only obtain robust measurements of ${T_{\rm dust}}$ or ${{\rm IR8}}$ for a fraction of the IR-detected galaxies. For the dust temperature, our sample reaches $80\%$ completeness only at $z<0.6$, however it can still probe galaxies with ${M_\ast}> 10^{11}\,{{\rm M}_\odot}$ at much higher redshifts (up to $z\sim$ 3) albeit with a lower completeness of $20\%$; this implies that selection effects are important and have to be studied carefully. The situation for the ${{\rm IR8}}$ is not as dramatic, as the sample reaches $80\%$ completeness at $z=1$ above ${M_\ast}> 8\times10^{10}\,{{\rm M}_\odot}$, and $50\%$ completeness at $z=2$ above the same mass. We display in [Fig. \[FIG:mz\_bias\]]{} the resulting selection biases on ${T_{\rm dust}}$, ${{\rm IR8}}$ and ${R_{\rm SB}}$. We found that our selection criteria bias our samples toward galaxies with higher ${{\rm SFR}}$ at fixed mass, hence to galaxies offset from the main sequence. For example, our sample with ${T_{\rm dust}}$ measurement will only probe galaxies a factor three above the main sequence for masses less than $10^{11}\,{{\rm M}_\odot}$ at $1<z<3$. Because ${T_{\rm dust}}$ and ${{\rm IR8}}$ both correlate with ${R_{\rm SB}}$ [@elbaz2011; @magnelli2014], this translates into a positive bias on both quantities: at a given mass and redshift, the observed average ${T_{\rm dust}}$ and ${{\rm IR8}}$ of our sample are higher than their true average. This is particularly the case for ${T_{\rm dust}}$, and is apparent also on the real data set ([Fig. \[FIG:indiv\]]{}). Interestingly, although our ${T_{\rm dust}}$ measurements require a detection in the SPIRE bands, which could bias our sample toward colder temperatures, we find that this is a negligible effect compared to the bias toward starbursts: on no occasion is the average measured ${T_{\rm dust}}$ lower than the true average. This analysis implies that the bias toward higher ${R_{\rm SB}}$ is the dominant source of bias on the measured dust properties. In the following, we focus on the trend of both ${T_{\rm dust}}$ and ${{\rm IR8}}$ with ${R_{\rm SB}}$, and therefore we will not be affected by this bias. Yet it is important to keep in mind that our results are based almost entirely on galaxies with masses larger than $3\times10^{10}\,{{\rm M}_\odot}$; studying galaxies at lower masses will require next generation instruments such as ALMA and [[JWST]{}]{}. ![image](mz.eps){width="\textwidth"} ![image](mz_tdust.eps){width="\textwidth"} ### Dust temperatures \[SEC:indiv\_tdust\_result\] ![image](irsed_tdust_fpah.eps){width="\textwidth"} In [Fig. \[FIG:indiv\]]{} (top left) we show the measured dust temperatures for individual [[Herschel]{}]{}detections, ALESS galaxies, and galaxies from the HRS. These temperatures match broadly the trend observed in the stacked SEDs, but with a tendency to show systematically larger values. As discussed in the previous section, this is to be expected: the requirement of a well-measured IR SED biases this sample towards strongly star-forming starburst galaxies (particularly at high redshifts, $z>0.5$), which have higher temperatures. Subtracting our redshift-dependent average from the measured ${T_{\rm dust}}$ values ([Eq. \[EQ:tdust\_ms\]]{}), we then observed how the residuals correlate with the offset from the main sequence. The result is shown in [Fig. \[FIG:indiv\]]{} (top right). We found a positive correlation and parametrized it with a linear relation (obtained as the bisector of the data): $${T_{\rm dust}}[{{\rm K}}] = {T_{\rm dust}}^{MS} - (0.77 \pm 0.04) + (10.1\pm0.6) \times \log_{10}({R_{\rm SB}}) \,. \label{EQ:tdust_sb}$$ where ${T_{\rm dust}}^{MS}$ is defined in [Eq. \[EQ:tdust\_ms\]]{}. The error bars were determined by bootstrapping. Such a trend was first observed in [@elbaz2011], and later quantified by [@magnelli2014] who stacked galaxies at various locations on the ${{\rm SFR}}$–${M_\ast}$ plane (see also @matsuki2017). They reported a linear relation between ${T_{\rm dust}}$ and $\log_{10}({R_{\rm SB}})$ — which they call $\Delta\log({{\rm sSFR}})$ — with a slope of $6.5\,{{\rm K}}$, which is shallower than the one we measure here but still roughly fits our data. The residual intrinsic scatter of the temperatures is presented in [Fig. \[FIG:tdust\_scatter\]]{} as a function of redshift, after subtracting the measurement and redshift uncertainties (assuming $\Delta z/(1+z) = 3\%$; @pannella2015). The absolute scatter (in Kelvins) increases mildly with redshift, and this evolution is consistent with a constant relative scatter of $12\%$, relative to ${T_{\rm dust}}^{\rm MS}$ as given in [Eq. \[EQ:tdust\_ms\]]{}. The evolution of this scatter beyond $z=2$ is poorly constrained, and larger samples with $870$ or $1200\,{\mu{\rm m}}$ coverage would be required to determine whether it keeps increasing at higher redshifts. Over the whole sample, the scatter in temperature was initially $17\%$ (relative to the median). After removing the evolution of the main sequence temperature with redshift, this scatter dropped to $15\%$ (relative to ${T_{\rm dust}}^{\rm MS}$), and removing the starburstiness trend further reduced the scatter to the final value of $12\%$. ### Temperature – luminosity relation \[SEC:lir\_tdust\] ![image](lir_tdust.eps){width="90.00000%"} Historically, the first studied correlation was that between the dust temperature and the infrared luminosity (see references in [section \[SEC:introduction\]]{}, §7). Whether this correlation or that involving the redshift and the starburstiness provides the best description of the observations has been repeatedly debated in the literature [see, e.g., @casey2012-a; @symeonidis2013; @magnelli2014]. Unfortunately, owing to the strong selection effects of flux-limited FIR samples, it is generally difficult to de-correlate the effect of redshift and luminosity. As shown in [Fig. \[FIG:lir\_tdust\]]{} (left), our sample is affected by this issue: while our galaxies do show a clear correlation between ${T_{\rm dust}}$ and ${L_{\rm IR}}$, with ${T_{\rm dust}}= 5.57\times{L_{\rm IR}}^{0.0638}$, ${L_{\rm IR}}$ also strongly correlates with the redshift. The residual scatter around the ${L_{\rm IR}}$–${T_{\rm dust}}$ relation is $13\%$, which is comparable to that found in the previous section; individual galaxies do not favor one description over the other. At present, only stacking has enough discriminative power to address this question. For example, [@magnelli2014] found, at fixed redshift, a tighter correlation between ${T_{\rm dust}}$ and ${R_{\rm SB}}$ than with ${L_{\rm IR}}$ in their stacks. In our stacks in bins of stellar mass and redshift, we found no evidence for a correlation between ${T_{\rm dust}}$ and ${L_{\rm IR}}$ at fixed redshift; for example at $z=1.5$, ${T_{\rm dust}}$ remains mostly constant while ${L_{\rm IR}}$ spans almost two orders of magnitude, as shown in [Fig. \[FIG:lir\_tdust\]]{} (left). This suggests that there exists no fundamental correlation between ${L_{\rm IR}}$ and ${T_{\rm dust}}$ (at a given redshift) beyond that induced by the starburstiness, which gets averaged-out in our stacks, and selection effects. To further demonstrate this, we show in [Fig. \[FIG:lir\_tdust\]]{} (right) the ${L_{\rm IR}}$–${T_{\rm dust}}$ relation arising in the mock catalog produced with EGG (see [section \[SEC:selection\_effects\]]{}). This relation is very similar to that observed in the real catalog, albeit with a slightly higher scatter (15%). This ${L_{\rm IR}}$–${T_{\rm dust}}$ relation was not imposed when generating the mock, instead it emerges naturally as a combination of selection effects and the relations discussed in the previous section. ### IR8 \[SEC:indiv\_ir8\_result\] We applied the same procedure to ${{\rm IR8}}$, and the results are shown in [Fig. \[FIG:indiv\]]{} (bottom left and right). Consistently with the results of [@elbaz2011] and [@nordon2012], we found a correlation between ${{\rm IR8}}$ and $\log_{10}({R_{\rm SB}})$, meaning that starburst galaxies have depressed PAH emission, which Elbaz et al. interpreted as a sign of increased compactness of the star-forming regions. We modeled this dependence with a linear relation (obtained as the bisector of the data): $${{\rm IR8}}= {{\rm IR8}}^{\rm MS} \times (0.81 \pm 0.02) \times {R_{\rm SB}}^{0.66 \pm 0.05}\,.\label{EQ:ir8_sb}$$ This relation is consistent with that derived by [@nordon2012], although we do not find the need for a separate regime for galaxies below the main sequence. Over the whole sample, the scatter in $\log_{10}({{\rm IR8}})$ is $0.28\,{{\rm dex}}$. This is reduced to $0.22$ and $0.18\,{{\rm dex}}$ after subtracting the redshift and starburstiness dependences, respectively. ![Residual intrinsic scatter of ${T_{\rm dust}}$, obtained after removing the redshift evolution ([Eq. \[EQ:tdust\_ms\]]{}) and starburstiness dependence ([Eq. \[EQ:tdust\_sb\]]{}), and subtracting statistically the uncertainty on the ${T_{\rm dust}}$ measurements and on the redshift. Measurements of the scatter in CANDELS and ALESS are shown with solid circles, and the HRS is shown with a solid triangle. The trend with redshift is modeled as a constant relative uncertainty (black line).[]{data-label="FIG:tdust_scatter"}](irsed_tdust_scatter.eps){width="48.00000%"} Optimal ${L_{\rm IR}}$ and ${M_{\rm dust}}$ measurements \[SEC:irsed\_recipe\] ============================================================================== Contrary to the standard FIR libraries from the literature , ours has three degrees of freedom: the normalization (either ${L_{\rm IR}}$ or ${M_{\rm dust}}$), the dust temperature, and the ${{\rm IR8}}$ (or ${f_{\rm PAH}}$). This requires particular care when the observed SEDs are poor and some of these parameters are unconstrained. To obtain a robust determination of ${L_{\rm IR}}$ for galaxies with variable wavelength coverage, the procedure we recommend is described in the following subsections. In the next section we quantify the accuracy of monochromatic ${L_{\rm IR}}$ or ${M_{\rm dust}}$ measurements (i.e, when only a single broadband flux is available for a galaxy) when this procedure is applied, and make predictions for [[JWST]{}]{}and ALMA. Selection of the free parameters -------------------------------- The dust temperature must be fixed if the peak of the FIR emission is not constrained. In practice, this requires at least one measurement at more than $3\sigma$ on either side of the peak [@hwang2010-a], and within the rest-frame $15\,{\mu{\rm m}}$ to $3\,{\rm mm}$ to avoid contribution from PAH or free-free emission. Here we defined the “peak” by first fitting the galaxy with a free ${T_{\rm dust}}$, and measuring $\lambda_{\rm max}$ from the resulting best-fit template. To fix the temperature, one will use [Eq. \[EQ:tdust\_ms\]]{} evaluated at the redshift of the galaxy. The library is then reduced to a single dust continuum and a single PAH template. Likewise, the ${{\rm IR8}}$ must be fixed if no measurement probes the rest-frame $5$ to $15\,{\mu{\rm m}}$ (to constrain ${L_8}$), or if no observation is available for wavelengths greater than $15\,{\mu{\rm m}}$ (to constrain ${L_{\rm IR}}$). In this case the value of ${{\rm IR8}}$ should be taken from [Eq. \[EQ:ir8\_ms\]]{} and evaluated at the redshift of the galaxy. Using [Eq. \[EQ:fpah\_ir8\_th\]]{}, ${{\rm IR8}}$ can then be translated to ${f_{\rm PAH}}$ and fix the relative amplitude of the PAH and continuum templates. Fitting the observed photometry ------------------------------- For each value of ${T_{\rm dust}}$ allowed in the fit, the templates (provided in rest-frame quantities) must be translated to the observer frame and integrated under the filter response curves of the available photometry. Using a linear solver, one can then fit the convolved templates to the observed photometry as a linear combination of the dust continuum and PAH emission: $$S_{\nu}^{\rm model} = {M_{\rm dust}}^{\rm cont}\,S_{\nu}^{\rm cont} + {M_{\rm dust}}^{\rm PAH}\,S_{\nu}^{\rm PAH}\,, \label{EQ:fit_model_full}$$ where ${M_{\rm dust}}^{\rm cont}$ and ${M_{\rm dust}}^{\rm PAH}$ are free parameters, and compute the $\chi^2$. Among all the templates in the library, one can then pick as the best-fit solution the ${T_{\rm dust}}$ value which produced the smallest $\chi^2$ (or the only available ${T_{\rm dust}}$ value if it is kept fixed). If the fit is performed with a standard linear solver, the amplitude of either component can become negative. This typically happens when the observed photometry requires an ${{\rm IR8}}$ value larger than what the library has to offer (which is rare by construction), and the fit uses a PAH component with a negative amplitude to reduce the mid-IR emission. In such cases, one could fix ${f_{\rm PAH}}=0$ (i.e., no PAH emission) or exclude the rest-frame $8\,{\mu{\rm m}}$ photometry altogether. Such situations can hint at the presence of obscured AGNs [@donley2012], or galaxies with strong silicate absorption [@magdis2011-a]. If ${{\rm IR8}}$ is fixed, the two components of the library are merged into one single template for each value of ${T_{\rm dust}}$. [Eq. \[EQ:fit\_model\_full\]]{} becomes $$S_{\nu}^{\rm model} = {M_{\rm dust}}\,\Big[(1-{f_{\rm PAH}}) \times S_{\nu}^{\rm cont} + {f_{\rm PAH}}\times S_{\nu}^{\rm PAH}\Big]\,,$$ where ${f_{\rm PAH}}$ is computed from ${{\rm IR8}}$ using [Eq. \[EQ:fpah\_ir8\_th\]]{}, and the only free parameter is ${M_{\rm dust}}$. Similarly to the procedure above, one can then vary ${T_{\rm dust}}$ and choose as the best-fit solution the value of ${T_{\rm dust}}$ that produced the smallest $\chi^2$. The dust mass is computed as ${M_{\rm dust}}= {M_{\rm dust}}^{\rm cont} + {M_{\rm dust}}^{\rm PAH}$, while the other parameters (${L_{\rm IR}}$ and ${L_8}$) can be obtained from the tabulated values corresponding to the best-fit ${T_{\rm dust}}$. Computing uncertainties on best-fit parameters ---------------------------------------------- The most straightforward and secure way to determine uncertainties is to perform a Monte Carlo simulation for each galaxy, where the observed fluxes are randomly perturbed with a Gaussian scatter of amplitude set by the flux uncertainties. This needs to be repeated at least $100$ times for reliable $1\sigma$ error bars. For each parameter, the probability distribution function can be determined from the distribution of best fit values among all random realizations. Accuracy of monochromatic measurements \[SEC:irsed\_mono\] ========================================================== ![image](monodisp.eps){width="\textwidth"} While our library provides three degrees of freedom, in most cases the observed SED of a galaxy will consist of only a couple of data points. The procedure we described in the previous section provides a simplified fit procedure where the number of degrees of freedom is progressively reduced until only one is left: the normalization of the template SED. At fixed ${T_{\rm dust}}$, this normalization can be translated either into ${L_{\rm IR}}$ or ${M_{\rm dust}}$, which are usually the quantities observers are after. Often, only one observed data point is available — typically either from [[Spitzer]{}]{}MIPS $24\,{\mu{\rm m}}$, ALMA $870\,{\mu{\rm m}}$, or $1.1\,{\rm mm}$, but also in the future from James Webb. In this case we dub the measurement of ${L_{\rm IR}}$ or ${M_{\rm dust}}$ as “monochromatic”. In the case of such monochromatic measurements, since the shape of the SED depends on ${T_{\rm dust}}$ in a strongly non-linear way, the uncertainty on ${T_{\rm dust}}$ will propagate into different uncertainties on ${L_{\rm IR}}$ and ${M_{\rm dust}}$ depending on which band is used to determine the normalization of the template (the case of the many ALMA bands in the millimeter regime is discussed in [section \[SEC:alma\_mdust\_lir\]]{}). We have shown in [section \[SEC:indiv\_tdust\_result\]]{} that fixing ${T_{\rm dust}}$ to the average value for galaxies at a given redshift ([Eq. \[EQ:tdust\_ms\]]{}) provides the correct temperature within $15\%$. In this section, we discuss the uncertainties on ${L_{\rm IR}}$ and ${M_{\rm dust}}$ resulting from this uncertainty on ${T_{\rm dust}}$, and show which bands are best suited for monochromatic measurements or either quantities. We also discuss how well the rest-frame $8\,{\mu{\rm m}}$ luminosity can be used to measure ${L_{\rm IR}}$, which is a situation arising for $z\sim1$ to $2$ galaxies too faint to be detected with [[Herschel]{}]{}but seen by [[Spitzer]{}]{}MIPS at $24\,{\mu{\rm m}}$ (or in the future with [[JWST]{}]{}, which is discussed more thoroughly in [sections \[SEC:conv\]]{} and \[SEC:delta\_z\]). Mock catalog ------------ Using a mock catalog built with EGG (see [section \[SEC:stack\_correct\]]{}), we simulated monochromatic measurements and compared them to the true values of ${L_{\rm IR}}$ and ${M_{\rm dust}}$. We produced a mock catalog spanning $10\,\deg^2$ and selected all the star-forming galaxies more massive than ${M_\ast}= 10^{10}\,{{\rm M}_\odot}$. We then created a flux catalog containing the fluxes of all galaxies in the simulation in all [[Spitzer]{}]{}and [[Herschel]{}]{}bands, as well as [[JWST]{}]{}MIRI and ALMA. For each galaxy in the mock catalog, we built its “average” expected SED from [Eqs. \[EQ:tdust\_ms\]]{} and \[EQ:ir8\_ms\] and used this SED to convert each flux into a value of ${L_{\rm IR}}$ and ${M_{\rm dust}}$. We note that we did not simulate the stellar continuum for this experiment; we assumed it can be properly constrained and subtracted using the shorter wavelengths. This will be particularly important for the $16$ and $24\,{\mu{\rm m}}$ bands at high redshifts. Likewise, the impact of AGNs was also ignored, and these will increase the uncertainty in these same bands at all redshifts [e.g., @mullaney2011]. Therefore the uncertainties we derived should be considered as lower limits. The resulting uncertainties are shown in [Fig. \[FIG:pred\_disp\]]{} as a function of redshift, for all [[Spitzer]{}]{}and [[Herschel]{}]{}bands, as well as two ALMA bands for illustration (band 7 at $8770{\mu{\rm m}}$, and band 6 at $1100\,{\mu{\rm m}}$); the case of [[JWST]{}]{}and ALMA are discussed further in [section \[SEC:conv\]]{}. Several interesting features come out of this figure and we describe them in the following sections. ### Infrared luminosity \[SEC:mono\_lir\] When measuring fluxes on the dust continuum, we found ${L_{\rm IR}}$ is best measured when the photometric measurement is close to the peak of the FIR SED: the optimal uncertainty is of the order of $0.05\,{{\rm dex}}$, using $100\,{\mu{\rm m}}$ at $z<0.2$, $160\,{\mu{\rm m}}$ at $0.2<z<1.5$, $250\,{\mu{\rm m}}$ at $1.5<z<3.9$, and $350\,{\mu{\rm m}}$ at $3.9<z<5$. These correspond respectively to rest-frame wavelengths of $90$, $86$, $68$ and $64\,{\mu{\rm m}}$, which are precisely the peak wavelengths of the dust SED at each redshift (see also @schreiber2015, Fig. 9). Rightward of the peak, the uncertainty rises continuously as the rest wavelength increases, since the emission beyond the rest-frame $250\,{\mu{\rm m}}$ is rather tracing the dust mass (see [sections \[SEC:mono\_dust\]]{}, \[SEC:alma\_mdust\_lir\], and @scoville2014), and fluctuations in the ${L_{\rm IR}}/{M_{\rm dust}}$ ratio (displayed in [Fig. \[FIG:pred\_disp\]]{}, top and bottom left) are driven by the adopted scatter in ${T_{\rm dust}}$. Leftward of the peak, the uncertainties rise significantly up to $0.22\,{{\rm dex}}$ when probing the rest-frame $5$ to $10\,{\mu{\rm m}}$, that is, for $16\,{\mu{\rm m}}$ at $0.5<z<2$ and $24\,{\mu{\rm m}}$ at $1.5<z<3$. This, in turn, is caused by variations of ${{\rm IR8}}$ (displayed in [Fig. \[FIG:pred\_disp\]]{}, top left), and shows that fluxes in this wavelength domain should be interpreted carefully. Outside of these ranges dominated by PAH emission, our model suggests that the short wavelengths can be excellent tracers of ${L_{\rm IR}}$, for example with $24\,{\mu{\rm m}}$ reaching the same accuracy as $350\,{\mu{\rm m}}$ at $z>3.9$. This is linked to our assumption of a constant fraction of very small grains. Little data can back up this assumption at present; in the local Universe [@chary2001] reported a scatter of $0.15\,{{\rm dex}}$ between ${L_{\rm IR}}$ and the $12$ or $16\,{\mu{\rm m}}$ luminosities, which is consistent with the values we obtained with our model and suggest the small grain population does not vary strongly from one galaxy to the next. On the other hand, the hotter ${T_{\rm dust}}$ and reduced PAH emission observed in distant galaxies suggest that small grains could get destroyed more efficiently at this epoch. In the near future, [[JWST]{}]{}will be able to detect distant galaxies at rest wavelengths less than $12\,{\mu{\rm m}}$ and check this assumption. Another source of uncertainty will be the subtraction of the stellar continuum which can start to dominate below $5\,{\mu{\rm m}}$, although this should be less of a problem at high redshifts where the specific SFRs are higher. Last but not least, we caution that this whole discussion is ignoring AGNs, which seem to be very common at least among massive galaxies at $z>3$ [@marsan2017]. Using a single photometric point, and therefore fixing ${{\rm IR8}}$ and ${T_{\rm dust}}$ to their redshift average, one will be systematically biased for those galaxies which have unusual ${{\rm IR8}}$ or ${T_{\rm dust}}$ values. As shown in [section \[SEC:irsed\_indiv\]]{}, this is the case for starburst galaxies. For this reason, we also display on [Fig. \[FIG:pred\_disp\]]{} (top right) the predicted value of this systematic bias for galaxies with ${R_{\rm SB}}> 2$ (i.e., at least one sigma away from the main sequence). The trend is for measurements leftward of the peak to barely overestimate the ${L_{\rm IR}}$ by no more than ${R_{\rm SB}}^{0.3}$. In other words, a galaxy which is truly a factor five above the main sequence will be observed instead at a factor eight. On the other hand, measurements rightward of the peak or those dominated by PAH emission can reach systematic underestimations by a factor of ${R_{\rm SB}}^{0.5}$, so a galaxy a factor five above the MS will be seen at only a factor $2.2$. These systematic errors will tend to bring starburst galaxies closer to the main sequence than what they are in reality, to the point where they can no longer be identified as such. Therefore, SFRs measured only from a single $24\,{\mu{\rm m}}$ or ALMA flux at $z=2$ should not be used to study starburst galaxies. ### Dust mass \[SEC:mono\_dust\] We present similar figures for ${M_{\rm dust}}$ in [Fig. \[FIG:pred\_disp\]]{} (bottom left and right) for submillimeter bands. The most striking fact to take out of this plot is that no band provides a measurement of ${M_{\rm dust}}$ at better than $0.1\,{{\rm dex}}$ [as observed in the local Universe in @groves2015]. This uncertainty rises steadily with redshift for the SPIRE bands, as rest-wavelengths get closer to the peak of the dust emission. This is partly compensated by the increase of the dust temperature with redshift ([Eq. \[EQ:tdust\_ms\]]{}), which shifts the peak toward shorter wavelengths. As a consequence we found that the $870\,{\mu{\rm m}}$ or $1.1\,{\rm mm}$ bands suffer roughly equivalent uncertainties, with only a minimal increase with redshift. [@scoville2014] recommended using rest-wavelengths larger than $250\,{\mu{\rm m}}$ to measure the dust mass, which should rule out the use of $870\,{\mu{\rm m}}$ beyond $z=2.5$. However they have assumed a constant and relatively cold average dust temperature of $25{{\rm K}}$: for $\left<{T_{\rm dust}}\right>=42{{\rm K}}$ as we observed at $z=4$, the equivalent wavelength limit (in terms of distance to the peak) becomes $145\,{\mu{\rm m}}$, which corresponds to $725\,{\mu{\rm m}}$ at $z=4$. This implies that high frequency ALMA bands can still be used to trace the dust mass at high redshifts with a similar accuracy as $500\,{\mu{\rm m}}$ at $z<1$ (see also [sections \[SEC:alma\_mdust\_lir\]]{} where this is developed further). The systematic bias on the dust mass of starburst galaxies is relatively constant with redshift, and most importantly never reaches zero. For these galaxies, we predict a systematic overestimation of the dust masses by a factor of at least ${R_{\rm SB}}^{0.2}$, so a galaxy a factor five above the main sequence will have its dust mass overestimated by at least $40\%$. When the dust mass is converted to a gas mass (see [section \[SEC:gas\]]{}), this implies that the gas fraction and depletion time of starbursts will be overestimated by a similar factor. Monochromatric conversion factors for ALMA and [[JWST]{}]{}\[SEC:conv\] ------------------------------------------------------------------------- ![image](lconv_alma.eps){width="48.00000%"} ![image](dconv.eps){width="48.00000%"} ![Conversion factor from observed monochromatic luminosity to ${L_{\rm IR}}$. These factors are provided for all [[JWST]{}]{}MIRI bands, at all redshifts where the band probes the rest-frame $>3\,{\mu{\rm m}}$ (i.e., where it may not be dominated by stellar continuum). The top panel gives the associated relative uncertainty on the conversion. These data are tabulated for easier access in [Table \[TAB:lconv\]]{}.[]{data-label="FIG:conv_jwst"}](lconv.eps){width="48.00000%"} We provide numerical factors to convert an observed ALMA flux into a dust mass and infrared luminosity in [Table \[TAB:lconv\_alma\]]{} and [Table \[TAB:dconv\]]{}, respectively, and conversion from a [[JWST]{}]{}MIRI luminosity into ${L_{\rm IR}}$ in [Table \[TAB:lconv\]]{}. The later is truncated beyond the redshift where the [[JWST]{}]{}bands probe the rest-frame $\lambda < 4\,{\mu{\rm m}}$, where the stellar continuum starts to dominate the emission. These tables also include the conversion uncertainty (in dex), as shown on [Fig. \[FIG:pred\_disp\]]{} for the [[Spitzer]{}]{}and [[Herschel]{}]{}bands. These data are also displayed in [Figs. \[FIG:conv\]]{} and \[FIG:conv\_jwst\]. It is clear from [Fig. \[FIG:conv\_jwst\]]{} that the accuracy of a [[JWST]{}]{}band in measuring ${L_{\rm IR}}$ will strongly depend on the redshift, as was already apparent for [[Spitzer]{}]{}$16$ and $24\,{\mu{\rm m}}$. Depending on the redshift range of interest, it will therefore be more profitable to observe with one band or the other: for example, using F2550W instead of F1800W at $z=1.5$ can reduce the uncertainty on ${L_{\rm IR}}$ from $0.23$ to $0.12\,{{\rm dex}}$. This has to be considered together with the telescope’s sensitivity in each band in determining the optimal observational setup (given, for example, that F2550W is expected to be seven time less sensitive than F1800W). ### Impact of redshift uncertainties for [[JWST]{}]{}\[SEC:delta\_z\] ![Uncertainty on the conversion of an observed [[JWST]{}]{}F1800W luminosity into ${L_{\rm IR}}$. This uncertainty is shown with three cases: no uncertainty on the redshift (solid line), $3\%$ uncertainty (dot-dashed line), and $10\%$ uncertainty (dotted line).[]{data-label="FIG:conv_jwst_dz"}](lconv_dz.eps){width="48.00000%"} A potentially important consideration related to [[JWST]{}]{}broadbands is the effect of redshift uncertainties. Because a large fraction of the flux in these bands comes from relatively narrow PAH emission lines, the conversion factor from flux to ${L_{\rm IR}}$ depends steeply on redshift as lines fall in and out of the bandpass. This can be precisely modeled if the spectroscopic redshift is known, but the case of photometric redshifts requires more care. To explore the impact of photometric redshift uncertainties, we have created two sets of “observed redshifts” for the galaxies in our mock catalog, obtained by randomly perturbing the true redshift with a Gaussian distribution of width $\Delta z$. We picked $\Delta z/(1+z)=3\%$ and $10\%$; $3\%$ is a typical (if not a conservative) value of the uncertainty in deep fields [@muzzin2013-a; @skelton2014; @pannella2015; @straatman2016], while $10\%$ is a more extreme value which applies only to rare outliers. The result is shown in [Fig. \[FIG:conv\_jwst\_dz\]]{}, and cannot be described as a simple increase in quadrature from the case with no redshift uncertainty. In the case of $\Delta z/(1+z)=3\%$, the impact of the redshift uncertainty is null at $0.9 < z < 1.2$, and shows a maximal increase (in quadrature) of $0.16\,{{\rm dex}}$ at $z=0.7$, $1.5$ and $2.2$. This has the net effect of shifting the domains of best and worst accuracy toward slightly higher redshifts. This can be explained through the so-called “negative $k$-correction”: when increasing the redshift, the decrease in flux caused by the larger distance can be compensated, partly or fully, if the intrinsic flux is higher at shorter rest-wavelengths. This happens when the broadband filter falls on the long-wavelength side of a PAH line. But globally, the uncertainty on the conversion is not dramatically increased, and the impact of the redshift uncertainty can be mostly ignored. In contrast, when $\Delta z/(1+z)=10\%$ the situation is much worse, with a minimal uncertainty of $0.25\,{{\rm dex}}$ around $z=1$, and $0.4\,{{\rm dex}}$ at other redshifts (i.e., essentially not a measurement). This highlights the sensitivity of the PAH region to redshift outliers, and suggests that multiple MIRI bands may be required to properly characterize the galaxies with the most uncertain redshifts (e.g., extremely dusty starbursts). ### ALMA: dust masses or infrared luminosities? \[SEC:alma\_mdust\_lir\] The dust mass and the infrared luminosity are the two main quantities one can measure from a set of ALMA fluxes; typically to infer gas masses and star formation rates, respectively. However, the conversion from an ALMA flux to either of these quantities depends on other factors, in particular on the dust temperature (see [Eq. \[EQ:lir\_mdust\]]{}). The most secure approach would thus be to fit for the dust temperature, and then derive ${L_{\rm IR}}$ and ${M_{\rm dust}}$. But when only a single flux measurement is available, the question arises: does this photometric point measure ${L_{\rm IR}}$, or ${M_{\rm dust}}$? Because both depend on the dust temperature, the correct answer is “neither”, which is not so helpful. However it is clear that one quantity will be constrained better than the other depending on the observed frequency (see previous sections and [Fig. \[FIG:pred\_disp\]]{}): one expects high frequency bands to better trace ${L_{\rm IR}}$, and low frequency bands to better trace ${M_{\rm dust}}$. We quantified this question in two ways using our mock catalog. We first determined, for each band, the redshift ranges in which ${L_{\rm IR}}$ or ${M_{\rm dust}}$ are measured at better than $0.2\,{{\rm dex}}$. We found that ${M_{\rm dust}}$ is measured at better than $0.2\,{{\rm dex}}$ at all redshifts, and for all bands except band 9 ($440\,{\mu{\rm m}}$, which is always worse than $0.2\,{{\rm dex}}$). In contrast, we found that ${L_{\rm IR}}$ can only be measured at this level of accuracy in band 9, 8, 7 and 6 and at $z>1$, $3.2$, $3.8$ and $5.7$, respectively. Therefore there are domains where either ${L_{\rm IR}}$ or ${M_{\rm dust}}$ can be measured at better than $0.2\,{{\rm dex}}$ from the same ALMA measurement. We note however that only one of the two quantities can really be “measured”, the other is then fully determined by the assumed ${T_{\rm dust}}$. Alternatively, for each redshift and band, we determined which of ${L_{\rm IR}}$ or ${M_{\rm dust}}$ is constrained with the lowest uncertainty. The answer is always ${M_{\rm dust}}$, except for band 9, 8 and 7 at $z>0.8$, $3.6$ and $5.7$, respectively, where ${L_{\rm IR}}$ takes over. Therefore, while band 7 ($870\,{\mu{\rm m}}$) can be used starting from $z=3.8$ to measure ${L_{\rm IR}}$, it is only at $z>5.7$ that it traces ${L_{\rm IR}}$ better than it traces ${M_{\rm dust}}$. These results depend strongly on the evolution of the average dust temperature with redshift. If it had remained constant at the $z=0$ value, high frequency ALMA bands would have traced ${L_{\rm IR}}$ better at lower redshifts. This highlights that a proper knowledge of the dust temperature is crucial to interpret sub-millimeter fluxes, especially in cases where a single band is used at multiple redshifts to determine, for example, luminosity functions or cosmic ${{\rm SFR}}$ densities, or to build scaling laws. Gas masses from dust masses \[SEC:gas\] --------------------------------------- While the dust mass in itself is only of moderate interest, it is often used as a proxy for determining gas masses, though the assumption of a gas-to-dust ratio. In [@schreiber2016], we have derived gas-to-dust ratios appropriate for our library: since systematic uncertainties on the dust masses are significant (see [section \[SEC:amorphous\]]{}), it is crucial to use a consistent calibration of gas-to-dust ratios, derived using the same dust model. We obtained $$\begin{aligned} \frac{{M_{\rm gas}}}{{M_{\rm dust}}} = (155 \pm 23)\times\frac{Z_{\sun}}{Z}\,, \label{EQ:gdr}\end{aligned}$$ where ${M_{\rm gas}}$ is the total mass of atomic and molecular hydrogen, including helium, and $Z$ is the metallicity. This relation assumes that the metallicity is inferred from the oxygen abundance $12 + \log({\rm O}/{\rm H})$, measured using the [@pettini2004] calibration (see @magdis2012), a solar metallicity of $Z_\sun=0.0134$ and a solar oxygen abundance of $8.73$ [@asplund2009]. The Pettini & Pagel calibration was used by [@mannucci2010] to build the Fundamental Metallicity Relation (FMR), therefore the above formula can be applied directly to metallicities estimated using the FMR. The uncertainty in the above formula is only statistical. Using galaxies in the HRS with well measured dust masses and independent measurement of gas masses from CO and H<span style="font-variant:small-caps;">i</span>, we determined that gas masses derived from dust masses were accurate at the level of $0.2\,{{\rm dex}}$. We emphasize that this gas-to-dust ratio was empirically calibrated using measured dust masses of nearby galaxies, and therefore the value of this ratio depends entirely on the model used to infer the dust masses. Using the DL07 model, for example, would produce higher dust masses by a factor two, hence the gas-to-dust ratio would have to be reduced by the same factor to remain consistent. Ultimately, the gas masses derived from dust masses of any model should be the same, provided the gas-to-dust ratios are properly calibrated. Conclusions =========== We have introduced a new library of infrared SEDs, publicly available on-line, with three degrees of freedom: the dust mass or infrared luminosity, the dust temperature ${T_{\rm dust}}$, and the mid-to-total infrared color ${{\rm IR8}}\equiv{L_{\rm IR}}/{L_8}$. Using this library, we fit stacked SEDs of complete samples of main sequence galaxies in the CANDELS fields and derived the redshift evolution of the average ${T_{\rm dust}}$ and ${{\rm IR8}}$, recovering and extending the trends previously identified in the literature. We observed that the most massive galaxies (${M_\ast}> 10^{11}\,{{\rm M}_\odot}$) at $z<1$ have a reduced ${T_{\rm dust}}$, sign of a reduced star formation efficiency, and found that low mass galaxies (${M_\ast}< 10^{10}\,{{\rm M}_\odot}$) have an increased ${{\rm IR8}}$, probably because of their lower metallicities. Aside from these two mass domains, we found the infrared SED of the stacked galaxies only depend on the redshift, confirming the existence of a universal dust SED for main sequence galaxies at each epoch of the Universe. We then used our new library to model galaxies individually detected in the [[Herschel]{}]{}images to determine how ${T_{\rm dust}}$ and ${{\rm IR8}}$ vary for galaxies located above the main sequence, that is, the starbursts, and measure for the first time the scatter of both quantities. We recovered previous claims of a positive correlation of ${T_{\rm dust}}$ and ${{\rm IR8}}$ with the offset from the main sequence. Both trends hint that the star forming regions in starbursts are more compact than in the typical main sequence galaxy. We observed a low residual intrinsic scatter of $12\%$ in ${T_{\rm dust}}$ and $0.18\,{{\rm dex}}$ for ${{\rm IR8}}$, confirming that most of the observed variations of these two parameters are captured by the relations we derived with redshift and offset from the main sequence. We have implemented these relations and scatters in the Empirical Galaxy Generator (EGG) to predict the accuracy of monochromatic measurements of ${L_{\rm IR}}$ and ${M_{\rm dust}}$, as provided now by [[Spitzer]{}]{}MIPS $24\,{\mu{\rm m}}$ and ALMA, and in the future with [[JWST]{}]{}. We found that ${L_{\rm IR}}$ is best measured by wavelengths close to the peak of the dust emission, with a minimal uncertainty of $0.05\,{{\rm dex}}$, while mid-IR bands such as those of [[JWST]{}]{}have a typical uncertainty of $0.1$ to $0.25\,{{\rm dex}}$. The highest frequency bands of ALMA can also be used to determine ${L_{\rm IR}}$, with an uncertainty of $0.2\,{{\rm dex}}$ or less at $z>0.9$, $3.2$, $3.8$, and $5.7$ for bands 9, 8, 7, and 6, respectively. Using randomly perturbed redshifts, we found these values to be only moderately increased in case of a typical redshift uncertainty of $3\%$. When measuring flux leftward of the peak (in wavelength), ${L_{\rm IR}}$ is only barely biased for starburst galaxies, however measurements rightward of the peak or using the rest-frame $8\,{\mu{\rm m}}$ will underestimate the ${L_{\rm IR}}$ of starburst galaxies to the point of artificially bringing them back within the upper envelope of the main sequence. Using the same mock catalog, we determined that the dust masses are best determined from the longest wavelength bands with an uncertainty of less than $0.15\,{{\rm dex}}$. High-frequency ALMA bands such as band $8$ and $7$ can also be used with an uncertainty of $0.2\,{{\rm dex}}$, however these bands will also tend to overestimate the dust (and gas) masses of starburst galaxies. Finally, we tabulated the coefficients to convert observed ALMA fluxes into ${M_{\rm dust}}$ and ${L_{\rm IR}}$ and [[JWST]{}]{}luminosities into ${L_{\rm IR}}$, and provided estimates of the uncertainty associated to this conversion. These results and our library can be used immediately to interpret the many observations in the ALMA archive, which most often consist of a single measurement per galaxy. Furthermore, we expect this will also be most useful for future proposals, either for ALMA or [[JWST]{}]{}, by providing accurate predictions for the expected flux range of individual galaxies at various epochs. The authors want to thank the anonymous referee for their comments that improved the consistency and overall quality of this paper. Most of the numerical analysis conducted in this work have been performed using [phy++]{}, a free and open source C++ library for fast and robust numerical astrophysics ([<http://cschreib.github.io/phypp/>]{}). This work is based on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA [[HST]{}]{}, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. This research was supported by the French Agence Nationale de la Recherche (ANR) project ANR-09-BLAN-0224 and by the European Commission through the FP7 SPACE project ASTRODEEP (Ref.No: 312725). Tabulated conversion factors for ${L_{\rm IR}}$ and ${M_{\rm dust}}$ ==================================================================== [cccccccc]{}\ $z$ & band 9 & band 8 & band 7 & band 6 & band 5 & band 4 & band 3\ & 678 GHz & 404 GHz & 343 GHz & 229 GHz & 202 GHz & 149 GHz & 96.3 GHz\ \ 0.25 & 0.008 (0.25) & 0.035 (0.29) & 0.057 (0.30) & 0.210 (0.31) & 0.295 (0.31) & 0.821 (0.32) & 3.290 (0.33)\ 0.37 & 0.015 (0.24) & 0.061 (0.28) & 0.099 (0.29) & 0.358 (0.31) & 0.512 (0.31) & 1.364 (0.32) & 5.797 (0.32)\ 0.49 & 0.023 (0.23) & 0.094 (0.27) & 0.150 (0.28) & 0.539 (0.30) & 0.780 (0.30) & 2.009 (0.31) & 8.936 (0.32)\ 0.62 & 0.033 (0.22) & 0.129 (0.27) & 0.209 (0.28) & 0.734 (0.30) & 1.087 (0.30) & 2.735 (0.31) & 12.75 (0.31)\ 0.76 & 0.046 (0.21) & 0.167 (0.26) & 0.272 (0.27) & 0.931 (0.29) & 1.392 (0.30) & 3.530 (0.30) & 16.79 (0.31)\ 0.92 & 0.059 (0.20) & 0.211 (0.26) & 0.337 (0.27) & 1.136 (0.29) & 1.701 (0.29) & 4.379 (0.30) & 19.72 (0.31)\ 1.09 & 0.075 (0.19) & 0.254 (0.25) & 0.399 (0.26) & 1.330 (0.29) & 1.978 (0.29) & 5.182 (0.30) & 22.12 (0.31)\ 1.28 & 0.093 (0.19) & 0.297 (0.25) & 0.466 (0.26) & 1.532 (0.28) & 2.238 (0.29) & 5.937 (0.30) & 24.36 (0.31)\ 1.48 & 0.115 (0.18) & 0.343 (0.25) & 0.536 (0.26) & 1.757 (0.28) & 2.530 (0.29) & 6.657 (0.30) & 27.21 (0.31)\ 1.70 & 0.140 (0.17) & 0.397 (0.24) & 0.610 (0.26) & 1.979 (0.28) & 2.865 (0.29) & 7.378 (0.30) & 30.70 (0.31)\ 1.94 & 0.174 (0.16) & 0.466 (0.24) & 0.708 (0.25) & 2.261 (0.28) & 3.319 (0.29) & 8.346 (0.30) & 35.68 (0.31)\ 2.20 & 0.217 (0.15) & 0.552 (0.23) & 0.836 (0.25) & 2.615 (0.27) & 3.832 (0.28) & 9.595 (0.29) & 41.80 (0.30)\ 2.49 & 0.270 (0.14) & 0.656 (0.22) & 0.980 (0.24) & 3.026 (0.27) & 4.367 (0.28) & 11.06 (0.29) & 48.33 (0.30)\ 2.80 & 0.332 (0.13) & 0.772 (0.21) & 1.140 (0.23) & 3.477 (0.26) & 5.018 (0.27) & 12.74 (0.28) & 55.34 (0.29)\ 3.14 & 0.398 (0.12) & 0.898 (0.20) & 1.302 (0.22) & 3.867 (0.25) & 5.604 (0.26) & 14.18 (0.27) & 60.88 (0.29)\ 3.51 & 0.472 (0.10) & 1.031 (0.19) & 1.473 (0.21) & 4.284 (0.24) & 6.164 (0.25) & 15.45 (0.26) & 65.69 (0.28)\ 3.91 & 0.561 (0.09) & 1.181 (0.18) & 1.662 (0.20) & 4.687 (0.23) & 6.674 (0.24) & 16.64 (0.25) & 69.96 (0.27)\ 4.35 & 0.678 (0.08) & 1.356 (0.17) & 1.881 (0.19) & 5.141 (0.22) & 7.290 (0.23) & 18.05 (0.24) & 75.17 (0.25)\ 4.82 & 0.822 (0.07) & 1.540 (0.16) & 2.125 (0.18) & 5.608 (0.21) & 7.931 (0.22) & 19.39 (0.23) & 81.47 (0.25)\ 5.34 & 0.994 (0.07) & 1.748 (0.15) & 2.389 (0.17) & 6.117 (0.20) & 8.553 (0.21) & 20.70 (0.23) & 87.66 (0.24)\ 5.91 & 1.199 (0.06) & 1.989 (0.14) & 2.708 (0.16) & 6.781 (0.20) & 9.419 (0.21) & 22.53 (0.22) & 94.57 (0.24)\ 6.52 & 1.438 (0.05) & 2.249 (0.14) & 3.065 (0.16) & 7.553 (0.19) & 10.40 (0.20) & 24.61 (0.22) & 101.3 (0.23)\ 7.19 & 1.748 (0.05) & 2.587 (0.14) & 3.519 (0.16) & 8.614 (0.19) & 11.77 (0.20) & 27.49 (0.22) & 111.9 (0.23)\ 7.92 & 2.135 (0.04) & 3.032 (0.13) & 3.990 (0.15) & 9.739 (0.19) & 13.20 (0.20) & 30.18 (0.22) & 124.3 (0.23)\ [cccccccc]{}\ $z$ & band 9 & band 8 & band 7 & band 6 & band 5 & band 4 & band 3\ & 678 GHz & 404 GHz & 343 GHz & 229 GHz & 202 GHz & 149 GHz & 96.3 GHz\ \ 0.25 & 0.085 (0.19) & 0.355 (0.15) & 0.588 (0.14) & 2.157 (0.13) & 3.036 (0.12) & 8.430 (0.12) & 33.62 (0.11)\ 0.37 & 0.137 (0.19) & 0.566 (0.15) & 0.914 (0.14) & 3.311 (0.13) & 4.741 (0.12) & 12.61 (0.12) & 53.35 (0.12)\ 0.49 & 0.191 (0.19) & 0.768 (0.15) & 1.230 (0.14) & 4.414 (0.13) & 6.398 (0.12) & 16.47 (0.12) & 73.09 (0.12)\ 0.62 & 0.244 (0.20) & 0.939 (0.15) & 1.520 (0.14) & 5.336 (0.13) & 7.918 (0.12) & 19.90 (0.12) & 92.96 (0.11)\ 0.76 & 0.294 (0.21) & 1.075 (0.16) & 1.749 (0.15) & 5.987 (0.13) & 8.951 (0.13) & 22.69 (0.12) & 108.0 (0.11)\ 0.92 & 0.334 (0.21) & 1.182 (0.16) & 1.891 (0.15) & 6.371 (0.13) & 9.545 (0.13) & 24.56 (0.12) & 110.6 (0.11)\ 1.09 & 0.369 (0.22) & 1.241 (0.16) & 1.954 (0.15) & 6.504 (0.13) & 9.679 (0.13) & 25.34 (0.12) & 108.2 (0.12)\ 1.28 & 0.391 (0.23) & 1.250 (0.17) & 1.966 (0.15) & 6.458 (0.13) & 9.430 (0.13) & 25.01 (0.12) & 102.6 (0.11)\ 1.48 & 0.410 (0.23) & 1.228 (0.17) & 1.922 (0.16) & 6.296 (0.13) & 9.067 (0.13) & 23.85 (0.12) & 97.50 (0.11)\ 1.70 & 0.418 (0.24) & 1.187 (0.17) & 1.824 (0.16) & 5.921 (0.14) & 8.579 (0.13) & 22.09 (0.12) & 91.84 (0.11)\ 1.94 & 0.413 (0.25) & 1.109 (0.18) & 1.685 (0.16) & 5.380 (0.14) & 7.893 (0.13) & 19.85 (0.12) & 84.81 (0.11)\ 2.20 & 0.398 (0.26) & 1.013 (0.18) & 1.533 (0.17) & 4.797 (0.14) & 7.029 (0.13) & 17.59 (0.12) & 76.64 (0.11)\ 2.49 & 0.377 (0.26) & 0.920 (0.19) & 1.375 (0.17) & 4.247 (0.14) & 6.131 (0.14) & 15.53 (0.12) & 67.83 (0.11)\ 2.80 & 0.352 (0.27) & 0.822 (0.19) & 1.213 (0.17) & 3.706 (0.14) & 5.347 (0.13) & 13.58 (0.12) & 58.99 (0.11)\ 3.14 & 0.330 (0.27) & 0.746 (0.19) & 1.083 (0.17) & 3.219 (0.14) & 4.663 (0.13) & 11.80 (0.12) & 50.66 (0.11)\ 3.51 & 0.307 (0.27) & 0.672 (0.19) & 0.960 (0.17) & 2.792 (0.14) & 4.017 (0.13) & 10.07 (0.12) & 42.83 (0.11)\ 3.91 & 0.288 (0.27) & 0.608 (0.19) & 0.856 (0.17) & 2.413 (0.14) & 3.434 (0.13) & 8.560 (0.12) & 35.99 (0.11)\ 4.35 & 0.270 (0.27) & 0.542 (0.18) & 0.753 (0.17) & 2.057 (0.14) & 2.917 (0.13) & 7.217 (0.12) & 30.07 (0.10)\ 4.82 & 0.256 (0.27) & 0.482 (0.18) & 0.665 (0.17) & 1.756 (0.13) & 2.487 (0.13) & 6.076 (0.12) & 25.53 (0.10)\ 5.34 & 0.241 (0.27) & 0.427 (0.18) & 0.584 (0.17) & 1.496 (0.13) & 2.091 (0.13) & 5.060 (0.11) & 21.43 (0.10)\ 5.91 & 0.221 (0.27) & 0.370 (0.18) & 0.504 (0.17) & 1.263 (0.13) & 1.753 (0.12) & 4.191 (0.11) & 17.60 (0.10)\ 6.52 & 0.201 (0.27) & 0.317 (0.19) & 0.431 (0.17) & 1.064 (0.13) & 1.463 (0.13) & 3.457 (0.11) & 14.27 (0.10)\ 7.19 & 0.175 (0.28) & 0.262 (0.19) & 0.356 (0.17) & 0.876 (0.13) & 1.190 (0.13) & 2.790 (0.11) & 11.38 (0.10)\ 7.92 & 0.158 (0.29) & 0.229 (0.20) & 0.300 (0.18) & 0.733 (0.14) & 0.993 (0.13) & 2.274 (0.12) & 9.301 (0.10)\ [cccccccc]{}\ $z$ & F777W & F1000W & F1280W & F1500W & F1800W & F2100W & F2550W\ & 7.66$\,{\mu{\rm m}}$ & 9.97$\,{\mu{\rm m}}$ & 12.8$\,{\mu{\rm m}}$ & 15.1$\,{\mu{\rm m}}$ & 18.0$\,{\mu{\rm m}}$ & 20.8$\,{\mu{\rm m}}$ & 25.4$\,{\mu{\rm m}}$\ \ 0.06 & 3.849 (0.23) & 10.15 (0.15) & 6.267 (0.18) & 9.897 (0.13) & 9.247 (0.12) & 9.947 (0.08) & 9.938 (0.06)\ 0.07 & 3.870 (0.21) & 9.508 (0.15) & 6.331 (0.17) & 9.568 (0.13) & 9.245 (0.12) & 9.969 (0.09) & 9.946 (0.07)\ 0.09 & 3.788 (0.22) & 8.741 (0.16) & 6.213 (0.18) & 9.206 (0.14) & 9.191 (0.12) & 9.822 (0.10) & 9.911 (0.07)\ 0.10 & 4.293 (0.22) & 8.415 (0.16) & 6.653 (0.17) & 9.256 (0.13) & 9.576 (0.11) & 9.973 (0.09) & 10.05 (0.07)\ 0.12 & 4.403 (0.24) & 7.280 (0.19) & 6.473 (0.19) & 8.640 (0.15) & 9.316 (0.13) & 9.520 (0.10) & 9.974 (0.08)\ 0.13 & 5.192 (0.21) & 6.863 (0.18) & 7.059 (0.17) & 8.741 (0.14) & 9.815 (0.11) & 9.734 (0.10) & 10.18 (0.07)\ 0.15 & 5.610 (0.22) & 5.802 (0.20) & 6.997 (0.18) & 8.233 (0.15) & 9.722 (0.12) & 9.427 (0.10) & 9.985 (0.07)\ 0.16 & 6.333 (0.21) & 5.074 (0.21) & 7.216 (0.17) & 8.006 (0.15) & 9.944 (0.12) & 9.435 (0.11) & 10.08 (0.08)\ 0.18 & 7.003 (0.21) & 4.469 (0.22) & 7.527 (0.18) & 7.697 (0.16) & 10.10 (0.12) & 9.349 (0.11) & 9.907 (0.07)\ 0.19 & 7.470 (0.19) & 4.015 (0.21) & 7.727 (0.17) & 7.223 (0.16) & 10.31 (0.11) & 9.393 (0.11) & 10.09 (0.07)\ 0.21 & 8.125 (0.20) & 3.799 (0.23) & 8.214 (0.17) & 6.953 (0.18) & 10.48 (0.12) & 9.523 (0.11) & 10.16 (0.08)\ 0.23 & 8.563 (0.20) & 3.651 (0.23) & 8.744 (0.16) & 6.814 (0.18) & 10.56 (0.12) & 9.594 (0.11) & 10.06 (0.08)\ 0.24 & 8.684 (0.19) & 3.492 (0.23) & 9.610 (0.15) & 6.636 (0.18) & 10.33 (0.12) & 9.532 (0.11) & 10.06 (0.08)\ 0.26 & 8.694 (0.20) & 3.364 (0.23) & 10.98 (0.14) & 6.443 (0.18) & 10.08 (0.12) & 9.442 (0.12) & 10.03 (0.08)\ 0.28 & 8.859 (0.20) & 3.370 (0.24) & 12.36 (0.13) & 6.420 (0.18) & 9.884 (0.13) & 9.463 (0.12) & 9.975 (0.08)\ 0.29 & 9.153 (0.20) & 3.425 (0.24) & 12.76 (0.12) & 6.510 (0.18) & 9.566 (0.13) & 9.468 (0.12) & 9.910 (0.09)\ 0.31 & 9.309 (0.19) & 3.408 (0.23) & 12.52 (0.13) & 6.624 (0.18) & 9.136 (0.14) & 9.543 (0.12) & 9.892 (0.09)\ 0.33 & 9.584 (0.19) & 3.432 (0.23) & 11.79 (0.13) & 6.864 (0.18) & 8.754 (0.14) & 9.611 (0.12) & 9.768 (0.09)\ 0.35 & 9.891 (0.19) & 3.473 (0.24) & 10.69 (0.15) & 7.084 (0.18) & 8.404 (0.15) & 9.628 (0.12) & 9.571 (0.10)\ 0.37 & 10.67 (0.19) & 3.700 (0.24) & 9.885 (0.16) & 7.431 (0.18) & 8.283 (0.15) & 9.821 (0.12) & 9.548 (0.10)\ 0.39 & 12.06 (0.18) & 3.937 (0.23) & 9.077 (0.16) & 7.565 (0.17) & 7.962 (0.15) & 9.809 (0.12) & 9.333 (0.10)\ 0.40 & 15.62 (0.16) & 4.396 (0.23) & 8.450 (0.17) & 7.779 (0.17) & 7.724 (0.16) & 9.869 (0.12) & 9.214 (0.11)\ 0.42 & 20.09 (0.14) & 4.994 (0.22) & 7.885 (0.18) & 8.034 (0.17) & 7.530 (0.16) & 9.908 (0.12) & 9.086 (0.11)\ 0.44 & 23.49 (0.12) & 5.598 (0.21) & 7.199 (0.18) & 8.424 (0.16) & 7.195 (0.17) & 9.838 (0.12) & 9.066 (0.11)\ 0.46 & 25.81 (0.11) & 5.963 (0.21) & 6.410 (0.19) & 9.179 (0.16) & 6.702 (0.17) & 9.619 (0.13) & 9.046 (0.11)\ 0.48 & 27.55 (0.10) & 6.221 (0.21) & 5.542 (0.20) & 10.52 (0.14) & 6.199 (0.18) & 9.314 (0.13) & 9.057 (0.11)\ 0.50 & 29.15 (0.09) & 6.468 (0.21) & 4.832 (0.21) & 11.74 (0.13) & 5.977 (0.18) & 9.067 (0.14) & 9.072 (0.12)\ 0.52 & 31.14 (0.08) & 6.794 (0.21) & 4.326 (0.22) & 12.32 (0.13) & 6.040 (0.18) & 8.934 (0.14) & 9.240 (0.12)\ 0.54 & 32.91 (0.08) & 6.964 (0.20) & 3.914 (0.22) & 12.06 (0.13) & 6.138 (0.18) & 8.696 (0.14) & 9.282 (0.12)\ 0.56 & 34.50 (0.07) & 7.103 (0.21) & 3.628 (0.23) & 11.28 (0.14) & 6.315 (0.18) & 8.446 (0.15) & 9.398 (0.12)\ 0.59 & 35.85 (0.07) & 7.259 (0.20) & 3.483 (0.23) & 10.47 (0.15) & 6.516 (0.18) & 8.210 (0.15) & 9.518 (0.11)\ 0.61 & 37.09 (0.06) & 7.403 (0.20) & 3.412 (0.23) & 9.770 (0.15) & 6.724 (0.18) & 7.912 (0.15) & 9.660 (0.11)\ 0.63 & 38.44 (0.06) & 7.512 (0.20) & 3.380 (0.23) & 9.072 (0.16) & 6.924 (0.18) & 7.423 (0.16) & 9.740 (0.11)\ 0.65 & 40.17 (0.06) & 7.724 (0.21) & 3.451 (0.23) & 8.486 (0.17) & 7.254 (0.18) & 7.022 (0.17) & 9.896 (0.11)\ 0.67 & 41.80 (0.05) & 7.902 (0.20) & 3.493 (0.23) & 7.779 (0.18) & 7.592 (0.17) & 6.754 (0.17) & 9.939 (0.12)\ 0.70 & 43.04 (0.05) & 8.144 (0.20) & 3.500 (0.23) & 6.950 (0.19) & 8.054 (0.17) & 6.632 (0.17) & 9.942 (0.12)\ 0.72 & 44.20 (0.05) & 8.832 (0.20) & 3.548 (0.23) & 6.147 (0.20) & 9.048 (0.16) & 6.643 (0.17) & 9.950 (0.12)\ 0.74 & 45.26 (0.04) & 10.59 (0.19) & 3.644 (0.23) & 5.449 (0.20) & 10.63 (0.14) & 6.679 (0.17) & 9.915 (0.12)\ 0.77 & 46.12 (0.04) & 14.02 (0.17) & 3.772 (0.23) & 4.812 (0.21) & 12.01 (0.13) & 6.645 (0.17) & 9.765 (0.12)\ 0.79 & 47.07 (0.04) & 18.20 (0.14) & 4.078 (0.23) & 4.339 (0.22) & 12.83 (0.12) & 6.693 (0.18) & 9.637 (0.12)\ 0.81 & 47.94 (0.04) & 21.21 (0.12) & 4.497 (0.22) & 3.998 (0.22) & 13.10 (0.12) & 6.822 (0.17) & 9.401 (0.13)\ 0.84 & 48.82 (0.04) & 23.39 (0.11) & 4.988 (0.22) & 3.767 (0.23) & 12.99 (0.12) & 7.058 (0.17) & 9.062 (0.13)\ 0.86 & 49.63 (0.04) & 25.18 (0.11) & 5.402 (0.22) & 3.590 (0.23) & 12.50 (0.12) & 7.285 (0.17) & 8.688 (0.14)\ 0.89 & 50.37 (0.04) & 26.93 (0.10) & 5.769 (0.21) & 3.495 (0.23) & 11.67 (0.13) & 7.464 (0.17) & 8.390 (0.14)\ 0.92 & 50.96 (0.04) & 28.86 (0.09) & 6.138 (0.21) & 3.451 (0.23) & 10.51 (0.15) & 7.615 (0.17) & 8.151 (0.15)\ 0.94 & – & 30.67 (0.08) & 6.449 (0.21) & 3.449 (0.23) & 9.363 (0.16) & 7.756 (0.17) & 7.890 (0.15)\ 0.97 & – & 32.37 (0.08) & 6.851 (0.21) & 3.544 (0.23) & 8.623 (0.17) & 8.059 (0.17) & 7.757 (0.15)\ 0.99 & – & 33.63 (0.07) & 7.072 (0.20) & 3.562 (0.23) & 7.941 (0.17) & 8.323 (0.16) & 7.520 (0.16)\ 1.02 & – & 35.15 (0.07) & 7.403 (0.20) & 3.693 (0.23) & 7.451 (0.18) & 8.986 (0.16) & 7.351 (0.16)\ 1.05 & – & 36.91 (0.06) & 7.713 (0.20) & 3.852 (0.23) & 6.880 (0.19) & 10.05 (0.15) & 7.046 (0.17)\ 1.08 & – & 38.82 (0.06) & 8.033 (0.20) & 4.070 (0.23) & 6.217 (0.20) & 11.20 (0.14) & 6.692 (0.17)\ 1.11 & – & 40.68 (0.05) & 8.432 (0.20) & 4.417 (0.23) & 5.536 (0.21) & 11.90 (0.13) & 6.522 (0.18)\ 1.13 & – & 42.14 (0.05) & 8.845 (0.20) & 4.853 (0.22) & 4.927 (0.21) & 11.85 (0.13) & 6.560 (0.18)\ 1.16 & – & 43.44 (0.04) & 9.297 (0.20) & 5.358 (0.22) & 4.454 (0.22) & 11.39 (0.14) & 6.762 (0.18)\ 1.19 & – & 44.60 (0.04) & 9.808 (0.20) & 5.933 (0.22) & 4.125 (0.23) & 10.94 (0.14) & 7.054 (0.17)\ 1.22 & – & 45.76 (0.04) & 10.70 (0.19) & 6.664 (0.21) & 3.989 (0.23) & 10.62 (0.15) & 7.470 (0.17)\ 1.25 & – & 46.80 (0.04) & 12.50 (0.18) & 7.341 (0.21) & 3.928 (0.23) & 10.15 (0.15) & 7.867 (0.17)\ 1.28 & – & 47.81 (0.03) & 16.26 (0.16) & 7.901 (0.20) & 3.949 (0.24) & 9.554 (0.16) & 8.331 (0.16)\ 1.31 & – & 48.89 (0.03) & 21.01 (0.13) & 8.452 (0.20) & 4.057 (0.24) & 8.912 (0.17) & 8.990 (0.16)\ 1.35 & – & 49.92 (0.03) & 24.54 (0.11) & 8.886 (0.20) & 4.159 (0.24) & 8.112 (0.18) & 9.707 (0.15)\ [cccccccc]{}\ $z$ & F777W & F1000W & F1280W & F1500W & F1800W & F2100W & F2550W\ & 7.66$\,{\mu{\rm m}}$ & 9.97$\,{\mu{\rm m}}$ & 12.8$\,{\mu{\rm m}}$ & 15.1$\,{\mu{\rm m}}$ & 18.0$\,{\mu{\rm m}}$ & 20.8$\,{\mu{\rm m}}$ & 25.4$\,{\mu{\rm m}}$\ \ 1.38 & – & 50.98 (0.03) & 27.08 (0.10) & 9.296 (0.20) & 4.306 (0.24) & 7.359 (0.19) & 10.47 (0.14)\ 1.41 & – & 52.01 (0.03) & 29.06 (0.09) & 9.701 (0.20) & 4.501 (0.24) & 6.730 (0.20) & 11.19 (0.14)\ 1.44 & – & 53.00 (0.03) & 30.87 (0.08) & 10.13 (0.19) & 4.778 (0.24) & 6.214 (0.21) & 11.91 (0.13)\ 1.48 & – & 54.01 (0.03) & 32.84 (0.07) & 10.66 (0.19) & 5.248 (0.23) & 5.865 (0.21) & 12.66 (0.12)\ 1.51 & – & – & 34.84 (0.07) & 11.18 (0.19) & 5.946 (0.22) & 5.668 (0.22) & 13.38 (0.12)\ 1.54 & – & – & 36.56 (0.06) & 11.69 (0.19) & 6.897 (0.22) & 5.570 (0.22) & 13.93 (0.11)\ 1.58 & – & – & 38.07 (0.06) & 12.29 (0.19) & 8.108 (0.21) & 5.564 (0.22) & 14.34 (0.11)\ 1.61 & – & – & 39.32 (0.05) & 13.03 (0.18) & 9.081 (0.20) & 5.526 (0.23) & 14.44 (0.11)\ 1.65 & – & – & 40.76 (0.05) & 14.81 (0.17) & 9.582 (0.19) & 5.620 (0.23) & 14.31 (0.11)\ 1.69 & – & – & 42.31 (0.04) & 18.25 (0.16) & 9.630 (0.20) & 5.734 (0.23) & 13.70 (0.12)\ 1.72 & – & – & 43.85 (0.04) & 22.77 (0.13) & 9.822 (0.20) & 5.904 (0.23) & 12.84 (0.13)\ 1.76 & – & – & 45.10 (0.03) & 26.51 (0.10) & 10.08 (0.20) & 6.021 (0.23) & 12.00 (0.14)\ 1.80 & – & – & 46.28 (0.03) & 29.31 (0.09) & 10.49 (0.20) & 6.208 (0.23) & 11.41 (0.15)\ 1.83 & – & – & 47.34 (0.03) & 31.37 (0.08) & 10.92 (0.19) & 6.407 (0.23) & 10.89 (0.16)\ 1.87 & – & – & – & 33.23 (0.07) & 11.38 (0.19) & 6.700 (0.23) & 10.35 (0.17)\ 1.91 & – & – & – & 35.04 (0.06) & 11.78 (0.19) & 7.038 (0.23) & 9.644 (0.18)\ 1.95 & – & – & – & 36.86 (0.06) & 12.22 (0.19) & 7.456 (0.22) & 8.833 (0.19)\ 1.99 & – & – & – & 38.50 (0.05) & 12.82 (0.19) & 7.995 (0.22) & 8.092 (0.20)\ 2.03 & – & – & – & 39.78 (0.05) & 13.47 (0.19) & 8.659 (0.22) & 7.342 (0.21)\ 2.07 & – & – & – & 40.85 (0.04) & 14.38 (0.18) & 9.493 (0.21) & 6.713 (0.22)\ 2.12 & – & – & – & 41.98 (0.04) & 16.29 (0.17) & 10.40 (0.20) & 6.292 (0.23)\ 2.16 & – & – & – & 43.21 (0.04) & 19.60 (0.15) & 11.18 (0.19) & 6.032 (0.23)\ 2.20 & – & – & – & 44.58 (0.03) & 23.68 (0.12) & 11.90 (0.19) & 5.988 (0.24)\ 2.24 & – & – & – & – & 26.92 (0.10) & 12.39 (0.18) & 6.027 (0.24)\ 2.29 & – & – & – & – & 29.31 (0.09) & 12.79 (0.18) & 6.137 (0.24)\ 2.33 & – & – & – & – & 31.12 (0.08) & 13.17 (0.18) & 6.330 (0.23)\ 2.38 & – & – & – & – & 32.80 (0.07) & 13.58 (0.18) & 6.634 (0.23)\ 2.43 & – & – & – & – & 34.44 (0.07) & 13.85 (0.18) & 6.995 (0.23)\ 2.47 & – & – & – & – & 36.05 (0.06) & 14.11 (0.18) & 7.460 (0.22)\ 2.52 & – & – & – & – & 37.51 (0.05) & 14.49 (0.18) & 8.036 (0.22)\ 2.57 & – & – & – & – & 38.74 (0.05) & 14.90 (0.18) & 8.615 (0.21)\ 2.62 & – & – & – & – & 39.87 (0.05) & 15.57 (0.18) & 9.198 (0.21)\ 2.66 & – & – & – & – & 41.08 (0.04) & 16.79 (0.17) & 9.663 (0.20)\ 2.71 & – & – & – & – & 42.44 (0.04) & 19.28 (0.15) & 9.927 (0.20)\ 2.76 & – & – & – & – & 43.74 (0.03) & 22.84 (0.13) & 9.923 (0.20)\ 2.82 & – & – & – & – & – & 26.47 (0.11) & 9.887 (0.21)\ 2.87 & – & – & – & – & – & 29.45 (0.09) & 10.15 (0.20)\ 2.92 & – & – & – & – & – & 31.60 (0.08) & 10.55 (0.20)\ 2.97 & – & – & – & – & – & 33.33 (0.07) & 11.03 (0.20)\ 3.03 & – & – & – & – & – & 34.99 (0.07) & 11.58 (0.19)\ 3.08 & – & – & – & – & – & 36.68 (0.06) & 12.25 (0.19)\ 3.14 & – & – & – & – & – & 38.13 (0.05) & 12.88 (0.19)\ 3.19 & – & – & – & – & – & 39.47 (0.05) & 13.80 (0.18)\ 3.25 & – & – & – & – & – & 40.67 (0.05) & 14.88 (0.18)\ 3.31 & – & – & – & – & – & 41.79 (0.04) & 16.04 (0.17)\ 3.37 & – & – & – & – & – & 43.00 (0.04) & 17.30 (0.16)\ 3.43 & – & – & – & – & – & 44.29 (0.04) & 18.73 (0.15)\ 3.49 & – & – & – & – & – & 45.48 (0.03) & 20.35 (0.14)\ 3.55 & – & – & – & – & – & – & 22.31 (0.13)\ 3.61 & – & – & – & – & – & – & 24.01 (0.12)\ 3.67 & – & – & – & – & – & – & 26.16 (0.11)\ 3.74 & – & – & – & – & – & – & 28.49 (0.10)\ 3.80 & – & – & – & – & – & – & 31.31 (0.08)\ 3.87 & – & – & – & – & – & – & 33.72 (0.07)\ 3.93 & – & – & – & – & – & – & 35.62 (0.06)\ 4.00 & – & – & – & – & – & – & 37.31 (0.06)\ 4.07 & – & – & – & – & – & – & 38.60 (0.05)\ 4.14 & – & – & – & – & – & – & 40.00 (0.05)\ 4.21 & – & – & – & – & – & – & 41.38 (0.04)\ [^1]: The dust library described in this paper is available publicly at <http://cschreib.github.io/s17-irlib/> [^2]: [Tables \[TAB:lconv\_alma\]]{} to \[TAB:lconv\] are only available in electronic form at the CDS via anonymous ftp to [cdsarc.u-strasbg.fr](cdsarc.u-strasbg.fr) `(130.79.128.5)` or via <http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/x/y> [^3]: The last two IRAC channels, at $5.8$ and $8\,{\mu{\rm m}}$, were not used to derive the stellar mass for two reasons. First, at low redshift these bands are contaminated by the dust emission and AGNs, which cannot be taken into account by FAST. Second, while the corresponding images are reasonably deep in the two GOODS fields, the observations in UDS and COSMOS are substantially shallower. Excluding these bands from the fit therefore prevents introducing field-to-field systematics.
--- abstract: \| We present an overview of some of the issues surrounding current models of galaxy formation, highlighting recent insights obtained from cosmological hydrodynamic simulations. Detailed examination of gas accretion processes show a hot mode of gas cooling from near the halo’s virial temperature, and a previously underappreciated cold mode where gas flows in along filaments on dynamical timescales, emitting its energy in line radiation. Cold mode dominates in systems with halo masses slightly smaller than the Milky Way and below, and hence dominates the global accretion during the heydey of galaxy formation. This rapid accretion path enables prompt assembly of massive galaxies in the early universe, and results in $z\sim 4$ galaxy properties in broad agreement with observations, with the most massive galaxies being the most rapid star formers. Massive galaxies today are forming stars at a much reduced rate, a trend called downsizing. The trend of downsizing is naturally reproduced in simulations, owing to a transition from cold mode accretion in the early growth phase to slower hot mode accretion once their halos grow large. However, massive galaxies at the present epoch are still observed to have considerably redder colors than simulations suggest, suggesting that star formation is not sufficiently truncated in models by the transition to hot mode, and that another process not included in current simulations is required to suppress star formation. author: - Romeel Davé - Kristian Finlator - Lars Hernquist - Neal Katz - Dušan Kereš - Casey Papovich - 'David H. Weinberg' title: Building Galaxies with Simulations --- Introduction ============ Galaxy formation is harder than it looks. The classic papers of the 1970’s outlined broad scenarios for how galaxies form in the presence of dark matter halos (e.g. [@whi78]), and together with the advent of hierarchical structure formation, the following conventional wisdom has dominated models for galaxy formation: Galaxies form within biased peaks in large-scale structure, by cooling out from a virial-temperature halo of gas onto a rotationally-supported central disk where stars form. The broad success of this scenario has spawned an increasingly sophisticated industry known as semi-analytic modeling, building on the above framework by introducing and tuning analytic parameters to describe the sundry gastrophysical processes that govern the appearance and evolution of galaxies. While promising, various semi-analytic methods have often found non-unique solutions in parameter space that are able to equivalently match available data, resulting in some ambiguity for the predictions of such models. The ultimate goal of galaxy formation theory is to understand the process thoroughly enough that semi-analytic (or equivalently simple and intuitive) models can successfully and uniquely reproduce the broad range of properties that galaxies exhibit across cosmic time. Unfortunately, it appears that our current level of understanding leaves us far from this goal. To better understand and characterize galaxy formation, it is useful to employ numerical simulations of the growth of structure that dynamically follow the processes required to form baryonic galaxies. Such techniques have been developing at an accelerated pace due to a combination of Moore’s Law, algorithmic improvements in modeling complex baryonic phyics, and the advent of a standard model of cosmology [@spe03]. Since galaxy formation connects scales from sub-parsec star formation to megaparsec structure formation, simulations are unrelentingly limited by dynamic range; hence the continuing exponential increase in hardware computing speed have not only enabled more accurate models, but have also spurred greater levels of insight. Modeling baryons better involves both developing more accurate algorithms to follow gas dynamics, as well as understanding how to model all the physical processes that are relevant to galaxy formation; both areas have seen significant progress in recent years. Finally, the concordance cosmology now favored by a variety of observations allows the evolution of large-scale structure and dark matter halos to be specified precisely (or precisely enough), so that efforts in exploring parameter space can now be expended towards constraining more uncertain baryonic processes rather than the underlying cosmology. These improvements together have propelled simulations into becoming an indispensible tool for studying the physics of galaxy formation and interpreting observations of galaxies. A successful theory of galaxy formation must be able to explain a wide range of multiwavelength observations at a variety of epochs. X-rays observations indicate that galaxies must have a significant impact on their environment, because intracluster gas shows elevated entropy levels and substantial enrichment. Optical surveys such as SDSS and 2dF have mapped the local galaxy distribution in exquisite detail, and deeper probes are now finding substantial populations of galaxies out to $z\sim 6$. Near infrared surveys, a focus of this meeting reflecting the recent detector-driven boom in this field, have enabled detailed stellar mass assembly studies out to $z\sim 3$, along with searches for galaxies into the reionization epoch. Far infrared studies have also received a large boost from [Spitzer]{}, revealing the evolution of the dust enshrouded universe. Sub-millimeter bolometers have detected galaxies at a range of redshifts thanks to fortuitous K-corrections, discovering some of the most actively star forming galaxies in the universe. Radio observations are useful for studying molecular gas out to $z\sim 6$, and the promise of 21 cm mapping of the high-redshift HI distribution looms on the horizon. Forefront work in many of these areas is represented in these proceedings. In order to model the formation of all these objects in the correct number and by the correct time, we have at our disposal the following: An understanding of the growth of the dominant mass component in the universe, an empirically tight yet vague understanding of how gas turns into stars from the Kennicutt (1998) relation, and some really big computers. Can we do it? Not surprisingly, the current answer is no, but in this review I hope to give at least a partial view of where we stand in the process. Galaxy formation can be thought of as having two phases: Gas enters into galaxies from the intergalactic medium (“accretion"), and gas is returned from galaxies, carrying energy and metals, back into the intergalactic medium (“feedback"). Our understanding of both components remains incomplete, significantly more so for the latter. In these proceedings I will discuss recent progress using cosmological hydrodynamic simulations towards understanding how gas gets into galaxies, present some comparisons to observations at high redshifts, mention the origin of galaxy downsizing and implications thereof, and discuss how massive early type galaxies reveal some outstanding issues that simulations must confront in the coming years. How Does Gas Get Into Galaxies? =============================== Recent simulations have provided some quantitative insights into the nature of gas accretion onto galaxies at various masses and epochs. The White & Rees [@whi78] scenario for gas accretion onto galaxies is clearly seen to occur in simulations; hot halos form around massive galaxies with central densities sufficiently high that a cooling flow is established. Indeed, later we will discuss the difficulty in preventing such a cooling flow, as appears to be required by observations. But this form of accretion does not appear to be the whole story, or indeed even the majority of the story. Kereš et al. [@ker05] found that a second mode of gas accretion dominates in halos with masses somewhat smaller than the Milky Way’s and below. To differentiate, they dubbed accretion from cooling of virial temperature gas as “hot mode", and this alternate mode as “cold mode". Cold mode occurs when the cooling time in the halo outskirts is sufficiently short that infall becomes regulated instead by the dynamical time. In cold mode, the infalling gas temperature never approaches the virial temperature of the halo, but instead its gravitational potential energy is radiated away as line emission in primordial and (if present) metal atomic cooling lines. The existence of cold mode accretion is not a new idea. [@bin77] recognized that such a mechanism should exist, and indeed all semi-analytic models contain such a mode that gets activated whenever the cooling time at the virial radius is shorter than the Hubble time (or some other relevant formation timescale). Hydrodynamic simulations have provided additional insights into the quantitative importance of cold mode, in particular demonstrating that owing to processes inherent in a three-dimensional fully dynamical model, cold mode is significantly more important than had been previously realized. To estimate the quantitative importance of cold mode versus hot mode, one can track the temperature history of each gas particle that ends up as a star in a galaxy, find the maximum temperature achieved, and determined the fraction of mass accreted at a given maximum temperature. Such a plot (from Kereš et al.) is shown in Figure \[fig:diff\_acc\_ns\], at a range of redshifts. The bimodality of accretion temperatures is evident, particularly at early times ($z>2$) when the universe is rapidly forming stars. Identifying the lower-temperature hump with cold mode and the higher-temperature one with hot mode, these simulations show a dividing temperature between modes of $\approx 2.5\times 10^5$ K. As can be seen, when star formation is most active in the universe at $z>2$, cold mode is actually the dominant mechanism for gas accretion into galaxies. Insights on the physical mechanism behind cold accretion come from investigating the dependence of the cold mode fraction (i.e. the fraction of star particles whose parent gas particle’s temperature never exceeded $2.5\times 10^5$ K) on galaxy and halo mass. The clearest trends occured with halo mass, in the sense that there appears to be a dividing mass of $\approx 10^{11.4}M_\odot$, independent of redshift (for $z<3$), below which accretion is predominantly cold mode, as shown in Figure \[fig:mass\_ratio\_halo\]. This value exceeds by a factor of a few the mass threshold obtained by simply equating the cooling time to the Hubble time at the virial radius. An independent investigation by [@bir03] using a one-dimensional galaxy formation model including primordial cooling and cosmological accretion found a threshold for stable accretion shock formation of below $10^{11}M_\odot$, comparable to the value from a simple virial radius cooling time argument. Hence it is the three-dimensional geometry of infall, particularly enhancement of density along filaments [@kat02], that is responsible for the increased importance of cold mode accretion. The exact temperature of the division between hot and cold accretion depends on a variety of parameters. The investigations above only included primordial cooling, whereas if metal line cooling is included the dividing mass will presumably rise. The exact value is also sensitive to the ratio of $\Omega_m/\Omega_b$, which was assumed to be 7.5 in Kereš et al.; a more canonical concordance value of 6 would likewise result in a higher dividing mass. It is probable therefore that cold mode is even more important overall than suggested in Kereš et al. In any case, more significant than the precise values are the qualitative notions that (1) Cold mode dominates in smaller halos, and hence is more important at earlier epochs; (2) Cold mode dominates in galaxies with masses up to $M_$ and higher at redshifts when the universe is most rapidly forming stars; and (3) Cold mode arises from the lack of virial shock pressure support in the presence of radiative cooling. An Early Epoch of Star Formation ================================ One consequence of having a rapid mode of accretion being effective at high redshift is that it enables massive galaxies to form their stars quickly in the early universe. Such an early epoch of star formation has been inferred from the observed stellar populations of early type galaxies out to $z\sim 2$. The galaxy population in the early universe therefore presents an interesting test of the overall simulation picture of stellar mass assembly and gas accretion in galaxies. To study this, Finlator et al. [@fin05] examined the population of galaxies in Gadget-2 [@spr03] simulations of $z=4$ galaxies, as would be seen in the color selected B-dropout sample of the Great Observatories Origins Deep Survey (GOODS). By calculating the photometric properties of galaxies using the star formation histories convolved with the Bruzual & Charlot (2003) population synthesis code, we predicted the luminosity function as would be observed using a Lyman break selection technique at $z\approx 4$. We also included dust reddening using a novel prescription calibrated from the local metallicity-extinction relation from SDSS. The results are shown in Figure \[fig:lf\_goods\]. The agreement is quite good around $L_$, but it appears the simulations produce too many brighter galaxies. Recent work from the VVDS Survey [@lef05] that does not use color selection suggests that many bright galaxies are missed by the color criteria typically used for Lyman break selection. Hence it remains to be seen if the excess actually indicates a substantial failing of the model, or merely an improper reckoning of selection effects. Another possibility is that the bright galaxies are actually much dustier than is assumed by the reddening prescription used (so that they would not be seen in UV-selected samples), and in fact that such galaxies should instead be associated with sub-millimeter sources detected by SCUBA. Figure \[fig:biggals\] shows the instantaneous star formation rates of these bright galaxies, which in some cases exceeds 1000 $M_\odot$/yr, similar to that inferred for SCUBA galaxies. The star formation histories of the most rapid star formers are shown in the top panel of Figure \[fig:biggals\], and show a time-averaged rate of $\sim 500 M_\odot/yr$ in the recent past. This means that these galaxies are undergoing only a mild burst of star formation, by a factor of a couple, over their “quiescent" state. This is shown more clearly the lower left panel of Figure \[fig:biggals\], plotting the instantaneous versus time-averaged star formation rates for all galaxies in the G6 run. Visual inspection indicates that these rapid star formers live at the centers of protoclusters, and exhibit a disturbed morphology due to the local high density of galaxies. Overall, the latest observations of galaxy properties at $z\sim 4$ are broadly reproduced in simulations. This is reassuring, both because it means that gas accretion processes are probably being modeled in a reasonable manner, and also because it appears that there are no physical processes that are unaccounted for in these models that have a large impact on the galaxy population. As we will see in the next section, this statement is not true at lower redshifts. What Stops Gas From Getting Into Galaxies? ========================================== While agreement at high redshifts is comforting, it is obtained in part because the data at those epochs is not of the same quality and quantity as is available locally. Galaxy observations in the local universe cover a wide range of objects at a wide range of wavelengths, and there are many outstanding issues that remain to be resolved theoretically. Here we focus on massive early-type systems that should, in principle, be among the easiest and simplest systems to model. As we consider the various observations of massive ellipticals out to $z\sim 1$, one common feature unifies these objects: They are virtually all “red and dead". This feature is best illustrated by the red sequence in a color-magnitude diagram. The uniformly old stellar populations together with a mass-metallicity correlation produce a tight locus in color-magnitude space in which more luminous galaxies are redder. The tightness of this relation is indicative of the lack of recent start formation. It is this feature that is among the most difficult to understand about galaxy formation. In simulations, massive galaxies sit at the bottom of large potential wells, where intergalactic gas is constantly infalling. This is the classic scenario for hot mode accretion, where the central gas density is high enough for cooling times to be rapid, resulting in a substantial amount of gas cooling and subsequent star formation. Shutting this gas supply off requires not only tremendous amounts of feedback energy, but also requires that that energy be distributed quasi-spherically among the gas elements in the cooling flow region. What physical process could be responsible for turning off gas accretion in massive galaxies? The current wisdom is to invoke AGN as the heating mechanism. This idea has a number of positives, in that there is plenty of energy in the central engine, it can be generated with a very small amount of accompanying gas infall or star formation, and there is a tight observed relation between central black hole and bulge masses. Indeed, Kauffmann in these proceedings has assembled a compelling set of observations showing that AGN are responsible for moving galaxies onto the red sequence. Simulations of galaxy mergers with AGN feedback [@dim05] reproduce many observed properties of quasars [@hop05], and succeed in heating gas sufficiently to prevent star formation. All in all, there is accumulating evidence pointing towards AGN shutting off star formation, although much of it remains circumstantial. The main remaining difficulty is trying to understand how energy from the black hole couples to the gas in such a uniform and widespread manner so as to shut off cooling flows. Whatever the mechanism, it is clear that star formation must be truncated, rapidly and at an early epoch, in massive galaxies. One approach is to identify what physical parameter governs when a galaxy will shut off its accretion. Is it the galaxy mass? Is it hot mode accretion? We can roughly test these ideas by turning off star formation in massive systems by post-processing their simulated star formation histories. To do so, we take the G6 Gadget-2 simulation of [@spr03] (described in [@fin05]), identify all galaxies at $z=0$, and compute each galaxy’s $U$ and $V$ absolute magnitudes. The top panel of Figure \[fig:redseq\] shows the resulting simulated color-magnitude diagram. As can be seen, there is no red sequence evident at $U-V\approx 1.5$ as observed for real galaxies, and more massive galaxies do not become progressively redder. This is because of residual star formation in massive systems due to continued cooling and infall. We now take each galaxy, track its growth back in time, and eliminate all star formation once its progenitor’s stellar mass exceed $2\times 10^{10} M_\odot$. The resulting star formation histories are then used to recompute the CMD, shown in the middle panel of Figure \[fig:redseq\]. This approach results in a decent looking red sequence (although it does not have the proper slope, but this may be due to metallicity effects not included here). However, it is now missing another important population: Massive blue spiral galaxies, such as our own Milky Way! This simple stellar mass cut apparently truncates star formation in all galaxies, rather than only the early types. Another idea, suggested by [@ker05] and [@dek04], is that whatever mechanism is responsible for stopping cooling flows preferentially does so by keeping virial temperature gas hot, and not allowing it to cool. Again, we can examine this in our simulation by tracking the temperature history of each particle, and removing all star formation events if the parent gas particle’s temperature had at any time exceeded $2.5\times 10^5$ K, i.e. if it is a hot mode accretion event. The resulting CMD is shown in the bottom panel. This scenario is at best only slightly more successful, as once again bright blue galaxies have mostly been eliminated, and larger galaxies are not particularly redder. It seems that while the hot/cold accretion scenario may play some role in the truncation of star formation in massive systems, it does not appear to be the governing factor. Semi-analytic models of [@som05] and [@cro05] have had some success producing a red sequence, in the former case by truncating star formation when the bulge mass (not total mass) exceeds a certain value, and in the latter case by shutting off gas accretion onto satellite galaxies and including a “radio mode" of feedback tied to AGN growth. These parameterizations serve as useful guides for constraining the phenomena that may be responsible. Birthrates and Downsizing ========================= The formation of a red sequence is part of a larger trend in galaxy formation known as downsizing [@cow96]. Downsizing states that massive galaxies form their star earlier and are comparatively quiescent now, while smaller galaxies have formed (or are forming) most of their stars today. This trend can be summarized using the birthrate parameter, which measure the current star formation rate normalized to the past averaged star formation rate. To study birthrates, we run a Gadget-2 simulation having $32{h^{-1}{\rm Mpc}}$ box size, $2.5{h^{-1}{\rm kpc}}$ softening, and $2\times 256^3$ particles, turning off superwind feedback so we can more cleanly study the basic processes of accretion and star formation as a function of galaxy mass. Figure \[fig:birthrate\] shows the birthrates from this simulation at $z=3$, 2, and 1.5, where the past averaged star formation rate has been computed over a Hubble time. A galaxy with a birthrate parameter of unity, for example, would double its stellar mass in another Hubble time (assuming no merging). Downsizing reflects the trend that massive galaxies have lower birthrates at the present time than less massive galaxies. Such a trend is evident in the top panel of Figure \[fig:birthrate\], at every epoch shown. Another way to characterize this plot is to say that the typical mass of actively star forming galaxies (e.g. those with birthrate parameters exceeding unity) reduces with time. Thus the observed trend of downsizing is in accord with predictions of hierarchical galaxy formation models. The sense of the trend can be easily understood in terms of hot and cold mode accretion: Big galaxies reside in large density fluctuations that collapse first in the early universe, and start forming stars vigorously owing to rapid cold mode accretion. As they grow in size, a virial shock is able to form which slows down the accretion, leading to a higher birthrate in the past than at the present day– i.e., downsizing. Smaller galaxies continue to form stars vigorously even today as they collapse later and continue to be dominated by cold mode accretion. Although the transition from hot mode to cold mode is fairly sharp in halo mass, when viewed as a function of galaxy stellar mass it is much more gradual, which is why no obvious feature exists in the birthrate plot marking the transition between hot and cold mode. Another statement of downsizing is that the formation time of stars is older in more massive galaxies. The bottom panel of Figure \[fig:birthrate\] shows a histogram of the median formation time of stars in galaxies subdivided into three mass bins. This plot is made at $z=1.5$, when the universe is 4.2 Gyr old ($\Omega=0.3$, $H_0=70{{\;{\rm km}\,{\rm s}^{-1}}\;{\rm Mpc}^{-1}}$, $\Lambda$CDM). This shows that large galaxies typically form their stars at earlier times. Here, galaxies with $M_\approx 2\times 10^{11}$ have a median star formation redshift of $z\approx 4$ (1.5 Gyr), while for galaxies with $M_\approx 3\times 10^9 M_\odot$ it is around $z\approx 2.5$. Note that the median star formation time is [inversely]{} correlated with the median halo formation time, since large halos assemble later in hierarchical models. Hence the “anti-hierarchical" appearance of downsizing is in fact a natural outcome of galaxy assembly within hierarchical structure formation models. Unfortunately, the qualitative trend of downsizing is only part of the story. Quantitatively, it appears that cosmological simulations cannot self-consistently reproduce the [strength]{} of downsizing in massive galaxies. While massive galaxies form stars more slowly relative to their stellar mass than smaller ones, they are still forming some stars, whereas observations indicate they don’t form stars at all (though see new GALEX results in these proceedings showing residual star formation in $\sim 20\%$ of present-day ellipticals). So some physics is still missing. A topical implication of downsizing is that massive galaxies must grow substantially by so-called dry merging, i.e. mergers of predominantly stellar systems. The argument for this simply goes that if today’s massive systems form their stars early while their masses are still small, then in order to grow into massive systems we see today they must merge with other large systems who must have also formed their stars early. These mergers will then be predominantly stellar. Indeed, there is growing evidence for ubiquitous dry merging locally (see van Dokkum, these proceedings), and out to $z\sim 0.7$ [@bel05]. In summary, the trend of downsizing is naturally expected in a hierarchical structure formation scenario, arising from the interplay of biased galaxy formation and gas accretion processes. This trend implies that dry merging is an important growth path for massive galaxies. However, the strength of the downsizing in massive galaxies seems to be underestimated in current models, requiring a new form of feedback possibly associated with black hole growth to sufficiently truncate star formation in today’s ellipticals. Conclusions =========== Galaxy formation remains an unsolved problem. Hydrodynamic simulations have enabled many advances in understanding the accretion processes by which galaxies obtain their fuel for growth, including a newfound appreciation for the importance of cold mode accretion. Observations of high-redshift galaxies provide a critical test for models, and current comparisons suggest that simulations are doing a reasonable job producing big galaxies in their youth. A critical test appears to be the formation of red sequence galaxies at low redshifts, something that is currently pointing towards new feedback processes such as AGN heating. The coming years will see continued dramatic advances in observations, and it is incumbent upon theoretical models to keep pace. A solid foundation is in place connecting primordial fluctuations and the objects we see today, but the details remain a work in progress. [ White, S. D. M., & Rees, M. J. 1978, MNRAS, 183, 341 Spergel, D. N. et al. 2003, , 148, 175 Kereš, D., Katz, N., Weinberg, D. H., & Davé, R. 2005, MNRAS, 363, 2 Binney, J. 1977, MNRAS, 215, 483 Birnboim, Y., Dekel, A. 2003, , 345, 349 Katz, N., Kereš, D., Davé, R., Weinberg, D. H. 2002, in proc. “IGM/Galaxy Connection- The Distribution of Baryons at z=0”, eds. 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--- abstract: 'We report significant anisotropies in the projected two-dimensional (2D) spatial distributions of Globular Clusters (GCs) of the giant Virgo elliptical galaxy NGC4649 (M60). Similar features are found in the 2D distribution of low-mass X-ray binaries (LMXBs), both associated with GCs and in the stellar field. Deviations from azimuthal symmetry suggest an arc-like excess of GCs extending north at 4-15 kpc galactocentric radii in the eastern side of major axis of NGC4649. This feature is more prominent for red GCs, but still persists in the 2D distribution of blue GCs. High and low luminosity GCs also show some segregation along this arc, with high-luminosity GCs preferentially located in the southern end and low-luminosity GCs in the northern section of the arc. GC-LMXBs follow the anisotropy of red-GCs, where most of them reside; however, a significant overdensity of (high-luminosity) field LMXBs is present to the south of the GC arc. These results suggest that NGC4649 has experienced mergers and/or multiple accretions of less massive satellite galaxies during its evolution, of which the GCs in the arc may be the fossil remnant. We speculate that the observed anisotropy in the field LMXB spatial distribution indicates that these X-ray binaries may be the remnants of a star formation event connected with the merger, or maybe be ejected from the parent red GCs, if the bulk motion of these clusters is significantly affected by dynamical friction. We also detect a luminosity enhancement in the X-ray source population of the companion spiral galaxy NGC4647. We suggest that these may be younger high mass X-ray binaries formed as a result of the tidal interaction of this galaxy with NGC4649.' author: - 'R. D’Abrusco, G. Fabbiano, S. Mineo, J. Strader, T. Fragos, D.-W. Kim, B. Luo & A. Zezas' title: 'The Two-Dimensional Spatial Distributions of the Globular Clusters and Low-Mass X-ray Binaries of NGC4649.' --- Introduction {#sec:intro} ============ Recent work is bringing forth a picture of complex and diverse Globular Cluster (GC) populations in elliptical galaxies, consistent with several merging episodes of which the different GC populations are the fossil remnant (@strader2011, for M87; @blom2012, for NGC4365). Our work on the elliptical galaxy NGC4261 has shown large-scale spiral-like features in the two-dimensional (2D) GC distribution [@bonfini2012; @dabrusco2013], which would have been unreported in the study of radial distributions. This continuing merging evolution would be expected in the framework of simulations based on the $\Lambda$CDM model of hierarchical galaxy formation [@white1978; @dimatteo2005]. The $\Lambda$CDM model of hierarchical galaxy formation [@white1978] has been successful in reproducing several results of both deep and large sky surveys (e.g., @navarro1994, @dimatteo2005, @navarro2010). However, direct observational validation of continuous galaxy evolution via satellite merging from observations of galaxies in the local universe is only recently appearing in the literature, and has been mostly limited to the Milky Way and Local Group. Large-scale surveys of the stellar population of the Milky Way have shown an increasing complex dynamical environment of satellite dwarf galaxies interacting gravitationally with and being accreted by our galaxy (e.g. the Sloan Digital Sky Survey results of @belokurov2006). The Sagittarius Stream [@ibata1994] is a well-studied example of this phenomenon. A wide-area survey of the M31-M33 region confirms this picture, detecting streams and dwarf galaxies in the halo of M31 and in the region between M31 and M33 [@mcconnachie2009]. Although these subtle details may not be perceivable beyond our Local Group, GCs may provide a marker of these interactions in farther away systems. Observationally, for example, the Sagittarius dwarf/stream is associated with several GCs (e.g., @salinas2012, @forbes2010). Although the parent dwarf galaxy may be disrupted, and therefore hard to be detected against the background of the more luminous dominant galaxy, GC may retain the information of the encounter. This is suggested by simulations - albeit limited to the tidal evolution of GCs in dwarf spheroidal dark matter halos - which show that GCs with combined high mass and high density will survive the encounter [@penarrubia2009]. The giant elliptical galaxy NGC4649 (M60) in the Virgo cluster was recently surveyed with several Chandra and HST exposures, completely covering the area within the D25 ellipse [@devaucouleurs1991] and extending with variable coverage to larger radii. The resulting catalogs of GCs [@strader2012] and Low Mass X-ray Binaries (LMXBs) [@luo2013] give us a unique opportunity to study the properties of these populations and to explore further the GC-LMXB connection [see review, @fabbiano2006], with full spatial coverage. NGC4649 is a relatively isolated galaxy, except for the close neighbor spiral NGC4647, with which NGC4649 may be tidally interacting [@lanz2013]. In the first follow-up paper based on these Chandra and HST data [@mineo2013], we studied the radial distributions of the GCs and LMXBs populations and compared them with that of the optical surface brightness, which is a good proxy of the stellar mass for the old stellar population of NGC4649. The radial distributions of red and blue GCs differ, as generally observed in other galaxies [see review, @brodie2006; @strader2012]. The blue GC radial profile is definitely wider than that of the stellar light. The red GCs profile is steeper, close to that of the stellar light, with a noticeable “dip” in the denser centermost region (r$<\!40^{\prime\prime}\!\sim$ 3 kpc at the NGC4649 distance of 16.5 Mpc [@blakeslee2009]), where tidal disruption of GCs may be more efficient. The LMXBs associated with GCs follow the same radial distributions as their parent GC populations. Roughly three times more of these LMXBs are found in red rather than in blue GCs, as generally observed in early-type galaxies (@kundu2002; @kim2009; @mineo2013; see [@ivanova2012] for a theoretical explanation). The field LMXBs - those without a GC counterpart - follow the stellar light, i.e., the mass distribution of old stars, within the $D_{25}$ ellipse [@kim2004; @gilfanov2004]. [@mineo2013] also report a departure of the LMXB radial distribution from that of the stellar mass at larger galactocentric radii, which could also be associated with more luminous sources. As discussed there, this over-luminosity may point to a younger field LMXB population: evolutionary models of native field LMXB populations have shown that LMXBs become fainter with the increasing age of the parent stellar population [@fragos2013a; @fragos2013b]. Here we complete our study of the spatial distribution of the GC populations in NGC4649, by examining their 2D observed distributions, to explore the presence of irregularities that may point to mergers or satellite accretion by NGC4649. Given the association and possible evolutionary relation between GCs and LMXBs [see @fabbiano2006 and refs. therein], we also extend this study to the LMXB population of NGC4649. In particular we seek to explore if the regularity observed in the radial distributions of these sources still persists in the azimuthal dimension. In Section 2 we describe the catalogs of GCs and LMXBs used in this paper. These data were analyzed with the method described in [@dabrusco2013]. This method is based on the K-Nearest Neighbor (KNN) density estimator [@dressler1980], augmented by Monte Carlo simulations, to estimate the statistical significance of any spatial feature (Section \[sec:method\]). The results are presented in Section \[sec:densityresidual\] and discussed in Section \[sec:discussion\]. We summarize our finding in Section \[sec:conclusions\]. In this paper we adopt a distance of 16.5 Mpc to NGC4649 [@blakeslee2009]. At this distance, an angular separation of 1$^{\prime}$ corresponds to a linear distance of $\sim$4.8 kpc. Data: GCs and LMXBs {#sec:data} =================== The GC and LMXB samples used in this work are the results of the joint Chandra-HST large-area mapping of NGC4649 (PI Fabbiano[^1]). The GC sample is based on the GC catalog extracted by [@strader2012] from six HST ACS pointings of NGC4649. It contains 1603 GCs with photometry in $g$ and $z$ filters, selected by requiring $z\!<\!24$ and $0.5\!<g-z\!<\!2.0$ and discarding all sources with size consistent with the PSF (presumed to be foreground stars). This catalog includes 841 red GCs with $g\!-\!z\!>\!1.18$, and 762 blue GCs with $g\!-\!z\!\leq\!1.18$. The color threshold was derived from [@strader2012] after verifying that significant color substructures are visible in the whole interval of galactocentric distances explored by the catalog of GCs. While the shapes and positions of the red and blue features in the color distribution of GCs (see Figure 6 in [@strader2012]) show a slight dependence on the galactocentric distance within NGC4649, the $g\!-\!z\!=\!1.18$ value allows a clear separation of the two GCs color classes over the whole range of distances. For our analysis we have also considered a split of the sample in luminosity, resulting in 467 high-luminosity GCs with $g\!\leq\!23$, and 1136 low-luminosity GCs with $g\!>\!23$. As discussed later in Section \[sec:method\], small variations of both color and luminosity boundaries do not alter the results of our analysis. In this particular case, the magnitude threshold used to define low- and high-luminosity GCs was chosen only to provide luminosity classes of similar sizes (see Table \[tab:summary\]). Figure \[fig:positions\] (left) shows the spatial distributions of the GCs in the plane of the sky. The HST pointings overlap marginally with the $D_{25}$ elliptical isophote of the neighboring spiral galaxy NGC4647. We have excluded all the GCs within the NGC4647 $D_{25}$ ellipse from the sample used to evaluate the observed density profile of GCs in NGC4649 (see Section \[sec:method\]), but we have kept them in the density maps. ![image](fig1.eps){height="8cm" width="8cm"} ![image](fig2.eps){height="8cm" width="8cm"} The LMXB sample is from the catalog of [@luo2013], where identifications with GCs are based on [@strader2012]. This catalog was derived from six [Chandra]{} ACIS-S3 pointings of NGC4649, reaching a total exposure of 299 ks and yielding a total of 501 X-ray sources with 0.3-8.0 keV luminosities ranging from $9.3\!\times\!10^{36}$ to $5.4\!\times\!10^{39}$ ergs s$^{-1}$. Out of the total 427 LMXBs within the HST combined footprint, 161 are GC-LMXBs and 266 are found in the stellar field of NGC4649 (field LMXBs). Moreover, 74 LMXBs were detected in the region external to the HST footprint (see @mineo2013). Of the X-ray sources detected in the area not overlapped by the HST observations used to extract the catalog of GCs used in this paper, twelve of them can be identified with GCs in the ground-based catalog of [@lee2008]; the other X-ray sources in this area are likely to be mostly background Active Galactic Nuclei (AGNs), based on the contamination estimates of [@luo2013]. To avoid the large uncertainties on the sample of X-ray sources detected at larger radii, in this paper we will derive the density reconstruction of only the GCs and field LMXBs located within the HST footprint. Only $\sim\!20$ background AGNs are expected within the $D_{25}$ elliptical isophote of NGC4649; these are likely to contaminate the field LMXB sample, because the GC identification excludes this contamination in the sample of GC-LMXBs. We have also explored luminosity classes based on a threshold of $L_{\mathrm{X}}\!=\!10^{38}$ erg s$^{-1}$ for the LMXBs. The two luminosity classes comprise 339 and 162 sources respectively. The higher luminosity class may contain a larger proportion of black hole binaries or of younger LMXBs (@fragos2013a [@tzanavaris2013]). The spatial distribution of the LMXBs is shown in Figure \[fig:positions\] (right). The number of members of each class of sources investigated in this paper is reported in Table \[tab:summary\]. [lccccc]{} & $N_{\mathrm{tot}}$ &$N_{\mathrm{red}}$ &$N_{\mathrm{blue}}$ & $N_{\mathrm{HighL}}$ & $N_{\mathrm{LowL}}$\ GCs[^2] & 1603 &841\[642/198\] & 762\[493/269\] & 467 & 1136\ & $N_{\mathrm{tot}}$ &$N_{\mathrm{GCs}}$ &$N_{\mathrm{field}}$ & $N_{\mathrm{HighL}}$ & $N_{\mathrm{LowL}}$\ LMXBs[^3] & 501 & 161\[28/133\] & 266(74) & 162 & 339\ \ \[tab:summary\] ![image](fig3.eps){height="8cm" width="8cm"} ![image](fig4.eps){height="8cm" width="8cm"} Method {#sec:method} ====== We determined the 2D spatial distributions of GCs and LMXBs by applying a method based on the KNN density estimator [@dressler1980]. We used simulations to evaluate the significance of the results. A detailed explanation of our application of the KNN method can be found in [@dabrusco2013]. The assumption behind the KNN method is that the density is locally constant. The uncertainty on the KNN density scales with the square root of $K$, where $K$ is the index of the nearest neighbor used to calculate the density and the fractional accuracy of the method increases with increasing K at the expense of the spatial resolution. For each sample discussed in this paper, we calculated the 2D surface densities for $K$ ranging from 2 to 9, and a regular grid with spacing $\Delta\!(\mathrm{R.A})$ $\sim\!0.005^{\circ}$ ($\sim18^{\prime\prime}$) and $\Delta\!(\mathrm{Dec})$ $\sim\!0.004^{\circ}$ ($\sim\!14^{\prime\prime}$) for the GCs, and $\Delta\!(\mathrm{R.A})$ $\sim\!0.006^{\circ}$ ($\sim\!22^{\prime\prime}$) and $\Delta\!(\mathrm{Dec})$ $\sim\!0.006^{\circ}$ ($\sim\!22^{\prime\prime}$) for the LMXBs. These grids define 2D cells of “pixels”. This choice of grids is justified by the different areas covered by the HST and [Chandra]{} observations and produces similar average number of source per pixel for both GCs and LMXBs. This choice permits to compare meaningfully the statistical reliability of the features determined in density and residual maps for both classes of sources. The density in the boundary pixels was weighted according to the fraction of the pixel within the observed region. To assess the statistical significance of features suggested by the KNN density maps (hereafter “observed” density maps), we employed Monte-Carlo simulations, using the observed radial distributions of each of the classes of GCs and LMXBs investigated, as seeds. The azimuthal distributions were independently extracted from a uniformly random distribution between 0$^{\circ}$ and $360^{\circ}$. A geometrical correction on the expected number of GCs per elliptical annulus was applied to take into account the eccentricity of the galaxy. We performed $N_{\mathrm{Sim}}\!=\!1000$ simulations for each of the samples described in Table \[tab:summary\], and analyzed the results as done with the observed data (see above). Figure \[fig:radialprofiles\] compares observed and simulated mean radial density profiles for the GC and LMXB samples, respectively. To characterize the deviations from the average 2D trends of the GC and LMXB distributions, we calculated maps of residuals by subtracting pixel-by-pixel the mean simulated density map from the corresponding observed density map, and normalizing to the value of the density in the average simulated map. The pixel-by-pixel distributions of the simulated KNN densities are well approximated by Gaussians, simplifying the calculation of the statistical significance of the residuals. To evaluate the latter, for each set of simulated density maps we calculated the fraction of pixels with values above the 90-th percentile of the densities in the observed maps (the “extreme” pixels). Since these extreme pixels in both observed density and residual maps tend to be spatially correlated, we also evaluated the fraction of simulations with at least one group of contiguous extreme pixels (with area equal to 18 pixels) as large as the observed. Since we did not impose any specific geometry to the groups of simulated contiguous extreme pixels, these fractions are upper limits to the fraction expected for a given spatial distribution of residuals. The results are given in Section \[sec:densityresidual\]. As discussed above, the radial distributions of the simulated distributions of sources follow the observed radial distributions, while the azimuthal distribution is uniform in the $[0,\!2\pi]$ interval. If the incompleteness of the observed distribution of sources is a function only of the radial distance from the center without azimuthal dependencies, it will affect in the same way both observed and simulated density maps. Accordingly, in this paper we have not included completeness corrections for either GC and LMXB samples because both populations are affected by incompleteness which only depends on the radial dependence from the center of the NGC4649 galaxy (see @mineo2013 [@luo2013] for GCs and LMXBs incompleteness maps respectively). [lccccccc]{} Density & & & & & &\ &$K\!=\!4$ &$K\!=\!5$ &$K\!=\!6$ &$K\!=\!7$ &$K\!=\!8$ &$K\!=\!9$\ All GCs (red$+$blue) &99.5%(1.8%) &49.5%(1.2%) &6%(0.3%) &0%(0%) &0%(0%) &0%(0%)\ Red GCs &96%(3.1%) &25.9%(2.5%) &2.3%(1.4%) &0%(0.3%) &0%(0%) &0%(0%)\ Blue GCs &100%(4.7%) &60.5%(1.2%) &4%(0%) &1.2%(0%) &0%(0%) &0%(0%)\ High-L GCs &100%(3.4%) &74%(0.9%) &15%(0.2%) &2.8%(0%) &0%(0%) &0%(0%)\ Low-L GCs &99.5%(1.3%) &49%(0.3%) &5.7%(0%) &0%(0%) &0%(0%) &0%(0%)\ All LMXBs &97%(73%) &86%(56.4%) &68%(34.8%) &45.5%(12.9%) &33.5%(8%) &21.5%(0.5%)\ GC-LMXBs &85.5%(54.6%) &64.5%(33.1%)&45.5%(12.2%) &33.5%(3.5%) &25.5%(1.4%) &20.5%(0.3%)\ Field LMXBs &85%(34%) &64%(28.8%) &52%(17.5%) &43%(11.8%) &12%(5.4%) &20%(0%)\ High-L LMXBs &99.5%(87.4%) &94%(76.9%) &88%(65.3%) &71%(43.9%) &60%(19.3%) &53.5%(0.8%)\ Low-L LMXBs &61%(17.9%) &28%(12.8%) &15.5%(3.7%) &7.5%(0.5%) &4%(0%) &3%(0%)\ Residuals & & & & & &\ &$K\!=\!4$ &$K\!=\!5$ &$K\!=\!6$ &$K\!=\!7$ &$K\!=\!8$ &$K\!=\!9$\ All GCs (red$+$blue) &65.2%(10.5%) &8.2%(4.8%) &1.1%(0.7%) &0%(0%) &0%(0%) &0%(0%)\ Red GCs &54.8%(8.9%) &4.2%(1.1%) &0.8%(0.5%) &0%(0.1%) &0%(0%) &0%(0%)\ Blue GCs &71.8%(0.2%) &9.1%(0%) &1.7%(0%) &0%(0.4%) &0%(0%) &0%(0%)\ High-L GCs &89.5%(12.2%) &16%(1.9%) &4.5%(0.6%) &0.5%(0%) &0%(0%) &0%(0%)\ Low-L GCs &88.6%(7.2%) &5.2%(5.4%) &0%(1.1%) &0%(0.4%) &0%(0%) &0%(0%)\ All LMXBs &78.2%(10.5%) &51.7%(4.8%) &22.1%(2.7%) &13.5%(0%) &2.3%(0%) &0%(0%)\ GC-LMXBs &64%(8.9%) &41.7%(3.1%) &30.4%(1.5%) &14.2%(0%) &2.5%(0%) &0.1%(0%)\ Field LMXBs &81%(12.2%) &59.4%(9.3%) &43.9%(4.7%) &28.7%(2.4%) &6.4%(1.3%) &0.5%(0%)\ High-L LMXBs &85.9%(17.2%) &78.4%(12.9%)&52.1%(9.6%) &34.7%(6.5%) &21%(5%) &9.7%(0.8%)\ Low-L LMXBs &57.4%(7.2%) &43.8%(5.4%) &28.5%(3.2%) &13.9%(1.6%) &3.2%(0%) &0.1%(0%)\ \[tab:statistics\] Results of the KNN analysis {#sec:densityresidual} =========================== For each GC and LMXB sample, Table \[tab:statistics\] summarizes the percentages of simulated maps with a number of extreme pixels exceeding that in the observed density and residual maps for values of $K$ ranging from 4 to 9. Given the size of the samples of GCs and LMXBs used in this paper, larger values of $K$, while enhancing possible low-contrast large-scale features, would degrade the spatial resolution of the density and residual maps. The fraction of extreme pixels in the simulated density and residual maps decreases for increasingly larger $K$ values: for GCs, it is zero for $K\!=\!8$; for LMXBs, very small and zero values are obtained for $K\!=\!9$ in the case of residual maps, while significantly positive values occur even for $K\!=\!9$ for the density maps. However, once the spatial clustering of the extreme pixels is taken into account, the LMXBs density maps have negligible fractions of extreme simulated pixels when $K\!=\!9$, making the asymmetries at these scales unlikely to be the results of random statistical fluctuations. In the following we will show and discuss the results obtained for $K\!=\!8$ for all GCs classes and $K\!=\!9$ for LMXBs. Table \[tab:numover\] gives the number of sources of each class located within pixels associated to over-density with significance greater than 1, 2, and 3$\sigma$ for the GC ($K\!=\!8$) and LMXB ($K\!=\!9$) samples. In Table \[tab:numover\], the fraction of sources located in the $>\!1$, $>\!2$ and $>\!3\sigma$ over-density pixels of the map and the number of sources in excess to the expected number (based on the radial profiles shown in Figure \[fig:radialprofiles\]) are shown in parentheses and square brackets, respectively. Density and Residual maps of GCs {#subsec:gcdensity} -------------------------------- ![image](fig5.eps){height="6cm" width="6cm"} ![image](fig6.eps){height="6cm" width="6cm"} ![image](fig7.eps){height="6cm" width="6cm"} Figure \[fig:2dmapsngc4649\_gc\] (left) shows the 2D KNN density map of the spatial distribution of the entire GC sample for $K\!=\!8$. GCs are concentrated in the center of the galaxy, with an over-density elongated along the major axis of NGC4649. Figure \[fig:2dmapsngc4649\_gc\] (center) shows the same density map where pixels are color-coded according to the average color of the GCs located in each pixel. As expected, high-density pixels, which tend to be more centrally located, contain on average redder GCs. ![Positions of the $K\!=\!8$ residuals with significance larger than 1$\sigma$, 2$\sigma$ and 3$\sigma$ obtained from the residual maps derived from the distribution of the whole catalog of GCs in NGC4649. All the negative residuals (blue pixels) have significance between 1 and 2 $\sigma$. Positive (orange) pixels $\!>\!2\sigma$ are indicated with “$-$” sign; $\!>\!3\sigma$ with a “$+$” sign. The alignment of the over-density and under-density pixels along the major and minor axes of the galaxy, respectively, is not an effect of the assumption of the circularly symmetric distribution of sources in the simulations.[]{data-label="fig:res2dmapsngc4649sigmas_gc"}](fig8.eps){height="8cm" width="8cm"} ![Density map of GCs distribution in the plane generated by the radial distance and the $g\!-\!z$ color obtained with the KNN method for $K\!=\!8$. The horizontal black line shows the color value used as threshold to separate red and blue GCs in this paper. The gray points represent the observed positions of the GCs used to reconstruct the density. The arbitrary isodensity contours are added to highlight the position of the main over-densities.[]{data-label="fig:dens2dradialgmzcolor_gc"}](fig9.eps){height="8cm" width="8cm"} A slight excess ($\sim\!1.5\sigma$) of red GCs is found in the over-density structure observed in the N-E quadrant of the galaxy and along the East section of the major axis. Figure \[fig:2dmapsngc4649\_gc\] (right) shows the residual map, which emphasizes over-densities along the major axis, extending into the N-E quadrant. For $K\!=\!8$, the number of all GCs located within the pixels associated to positive residuals is 438 ($\sim\!27\%$), 208 ($13\%$) and 127 ($\sim\!8\%$) for 1$\sigma$, 2$\sigma$ and 3$\sigma$ significance respectively (Table \[tab:numover\]). Figure \[fig:res2dmapsngc4649sigmas\_gc\] shows that the majority of the pixels associated to positive residual values with significance larger than 2$\sigma$ and 3$\sigma$ are located in the E and to the N-E of the main axis of NGC4649. Figure \[fig:dens2dradialgmzcolor\_gc\] shows the $K\!=\!8$ density map in the radial distance vs $g\!-\!z$ plane. The bi-modality of the distribution of GCs in color is clear for $r\!<\!3.4^{\prime}$ which corresponds to the major axis of the $D_{25}$ isophote, in agreement with the finding of [@strader2012]. For $r\!>\!3.4^{\prime}$ the red, metal-rich GCs become less numerous, confirming that the average $g\!-\!z$ color in pixels outside the $D_{25}$ is significantly bluer than inner regions of the galaxy (see Figure \[fig:2dmapsngc4649\_gc\], center). [lccc]{} & $1\sigma$&$2\sigma$&$3\sigma$\ All GCs[^4] &438($27.3\%$)\[190\] &208($13\%$)\[109\] &127($7.9\%$)\[82\]\ Red GCs &388($46.1\%$)\[258\] &184($21.9\%$)\[135\]&111($13.2\%$)\[82\]\ Blue GCs &388($50.9\%$)\[266\] &161($21.1\%$)\[117\]&74($9.7\%$)\[57\]\ High-L GCs &307($65.7\%$)\[240\] &109($23.3\%$)\[87\]&45($9.6\%$)\[37\]\ Low-L GCs &430($37.9\%$)\[243\] &180($15.8\%$)\[111\]&82($7.2\%$)\[55\]\ All LMXBs[^5] &153($30.5\%$)\[74\] &72($14.4\%$)\[40\] &47($9.4\%$)\[31\]\ GC LMXBs &60($37.3\%$)\[30\] &43($26.7\%$)\[25\]&31($19.3\%$)\[20\]\ Field LMXBs &83($31.2\%$)\[38\] &39($14.7\%$)\[20\]&19($7.1\%$)\[12\]\ High-L LMXBs &49($30.2\%$)\[19\] &22($13.6\%$)\[11\]&10($6.2\%$)\[5\]\ Low-L LMXBs &120($35.4\%$)\[57\]&63($18.6\%$)\[37\]&41($12.1\%$)\[29\]\ \[tab:numover\] The $K\!=\!8$ density maps of red and blue GC subsamples do not show significant differences in the location of the main over-densities (upper panels in Figure \[fig:redbluengc4649\_gc\]), except for red GCs being more centrally concentrated (see also Figure \[fig:2dmapsngc4649\_gc\], center). The residuals map derived from the red GCs distribution (lower left panel in Figure \[fig:redbluengc4649\_gc\]) highlights a large contiguous region of positive residuals in the N-E quadrant, with two smaller regions of positive residuals located in the S-W quadrant. The $K\!=\!8$ residual map of blue GCs (lower right panel in Figure \[fig:redbluengc4649\_gc\]) displays a more scattered distribution of positive residuals: a marginal enhancement corresponds to the region occupied by the large N-W positive residual structure seen in the red GCs map, but small significant clusters of positive residual pixels are located also in other regions. ![image](fig10.eps){height="8cm" width="8cm"} ![image](fig11.eps){height="8cm" width="8cm"}\ ![image](fig12.eps){height="8cm" width="8cm"} ![image](fig13.eps){height="8cm" width="8cm"}\ Figure \[fig:highlowlngc4649\_gc\] gives the corresponding results for the high and low luminosity GCs subsamples. The density distributions indicate that both luminosity classes peak in the center of the galaxy, with high-L GCs more centrally concentrated than low-L ones. The residual maps (lower panels in Figure \[fig:highlowlngc4649\_gc\]) emphasize the differences between the two classes. The high-L GCs residual distribution is characterized by a significant cluster of spatially correlated pixels located along the E section of the major axis of the galaxy. Two other smaller structures are visible in the S-W quadrant within and outside of the $D_{25}$ isophote respectively. The residual map of the low-L GCs differs significantly from the high-L map in the E side: a large group of nearby pixels with highly significant positive residuals is located almost entirely in the N-E quadrant. This structures occupies a region devoid of positive high-L pixels and overlaps only partially with the high-L main over-density structure aligned along the major galaxy axis in the E direction. ![image](fig14.eps){height="8cm" width="8cm"} ![image](fig15.eps){height="8cm" width="8cm"}\ ![image](fig16.eps){height="8cm" width="8cm"} ![image](fig17.eps){height="8cm" width="8cm"}\ Following [@dabrusco2013], we have verified that the differences between the spatial distributions of Red and Blue GCs, and high-L and low-L GCs are not the results of a particular choice of the thresholds values by reconstructing the density and residual maps of the two classes for several values of the thresholds, and checking that the qualitative results do not change. For the color classes, we have used ten regularly spaced threshold values around the threshold value $g\!-\!z\!=\!1.18$ in the interval $g\!-\!z\!=\![1, 1.3]$. The results are consistent throughout this range of colors. For values outside this interval, the significance of the residual map for one of the two classes degrades rapidly because of the small number of GCs in that class. The same qualitative conclusions discussed above for the characterization of the spatial distributions of low and high luminosity classes of GCs in NGC4649 hold true using ten regularly spaced values of the magnitude threshold within the interval $g\!=\![22.5, 23.5]$. Other values of the $g$ magnitude threshold have not been used since they would generate luminosity classes too unbalanced to correctly estimate the significance of the results. Density and residual maps of LMXBs {#subsec:lmxbdensity} ---------------------------------- The distribution of LMXBs is centrally concentrated (Figure \[fig:2dmapsngc4649\_lmxb\], left), except for an over-density located within the $D_{25}$ elliptical isophote of the companion spiral galaxy NGC4647 and a less significant enhancement in the S-E quadrant within the $D_{25}$ NGC4649 isophote. The residual map enhances these over-densities (Figure \[fig:2dmapsngc4649\_lmxb\], center, right). A significant association of positive residuals pixels can be seen along the major axis of NGC4649. Two other significant regions composed of multiple spatially clustered positive residual pixels are visible in the N-E and S-W corners of the footprint of the [Chandra]{} observations. Although the footprint of the [@strader2012] does not include the regions of the [Chandra]{} field occupied by these two structures, three X-ray sources in the N-E over-density and four in the S-W density enhancement have been associated by [@luo2013] to GCs from the catalog of [@lee2008]. The remaining X-ray sources in the two structures do not have an optical counterpart that could be identified by our cross-correlation with the Nasa Extragalactic Database[^6] (NED), Simbad Astronomical Database[^7], Sloan Digital Sky Survey[^8] (SDSS) DR10, Two Micron All Sky Survey[^9] (2MASS) and Wide-Field Infrared Survey Explorer[^10] (WISE) ALLWise archives. Nine Ultra-Luminous X-ray sources (ULXs) with $L_{X}\!>\!10^{39}$ ergs s$^{-1}$ were identified by [@luo2013], but they are not located within the boundaries of the N-E and S-W structures. The presence of LMXBs associated to GCs [@lee2008] suggests that these two over-densities could be the remnants of past interactions between NGC4649 and small accreted satellite galaxies. However, considering the background AGN density calculated in [@luo2013], we cannot rule out contribution from background AGNs. ![image](fig18.eps){height="6cm" width="6cm"} ![image](fig19.eps){height="6cm" width="6cm"} ![image](fig20.eps){height="6cm" width="6cm"} Differences are found in the distribution of residuals for field vs GCs-LMXBs and low and high X-ray luminosities LMXBs. These will be discussed in Section \[sec:discussion\]. Comparison with the distribution of diffuse stellar light {#subset:diffuselight} --------------------------------------------------------- We have searched for structures in the distribution of the diffuse stellar light of NGC4649 observed in the HST data used by [@strader2012] that could be spatially overlapping with the over-density structures observed in the spatial distribution of GCs. We have fitted and subtracted elliptical isophotal models to the images of NGC4649 galaxy in both $g$ and $z$ filters, without finding significant residuals that may be indication of merging (cp. with the results discussed by [@tal2009; @janowiecki2010]). Also, we searched for locally enhanced stellar formation in the color map of NGC4649 obtained by combining the HST g and z, using the method based on the distribution of the pixels images in the color-magnitude (CM) diagram [see @degrijs2006]. We do not observe significant differences in the distribution of pixels in the CM diagram derived for different azimuthal regions of the galaxy containing the GC over-density structures. The only feature in the pixel CM diagram is the “blue” cloud of pixels in the region of NGC4649 overlapping with NGC4647, already observed by [@degrijs2006]). While our results do not suggest the existence of significant deviations from a smooth model of the diffuse light and color distributions in NGC4649, more detailed analysis will be necessary to confirm the lack of faint structures in the diffuse light 2D model of this galaxy. In Figure \[fig:stellarmassdensity\_ngc4649\] we plot the isodensity contours of the over-density structures observed in the spatial distribution of LMXBs in NGC4649 over the stellar mass map from [@mineo2013]. This stellar mass map is smooth, with a steep positive gradient towards the core of NGC4649. The only significant enhancement is associated to the central region of the companion galaxy NGC4647, where some granularity, possibly due to the spiral arms, is visible (see Figure 10 in [@mineo2013]). The position of the positive residual structures on the E side of NGC4649 does not correlate with any feature in the stellar mass map, while on the W side the LMXBs over-density contours clearly follows the stellar mass density enhancement associated to the spiral galaxy NGC4647. The different spatial resolutions of the residual maps of the distribution of LMXBs in NGC4649 and of the stellar mass density map do not permit a direct comparison of the shape of the LMXBs over-density contours to the position of the granularities of the map though. ![Stellar mass density map for NGC4649 and NGC4647, as calculated by [@mineo2013]. Isodensity contours from the residual map distribution of all LMXBs from Figure \[fig:2dmapsngc4649\_lmxb\] are also shown.[]{data-label="fig:stellarmassdensity_ngc4649"}](fig21.eps){height="8cm" width="8cm"} Discussion {#sec:discussion} ========== Our analysis of the two-dimensional projected distribution of the GC and LMXB populations of the giant Virgo elliptical NGC4649 (M60) confirms a well-known feature of this GC system, that red GCs are more centrally concentrated than blue GCs, as usually observed in elliptical galaxies (e.g., @brodie2006; see also @mineo2013 for NGC4649), but also finds unexpected significant and complex 2D asymmetries in their projected distributions. The GC population of NGC4649 {#subsec:gcngc4649} ---------------------------- NGC4649 is the third most luminous galaxy in the Virgo Cluster, and resides in a galaxy-dense environment, where gravitational interactions and accretion of satellites may be frequent. Kinematics evidence, suggesting a merging and accretion past for this galaxy, was recently found in studies of Planetary Nebulae (PN) and GCs [@teodorescu2011; @das2011; @coccato2013]. In particular, [@das2011] show that at galactocentric radii larger than 12 kpc, PNs and GCs may belong to separate dynamical systems. A similar conclusion was reached by [@coccato2013], who propose either tidal stripping of GCs for less massive companion galaxies, or a combination of multiple mergers and dwarf galaxy accretion events to explain the observed kinematics. Besides experiencing continuing evolution at the outer radii via interaction with and accretion of companions, NGC4649 itself may be the result of a major dry merger. This is suggested by recent kinematical measurements of the diffuse stellar light that revealed disk-like outer rotation, as it may have stemmed from a massive lenticular galaxy progenitor [@arnold2013]. The anisotropies we find in the 2D distribution of GCs may be a different pointer to this complex evolution. We have found that in NGC4649, the 2D distribution of GCs shows strong positive residuals along the eastern major axis of the galaxy, with a northward arc-like curvature beginning at a galacto-centric radius of $\sim\!4$ kpc and extending out to $\sim\!15$ kpc. This large-scale feature is more prominent in the distribution of red GCs, but can still be seen in the 2D distribution of blue GCs, which is overall noisier. High-luminosity GCs tend to concentrate at the southern end of this feature, while low-luminosity GCs are found in the northern portion. These trends and differences are highlighted in Figure \[fig:sigma\_pixels\_maps\_gc\], which displays the most significant single pixels ($>\!1\sigma,2\sigma,3\sigma$) of each residual distribution. The extreme pixel distributions of red GCs (upper left) and low-luminosity GCs (lower right) are similar, and differ from those of high-L GCs. The reason for this similarity resides in the choice of the particular magnitude threshold used to define luminosity classes (see discussion in Section \[sec:data\] and Table \[tab:summary\]). With the $g\!\leq\!23$ threshold, red and blue GCs comprise $\sim\!76\%$ and $\sim\!65\%$ of low-L GCs, respectively. Using $g\!\leq\!22$ as a threshold instead, the residual maps of low-L and high-L GCs are more similar, although high-L and low-L still are spatially segregated in the N side of the “arc”. However, this magnitude threshold yields 140 high-L GCs and 1463 low-L GCs. ![image](fig22.eps){height="8cm" width="8cm"} ![image](fig23.eps){height="8cm" width="8cm"}\ ![image](fig24.eps){height="8cm" width="8cm"} ![image](fig25.eps){height="8cm" width="8cm"}\ The color and luminosity differences suggest that we may not simply be witnessing the disruption of a satellite dwarf galaxy, and observing the stream of its GC system. Perhaps some additional GC formation resulted from this merger, increasing the population of red, high-metallicity GCs. GC formation during galaxy merging has been suggested as an important mechanism for the assembly of GC systems, especially in the case of large elliptical galaxies [@ashman1992], and is supported by high-resolution simulations of galaxy mergers [@bournaud2008]. Numerous young massive stellar clusters, which may evolve into GCs, are for example seen in the advanced merger remnant NGC7252 [@bastian2013]. This picture is being confirmed by other observational studies. A kinematical study of the GC system of the central galaxy of the Virgo Cluster, M87, shows the presence of significant sub-structures, pointing to active galaxy assembly [@strader2011]. In the Virgo elliptical NGC4365, three separate rotating GC systems have been reported associated with the three color-families of GCs, in addition to a stream-like system [@blom2012]. With the exception of the large-scale arc structure observed in NGC4261 [@dabrusco2013], little evidence is available in the literature about the 2D spatial distribution of GCs in early-type galaxies, let alone about the differences in the spatial distributions of GC color and luminosity classes due to major mergers. [@romanowsky2012] discussed the existence of a phase space shell composed of GCs located in the inner halo of M87, which is possibly due to a significant merger, but there is no evidence of similar structures in the GC spatial distribution in M87. Based on these results, we deduce that the differences in the spatial distribution of high- and low-luminosity GCs in the N section of the NGC4649 arc suggests a more complex history involving both external minor accretion events and internal mechanisms that could influence the evolution of the NGC4649 GC system (i.e., dynamical friction and/or disk shocking). Overall, the emerging picture is a perturbed GC system, suggesting a still evolving NGC4649, perhaps caught in the moment of accreting some smaller satellite galaxy. This interpretation is not straightforward because other supporting evidence is still lacking other than the overall kinematics of PN, GCs and the stellar component of NGC4649 [@das2011; @arnold2013]. We have searched for the kinematical signature of gravitational interactions and accretion of satellites undergoing in NGC4649, by comparing the GC over-densities with the radial velocities for the sample of 121 GCs discussed by [@lee2008] with spectroscopic observations. We do not find any obvious correlation. Moreover, [@pierce2006] obtained spectra of 38 GCs in NGC4649 (all included in our sample) and did not find evidence that red GCs are significantly younger than blue GCs. However, the GC samples of [@lee2008] and [@pierce2006] are both sparse and do not provide the optimal coverage of the over-density feature. The similarity of the radial density profiles of red and blue GCs in NGC4649 (per Figure \[fig:radialprofiles\], left) would be in principle consistent with NGC4649 being the result of a major dry merger [see @arnold2013; @shin2009]. However, our results suggest that the flattening of the radial density profiles of both color classes, instead than being the result of a global process with no azimuthal dependence, could be driven by the existence of major features in the spatial distribution of GCs. In the case of NGC4649, the arc extending from the center of the galaxy to the $D_{25}$ isophote and more significant for red GCs than for blue GCs, could be the responsible for the overall similarity of the radial density profiles of the two GC color classes. The LMXB population of NGC4649 {#subsec:lmxbngc4649} ------------------------------ It has been recognized since their early discovery in the Milky Way that dynamical formation of LMXBs in GCs is highly efficient [@clark1975]; some of these binaries could then be either kicked out from the parent cluster or be left in the stellar field after cluster disruption [@grindlay1984] [see also @kundu2002 for early-type galaxies]. Therefore at least some field LMXBs could have been formed in GCs. However, LMXBs can also evolve from native binary systems in the stellar field [see review from @verbunt1995]. The effectiveness of either or both formation channels for the LMXB populations of both the Milky Way and external galaxies is still a matter of debate. Based on statistical considerations on the properties of the LMXB populations detected with Chandra in early-type galaxies, there is some indication that field evolution is a viable formation channel [@juett2005; @irwin2005; @kim2009], and that the LMXBs detected in the stellar field may be of mixed origin [@kim2009; @mineo2013]. As is the case for GCs, LMXBs are also objects that can be detected individually in elliptical galaxies with the resolution of Chandra [see @fabbiano2006]. Therefore, their spatial distribution may provide some constraints on their origin. In particular, native field LMXBs should - within statistics - trace the stellar surface brightness distribution of their parent galaxy. Although there has been some debate on the radial profiles of LMXBs and their comparison with those of the stellar surface brightness and of GCs [@kim2006; @kundu2007], in NGC4649 there is clear evidence [@mineo2013] that GC-LMXBs in red and blue GCs follow the same radial profiles as their parent GC populations, while field LMXBs in NGC4649 are radially distributed like the stellar surface brightness. However, within the limited sample statistics, LMXBs in red GCs are also distributed as the stellar light, with the exception of a marked lack of both red GCs and associated LMXBs in the centermost region. Therefore these results are still consistent with a mixed origin for the field LMXB population. ![image](fig26.eps){height="8cm" width="8cm"} ![image](fig27.eps){height="8cm" width="8cm"}\ ![image](fig28.eps){height="8cm" width="8cm"} ![image](fig29.eps){height="8cm" width="8cm"}\ We have found (Section \[subsec:lmxbdensity\]) that the 2D distribution of LMXBs shows a positive excess over a simple radial distribution with no azimuthal dependences, in the region near to the eastern major axis of NGC4649, where the excess feature of GCs is also observed (see Figure \[fig:sigma\_pixels\_maps\_lmxb\], which displays the higher significance individual pixels for the residual distribution of different classes of LMXBs). Not surprisingly, the excess residuals of the distribution of LMXBs in GC, which are overwhelmingly associated with red GCs [@luo2013; @mineo2013], resemble those of the red GCs. However, the 2D distribution of field LMXBs also shows large-scale anisotropy, which would not be expected if these sources were solely associated with native field binaries. The excess residuals appear to be adjacent to, but not overlapping, those seen in the 2D distribution of GC-LMXBs. This excess of field LMXBs is dominated by high luminosity LMXBs ($L_{X}\!>\!10^{38} \mathrm{erg\ s}^{-1}$). A second excess clump of these high-luminosity LMXBs is instead associated with the eastern end of the major axis; this over-density of high-luminosity LMXBs is composed by equal number of GC-LMXBs and field LMXBs (see right plot in Figure \[fig:positions\]). Positive residuals in the low luminosity LMXB 2D distribution are found extending along the eastern major axis, mostly associated to the positions of field LMBXs (compare right plot in Figure \[fig:positions\]); given their spatial distribution, though, these are also likely to have been originated in GCs. Of the two over-densities visible in the residual map generated with all the LMXBs (Figure \[fig:2dmapsngc4649\_lmxb\]) in the N-E and S-W corners of the [Chandra]{} field, only the S-W structure can be seen in the residual map generated by low-L$_{X}$ LMXBs; neither is see in the high-L$_{X}$ residuals (Figure \[fig:sigma\_pixels\_maps\_lmxb\]). The differences in the significance of these two LMXB over-density structures are due to the higher values of the radial density profile of high-L LMXBs at larger radii compared to that of low-L at large radii (see Figure \[fig:radialprofiles\]), while both over-densities are composed of similar fractions of high and low luminosity X-ray sources. The N-E structure contains a total of 23 X-ray sources (12 low-luminosity and 11 high-luminosity), while the S-W region contains 45 total sources (27 low-luminosity and 18 high-luminosity). The presence of this high-L excess of field sources, to the south of the GC (and GC-LMXB) excess feature, suggests that these sources cannot have been formed in GCs and ejected in the field, unless the streaming motion of the GCs and of the ejected LMXBs are subject to different accelerations. The escape velocity for a typical GC can be as low as $\sim$30 km s$^{-1}$ [@morscher2013], where the range of line of sight velocities observed for GCs in massive elliptical galaxies extends from few to few thousands km s$^{-1}$ [@strader2011]. This suggests that lacking other forces, the LMXBs ejected should settle into a cloud surrounding their parent GCs. However, dynamical friction may cause the observed positional offset between the over-density of red GCs observed along the E side of the major axis (upper right plot in Figure \[fig:sigma\_pixels\_maps\_lmxb\]) and the field LMXBs structure observed along the $D_{25}$ isophote south of the main GCs over-density (upper left plot in Figure \[fig:sigma\_pixels\_maps\_lmxb\]). We assume a standard mass for the LMXBs-hosting GCs $M_{GC}\!=\!10^{7}M_{\sun}$ consistent with their average absolute magnitudes [@strader2012]. We further assume a circular velocity $v_{circ}\!\sim\!450$ km s$^{-1}$ based on the results from [@humphrey2008]. Using as starting radial distance half the length of the major axis of the galaxy equivalent to $r\!=\!15$ kpc, we estimate the time required to recreate the angular and radial separation of the field LMXBs and GCs LMXBs over-densities with equation (7-26) of [@binney2008], that evaluates the time required for a GC moving on a circular orbit of radius $r_{i}$ with velocity $v_{c}$ to reach the center of the galaxy ($\ln{\Lambda}\!=\!10$ is assumed, based on typical values of the GC physical parameters): $$t_{\mathrm{fric}}\!=\!\frac{1.17}{\ln{\Lambda}}\frac{r_{i}^{2}v_{c}}{GM}$$ We obtain a time scale of $\sim\!5$ Gyr. This timespan is shorter than the average GC age of $\sim\!$10 Gyr estimated by [@pierce2006], and within the range of single GCs measurements by these authors. Accurate kinematic measurements for GC population in NGC4649 would be needed to test this hypothesis. Also, we point out that our simple evaluation of the effects of the dynamical friction should only be considered indicative. A full set of dynamical simulations of the evolution of the GC system of NGC4649 as the results of satellite accretion and mergers is required. Alternatively, one could speculate that the merging event originating the GC over-density may have also resulted in a compressed gaseous stream from the disrupted satellite galaxy, with subsequent star and X-ray binary formation. The higher luminosity of the over-dense field X-ray sources would be consistent with a younger age than the overall LMXB population (@fragos2013a [@fragos2013b]). The X-ray binary population of NGC4647 {#subsec:lmxbngc4647} -------------------------------------- NGC4647 is a spiral galaxy in Virgo that can be seen to the N-W of NGC4649. Based on its aspect and photometric properties, [@lanz2013] conclude that it is tidally interacting with NGC4649. There is a concentration of X-ray sources (positive residuals) associated with this spiral galaxy. The high-luminosity X-ray sources concentrate in the southern half of NGC4647. This is the part of the galaxy showing a prominent asymmetric arm suggesting interaction with NGC4649, and exhibiting more intense star formation [@lanz2013; @mineo2013]. These luminous sources are likely younger HMXBs, associated with the ongoing intense star formation [e.g. @mineo2012]. Our results agree with the overall picture of tidal interaction between NGC4647 and NGC4649. NGC4647 may well be the next accreted satellite. Summary and Conclusions {#sec:conclusions} ======================= Our analysis of the 2D projected distribution of the GC and LMXB populations of the giant Virgo elliptical NGC4649 (M60) has led to the following results: - There are significant 2D asymmetries in the projected distributions of both GCs and LMXBs. - Red GCs are more centrally concentrated than blue GCs, as usually observed in elliptical galaxies (e.g., @brodie2006; see also @mineo2013 for NGC4649). The same is also true for high luminosity GCs (which tend to be red; see @fabbiano2006). - The 2D distribution of GCs shows strong positive residuals along the major axis of NGC4649, especially on the E side, with an arc-like curvature towards the N, beginning at a galactocentric radius of $\sim\!4$ kpc and extending out to $\sim\!15$ kpc. This large-scale feature is more prominent in the distribution of red GCs, but can still be seen in the 2D distribution of blue GC, which is overall noisier. High-luminosity GCs tend to concentrate at the S end of the “arc”, while low-luminosity GCs are found in the Northern portion. - The 2D distribution of LMXBs also shows positive excess in the E major axis side of NGC4649. - The anisotropies of the distribution of LMXBs in GC, which are overwhelmingly associated with red GCs, resembles that of red GCs. However, field LMXBs also show large-scale anisotropies, with positive excess to the S of the red GC feature; this excess is composed of high luminosity LMXB ($L_{X}\!>\!10^{38} \mathrm{erg\ s}^{-1}$), which are also associated with the E end of the major axis (these are partially associated to the position of GC-LMXBs). Positive excess in low luminosity LMXB, mostly composed of field LMXBs, is found extending along the E major axis; given their position, we speculate that these sources are likely to have formed in GCs and subsequently ejected in the field. - There is a definite concentration of LMXBs (positive residuals) associated with the spiral galaxy NGC4647 in the N-W side of NGC4649. The higher X-ray luminosity portion of these sources tends to concentrate in the southern half of NGC4647. These results have important implications for the evolution of NGC4649. In particular, they suggest accretion and merging of satellite galaxies, of which the GC anisotropic distribution is the fossil remnant. LMXB over-densities are found associated with those of red GCs, consistent with dynamic LMXB formation in these clusters. Surprisingly, there is also an over-density in the distribution of field LMXBs, to the south of the red GC over-density. This over density suggests that these LMXBs should be somewhat connected with the GC overdensity. The displacement, however, implies either (1) GC formation followed by ejection and dynamical friction drag on the parent GCs, or (2) field formation stimulated by compression of the ISM during a merging event. We also clearly detect the over-density of X-ray sources connected with the X-Ray Binaries (XRB) population of the companion spiral galaxy NGC4647. We note that the XRBs sources located in the southern half of the NGC4647, where the star formation rate appears to be more intense (see [@mineo2013]), are significantly more luminous that the X-ray sources observed in northern region of NGC4647, with $L_{X}\!>\!10^{38} \mathrm{erg\ s}^{-1}$. This result may suggest a younger XRB population. We speculate that this asymmetry may indicate the beginning of the tidal interaction of NGC4647 with NGC4649. Our analysis and the kinematics of NGC4649 [@das2011; @coccato2013; @arnold2013] suggest a complex assembly history for this galaxy, including one major merger and a sequence of satellite accretion events, with possibly continuous tidal stripping of GCs and PNs from nearby low-luminosity galaxies. Explaining the origin of the over-density structures in the spatial distribution of GCs and of the luminosity and color segregations within such features in NGC4649 will require both detailed numerical simulations with a realistic model of the galaxy potential and extensive campaign for the spectroscopic observation required to reconstruct accurately the kinematics of the whole GC population. The detailed characterization of the 2D spatial distribution of GCs and LMXBs in NGC4649 that we have achieved will provide a new benchmark for future work aimed at the understanding of the dynamical evolution of this interesting system. We thank Jean Brodie for comments that have helped to improve the paper. GF thanks the Aspen Center for Physics for hospitality (NSF grant 1066293). TF acknowledges support from the CfA and the ITC prize fellowship programs. SM acknowledges funding from the STFC grant ST/K000861/1. This work was partially supported by the Chandra GO grant GO1-12110X and the associated HST grant GO-12369.01A, and the [Chandra]{} X-ray Center (CXC), which is operated by the Smithsonian Astrophysical Observatory (SAO) under NASA contract NAS8-03060. Angelini, L., Loewenstein, M., & Mushotzky, R. F. 2001, , 557, L35 Arnold, J. A., Romanowsky, A. J., Brodie, J. P., et al. 2013, arXiv:1310.2607 Ashman, K. M., & Zepf, S. E. 1992, , 384, 50 Bastian, N., Schweizer, F., Goudfrooij, P., Larsen, S. S., & Kissler-Patig, M. 2013, , 431, 1252 Belokurov, V., Zucker, D. B., Evans, N. W., et al. 2006, , 647, L111 Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition, by James Binney and Scott Tremaine. 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[^2]: In square brackets, the number of low-luminosity and high-luminosity GCs for red and blue classes, respectively. [^3]: In square brackets, the number of LMXBs associated to blue and red GCs, respectively. In parenthesis, the number of X-ray sources located outside the HST footprint. [^4]: Maps obtained for $K\!=\!8$. [^5]: Maps obtained for $K\!=\!9$. [^6]: http://ned.ipac.caltech.edu/ [^7]: http://simbad.u-strasbg.fr/simbad/ [^8]: http://www.sdss3.org/dr10/ [^9]: http://www.ipac.caltech.edu/2mass/ [^10]: http://wise2.ipac.caltech.edu/docs/release/allwise/
--- abstract: 'We look at price formation in a retail setting, that is, companies set prices, and consumers either accept prices or go someplace else. In contrast to most other models in this context, we use a two-dimensional spatial structure for information transmission, that is, consumers can only learn from nearest neighbors. Many aspects of this can be understood in terms of generalized evolutionary dynamics. In consequence, we first look at spatial competition and cluster formation without price. This leads to establishement size distributions, which we compare to reality. After some theoretical considerations, which at least heuristically explain our simulation results, we finally return to price formation, where we demonstrate that our simple model with nearly no organized planning or rationality on the part of any of the agents indeed leads to an economically plausible price.' address: - 'Dept. of Computer Science, ETH Zürich, Switzerland' - 'Cowles Foundation for Research in Economics, Yale University, New Haven, Connecticut' - 'Niels Bohr Institute, University of Copenhagen, Denmark' author: - 'Kai Nagel,' - 'Martin Shubik,' - 'Maya Paczuski,' - Per Bak bibliography: - 'ref.bib' - 'kai.bib' - 'xref.bib' title: Spatial competition and price formation --- econophysics; economics; simulation; markets Introduction ============ There are several basic concepts which lie at the heart of economic theory. They are the “economic atom” which is usually considered to be the individual, profits, money, price and markets and the more complex organism the firm. Much of economic theory is based on utility maximizing individuals and profit maximizing firms. The concept of a utility function attributes to individuals a considerable amount of sophistication. The proof of its existence poses many difficult problems in observation and measurement. In this study of market and price formation we consider simplistic social individuals who must buy to eat and who look for where to shop for the best price. In this foray into dynamics we opt for a simple model of consumer price formation. Our firms are concerned with survival rather than a sophisticated profit maximization. Yet we relate these simple behaviors to the more conventional and complex ones. A natural way to approach the economic physics of monopolistic competition is to introduce space explicitly. For much of economic analysis of competition space and information are critical factors. The basic aspects of markets involve an intermix of factors, such as transportation costs and delivery times which depend explicitly on physical space. But for pure information, physical distance is less important than direct connection. For questions concerning the growth of market areas, the spatial representation is appropriate. Consideration of space is sufficient to provide a justification of Chamberlin’s model of monopolistic competition as is evident from the work of Hotelling [@Hotelling]. Furthermore it is reasonably natural to consider space on a grid with some form of minimal distance. Many of the instabilities found in economic models such as the Bertrand model are not present with an appropriate grid. When investigating these topics, one quickly finds that many aspects of price formation can be understood in terms of generalized evolutionary dynamics. In consequence, our first models in this paper study spatial competition and cluster formation without the generation of price (Sec. \[sec:spatial\]). This generates cluster size distributions, which can be compared to real world data. We spend some time investigating theoretical models which can explain our simulation data (Sec. \[sec:theo\]). We then, finally, move on to price formation, where we implement the price dynamics “on top” of the already analyzed spatial competition models (Sec. \[sec:prices\]). The paper is concluded by a discussion and a summary. Related work ============ The model is an open one related to the partial equilibrium models of much of micro-economics. In particular money and its acceptance in trade is taken as a primitive concept. There is a literature on the acceptance of money both in a static equilibrium context (see for example [@Kiyotaki:Wright:money]) and in a “bootstrap” or dynamic context (see for example [@Bak:etc:money; @Sneppen:money]). These are extremely simple closed models of the economy where each individual is both a buyer and seller. Eventually we would like to construct a reasonable model where the acceptance of money, the emergence of competitive price and the emergence of market structure all arise from the system dynamics. This will call for an appropriate combination of the features of the model presented here with the closed models noted above. We do not pursue this further here. Instead by taking the acceptance of money as given our observations are confined to the emergence of markets and the nature of price. The static economic theories of monopoly and mass homogeneous competitive equilibrium provide natural upper and lower benchmarks to gauge market behavior. The intermediate zone between $n=2$ and very many is covered in the economic literature by various oligopoly models, of which those of Cournot [@Cournot:book], Bertrand [@Bertrand] and Chamberlin [@Chamberlin:book] serve as exemplars. The Chamberlin model unlike the earlier models stresses that all firms trade in differentiated goods. They are all in part differentiated or partially monopolistic. When one considers both information and physical location this is a considerable step towards greater realism. Other work on evolutionary or behavioral learning in price formation are Refs. [@Hehenkamp:etc:note; @Hehenkamp:sluggish; @Brenner:prices]. Spatial competition {#sec:spatial} =================== As mentioned in the introduction, we will start with spatial models without price. We will add price dynamics later. Basic spatial model (domain coarsening) {#sec:basic} --------------------------------------- We use a 2-dimensional $N = L \times L$ grid with periodic boundary conditions. Sites are numbered $i=1..N$. Each site belongs to a cluster, denoted by $c(i)$. Initially, each site belongs to “itself”, that is, $c(i) = i$, and thus cluster numbers also go from $1$ to $N$. The dynamics is such that in each time step we randomly pick a cluster, delete it, and the corresponding sites are taken over by neighboring clusters. Since the details, in particular with respect to the time scaling, make a difference, we give a more technical version of the model. In each time step, we first select a cluster for deletion by randomly picking a number $C$ between $1$ and $N$. All sites belonging to the cluster (i.e. $c(i) = C$) are marked as “dead”. We then let adjoining clusters grow into the “dead” area. Because of the interpretation later in the paper, in our model the “dead” sites play the active role. In parallel, they all pick randomly one of their four nearest neighbors. If that neighbor is not dead (i.e. belongs to a cluster), then the previously dead site will join that cluster. This step is repeated over and over, until no dead sites are left. Only then, time is advanced and the next cluster is selected for deletion. In physics this is called a domain coarsening scheme (e.g. [@Flyvbjerg:foams]): Clusters are selected and deleted, and their area is taken over by the neighbors. This happens with a total separation of time scales, that is, we do not pick another cluster for deletion before the distribution of the last deleted cluster has finished. Fig. \[fig:basic-snapshot\] shows an example. We will call a cluster of size larger than zero “active”. ![Snapshot of basic domain coarsening process. LEFT: The black space comes from a cluster that has just been deleted. RIGHT: The black space is being taken over by the neighbors. — Colors/grayscales are used to help the eye; clusters which have the same color/grayscale are still different clusters. System size $256^2$.[]{data-label="fig:basic-snapshot"}](basic-1-gz.eps "fig:"){width="40.00000%"} ![Snapshot of basic domain coarsening process. LEFT: The black space comes from a cluster that has just been deleted. RIGHT: The black space is being taken over by the neighbors. — Colors/grayscales are used to help the eye; clusters which have the same color/grayscale are still different clusters. System size $256^2$.[]{data-label="fig:basic-snapshot"}](basic-2-gz.eps "fig:"){height="40.00000%"} Note that it is possible to pick a cluster that has already been deleted. In that case, nothing happens except that the clock advances by one. This implies that there are two reasonable definitions of time: - Natural time $t$: This is the definition that we have used above. In each time step, the probability of any given cluster to be picked for deletion is a constant $1/N$, where $N = L^2$ is the system size. Note that it is possible to pick a cluster of size zero, which means that nothing happens except that time advances by one. - Cluster time $\tilde t$: An alternative is to chose between the active clusters only. Then, in each time step, the probability of any given cluster to be picked for deletion is $1/n(\tilde t)$, where $n(\tilde t) = N - \tilde t$ is the number of remaining active clusters in the system. Although the dynamics can be described more naturally in cluster time, we prefer natural time because it is closer to our economics interpretation. At any particular time step, there is a typical cluster size. In fact, in cluster time, since there are $n(\tilde t) = N - \tilde t$ clusters, the average cluster size as a function of cluster time is $\overline S(\tilde t) = N / n(\tilde t) = 1 / (1 - \tilde t/N)$. However, if one averages over all time steps, we find a scaling law. In cluster time, it is numerically close to $ \label{domain:coarsening} \tilde n(s) \sim s^{-3} \hbox{ or } \tilde n(>\!s) \sim s^{-2} \ , $ where $s$ is the cluster size, $n(s)$ is the number of clusters of size $s$, and $n(>\!s)$ is the number of clusters with size larger than $s$.[^1] In natural time, the large clusters have more weight since time moves more slowly near the end of the coarsening process. The result is again a scaling law (Fig. \[fig:basic-scaling\] (left)), but with exponents increased by one: $$\label{domain:coarsening:natural} n(s) \sim s^{-2} \hbox{ or } n(>\!s) \sim s^{-1} \ .$$ It is important to note that this is not a steady state result. The result emerges when averaging over the whole time evolution, starting with $N$ clusters of size one and ending with one cluster of size $N$. Random injection with space {#sec:rnd} --------------------------- In view of evolution, for example in economics or in biology, it is realistic to inject new small clusters. A possibility is to inject them at random positions. So in each time step, before the cluster deletion described above, in addition with probability $p_{\it inj}$ we pick one random site $i$ and inject a cluster of size one at $i$. That is, we set $c(i)$ to $i$. This is followed by the usual cluster deletion. It will be explained in more detail below what this means in terms of system-wide injection and deletion rates. This algorithm maintains the total separation of time scales between the cluster deletion (slow time scale) and cluster growth (fast time scale). That is, no other cluster will be deleted as long as there are still “dead” sites in the system. Note that the definition of time in this section corresponds to natural time. The probability that the injected cluster is really new is reduced by the probability to select a cluster that is already active. The probability of selecting an already active cluster is $n(t)/N$, where $n(t)$ is again the number of active clusters. In consequence, the effective injection rate is $$r_{\it inj,eff} = p_{\it inj} - n(t)/N \ .$$ Similarly, the effective cluster deletion depends on the probability of picking an active cluster, which is $n(t)/N$. In consequence, the effective deletion rate is $$r_{\it del,eff} = n(t)/N \ .$$ This means that, in the steady state, there is a balance of injection and deletion, $n_/N = p_{\it inj} - n_ / N$, and thus the steady state average cluster number is $$n_* = N \, p_{\it inj} / 2 \ .$$ In consequence, the steady state average cluster size is $$s_* = N/n_* = 2 / p_{\it inj} \ .$$ The cluster size distribution for the model of this section is numerically close to a log-normal distribution, see Fig. \[fig:basic-scaling\] (right). Indeed, the position of the distribution moves with $1/p_{\it inj}$ (not shown). In contrast to Sec. \[sec:basic\], this is now a steady state result. Injection on a line ------------------- It is maybe intuitively clear that the injection mechanism of the model described in Sec. \[sec:rnd\] destroys the scaling law from the basic model without injection (Sec. \[sec:basic\]), since injection at random positions introduces a typical spatial scale. One injection process that actually generates steady-state scaling is injection along a 1-d line. Instead of the random injection of Sec. \[sec:rnd\], we now permanently set $$c(i) = i$$ for all sites along a line. Fig. \[fig:snap:line\] (left) shows a snapshot of this situation. In this case, we numerically find a stationary cluster size distribution (Fig. \[fig:snap:line\] (right)) with $$n(s) \sim s^{-1.5} \hbox{ or } n(>\!s) \sim s^{-0.5} \ .$$ Since the injection mechanism here does not depend on time, and since the cluster size distribution itself is stationary, it is independent from the specific definition of time. ![ LEFT: Injection along a line. System size $256^2$. RIGHT: Scaling plot for basic model plus injection on a line. Number of clusters per logarithmic bin, divided by number of clusters in first bin. The straight line has slope $-1/2$ corresponding to $n(s) \sim s^{-3/2}$. System size $1024^2$. This is a steady state distribution. []{data-label="fig:snap:line"}](snap-line-gz.eps "fig:"){width="40.00000%"} ![ LEFT: Injection along a line. System size $256^2$. RIGHT: Scaling plot for basic model plus injection on a line. Number of clusters per logarithmic bin, divided by number of clusters in first bin. The straight line has slope $-1/2$ corresponding to $n(s) \sim s^{-3/2}$. System size $1024^2$. This is a steady state distribution. []{data-label="fig:snap:line"}](line-scaling-gpl.eps "fig:"){width="55.00000%"} Random injection without space {#sec:non-sptl} ------------------------------ One could ask what would happen without space. A possible translation of our model into “no space” is: Do in parallel: Instead of picking one of your four nearest neighbors, you pick an arbitrary other agent (random neighbor approximation). If that agent is not dead, copy its cluster number. Do this over and over again in parallel, until all agents are part of a cluster again. A cluster is now no longer a spatially connected structure, but just a set of agents. In that case, we obtain again power laws for the size distribution, but this time with slopes that depend on the injection rate $p_{\it inj}$ (Fig. \[fig:non-spatial-gpl\]); see Sec. \[sec:theo-wo-space\] for details. ![ Steady state cluster size distributions for different non-spatial simulations. Number of clusters per logarithmic bin, divided by number of clusters is first bin. System sizes $64^2$ to $512^2$. LEFT: $p_{\it inj}=0.1$. RIGHT: $p_{\it inj}=0.01$. []{data-label="fig:non-spatial-gpl"}](nsptl-p10-gpl.eps "fig:"){width="0.49\hsize"} ![ Steady state cluster size distributions for different non-spatial simulations. Number of clusters per logarithmic bin, divided by number of clusters is first bin. System sizes $64^2$ to $512^2$. LEFT: $p_{\it inj}=0.1$. RIGHT: $p_{\it inj}=0.01$. []{data-label="fig:non-spatial-gpl"}](nsptl-p01-gpl.eps "fig:"){width="0.49\hsize"} Real world company size distributions ------------------------------------- Fig. \[fig:sales\] shows actual retail company size distributions from the 1992 U.S. economic census [@econ:census:92], using annual sales as a proxy for company size. We use the retail sector because we think that it is closest to our modelling assumptions — this is discussed at the end of Sec. \[sec:discussion\]. We show two curves: establishment size, and firm size.[^2] It is clear that in order to be comparable with our model assumptions, we need to look at establishment size rather than at company size. Census data comes in unequally spaced bins; the procedure to convert it into useable data is described in the appendix. Also, the last four data points for firm size (not for the establishment size, however) were obtained via a different method than the other data points; for details, again see the appendix. From both plots, one can see that there is a typical establishment size around \$400000 annual sales; and the typical firm size is a similar number. This number intuitively makes sense: With, say, income of 10% of sales, smaller establishments will not provide a reasonable income. One can also see from the plots that the region around that typical size can be fitted by a log-normal. We also see, however, that for larger numbers of annual sales, such a fit is impossible since the tail is much fatter. A scaling law with $$n(>\!s) \sim s^{-1} \hbox{\ \ corresponding to \ \ } n(s) \sim s^{-2}$$ is an alternative here.[^3] This is, however, at odds with investigations in the literature. For example, Ref. [@Stanley:sizes:econ-letters] find a log-normal, and by using a Zipf plot they show that for large companies the tail is less fat than a log-normal. However, there is a huge difference between our and their data: They only use publicely traded companies, while our data refers to all companies in the census. Indeed, one finds that their plot has its maximum at annual sales of $\$10^8$, which is already in the tail of our distribution. This implies that the small scale part of their distribution comes from the fact that small companies are typically not publicely traded. In consequence, it reflects the dynamics of companies entering and exiting from the stock market, not entry and exit of the company itself. We conclude that from available data, company size distributions are between a log-normal and a power law with $n(s) \sim s^{-2}$ or $n(>\!s) \sim s^{-1}$. Further investigation of this goes beyond the scope of this paper. ![ 1992 U.S. Economic Census data. LEFT: Number of retail establishments/retail firms per logarithmic bin as function of annual sales. RIGHT: Number of establishments/firms which have more sales than a certain number. []{data-label="fig:sales"}](sales-loglog-gpl.eps "fig:"){width="49.00000%"} ![ 1992 U.S. Economic Census data. LEFT: Number of retail establishments/retail firms per logarithmic bin as function of annual sales. RIGHT: Number of establishments/firms which have more sales than a certain number. []{data-label="fig:sales"}](sales-accum-gpl.eps "fig:"){width="49.00000%"} Theoretical considerations {#sec:theo} ========================== Spatial coarsening model (slope -2 in natural time) --------------------------------------------------- We are looking again at the “basic model”. In cluster time this was: randomly pick one of the clusters, and give it to the neighbors. The following heuristic model gives insight: 1. We start with $N$ clusters of size 1. 2. We need $N/2$ time steps to delete $N/2$ of them and with that generate $N/2$ clusters of size 2. 3. In general, we need $N/2^k$ time steps to move from $N/2^{k-1}$ clusters of size $2^{k-1}$ to $N/2^k$ clusters of size $2^k$. 4. If we sum this over time, then in each logarithmic bin at $s=2^k$ the number of contributions is $N/2^k \times N/2^k$, i.e. $\sim s^{-2}$. 5. Since these are logarithmic bins, this corresponds to $\tilde n(s) \sim s^{-3} \hbox{ \ \ or \ \ } \tilde n(>\!s) \sim s^{-2} \ , $ which was indeed the simulation result in cluster time. 6. In natural time, we need a constant amount of time to move from $k-1$ to $k$, and thus obtain via the same argument $ n(s) \sim s^{-2} \hbox{ \ \ or \ \ } n(>\!s) \sim s^{-1} \ , $ which was the simulation result in natural time. Random injection in space (log-normal) -------------------------------------- At the moment, we do not have a consistent explanation for the log-normal distribution in the spatial model. A candidate is the following: Initially, most injected clusters of size one are within the area of some larger and older cluster. Eventually, that surrounding cluster gets deleted, and all the clusters of size one spread in order to occupy the now empty space. During this phase of fast growth, the speed of growth is proportional to the perimeter, and thus to $\sqrt{s}$, where $s$ is the area. Therefore, $\sqrt{s}$ follows a biased multiplicative random walk, which means that $\log(\sqrt{s}) = \log(s)/2$ follows a biased additive random walk. In consequence, once that fast growth process stops, $\log(s)$ should be normally distributed, resulting in a log-normal distribution for $s$ itself. In order for this to work, one needs that this growth stops at approximately the same time for all involved clusters. This is apprixomately true because of the “typical” distance between injection sites which is inversely proportional to the injection rate. More work will be necessary to test or reject this hypothesis. Injection on a line (slope -3/2) -------------------------------- If one looks at a snapshot of the 2D picture for “injection on a line” (Fig. \[fig:snap:line\]), one recognizes that one can describe this as a structure of cracks which are all anchored at the injection line. There are $L$ such cracks (some of length zero); cracks merge with increasing distance from the injection line, but they do not branch. According to Ref. [@Sneppen:etc:fractal-cracks], this leads naturally to a size exponent of $-3/2$, as found in the simulations. The argument is the following: The whole area, $L^2$, is covered by $$\int ds \, s \, n(s) \ ,$$ where $n(s)$ is the number of clusters of size $s$ on a linear scale. We assume $n(s) \sim s^{-\tau}$, however the normalization is missing. If all clusters are anchored at a line of size $L$, then a doubling of the length of the line will result in twice as many clusters. In consequence, the normalization constant is $\propto L$, and thus $n(s) \sim L \, s^{-\tau}$. Now we balance the total area, $L^2$, with what we just learned about the covering clusters: $$L^2 \sim \int ds \, s \, L \, s^{-\tau} % = L \, \int ds \, s^{1-\tau} % \sim L \, s^{2-\tau} \big\|_0^S \ .$$ Assuming that $\tau < 2$, then the integral does not converge for $S \to \infty$, and we need to take into account how the cut-off $S$ scales with $L$. This depends on how the cracks move in space as a function of the distance from the injection line. If the cracks are roughly straight, then the size of the largest cluster is $\sim L^2$. If the cracks are random walks, then the size of the largest cluster is $\sim L^{3/2}$. In consequence: - For “straight” lines: $L^2 \sim L \; (L^2)^{2-\tau}$ $\ \Rightarrow\ $ $ 2 = 1 + 2 \, (2-\tau) %= 1 + 4 - 2 \tau $ $\ \Rightarrow\ $ $ \tau = 3/2 \ . $ - For random walk: $ 2 = 1 + 3/2 \, (2-\tau) % = 1 + 3 - 3 \tau/2 % \ \ \Rightarrow \ \ % \tau = 4/3 \ . $ Since our simulations result in $\tau \approx 3/2$, we conclude that our lines between clusters are not random walks. This is intuitively reasonable: When a cluster is killed, then the growth is biased towards the center of the deleted cluster, thus resulting in random walks which are all differently biased. This bias then leads to the “straight line” behavior. — This implies that the $\sim s^{-3/2}$ steady state scaling law hinges on two ingredients (in a 2D system): (i) The injection comes from a 1D structure. (ii) The boundaries between clusters follow something that corresponds to straight lines. As we have seen, the biasing of a random walk is already enough to obtain this effect. Injection without space (variable slope) {#sec:theo-wo-space} ---------------------------------------- Without space, clusters do not grow via neighbors, but via random selection of one of their members. That is, we pick a cluster, remove it from the system, and then give its members to the other clusters one by one. The probability that the agent choses a cluster $i$ is proportional to that cluster’s size $s_i$. If for the moment we assume that time advances with each member which is given back, we obtain the rate equation $${d n(s) \over dt} % = (s\!-\!1) \, n(s\!\!-\!\!1) - \epsilon \, n(s) - s \, n(s) % - \epsilon \, p_{\it inj} \, n(s) % + \epsilon \, p_{\it inj} \, n(s\!\!+\!\!1) \ .$$ The first and second term on the RHS represent cluster growth by addition of another member; the third term represents random deletion; the fourth and fifth term the decrease by one which happens if one of the members is converted to a start-up via injection. $\epsilon$ is the rate of cluster deletion; since we first give all members of a deleted cluster back to the population before we delete the next cluster, it is proportional to the inverse of the average cluster size and thus to the injection rate: $\epsilon \sim 1/\langle s \rangle \sim p_{\it inj}$. This is similar to an urn process with additional deletion. Via the typical approximations $s \, n(s) - (s-1) \, n(s-1) \approx {d \over ds} (s \, n(s))$ etc. we obtain, for the steady state, the differential equation $$0 = - N - s \, {dN \over ds} - \epsilon \, N % + \epsilon \, p_{\it inj} \, {dN \over ds} \ . % %dN/N = - (1+\epsilon) \, ds / (s - \epsilon p_{\it inj}) \ .$$ This leads to $$n(s) \propto (s - \epsilon p_{\it inj})^{-(1+\epsilon)} % \sim s^{-(1+\epsilon)} \ .$$ That is, the exponent depends on the injection rate, and in the limit of $p_{\it inj} \to 0$ it goes to $-1$. This is indeed the result from Sec. \[sec:non-sptl\] (see Fig. \[fig:non-spatial-gpl\]).[^4] Price formation {#sec:prices} =============== What we will do now is to add the mechanism of price formation to our spatial competition model. For this, we identify sites with consumers/customers. Clusters correspond to domains of consumers who go to the same shop/company. Intuitively, it is clear how this should work: Companies which are not competitive will go out of business, and their customers will be taken over by the remaining companies. The reduction in the number of companies is balanced by the injection of start-ups. Companies can go out of business for two reasons: losing too much money, or losing too many customers. The first corresponds to a price which is too low; the second corresponds to a price which is too high. We model these aspects as follows: We again have $N$ sites on an $N= L \times L$ grid with periodic boundary conditions (torus). On each site, we have a consumer and a firm. These are not connected in any way except by the spatial position – one can imagine that the firm is located “downstairs” while the consumer lives “upstairs”. Firms with customers are called “active”, the other ones “inactive”. A time step consists of the following sub-steps: - Trades are executed. - Companies with negative profit go out of business. - Companies change prices. - New companies are injected. - Consumers can change where they shop. These steps are described in more detail in the following: Trade: All customers have an initial amount $M$ of money, which is completely spent in each time step and replenished in the next. Every customer $i$ also knows which firm $j=f(i)$ he/she buys from. Thus, he/she orders an amount $Q_i = M/P_j$ at his/her company, where $P_j$ is that company’s price. The companies produce to order, and then trades are executed. That is, a company that has $n_j$ customers and price $P_j$ will produce and sell $Q_j = n_j M/P_j$ units and will collect $n_j M$ units of money. Company exit: We assume an externally given cost function for production, $C(Q)$, which is the same for everybody. If profit $\Pi_j := n_j \, M - C(n_j \, M/P_j)$ is less than zero, then the company is losing money and will immediately go out of business.[^5] The prices of such a company is set to infinity. We will use $C(Q)=Q$, corresponding to a linear cost of production. With this choice, companies with prices $P_j<1$ will exit according to this rule as soon as they attract at least one customer. Price changes: With probability one, pick a random integer number between $1$ and $N$. If there is an active company with that number, its price is randomly increased or decreased by $\delta$. Company injection: Companies are made active by giving them one customer: With probability $p_{\it inj}$, pick a random site $i$ and make the consumer $i$ go shopping at company $i$. The price of the injected company is set to the price that the customer has paid before, randomly increased or decreased by $\delta$. Consumer adaptation: All customers whose prices got increased (either via “company exit” or via “price changes”) will search for a new shop.[^6] These “searching” consumers correspond to dead sites in the basic spatial models (Sec. 3), and the dynamics is essentially a translation of that: All searching consumers in parallel pick a random nearest neighbor. If that neighbor is also searching, nothing happens. If that neighbor is however not searching, and if that neighbor is paying a lower price, our consumer will accept the neighbor’s shop. Otherwise the customer will remain with her old shop, and she will no longer search. We keep repeating this until no consumer is searching any more. This model does not invest much in terms of rational or organized behavior by any of the entities. Firms change prices randomly; and they exit without warning when they lose money. New companies are injected as small variations of existing companies. Consumers only make moves when they cannot avoid it (i.e. their company went out of business and they need a new place to go shopping) or when prices just went up. Only in the last case they actively compare some prices. It will turn out (see below) that even that price comparison is not necessary. In the above model, price converges to the unit cost of production, which is the competetive price. In Fig. \[fig:adjustment-1\] (left, bottom curve) we show how an initially higher price slowly decreases towards a price of one. The reason for this is that, as long as prices are larger than one, there will be companies that, via random changes or injection, have a lower price than their neighbors. Eventually, these neighbors raise prices, thus driving their customers away and to the companies with lower prices. If, however, a company lowers its price below one, then it will immediately exit after it has attracted at least one customer.[^7] ![LEFT: Price adjustment. Bottom curve: when searching consumers compare prices. Top curve: when searching consumers accept prices no matter what they are. RIGHT: Prices tracking the cost of production. []{data-label="fig:adjustment-1"}](series-gpl.eps "fig:"){width="49.00000%"} ![LEFT: Price adjustment. Bottom curve: when searching consumers compare prices. Top curve: when searching consumers accept prices no matter what they are. RIGHT: Prices tracking the cost of production. []{data-label="fig:adjustment-1"}](track-gpl.eps "fig:"){width="49.00000%"} As already mentioned above, it turns out that the price comparison by the consumers is not needed at all. We can replace the rule “if price goes up, try to find a better price” by “if price goes up, go to a different shop no matter what the price there”. In both cases, we find the alternative shop via our neighbors, as we have done throughout this paper. The top curve in Fig. \[fig:adjustment-1\] shows the resulting price adjustment. Clearly, the price still moves towards the critical value of one, but it moves more slowly and the trajectory displays more fluctuations. This is what one would expect, and we think it is typical for the situation: If we reduce the amount of “rationality”, we get slower convergence and larger fluctuations. In terms of cluster size distribution, the price model is similar to the earlier spatial competition model with random injection. They would become the same if we separated bankruptcy and price changes. In Fig. \[fig:adjustment-1\] (right) we also show that our model is able to track slowly varying costs of production. For this, we replace $C(Q) = Q$ by a sinus-function which oscillates around $Q$. The plot implies that prices lag behind the costs of production. ![ Crosscorrelation function between $R_{\it price}$ and $R_{\it cost}$: $R_x := x(t) / x(t\!-\!1)$; ${\it Xcorr}(\tau) := \langle R_P(t) R_C(t\!-\!\tau) \rangle$. LEFT: Simulation. The crosses show the crosscorrelation values mirrored at the $\tau=0$ axis, in order to stress the asymmetry. RIGHT: U.S. Consumer price index for price and Producer price index for cost. Filled boxes are the crosscorrelation values; the smooth line is an interpolating spline for the filled boxes. The crosses show the crosscorrelation values mirrored at the $\tau=0$ axis. []{data-label="fig:xcorr"}](s-corr-gz.eps "fig:"){width="0.49\hsize"} ![ Crosscorrelation function between $R_{\it price}$ and $R_{\it cost}$: $R_x := x(t) / x(t\!-\!1)$; ${\it Xcorr}(\tau) := \langle R_P(t) R_C(t\!-\!\tau) \rangle$. LEFT: Simulation. The crosses show the crosscorrelation values mirrored at the $\tau=0$ axis, in order to stress the asymmetry. RIGHT: U.S. Consumer price index for price and Producer price index for cost. Filled boxes are the crosscorrelation values; the smooth line is an interpolating spline for the filled boxes. The crosses show the crosscorrelation values mirrored at the $\tau=0$ axis. []{data-label="fig:xcorr"}](f-corr-gz.eps "fig:"){width="0.49\hsize"} This is also visible in the asymmetry of the cross correlation function between both series. In order to be able to compare with non-stationary real world series, we look at relative changes, $R_x(t) = x(t) / x(t-1)$. The cross correlation function between price increases and cost increases then is $${\it Xcorr}(\tau) = \langle R_P(t) \, R_C(t-\tau) \rangle \ ,$$ where $\langle . \rangle$ averages over all $t$. In Fig. \[fig:xcorr\] (left) one can clearly see that prices are indeed lagging behind costs for our simulations. In order to stress the asymmetry, we also plot ${\it XCorr}(-\tau)$. In Fig. \[fig:xcorr\] (right) we show the same analysis for the Consumer Price Index vs. the Production Price Index (seasonally adjusted). Although the data is much more noisy, it is also clearly asymmetric. Discussion and outlook {#sec:discussion} ====================== The modelling approach with respect to economics in this paper is admittedly simplistic. Some obvious and necessary improvements concern credit and bankruptcy (i.e. rules for companies to operate with a negative amount of cash). Instead of those, we want to discuss some issues here that are closer to this paper. These issues are concerned with time, space, and communication. In this paper, in order to reach a clean model with possible analytic solutions, we have described the models in a language which is rather unnatural with respect to economics. For example, instead of “one company per time step” which changes prices one would use rates (for example a probability of $p_{\it ch}$ for each company to change prices in a given time step). However, in the limiting case of $p_{\it ch} \to 0$, at most one and usually zero companies change prices in a given time step. If one also assumes that consumers adaptation is fast enough so that it is always completed before the next price change occurs, then this will result in the same dynamics as our model. Thus, our model is not “different” from reality, but it is a limiting case for the limit of fast customer adaptation and slow company adaptation. Our approach is to understand these limiting cases first before we move to the more general cases. Similar comments refer to the utilization of space. We have already seen that moving from a spatial to a non-spatial model is rather straightforward. There is an even more systematic way to make this transition, which is the increase of the dimensions. In two dimensions on a square grid, every agent has four nearest neighbors. In three dimensions, there are six nearest neighbors. In general, if $D$ is the dimension, there are $2D$ nearest neighbors. If we leave the number $N$ of agents constant and keep periodic boundary conditions ($D$-dimensional torus), then at $D=(N-1)/2$ everybody is a nearest neighbor of everybody. Thus, a non-spatial model is the $D \to \infty$ limiting case of a spatial model.[^8] These concepts can be generalized beyond grids and nearest neighbors – the only two ingredients one needs is that (i) the probability to interact with someone else decreases fast enough with distance, and that (ii) if one doubles distance from $r$ to $2r$, then the number of interactions up to $2r$ is $2^D$ times the number of interactions up to distance $r$. This should also make clear that space can be seen in a generalized way if one replaces distance by generalized cost. For example, how many more people can you call for “20 cents a minute or less” than for “10 cents a minute or less”? If the answer to this is “two times as many”, then for the purposes of this discussion you live in a one-dimensional world. Given this, it is important to note that we have used space only for the communication structure, i.e. the way consumers aquire information (by asking neighbors). This is a rather weak influence of space, as opposed to, for example, transportation costs[@Hotelling]; it however also assumes a not very sophisticated information structure, as for example contrast to today’s internet. The details of this need to be left to future work. Last, one needs to consider which part of the economy one wants to model. For example, a stockmarket is a centralized institution, and space plays a weak role at best. In contrast, we had the retail market in mind when we developed the models of this paper. In fact, we implicitely assume perishable goods, since agents have no memory of what they bought and consumed the day before. Also, we assume that consumers spend little effort in selecting the “right” place to shop, which excludes major personal investments such as cars or furniture. Also note that our companies have no fixed costs, which implies that there are no capital investments, which excludes for example most manufacturing. Summary ======= Price formation is an important aspect of economic activity. Our interest was in price formation in “everyday” situations, such as for retail prices. For this, we assumed that companies are price setters and agents are price takers, in the sense that their only strategy option is to go someplace else. In our abstracted situation, this means that companies with too low prices will exit because they cannot cover costs, while companies with too high prices will exit because they lose their customers. We use space in order to simplify and structure the way in which information about alternative shopping places is found. This prevents the singularity of “Bertrand-style” models, where the market share of each company is independent from history, leading to potentially huge and unrealistic fluctuations. By doing this, one notices that the spatial dynamics can be separated from the price formation dynamics itself. This makes intuitively sense since, in generalized terms, we are dealing with evolutionary dynamics, which often does not depend on the details of the particular fitness function. We have therefore started with an investigation of a spatial competition model without prices. For this model, we have looked at cluster size distributions, and compared them with real world company size distributions. In contrast to investigations in the literature, which find log-normal distributions, we find a scaling law a better fit of our data. In the models, we find log-normal distributions or scaling laws, depending on the particular rules. We then added price formation to our spatial model. We showed that the price, in simple scenarios, converges towards the competitive price (which is here the unit cost of production), and that it is able to track slowly varying production costs, as it should. This predicts that prices should lag behind costs of production. We indeed find this in the data of consumer price index vs. production price index for the United States since 1941. Acknowledgments {#acknowledgments .unnumbered} =============== KN thanks Niels Bohr Institute for hospitality during the summer 1999, where this work was started. All of the authors thank Santa Fe Institute, where some of the authors met, which provided a platform for continuous discussion, and where some of the work was done in spring 2000. We also thank H. Flyvbjerg and K. Sneppen for invaluable hints and discussions. Converting the aggregated census data ===================================== #### Non-equidistant bins {#non-equidistant-bins .unnumbered} The size data in the 1992 U.S. economic census comes in non-equidistant bins. For example, we obtain the number of establishments with annual sales above 25000 k\$, between 10000 k\$ and 25000 k\$, etc. For an accumulated function, such as Fig. \[fig:sales\] (right), this is straightforward to use. For distributions, such as Fig. \[fig:sales\] (left), this needs to be normalized. We have done this in the following way: (1) We first divide by the weight of each bin, which is its width. In the above example, we would divide by $(25\,000~k\$ - 10\,000~k\$) = 15\,000~k\$$. Note that this immediately implies that we cannot use the data for the largest companies since we do not know where that bin ends. (2) For the log-normal distribution $$\rho(x) % \propto {1 \over x} \, \exp\big[ - ( \ln(x) - \ln(\mu) )^2 \big]$$ (note the factor $1/x$), one typically uses logarithmic bins, since then the factor $1/x$ cancels out. This corresponds to a weight of $x$ of each census data point. (3) Now we have to decide where we plot the data for a specific bin. We used the arithmic mean between the lower and the upper end. In our example case, $17\,500k\$$. (4) In summary, say the number of establishments between $s_i$ and $s_{i+1}$ is $N_i$. Then the transformed number $\tilde N_i$ is calculated according to $$\tilde N_i = {N_i \over s_{i+1} - s_i} \, {s_i + s_{i+1} \over 2} \ .$$ #### The largest firms {#the-largest-firms .unnumbered} For the largest firms (but not for the large establishments), the census also gives the combined sales of the four (eight, twenty, fifty) largest firms. We used the combined sales of the four largest firms divided by four as a (bad) proxy for the sales of each of these four companies. We then substracted the sales of the four largest firms from the sales of the eight largest firms, divided again, etc. Those data points should thus be seen as an indication only, and it probably explains the “kink” near $2 \times 10^9$ in Fig. \[fig:sales\]. [^1]: In this paper, we will also use $N(s) = s \, n(s)$ for the cluster size distribution in logarithmic bins, in particular for the figures. [^2]: An establishment is “a single physical location at which business is conducted. It is not necessarily identical with a company or enterprise, which may consist of one establishment or more.” [@econ:census:92]. [^3]: Remember again, that slopes from log-log plots in logarithmic bins are different by one from the exponent in the distribution. So $n(s) \sim s^{-2}$ corresponds to a slope $-1$ both in the accumulated distribution $n(>\!s)$ and when plotting logarithmic bins $N(s)/N(1)$. [^4]: Note that the approach in this section corresponds to measuring the cluster size distribution every time we give an agent back to the system, while in the simulations we measured the cluster size distribution only just before a cluster was picked for deletion. In how far this is important is an open question; preliminary simulation results indicate that it is important for the spatial case with injection but not important for the non-spatial case in this section. [^5]: In this model no accumulation of assets is allowed. This simplification will be relaxed in future work. [^6]: The simplification that customers react to price changes only is useful because it leads to the separation of time scales between consumer behavior and firm behavior. [^7]: If all prices in the system are more than $\delta$ below one, then the model is not well-defined. In the limit of large systems and when starting with prices above one, such a state cannot be reached via the dynamics. – Also note that if the model allowed credit, the exit of such a company would be delayed, allowing losses for limited periods of time. [^8]: Furthermore, models such as the ones discussed in this paper often have a so-called upper critical dimension, where some aspects of the model become the same as in infinite dimensions. This upper critical dimension often is rather low (below 10).
--- abstract: 'Molecular dynamics simulations are used to investigate the influence of molecular-scale surface roughness on the slip behavior in thin liquid films. The slip length increases almost linearly with the shear rate for atomically smooth rigid walls and incommensurate structures of the liquid/solid interface. The thermal fluctuations of the wall atoms lead to an effective surface roughness, which makes the slip length weakly dependent on the shear rate. With increasing the elastic stiffness of the wall, the surface roughness smoothes out and the strong rate dependence is restored again. Both periodically and randomly corrugated rigid surfaces reduce the slip length and its shear rate dependence.' author: - 'Nikolai V. Priezjev' title: 'Effect of surface roughness on rate-dependent slip in simple fluids' --- Introduction ============ The description of the fluid flow in confined geometry requires specification of the boundary condition for the fluid velocity at the solid wall. Usually the fluid is assumed to be immobile at the boundary. Although this assumption is successful in describing fluid flow on macroscopic length scales, it needs a revision for the microscopic scales due to possible slip of the fluid relatively to the wall [@KarniBeskok]. The existence of liquid slip at the solid surfaces was established in many experiments on the pressure driven flow in narrow capillaries [@Schnell56; @Churaev84; @Breuer03] and drainage of thin liquid films in the surface force apparatus [@MackayVino; @Charlaix01; @Granick01]. The most popular Navier model relates the fluid slip velocity to the interfacial shear rate by introducing the slip length, which is assumed to be rate-independent. The slip length is defined as a distance from the boundary where the linearly extrapolated fluid velocity profile vanishes. Typical values of the slip length inferred from the experiments on fluids confined between smooth hydrophobic surfaces is of the order of ten nanometers [@Granick01; @MackayVino; @Charlaix05; @BocquetPRL06; @Vinograd06]. Despite the large amount of experimental data on the slip length [@BonaccursoRev05], the underlying molecular mechanisms leading to slip are still poorly understood because it is very difficult to resolve the fluid velocity profile in the region near the liquid/solid interface at these length scales. Over the last twenty years, molecular dynamics (MD) simulations were extensively used to investigate the correlation between the structure of simple fluids in contact with atomically smooth surfaces and slip boundary conditions [@KB89; @Thompson90; @Barrat94; @Barrat99; @Travis00; @Cieplak01; @Quirke01; @Khare06]. The advantage of the MD method is that the fluid velocity profile can be resolved at the molecular level and no assumptions about the slip velocity at the interface are required. The main factors affecting slip for atomically smooth surfaces are the wall-fluid interaction, the degree of commensurability of liquid and solid structures at the interface, and diffusion of fluid molecules near the wall. The slip length was found to correlate inversely with the wall-fluid interaction energy and the amount of structure induced in the first fluid layer by the periodic surface potential [@Thompson90]. For weak wall-fluid interactions and smooth surfaces, the slip length is proportional to the collective relaxation coefficient of the fluid molecules near the wall [@Barrat99fd]. The thermal fluctuations of the wall atoms under the strong harmonic potential reduce the degree of the in-plane fluid ordering and result in larger values of the slip length [@Thompson90]. On the other hand, an excessive penetration of the wall atoms into the fluid phase reduces the slip velocity for soft thermal walls [@Tanner99]. Nevertheless, the effect of thermal surface roughness on the slip length in the shear-rate-dependent regime was not systematically explored even for atomically smooth walls. In the original MD study by Thompson and Troian [@Nature97] on boundary driven shear flow of simple fluids past atomically smooth rigid walls, the slip length was found to increase nonlinearly with the shear rate for weak wall-fluid interactions. A similar dynamic behavior of the slip length has been reported in thin polymer films [@Priezjev04]. The rate-dependent slip was also observed for the planar Poiseuille flow of simple fluids confined between hydrophobic surfaces with variable size of the wall atoms [@Fang05; @Yang06]. The variation of the slip length (from negative to positive values) with increasing shear rate for hydrophilic surfaces [@Fang05] can be well described by the power law function proposed in Ref. [@Nature97]. In the recent paper [@Priezjev07], we have reported a gradual transition in the shear rate dependence of the slip length, from linear to highly nonlinear function with pronounced upward curvature, by decreasing the strength of the wall-fluid interaction. Remarkably, in a wide range of shear rates and surface energies, the slip length is well fitted by a power law function of a single variable, which is a combination of the structure factor, contact density, and temperature of the first fluid layer. One of the goals of the present study is to investigate how the rate-dependent slip is affected by the presence of the molecular-scale surface roughness. Molecular scale simulations of simple [@Attard04; @Priezjev06] and polymeric [@Gao2000; @Jabb00] fluids (as well as recent experiments [@Granick02; @Archer03; @Leger06]) have shown that the slip length is reduced in the presence of the surface roughness. The effect is enhanced for smaller wavelengths and larger amplitudes of the surface corrugation [@Jabb00; @Priezjev06]. At low shear rates, the reduction of the effective slip length is caused by the local curvature of the fluid flow above macroscopic surface corrugations [@Richardson73; @Panzer90; @Priezjev06] or by more efficient trapping of the fluid molecules by atomic-scale surface inhomogeneities [@Thompson90; @Barrat94; @Attard04; @Priezjev05; @Priezjev06]. The analysis of more complex systems with combined effects of surface roughness and rate dependency poses certain difficulties in the interpretation of the experimental results because the exact dependence of the local slip length on shear rate is often not known. In this paper, we explore the influence of molecular-scale surface roughness on the slip behavior in a flow of simple fluids driven by a constant force. We will show that the functional form of the rate-dependent slip length is considerably modified by the presence of the thermal, random and periodic surface roughness. The growth of the slip length with increasing shear rate, which is observed for atomically smooth rigid walls, is strongly reduced by periodic and random surface roughness. Soft thermal walls produce very weak rate dependence of the slip length, while the linear behavior is restored for stiffer walls. The paper is organized as follows. The details of molecular dynamics simulations are described in the next section. Results for the shear rate dependence of the slip length on the thermal and random surface roughness are presented in Section \[sec:Results\_thermo\]. The effect of periodic wall roughness on the rate-dependent slip is discussed in Section \[sec:Results\_corrugated\]. The summary is given in the last section. MD Simulation model {#sec:Model} =================== The simulation setup consists of $N\,{=}\,3456$ fluid molecules confined between two stationary atomistic walls parallel to the $xy$ plane. The molecules interact through the truncated Lennard-Jones (LJ) potential $$V_{LJ}(r)\!=4\,\varepsilon\,\Big[\Big(\frac{\sigma}{r}\Big)^{12}\!-\Big(\frac{\sigma}{r}\Big)^{6}\,\Big],$$ where $\varepsilon$ and $\sigma$ represent the energy and length scales of the fluid phase with density $\rho\,{=}\,0.81\,\sigma^{-3}$. The interaction between wall atoms and fluid molecules is also modeled by the LJ potential with the energy $\varepsilon_{\rm wf}$ and length scale $\sigma_{\rm wf}$ measured in units of $\varepsilon$ and $\sigma$. In all our simulations, wall atoms do not interact with each other and $\sigma_{\rm wf}\,{=}\,\sigma$. The cutoff distance is set to $r_c\,{=}\,2.5\,\sigma$ for fluid-fluid and wall-fluid interactions. -------------------------------------------------- --------- --------- --------- -------- -------- Spring stiffness $\kappa\,[\varepsilon/\sigma^2]~~~~~$ $400\,$ $600\,$ $800\,$ $1200$ $1600$ \[3pt\] \[-5pt\] $\sqrt{\langle\delta u^2\rangle}/d~~~~$ $0.11$ $0.09$ $0.08$ $0.07$ $0.06$ \[3pt\] $2\pi\sqrt{m_w/\kappa}\,\,[\tau]$ $0.63$ $0.51$ $0.44$ $0.36$ $0.31$ \[2pt\] -------------------------------------------------- --------- --------- --------- -------- -------- : Root mean-square displacement, $\langle\delta u^2\rangle\,{=}\,3\,k_BT/\kappa$, divided by the nearest-neighbor distance $d\,{=}\,0.8\,\sigma$ and the typical oscillation time of the wall atoms tethered about their equilibrium lattice positions as a function of the spring stiffness for $m_w\,{=}\,4\,m$. \[tabela\] The steady-state flow was induced by a constant force $\text{f}_{\text{x}}$ in the $\hat{x}$ direction, which acted on each fluid molecule. The heat exchange with external reservoir was controlled by a Langevin thermostat with a random, uncorrelated force and a friction term, which is proportional to the velocity of the fluid molecule. The value of the friction coefficient $\Gamma\,{=}\,1.0\,\tau^{-1}$ is small enough not to affect significantly the self-diffusion coefficient of the fluid molecules [@Grest86; @GrestJCP04]. The Langevin thermostat was applied only along the $\hat{y}$ axis to avoid a bias in the flow direction [@Thompson90]. The equations of motion for a fluid monomer of mass $m$ are given by $$\begin{aligned} \label{Langevin_x} m\ddot{x}_i & = & -\sum_{i \neq j} \frac{\partial V_{ij}}{\partial x_i} + \text{f}_{\text{x}}\,, \\ \label{Langevin_y} m\ddot{y}_i + m\Gamma\dot{y}_i & = & -\sum_{i \neq j} \frac{\partial V_{ij}}{\partial y_i} + f_i\,, \\ \label{Langevin_z} m\ddot{z}_i & = & -\sum_{i \neq j} \frac{\partial V_{ij}}{\partial z_i}\,, %\end{aligned}$$ where $f_i$ is a randomly distributed force with $\langle f_i(t)\rangle\,{=}\,0$ and variance $\langle f_i(0)f_j(t)\rangle\,{=}\,\,2mk_BT\Gamma\delta(t)\delta_{ij}$ obtained from the fluctuation-dissipation theorem. The temperature of the thermostat is fixed to $T\,{=}\,1.1\,\varepsilon/k_B$, where $k_B$ is the Boltzmann constant. The equations of motion of the fluid molecules and wall atoms are integrated using the fifth-order gear-predictor method [@Allen87] with a time step $\triangle t\,{=}\,0.002\,\tau$, where $\tau\,{=}\,\sqrt{m\sigma^2/\varepsilon}$ is the LJ time. The wall atoms were allowed to oscillate about their equilibrium lattice sites under the harmonic potential $V_{sp}\,{=}\,\frac{1}{2}\,\kappa\,r^2$. The spring stiffness $\kappa$ was chosen so that the ratio of the root mean-square displacement of the wall atoms and their nearest-neighbor distance was less than the Lindemann criterion for melting, $\sqrt{\langle\delta u^2\rangle}/d\lesssim 0.15$, e.g. see Ref. [@Barrat03]. At the same time, the parameter $\kappa$ should be small enough so that the dynamics of the wall atoms can be accurately resolved with the MD integration time step. The mass of the wall atoms $m_w$ was chosen to be four times that of the fluid molecule to make their oscillation times comparable. The Langevin thermostat was also applied to all three components of the wall atoms equations of motion, e.g. for the $\hat{x}$ component $$\begin{aligned} \label{Langevin_wall_x} m_w\,\ddot{x}_i + m_w\,\Gamma\dot{x}_i & = & -\sum_{i \neq j} \frac{\partial V_{ij}}{\partial x_i} - \frac{\partial V_{sp}}{\partial x_i} + f_i\,, %\\ % \label{Langevin_wall_y} m_w\,\ddot{y}_i + m_w\,\Gamma\dot{y}_i & = & % -\sum_{i \neq j} \frac{\partial V_{ij}}{\partial y_i} % - \frac{\partial V_{sp}}{\partial y_i} + f_i\,, \\ % \label{Langevin_wall_z} m_w\,\ddot{z}_i + m_w\,\Gamma\dot{z}_i & = & % -\sum_{i \neq j} \frac{\partial V_{ij}}{\partial z_i} - % \frac{\partial V_{sp}}{\partial z_i} + f_i\,,%\end{aligned}$$ where the sum is taken over the fluid molecules within the cutoff radius $r_c\!\,\,{=}\,\,2.5\,\sigma$. Mean displacement of the wall atoms and their typical oscillation time are summarized in Table \[tabela\] for the values of the spring constant considered in this study. Each wall is constructed out of $648$ atoms distributed between two (111) planes of the face-centered cubic (fcc) lattice with density $\rho_w\,{=}\,2.73\,\sigma^{-3}$. The walls are separated by a distance $24.58\,\sigma$ along the $\hat{z}$ axis. The lateral dimensions of the computational domain in the $xy$ plane are measured as $L_x\,{=}\,25.03\,\sigma$ and $L_y\,{=}\,7.22\,\sigma$. Periodic boundary conditions are imposed along the $\hat{x}$ and $\hat{y}$ directions. After an equilibration period of at least $2\times10^4\tau$, the fluid velocity and density profiles were computed for a time interval up to $2\times10^5\tau$ within bins of thickness $\Delta z\,{=}\,0.2\,\sigma$ and $\Delta z\,{=}\,0.01\,\sigma$, respectively [@Priezjev07]. The shear viscosity, $\mu\,{=}\,(2.2\pm0.2)\,\,\varepsilon\tau\sigma^{-3}$, remained independent of the external force [@Priezjev04; @Priezjev07]. ![(Color online) Averaged fluid density profiles near the thermal $\kappa\,{=}\,600\,\varepsilon/\sigma^2$ ($\circ$) and rigid ($\diamond$) walls for $\varepsilon_{\rm wf}/\varepsilon\,{=}\,0.9$ and $\text{f}_{\text{x}}\,{=}\,0.001\,\varepsilon/\sigma$. Left vertical axis coincides with the location of the liquid/solid interface at $z=-11.79\,\sigma$.[]{data-label="mol_dens"}](Fig1.eps){width="10.4cm" height="7.6cm"} Results for the thermal walls {#sec:Results_thermo} ============================= Fluid density and velocity profiles ----------------------------------- Examples of the averaged fluid density profiles near the thermal and rigid walls are presented in Fig.\[mol\_dens\] for a small value of the external force $\text{f}_{\text{x}}\,{=}\,0.001\,\varepsilon/\sigma$. The first peak in the density profile is slightly broader for the thermal walls because the fluid molecules can move closer to the fcc lattice plane due to finite spring stiffness of the wall atoms. The maximum value of the first peak defines a contact density, which is larger for the rigid walls because of the higher in-plane fluid ordering. In both cases, the fluid density oscillations gradually decay to a uniform bulk profile within $5\!-\!6 \,\sigma$ away from the wall (not shown). The contact density decreases slightly for larger values of the applied force. A correlation between the fluid structure near the wall and the slip length will be examined in the next section. ![(Color online) Averaged velocity profiles, $\langle \text{v} \rangle\, \tau/\sigma$, for the indicated values of the applied force per fluid molecule for the rigid walls (left) and the thermal walls with $\kappa\,{=}\,600\,\varepsilon/\sigma^2$ (right). The wall-fluid interaction energy is fixed to $\varepsilon_{\rm wf}/\varepsilon\,{=}\,0.9$. The solid lines represent a parabolic fit to the data. The dashed lines indicate the location of liquid/solid interfaces at $z=\pm\,11.79\,\sigma$. Vertical axes denote the position of the fcc lattice planes at $z=\pm 12.29\,\sigma$.[]{data-label="parab_velo"}](Fig2.eps){width="10.4cm" height="7.6cm"} The solution of the Navier-Stokes equation for the force driven flow with slip boundary conditions at the confining parallel walls, $\textrm{v}(\pm h)\,{=}\,V_s$, is given by [@KarniBeskok] $$\text{v}(z)=\frac{\rho\,\text{f}_{\text{x}}}{2\mu}\,(h^2-z^2)+V_s\,, \label{velo_hydro}$$ where $2h$ is the distance between the walls and $\mu$ is the shear rate independent viscosity. The slip length is defined as an extrapolated distance relative to the position of the liquid/solid interface where the tangential component of the fluid velocity vanishes $$\Big\|\frac{\partial \text{v}}{\partial z}\,(\pm h)\Big\|=\frac{V_s}{L_s}\,.$$ Figure\[parab\_velo\] shows representative velocity profiles in steady-state flow for three different values of the external force $\text{f}_{\text{x}}$ and fixed wall-fluid interaction energy $\varepsilon_{\rm wf}/\varepsilon\,{=}\,0.9$. The data for thermal and rigid walls are presented only in half of the channel because of the symmetry with respect to the mid-plane of the fluid phase, i.e. $\text{v}(z)\!=\text{v}(-z)$. Fluid velocity profiles are well fitted by a parabola with a shift by the value of the slip velocity, as expected from the continuum predictions \[see Eq.(\[velo\_hydro\])\]. The simulation results presented in Fig.\[parab\_velo\] show that slip velocity $V_s$ increases with the applied force. The degree of slip depends on the wall stiffness and the interfacial shear rate. The fluid slip velocity is larger for the thermal walls and small forces $\text{f}_{\text{x}}\,{\leqslant}\,0.012\,\varepsilon/\sigma$. By contrast, rigid walls produce more slippage for the large value of the external force $\text{f}_{\text{x}}\,{=}\,0.024\,\varepsilon/\sigma$. In both cases, the slip velocity is greater than the difference between the fluid velocities at the center of the channel and near the walls. The upper bound for the Reynolds number is $Re\approx10$ [@Priezjev07], ensuring laminar flow conditions throughout. Effect of thermal wall roughness on slip length ----------------------------------------------- Slip boundary conditions for a fluid flow past atomically smooth rigid walls are determined by the molecular-scale surface roughness due to the wall atoms fixed at their equilibrium lattice sites. The thermal fluctuations of the wall atoms, being unavoidable in real surfaces, modify the effective coupling between liquid and solid phases. Depending on the stiffness of the surface, thermal walls can either reduce slip (due to the deep penetration of the wall atoms into the fluid phase) or increase slip because of the reduction of the surface induced structure in the adjacent fluid layer. In this section, the shear rate dependence of the slip length is investigated in a wide range of values of the parameter $\kappa$ that satisfy the Lindemann criterion for melting (see Table \[tabela\]). In the previous MD study on shear flow near solids by Thompson and Robbins [@Thompson90], it was shown that the thermal surface roughness reduces the degree of ordering in the adjacent fluid layer. The thermal fluctuations of the wall atoms produced slip lengths of about $0.5\,\sigma$ larger than their values for the rigid walls. In a wide range of wall-fluid interaction energies $0.2\,{\leqslant}\,\varepsilon_{\rm wf}/\varepsilon\,{\leqslant}\,25$, the slip length was found to be rate-independent and less than $3.5\,\sigma$. In our simulations, the surface potential is less corrugated because of the higher wall density; and, therefore, the effect of thermal surface roughness on the slip length is greater. ![(Color online) Slip length, $L_s/\sigma$, as a function of the shear rate at the interface for $\varepsilon_{\rm wf}/\varepsilon\,{=}\,0.9$. The values of the spring constant $\kappa$ for the thermal walls are listed in the inset. The dashed lines represent the best fit to the data for the thermal walls with $\kappa\,{=}\,1600\,\varepsilon/\sigma^2$ ($\triangledown$) and the rigid fcc walls ($\circ$). The slip length for rough rigid walls with $\langle\delta u^2\rangle^{1/2}\!\simeq0.07\,\sigma$ ($\triangleright$). The solid curves are a guide for the eye.[]{data-label="thermo"}](Fig3.eps){width="10.4cm" height="7.6cm"} The variation of the slip length with increasing shear rate for different values of the spring stiffness $\kappa$ and $\varepsilon_{\rm wf}/\varepsilon\,{=}\,0.9$ is presented in Fig.\[thermo\]. The data for smooth rigid walls, fitted by a straight line, are also shown in Fig.\[thermo\] for comparison with the results for the thermal walls. In the range of accessible shear rates, the slip length is larger for stiffer thermal walls. The surface becomes effectively smoother because the average displacement of the wall atoms with respect to their equilibrium sites is reduced at larger values of $\kappa$. A similar reduction in slip velocity for the soft thermal wall atoms was reported in recent MD simulations of thin films of hexadecane [@Tanner99]. For soft walls with $\kappa\,{=}\,400\,\varepsilon/\sigma^2$, the wall atoms penetrate deeper into the fluid phase, which makes the slip length smaller and weakly dependent on the shear rate. For $\kappa\,{=}\,600\,\varepsilon/\sigma^2$, the slip length increases slightly at low shear rates and then saturates at $\dot{\gamma}\tau\!\gtrsim0.063$, where it becomes smaller than $L_s$ for the rigid walls. This behavior is consistent with the results for the fluid velocity profiles presented in Fig.\[parab\_velo\] for the thermal and rigid walls. In the case of the largest spring constant $\kappa\,{=}\,1600\,\varepsilon/\sigma^2$, the slip length increases monotonically with the shear rate and its dependence can also be fitted well by a straight line. The slope of the fitted line is slightly larger than one for the rigid walls (see Fig.\[thermo\]). For a finite spring stiffness, a small downward curvature appears at $\dot{\gamma}\tau\!\gtrsim0.05$ because of the higher temperature and, as a consequence, larger mean displacement of the wall atoms. The maximum increase in temperature of the wall atoms and the adjacent fluid layer is about $10\%$ at the highest shear rates reported in Fig.\[thermo\]. The fluid temperature at the center of the channel increases up to $1.106\,\varepsilon/k_B$ at the highest $\dot{\gamma}$. The upper bound for the shear rate is determined by the maximum shear stress the liquid/solid interface can support [@Priezjev07]. The simulation results for the thermal walls presented in Fig.\[thermo\] demonstrate that the spring stiffness in the model of harmonic oscillators can be an important factor in determining the degree of slip in the rate-dependent regime. ![(Color online) (a) Behavior of the slip length $L_s/\sigma$ as a function of the inverse value of the in-plane structure factor, $1/S(\mathbf{G}_1)$, evaluated at the first reciprocal lattice vector $\mathbf{G}_1\,{=}\,(9.04\,\sigma^{-1},0)$. The wall-fluid interaction energy is $\varepsilon_{\rm wf}/\varepsilon\,{=}\,0.9$. (b) The same data for the indicated values of the spring constant are replotted versus $[S(\mathbf{G}_1)\,\rho_c\,\sigma^3]^{-1}$.[]{data-label="ls_S9_ro"}](Fig4.eps){width="10.4cm" height="7.6cm"} It is interesting to note that the surface roughness due to immobile wall atoms with random displacement of only a fraction of a molecular diameter significantly reduces slip length and leads to a slight upward curvature in the rate dependence (see Fig.\[thermo\]). The rough surfaces were constructed by fixing the instantaneous positions of initially equilibrated wall atoms with the spring stiffness $\kappa\,{=}\,\,600\,\varepsilon/\sigma^2$ in the absence of the flow. The parabolic velocity profiles for different values of the applied force were computed for the same realization of disorder. These random perturbations of the surface potential lead to the difference in the slip length of about $7\,\sigma$ in comparison with its values for the thermal walls with the spring stiffness $\kappa\,{=}\,\,600\,\varepsilon/\sigma^2$. The degree of slip at the liquid/solid interface correlates well with the amount of the surface induced order in the adjacent fluid layer [@Thompson90; @Barrat99fd; @Priezjev04; @Priezjev05]. The in-plane fluid structure factor is defined as $S(\mathbf{k})\,{=}\,1/N_{\ell}\,\,\|\sum_j e^{i\,\mathbf{k}\cdot\mathbf{r}_j}\|^2$, where $\mathbf{r}_j\,{=}\,(x_j,y_j)$ is the two-dimensional position vector of the $j$-th molecule and the sum is taken over $N_{\ell}$ molecules within the first layer. The effect of periodic surface potential on the structure of the adjacent fluid becomes more pronounced at the reciprocal lattice vectors. In the previous MD study [@Priezjev07] for similar parameters of the wall and fluid phases, it was shown that the slip length scales as $L_s\!\sim\!(\,T_1/S(\mathbf{G}_1)\,\rho_c)^{\alpha}$, where $\mathbf{G}_1$ is the the first reciprocal lattice vector in the flow direction, $T_1$ is temperature of the first fluid layer and $\alpha\,{=}\,1.44\pm0.10$. This scaling relation was found to hold in a wide range of shear rates and wall-fluid interactions for atomically smooth rigid walls and incommensurate structures of the liquid/solid interface [@Priezjev07]. The correlation between the inverse value of the fluid structure factor evaluated at the first reciprocal lattice vector $\mathbf{G}_1\,{=}\,(9.04\,\sigma^{-1},0)$ and the slip length is presented in Fig.\[ls\_S9\_ro\](a). Except for the rough rigid walls, the surface induced structure in the first fluid layer $S(\mathbf{G}_1)$ is reduced at higher shear rates and smaller values of $\kappa$. The slip length increases approximately linearly with $1/S(\mathbf{G}_1)$ for the rigid and stiff walls with $\kappa\geqslant1200\,\varepsilon/\sigma^2$. A gradual transition to a weak dependence of the slip length on $S(\mathbf{G}_1)$ is observed upon reducing the wall stiffness. In Figure \[ls\_S9\_ro\](b) the same data for the slip length are replotted as a function of the inverse product $[S(\mathbf{G}_1)\,\rho_c\,\sigma^3]^{-1}$, where $\rho_c$ is a contact density of the first fluid layer. Although the contact density decreases slightly with increasing the slip velocity, the functional form of the slip length is similar in both cases \[see Figs.\[ls\_S9\_ro\](a)–\[ls\_S9\_ro\](b)\]. The results shown in Fig.\[ls\_S9\_ro\] indicate that the dependence of the slip length on the fluid structure of the first layer is less pronounced for $\kappa\leqslant800\,\varepsilon/\sigma^2$ due the thermal surface roughness in the rate-dependent regime. Whether the scaling relation for the slip length [@Priezjev07] holds in the presence of the thermal roughness for different wall-fluid interaction energies will be the subject of the future research. Results for periodically corrugated walls {#sec:Results_corrugated} ========================================= Next, the results for the rate dependence of the slip length are compared for atomically smooth rigid walls and periodically roughened surfaces. The periodic surface roughness of the upper and the lower walls was modeled by introducing a vertical offset to the positions of the wall atoms $\Delta z(x)\,{=}\,a\sin(2\pi x/\lambda)$ with the wavelength $\lambda\,{=}\,4.17\,\sigma$. In this part of the study, the wall atoms are rigidly fixed with respect to their equilibrium sites. To properly compare the results for atomically smooth and roughened surfaces, both the local shear rate and the slip length were estimated from a parabolic fit of the velocity profiles at the same location of the interface, i.e. $z=\pm\,11.79\,\sigma$. A weaker wall-fluid interaction, $\varepsilon_{\rm wf}/\varepsilon\,{=}\,0.5$, was chosen to obtain larger values of $L_s$ in the absence of the imposed corrugation, since it is expected that the surface roughness strongly reduces the slip length [@Priezjev06]. The dynamic response of the slip length with increasing shear rate is presented in Fig.\[rough\] for atomically smooth and periodically corrugated walls. The data for the flat walls ($a\,{=}\,0$) are the same as in Ref. [@Priezjev07]. At low shear rates, the slip length $L_s^{\circ}$ (defined by the leftmost point on each curve in Fig.\[rough\]) decreases monotonically with increasing the amplitude $a$. The no-slip boundary condition is achieved for $a\gtrsim0.3\,\sigma$. This behavior for a similar geometry and interaction parameters was examined in detail in the previous paper [@Priezjev06]. A direct comparison between continuum analysis and MD simulations showed that there is an excellent agreement between the velocity profiles and the slip length for the large wavelengths $\lambda\gtrsim20\,\sigma$ and small values of $a/\lambda\lesssim0.05$. The continuum results overestimate the slip length when $\lambda$ approaches molecular dimensions [@Priezjev06]. ![(Color online) Variation of the slip length $L_s/\sigma$ as a function of the local shear rate for Poiseuille flows and $\varepsilon_{\rm wf}/\varepsilon\,{=}\,0.5$. The wavelength and amplitudes of the wall modulation are tabulated in the insets. The local shear rate and the slip length are extracted from a parabolic fit of the velocity profiles at $z=\pm\,11.79\,\sigma$.[]{data-label="rough"}](Fig5.eps){width="10.4cm" height="7.6cm"} At higher shear rates, the slope of the rate-dependent slip length is gradually reduced with increasing the amplitude of the surface corrugation (see Fig.\[rough\]). For the largest amplitude $a\,{=}\,0.3\,\sigma$, the slip length weakly depends on shear rate and its magnitude becomes smaller than the molecular diameter. As apparent from the set of curves shown in Fig.\[rough\], the same value of the slip length can be obtained by increasing simultaneously the amplitude of the surface corrugation and the shear rate. Analogous behavior of the slip length was observed experimentally for flows of Newtonian liquids past surfaces with variable nano-roughness [@Granick02]. We note, however, that the MD simulations of simple fluids reported in this study do not show any threshold in the rate dependence of the slip length for the amplitudes of the surface corrugation $a\,{\leqslant}\,0.3\,\sigma$. Summary {#sec:Conclusions} ======= In this paper the effect of molecular-scale surface roughness on the slip length in a flow of simple fluids was studied by molecular dynamics simulations. The parabolic fit of the steady-state velocity profiles induced by a constant force was used to define the values of interfacial shear rate and slip length. For atomically smooth rigid surfaces and weak wall-fluid interactions, the slip length increases approximately linearly with the shear rate. Three types of surface roughness were considered: thermal, random and periodic. The thermal surface roughness due to finite spring stiffness of the wall atoms significantly modifies the slip behavior. The large penetration of the wall atoms into the fluid phase observed for soft walls causes weak rate dependence of the slip length below its values for atomically smooth rigid walls. Increasing the wall stiffness produces effectively smoother surfaces and leads to the linear rate dependence of the slip length. Periodically and randomly corrugated rigid surfaces, with the amplitude below the molecular diameter, strongly reduce the slip length and its shear rate dependence. These findings open perspectives for modeling complex systems with combined effects of surface roughness, wettability and rate dependency. Acknowledgments {#acknowledgments .unnumbered} =============== Financial support from the Michigan State University Intramural Research Grants Program is gratefully acknowledged. Computational work in support of this research was performed at Michigan State University’s High Performance Computing Facility. [80]{} G. E. Karniadakis, A. Beskok, and N. 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--- abstract: 'We prove a four dimensional version of the Bernstein Theorem, with complex polynomials being replaced by quaternionic polynomials. We deduce from the theorem a quaternionic Bernstein’s inequality and give a formulation of this last result in terms of four-dimensional zonal harmonics and Gegenbauer polynomials. [^1]' author: - \| Alessandro Perotti\ Department of Mathematics, University of Trento\ Via Sommarive 14, I-38123 Povo Trento, Italy\ [email protected] date: title: 'A four dimensional Bernstein Theorem' --- Introduction ============ In 1930, S. Bernstein [@Bernstein] proved the following result: Let $p(z)$ and $q(z)$ be two complex polynomials with degree of $p(z)$ not exceeding that of $q(z)$. If $q(z)$ has all its zeros in $\{\|z\|\le1\}$ and $\|p(z)\|\le \|q(z)\|$ for $\|z\|=1$, then $\|p'(z)\|\le \|q'(z)\|$ for $\|z\| = 1$. From this result, the famous Bernstein’s inequality (first established in this form by M. Riesz in 1914) can be deduced. Taking $q(z)=Mz^n$, one obtains the following If $p(z)$ is a complex polynomial of degree $d$ and $\max_{\|z\|=1}\|p(z)\|=M$, then $\|p'(z)\|\le dM$ for $\|z\| = 1$. This note deals with a four dimensional version of such classic results, with complex polynomials being replaced by quaternionic polynomials. The extension of Bernstein’s inequality to the quaternionic setting has already appeared in [@GalSabadini]. The proof given there is based on a quaternionic version of the Gauss-Lucas Theorem. Unfortunately, this last result is valid only for a small class of quaternionic polynomials, as it has been recently showed in [@GaussLucas], where another version of the Gauss-Lucas Theorem, valid for every polynomial, has been proved. Recently, a different proof of the quaternionic Bernstein’s inequality has been given in [@Xu], using the Fejér kernel and avoiding the use of the Gauss-Lucas Theorem. We refer the reader to [@GeStoSt2013] and [@DivisionAlgebras] for definitions and properties concerning the algebra ${{\mathbb{H}}}$ of quaternions and many aspects of the theory of quaternionic slice regular functions, a class of functions which includes polynomials and convergent power series. The ring ${{\mathbb{H}}}[X]$ of quaternionic polynomials is defined by fixing the position of the coefficients with respect to the indeterminate $X$ (e.g. on the right) and by imposing commutativity of $X$ with the coefficients when two polynomials are multiplied together (see e.g. [@Lam §16]). Given two polynomials $P,Q\in{{\mathbb{H}}}[X]$, let $P{\cdot}Q$ denote the product obtained in this way. A direct computation (see [@Lam §16.3]) shows that if $P(x)\ne0$, then $$\label{product} (P{\cdot}Q)(x)=P(x)Q(P(x)^{-1}xP(x)),$$ while $(P{\cdot}Q)(x)=0$ if $P(x)=0$. In particular, if $P$ has real coefficients, then $(P{\cdot}Q)(x)=P(x)Q(x)$. In this setting, a [(left) root or zero]{} of a polynomial $P(X)=\sum_{h=0}^dX^h a_h$ is an element $x\in{{\mathbb{H}}}$ such that $P(x)=\textstyle\sum_{h=0}^dx^h a_h=0$. A subset $A$ of ${{\mathbb{H}}}$ is called circular if, for each $x\in A$, $A$ contains the whole set (a 2-sphere if $x\not\in{{\mathbb{R}}}$, a point if $x\in{{\mathbb{R}}}$) $$\label{sx} {{\mathbb{S}}}_x=\{pxp^{-1}\in{{\mathbb{H}}}\;\|\;p\in{{\mathbb{H}}}^\},$$ where ${{\mathbb{H}}}^:={{\mathbb{H}}}\setminus\{0\}$. In particular, for any imaginary unit $I\in{{\mathbb{H}}}$, ${{\mathbb{S}}}_I={{\mathbb{S}}}$ is the 2-sphere of all imaginary units in ${{\mathbb{H}}}$. It it is well-known (see e.g. [@GeStoSt2013 §3.3]) that if $P\not\equiv0$, the zero set $V(P)$ consists of isolated points or isolated 2-spheres of the form . We show that the quaternionic version of Theorem (A) holds true after imposing a necessary assumption on the second polynomial. We require that $Q\in{{\mathbb{H}}}[X]$ has every coefficients belonging to a fixed subalgebra of ${{\mathbb{H}}}$. This restricted version of the Bernstein Theorem is however sufficient to deduce the quaternionic Bernstein’s inequality, i.e. the analog of Theorem (B). In Section \[sec:zonal\], we restate the inequality in terms of four-dimensional zonal harmonics and Gegenbauer polynomials. To obtain this form, we use results of [@Harmonicity] to obtain an Almansi type decomposition of a quaternionic polynomial. Bernstein Theorem and inequality ================================ Let $I\in{{\mathbb{S}}}$ and let ${{\mathbb{C}}}_I\subset{{\mathbb{H}}}$ be the real subalgebra generated by $I$, i.e. the complex plane generated by 1 and $I$. If ${{\mathbb{C}}}_I$ contains every coefficient of $P\in{{\mathbb{H}}}[X]$, then we say that $P$ is a ${{\mathbb{C}}}_I$-polynomial. Every ${{\mathbb{C}}}_I$-polynomial $P$ is one-slice-preserving, i.e. $P({{\mathbb{C}}}_I)\subseteq{{\mathbb{C}}}_I$. If this property holds for two imaginary units $I,J$, with $I\ne\pm J$, then it holds for every unit and $P$ is called slice-preserving. This happens exactly when all the coefficients of $P$ are real. Let $P(X)=\sum_{k=0}^dX^ka_k\in{{\mathbb{H}}}[X]$ of degree $d\geq1$. Let $P'(X)=\sum_{k=1}^dX^{k-1}ka_k$ be the derivative of $P$. For every $I\in{{\mathbb{S}}}$, let $\pi_I:{{\mathbb{H}}}\to{{\mathbb{H}}}$ be the orthogonal projection onto ${{\mathbb{C}}}_I$ and $\pi_I^\bot=id-\pi_I$. Let $P^I(X):=\sum_{k=1}^dX^ka_{k,I}$ be the ${{\mathbb{C}}}_I$-polynomial with coefficients $a_{k,I}:=\pi_I(a_k)$. We denote by ${{\mathbb{B}}}=\{x\in{{\mathbb{H}}}\,\|\,\|x\|<1\}$ the unit ball in ${{\mathbb{H}}}$ and by ${{\mathbb{S}}}^3=\{x\in{{\mathbb{H}}}\,\|\,\|x\|=1\}$ the unit sphere. \[thm:Bernstein\] Let $P,Q\in{{\mathbb{H}}}[X]$ be two quaternionic polynomials with degree of $P$ not exceeding that of $Q$. Assume that there exists $I\in{{\mathbb{S}}}$ such that $Q$ is a ${{\mathbb{C}}}_I$-polynomial. If $V(Q)\subseteq\overline{{\mathbb{B}}}$ and $\|P(x)\|\le\|Q(x)\|$ for $x\in{{\mathbb{S}}}^3$, then $\|P'(x)\|\le\|Q'(x)\|$ for $x\in{{\mathbb{S}}}^3\cap{{\mathbb{C}}}_I$. Let $\lambda\in{{\mathbb{H}}}$ with $\|\lambda\|>1$ and set $R:=Q-P\lambda^{-1}\in{{\mathbb{H}}}[X]$. The polynomials $Q$ and $R^I=Q-(P\lambda^{-1})^I$ are ${{\mathbb{C}}}_I$-polynomials and then they can be identified with elements of ${{\mathbb{C}}}_I[X]$, with $\deg(R^I)\le\deg(Q)$. For every $x\in{{\mathbb{C}}}_I$, it holds $$\|R^I(x)-Q(x)\|=\|(P\lambda^{-1})^I(x)\|=\|\pi_I((P\lambda^{-1})(x))\|\le\|(P\lambda^{-1})(x)\|=\frac{\|P(x)\|}{\|\lambda\|}.$$ If $x\in{{\mathbb{S}}}^3\cap{{\mathbb{C}}}_I=\{x\in{{\mathbb{C}}}_I\,\|\,\|x\|=1\}$, then $$\label{eq:inequality} \|R^I(x)-Q(x)\|\le\frac{\|P(x)\|}{\|\lambda\|}\le\frac{\|Q(x)\|}{\|\lambda\|}\le \|Q(x)\|.$$ In view of Rouché’s Theorem for polynomials in ${{\mathbb{C}}}_I[X]$, $R^I$ and $Q$ have the same zeros in the disc $\{x\in{{\mathbb{C}}}_I\,\|\,\|x\|<1\}$. Moreover, if $\|x\|=1$ and $Q(x)=0$, the inequality gives $R^I(x)=0$. Since $\deg(R^I)\le\deg(Q)$ and $V(Q)\subseteq\overline{{\mathbb{B}}}$, we get that $V(R^I)\cap{{\mathbb{C}}}_I\subseteq \overline{{\mathbb{B}}}\cap{{\mathbb{C}}}_I$. From the classic Gauss-Lucas Theorem, we get $V(R')\cap{{\mathbb{C}}}_I\subseteq V((R^I)')\cap{{\mathbb{C}}}_I\subseteq \overline{{\mathbb{B}}}\cap{{\mathbb{C}}}_I$. Now let $x\in{{\mathbb{C}}}_I$ with $\|x\|>1$ be fixed and define $\lambda:=Q'(x)^{-1}P'(x)\in{{\mathbb{H}}}$. Observe that $Q'(x)\ne0$ again from the classic Gauss-Lucas Theorem applied to the polynomial $Q$ considered as element of ${{\mathbb{C}}}_I[X]$. If $\|\lambda\|>1$, the polynomial $R=Q-P\lambda^{-1}\in{{\mathbb{H}}}[X]$ defined as above has zero derivative at $x$: $R'(x)=Q'(x)-P'(x)\lambda^{-1}=0$, contradicting what obtained before. Therefore it must be $\|\lambda\|\le1$, i.e. $\|P'(x)\|/\|Q'(x)\|\le1$ for all $x\in{{\mathbb{C}}}_I$ with $\|x\|>1$. By continuity, $\|P'(x)\|\le\|Q'(x)\|$ for all $x\in{{\mathbb{C}}}_I$ with $\|x\|=1$. We recall that a quaternionic polynomial, as any slice regular function, satisfies the maximum modulus principle [@GeStoSt2013 Theorem 7.1]. Let $$\\|P\\|=\max_{\|x\|=1}\|P(x)\|=\max_{\|x\|\le1}\|P(x)\|$$ denote the sup-norm of the polynomial $P\in{{\mathbb{H}}}[X]$ on ${{\mathbb{B}}}$. \[cor:inequality\] If $P\in{{\mathbb{H}}}[X]$ is a quaternionic polynomial of degree $d$, then $\\|P'\\|\le d\\|P\\|$. Let $M=\\|P\\|$ and apply the previous theorem to $P(X)$ and $Q(X)=MX^d$. Since $Q$ is slice-preserving, the thesis of Theorem \[thm:Bernstein\] holds for every $I\in{{\mathbb{S}}}$. The inequality of Corollary \[cor:inequality\] is best possible with equality holding if and only if $P$ is a multiple of the power $X^d$. If $P\in{{\mathbb{H}}}[X]$ is a quaternionic polynomial of degree $d$, and $\|P'(y)\|= d\\|P\\|$ at a point $y\in{{\mathbb{S}}}^3$, then $P(X)=X^da$, with $a\in{{\mathbb{H}}}$, $\|a\|=\\|P\\|$. We can assume that $P(X)$ is not constant. Let $b=P'(y)^{-1}$ and set $Q(X):=P(X)b=\sum_{k=1}^dX^ka_k$. Then $Q'(y)=1$, $\\|Q\\|=1/d$ and $\\|Q'\\|\le1$. Let $I\in{{\mathbb{S}}}$ such that ${{\mathbb{C}}}_I\ni y$. Then $$\textstyle 1=Q'(y)=\sum_k ky^{k-1}a_k=\pi_I(Q'(y))=\sum_k ky^{k-1}\pi_I(a_k)=(Q^I)'(y).$$ If $x\in{{\mathbb{C}}}_I\cap{{\mathbb{S}}}^3$, it holds $$\textstyle\big\|(Q^I)'(x)\big\|=\big\|\sum_k kx^{k-1}\pi_I(a_k)\big\|=\big\|\pi_I\big(\sum_k kx^{k-1}a_k\big)\big\|\le\big\|\sum_k kx^{k-1}a_k\big\|=\|Q'(x)\|\le1.$$ This means that the ${{\mathbb{C}}}_I$-polynomial $Q^I$, considered as an element of ${{\mathbb{C}}}_I[X]$, satisfies the equality in the classic Bernstein’s inequality. The same inequality implies that $$1=\max_{x\in{{\mathbb{C}}}_I\cap{{\mathbb{S}}}^3}\|(Q^I_{\|{{\mathbb{C}}}_I})'(x)\|\le d\max_{x\in{{\mathbb{C}}}_I\cap{{\mathbb{S}}}^3}\|Q^I_{\|{{\mathbb{C}}}_I}(x)\|\le d\\|Q\\|=1,$$ i.e. $\max_{x\in{{\mathbb{C}}}_I\cap{{\mathbb{S}}}^3}\|Q^I_{\|{{\mathbb{C}}}_I}(x)\|=1/d$. Therefore the restriction of $Q^I$ to ${{\mathbb{C}}}_I$ coincides with the function $x^dc$, with $c\in{{\mathbb{C}}}_I$, $\|c\|=1/d$: $$Q^I(x)=\sum_{k=1}^d x^k\pi_I(a_k)=x^dc \text{\quad for every $x\in{{\mathbb{C}}}_I$}.$$ This implies that $\pi_I(a_d)=c$, $\pi_I(a_k)=0$ for each $k=1,\ldots,d-1$ and $Q$ can be written as $Q(X)=X^dc+\widetilde Q(X)$, with the coefficients of $\widetilde Q$ belonging to ${{\mathbb{C}}}_I^\bot=\pi_I^\bot({{\mathbb{H}}})$. When $x\in{{\mathbb{C}}}_I\cap{{\mathbb{S}}}^3$, $\widetilde Q(x)\in{{\mathbb{C}}}_I^\bot$, and then $$\frac1{d^2}\ge\|Q(x)\|^2=\|x^dc\|^2+\|\widetilde Q(x)\|^2=\frac1{d^2}+\|\widetilde Q(x)\|^2.$$ This inequality forces $\widetilde Q$ to be the zero polynomial and then $P(X)=Q(X)b^{-1}=X^d cb^{-1}$. We now show that in Theorem \[thm:Bernstein\], the assumption on $Q$ to be one-slice-preserving is necessary. \[counterexample\] Let $$P(X)=(X-i)\cdot(X-j)\cdot(X-k),\quad Q(X)=2X\cdot(X-i)\cdot (X-j).$$ Then $V(Q)=\{0,i\}\subseteq\overline{{\mathbb{B}}}$ and $\|P(x)\|\le\|Q(x)\|$ for every $x\in{{\mathbb{S}}}^3$, but there exists $y\in{{\mathbb{S}}}^3$ such that $\|P'(y)\|>\|Q'(y)\|$. By a direct computation we obtain: $$\begin{aligned} P(X)&=X^3-X^2(i+j+k)+X(i-j+k)+1,\quad Q(X)=2X^3-2X^2(i+j)+2Xk,\\ P'(X)&=3X^2-2X(i+j+k)+i-j+k,\quad Q'(X)=6X^2-4X(i+j)+2k.\end{aligned}$$ Let $P_1(X)=X-k$, $Q_1(X)=2X$, $P_2(X)=(X-j)\cdot P_1(X)$, $Q_2(X)=(X-j)\cdot Q_1(X)$. Then $P(X)=(X-i)\cdot P_2(X)$ and $Q(X)=(X-i)\cdot Q_2(X)$. For every $x\in{{\mathbb{S}}}^3\setminus\{j\}$, using formula we get $$\|P_2(x)\|=\|x-j\|\|(x-j)^{-1}x(x-j)-k\|\le2\|x-j\|=\|x-j\|\|2x\|=\|Q_2(x)\|.$$ Since $P_2(j)=Q_2(j)=0$, the inequality holds also at $j$. From this we obtain, for each $x\in{{\mathbb{S}}}^3\setminus\{i\}$, $$\|P(x)\|=\|x-i\|\|P_2((x-i)^{-1}x(x-i))\|\le \|x-i\|\|Q_2((x-i)^{-1}x(x-i))\|=\|Q(x)\|.$$ Since $P$ and $Q$ vanish at $i$, $\|P(x)\|\le\|Q(x)\|$ for every $x\in{{\mathbb{S}}}^3$. Let $y=\frac1{10}\left(1+9i+4j-\sqrt2k\right)\in{{\mathbb{S}}}^3$. An easy computation gives $$\|P'(y)\|^2=\frac7{25}(5+\sqrt2)\simeq 1.80, \quad \|Q'(y)\|^2=\frac4{25}(10-3\sqrt2)\simeq 0.92.$$ Bernstein inequality and zonal harmonics {#sec:zonal} ======================================== Since the restriction of a complex variable power $z^m$ to the unit circumference is equal to $\cos(m\theta)+i\sin(m\theta)$, the classic Bernstein inequality for complex polynomials can be restated in terms of trigonometric polynomials. In this section we show that a similar interpretation is possible in four dimensions, by means of an Almansi type decomposition of quaternionic polynomials and its relation with zonal harmonics in ${{\mathbb{R}}}^4$. Quaternionic polynomials, as any slice regular function, are biharmonic with respect to the standard Laplacian of ${{\mathbb{R}}}^4$ [@Harmonicity Theorem 6.3]. In view of Almansi’s Theorem (see e.g. [@Aronszajn Proposition 1.3]), the four real components of such polynomials have a decomposition in terms of a pair of harmonic functions. The results of [@Harmonicity] can be applied to obtain a refined decomposition of the polynomial in terms of the quaternionic variable. Let $\mathcal Z_{k}(x,a)$ denote the four-dimensional (solid) zonal harmonic of degree $k$ with pole $a\in{{\mathbb{S}}}^3$ (see e.g. [@HFT Ch.5]). The symmetry properties of zonal harmonics imply that $\mathcal Z_{k-1}(x,a)=\mathcal Z_{k-1}(x\overline a,1)$ for every $a\in{{\mathbb{H}}}$ and any $a\in{{\mathbb{S}}}^3$. Moreover it holds [@Harmonicity Corollary 6.7(d)] $$\label{eq:powers} x^k=\widetilde{\mathcal Z}_k(x)-\overline x\, \widetilde{\mathcal Z}_{k-1}(x)\text{\quad for every $x\in{{\mathbb{H}}}$ and $k\in{{\mathbb{N}}}$},$$ where ${\widetilde{\mathcal Z}}_k(x)$ is the real-valued zonal harmonic defined by ${\widetilde{\mathcal Z}}_k(x):=\frac1{k+1}{\mathcal Z}_k(x,1)$ for any $k\ge0$ and by $\widetilde{\mathcal Z}_{-1}:=0$. In the following we will consider polynomials in the four real variables $x_0,x_1,x_2,x_3$ of the form $A(x)=\sum_{k=0}^d\widetilde{\mathcal Z}_k(x) a_k$, with quaternionic coefficients $a_k\in{{\mathbb{H}}}$. They will be called zonal harmonic polynomials with pole 1. All these polynomials have an axial symmetry with respect to the real axis: for every orthogonal transformation $T$ of ${{\mathbb{H}}}\simeq{{\mathbb{R}}}^4$ fixing 1, it holds $A\circ T=A$. \[teo:Almansi\] Let $P\in{{\mathbb{H}}}[X]$ be a quaternionic polynomial of degree $d$. There exist two zonal harmonic polynomials $A$, $B$ with pole $1$, of degrees $d$ and $d-1$ respectively, such that $$P(x)=A(x)-\overline x B(x)\text{\quad for every $x\in{{\mathbb{H}}}$.}\label{eq:Almansi}$$ The restrictions of $A$ and $B$ to the unit sphere ${{\mathbb{S}}}^3$ are spherical harmonics depending only on $x_0={\operatorname{Re}}(x)$. Let $P(X)=\sum_{k=0}^dX^kc_k$. Formula follows immediately from setting $$A(x)=\sum_{k=0}^d\widetilde{\mathcal Z}_k(x)c_k\text{\quad and\quad}B(x)=\sum_{k=0}^{d-1}\widetilde{\mathcal Z}_{k}(x)c_{k+1}.$$ The restriction of $\widetilde{\mathcal Z}_k(x)$ to the unit sphere ${{\mathbb{S}}}^3$ is equal to the Gegenbauer (or Chebyshev of the second kind) polynomial $C^{(1)}_{k}(x_0)$, where $x_0={\operatorname{Re}}(x)$ (see [@Harmonicity Corollary 6.7(e)]). This property implies immediately the last statement. The zonal harmonics $A$ and $B$ of the previous decomposition can be obtained from $P$ through differentiation. Since $P(x)-P(\overline x)=A(x)-\overline xB(x)-A(\overline x)+xB(\overline x)=2{\operatorname{Im}}(x)B(x)$, the function $B$ is the spherical derivative of $P$, defined (see [@GhPe_AIM]) on ${{\mathbb{H}}}\setminus{{\mathbb{R}}}$ as $P'_s(x)=(2{\operatorname{Im}}(x))^{-1}(P(x)-P(\overline x))$. In [@Harmonicity] it was proved that the spherical derivative of a slice regular function, in particular of a quaternionic polynomial, is indeed the result of a differential operation. Given the Cauchy-Riemann-Fueter operator $${\overline{\partial}_{\scriptscriptstyle CRF}}=\dd{}{x_0}+i\dd{}{x_1}+j\dd{}{x_2}+k\dd{}{x_3},$$ it holds ${\overline{\partial}_{\scriptscriptstyle CRF}}P=-2{P'_s}$. Therefore $$\label{eq:AB} A(x)=P(x)-\frac12{\overline x}\,{\overline{\partial}_{\scriptscriptstyle CRF}}P(x),\quad B(x)=-\frac12{\overline{\partial}_{\scriptscriptstyle CRF}}P(x).$$ Defining $A$ and $B$ by formulas and using results from [@Harmonicity], it can be easily seen that the Almansi type decomposition $f(x)=A(x)-\overline xB(x)$ holds true for every slice regular function $f$, with $A$ and $B$ harmonic and axially symmetric w.r.t. the real axis. Observe that $B=f'_s$ is the spherical derivative of $f$ and $A=f^\circ_s+x_0f'_s$, where $f^\circ_s(x)=\frac12(f(x)+f(\overline x))$ is the spherical value of $f$ (see [@GhPe_AIM]). Thanks to the previous decomposition, the quaternionic Bernstein inequality of Corollary can be restated in terms of Gegenbauer polynomials $C^{(1)}_{k}(x_0)$. Let $d\in{{\mathbb{N}}}$. For any $(d+1)$-uple $\alpha=(a_0,\ldots,a_d)\in{{\mathbb{H}}}^{d+1}$, let $Q_\alpha:{{\mathbb{S}}}^3\to{{\mathbb{H}}}$ be defined by $$Q_\alpha(x):=\sum_{k=0}^d (C^{(1)}_{k}(x_0)-\overline x\,C^{(1)}_{k-1}(x_0))a_k$$ for any $x=x_0+ix_1+jx_2+kx_3\in{{\mathbb{S}}}^3$ (where we set $C^{(1)}_{-1}:=0$). Being the restriction to ${{\mathbb{S}}}^3$ of the quaternionic polynomial $P(X)=\sum_{k=0}^dX^ka_k$, which has biharmonic real components on ${{\mathbb{H}}}$, $Q_\alpha$ is a quaternionic valued spherical biharmonic of degree $d$ (see e.g. [@GrzebulaMichalik]). Let $\alpha=(a_0,\ldots,a_d)$ and $\alpha'=(a_1,2a_2,\ldots,ka_k,\ldots,da_d,0)\in{{\mathbb{H}}}^{d+1}$. Then it holds: $$\text{if \quad}\|Q_\alpha(x)\|=\left\|\sum_{k=0}^d \left(C^{(1)}_{k}(x_0)-\overline x\,C^{(1)}_{k-1}(x_0)\right)a_k\right\|\le M\text{\quad for every $x\in{{\mathbb{S}}}^3$},$$ $$\text{then\quad} \|Q_{\alpha'}(x)\|=\left\|\sum_{k=0}^{d-1} \left(C^{(1)}_{k}(x_0)-\overline x\,C^{(1)}_{k-1}(x_0)\right) (k+1)a_{k+1}\right\|\le dM\text{\quad for every $x\in{{\mathbb{S}}}^3$}.$$ Let $P(X)=\sum_{k=0}^dX^ka_k$. From formula it follows that the restriction of $P'$ to the unit sphere is the spherical biharmonic $Q_{\alpha'}$. Corollary \[cor:inequality\] permits to conclude. \[rem:max\] Let $P\in{{\mathbb{H}}}[X]$ be a polynomial with Almansi type decomposition $P(x)=A(x)-\overline xB(x)$ and let $y=\alpha+J\beta\in{{\mathbb{S}}}^3$, $\alpha,\beta\in{{\mathbb{R}}},\beta>0$. Let $v=A(y)\overline{B(y)}$. It follows from general properties of slice functions [@DivisionAlgebras Lemma 5.3] that if $v\in{{\mathbb{R}}}$, then $\|P\|_{\|{{\mathbb{S}}}_y}$ is constant, while if $v\not\in{{\mathbb{R}}}$, then the maximum modulus of $P$ on the 2-sphere ${{\mathbb{S}}}_y\subset{{\mathbb{S}}}^3$ is attained at the point $\alpha+I\beta$, with $I={\operatorname{Im}}(v)/\|{\operatorname{Im}}(v)\|$, while the minimum modulus is attained at $\alpha-I\beta$. In principle, this reduces the problem of maximizing or minimizing the modulus of $P$ on the unit sphere (or ball) to a one-dimensional problem. Consider the polynomial $P(X)=(X-i)\cdot(X-j)\cdot(X-k)$ of Proposition \[counterexample\]. Since the first four zonal harmonics are $${\widetilde{\mathcal Z}}_0(x)=1,\ {\widetilde{\mathcal Z}}_1(x)= 2 x_0,\ {\widetilde{\mathcal Z}}_2(x)=3 x_0^2 - x_1^2 - x_2^2 - x_3^2,\ {\widetilde{\mathcal Z}}_3(x)=4 x_0 (x_0^2 - x_1^2 - x_2^2 - x_3^2),$$ the Almansi type decomposition of $P$ is $P(x)=A(x)-\overline xB(x)$, with $$\begin{aligned} A(x)&=(1 + 4 x_0^3 - 4 x_0 x_1^2 - 4 x_0 x_2^2 - 4 x_0 x_3^2)+(i+j+k)(2 x_0 - 3 x_0^2 + x_1^2 + x_2^2 + x_3^2), \\ B(x)&=(3 x_0^2 - x_1^2 - x_2^2 - x_3^2)+i(1-2x_0)-j (1 + 2 x_0) +k (1 - 2 x_0)\end{aligned}$$ harmonic polynomials. Their restrictions to ${{\mathbb{S}}}^3$ are the spherical harmonics $$\begin{aligned} A_{\|{{\mathbb{S}}}^3}(x)&=(1 - 4 x_0 + 8 x_0^3)+i(1 + 2 x_0 - 4 x_0^2)+j(1 - 2 x_0 - 4 x_0^2)+k(1 + 2 x_0 - 4 x_0^2) , \\ B_{\|{{\mathbb{S}}}^3}(x)&=(-1+ 4 x_0^2)+i(1 - 2 x_0)-j(1 + 2 x_0)+k(1 - 2 x_0).\end{aligned}$$ Following the observation made in Remark \[rem:max\], since ${\operatorname{Im}}(A(y)\overline{B(y)})=4((\alpha-1)i+\alpha k)$, where $\alpha={\operatorname{Re}}(y)$, $y\in{{\mathbb{S}}}^3$, one can find the 2-sphere ${{\mathbb{S}}}_y\subset{{\mathbb{S}}}^3$ where the maximum modulus of $P$ is attained. A direct computation gives ${\operatorname{Re}}(y)=(1-\sqrt{19})/6\sim-0.56$ and the corresponding maximum value $\\|P\\|\sim4.70$ attained at the point $\tilde y=(1-\sqrt{19})/6-i(5+\sqrt{19})/12+k(1-\sqrt{19})/12$ of ${{\mathbb{S}}}^3$. Some of the results presented in this note can be generalized to the general setting of real alternative \-algebras, where polynomials can be defined and share many of the properties valid on the quaternions (see [@GhPe_AIM]). The polynomials of Proposition \[counterexample\] can be defined every time the algebra contains an Hamiltonian triple $i,j,k$, i.e. when the algebra contains a subalgebra isomorphic to ${{\mathbb{H}}}$ (see [@Numbers §8.1]). This is true e.g. for the algebra of octonions and for the Clifford algebras with signature $(0,n)$, with $n\ge2$. In all such algebras we can repeat the previous proofs and get the analog of Theorem \[thm:Bernstein\], as well as of the Bernstein inequality (see also [@Xu] for this last result). [1]{} N. Aronszajn, T. M. Creese and L. J. Lipkin. , The Clarendon Press, Oxford University Press, New York, 1983. S. Axler, P. Bourdon, and W. Ramey, , , Vol. 137, [Springer-Verlag, New York]{}, 1992. S. N. Bernstein. Sur la limitation des derivees des polynômes. , 190:338–340, 1930. H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, and R. Remmert. , volume 123 of [Graduate Texts in Mathematics]{}. Springer-Verlag, New York, 1990. S. G. Gal, I. Sabadini, On Bernstein and Erdös-Lax’s inequalities for quaternionic polynomials, , 353(1):5–9, 2015. G. Gentili, C. Stoppato, and D. C. Struppa. . Springer Monographs in Mathematics. Springer, 2013. R. Ghiloni and A. Perotti. Slice regular functions on real alternative algebras. , 226(2):1662–1691, 2011. R. Ghiloni and A. Perotti. The quaternionic Gauss-Lucas Theorem. , 197(6):1679–1686, 2018. R. Ghiloni, A. Perotti, and C. Stoppato. ivision algebras of slice functions. To appear in [Proceedings A of the Royal Society of Edinburgh]{}. , 2019, <http://arxiv.org/abs/1711.06604>, H. Grzebuła and S. Michalik. Spherical polyharmonics and Poisson kernels for polyharmonic functions. , 64:3, 420–442, 2019. T. Y. Lam. , volume 131 of [ Graduate Texts in Mathematics]{}. Springer-Verlag, New York, 1991. A. Perotti. , to appear in , Trends Math. Birkhäuser, Basel, 2019. , <http://arxiv.org/abs/1801.03045> Z. Xu. The Bernstein inequality for slice regular polynomials. . [^1]: [Mathematics Subject Classification (2010)]{}. Primary 30G35; Secondary 26D05, 33C50.\ [Keywords:*]{} Bernstein Theorem, Bernstein inequality, Quaternionic polynomials, Zonal harmonics
--- abstract: 'In this paper we present the simplest individual level model of predator-prey dynamics and show, via direct calculation, that it exhibits cycling behavior. The deterministic analogue of our model, recovered when the number of individuals is infinitely large, is the Volterra system (with density-dependent prey reproduction) which is well-known to fail to predict cycles. This difference in behavior can be traced to a resonant amplification of demographic fluctuations which disappears only when the number of individuals is strictly infinite. Our results indicate that additional biological mechanisms, such as predator satiation, may not be necessary to explain observed predator-prey cycles in real (finite) populations.' author: - 'A. J. McKane' - 'T. J. Newman' title: 'Predator-prey cycles from resonant amplification of demographic stochasticity' --- Predator-prey cycles are one of the most striking phenomena observed in population biology, and as such, inspire intense discussion among ecologists [@ber02; @tur03]. Cycles are also seen in a wide variety of other “host-natural enemy” systems, such as host-pathogen [@and86] systems – one of the most well known examples is measles, epidemics of which have been studied for many years [@and91]. In this paper, we will be concerned with modeling the phenomenon of cycles, and will focus on predator-prey systems, for concreteness, but our main results will have direct applicability to other host-natural enemy systems, since they can be modeled in a similar way, often using identical equations. We also believe that, since the phenomenon we describe is quite generic in certain classes of stochastic systems, it should be found outside population dynamics. It seems that the precise mechanism underlying the existence of the cycles has not so far been elucidated because it involves the analysis of stochastic systems with a large, but finite, number of constituents, and also because it involves concepts such as resonance, which are more familiar to physicists than biologists. Among the numerous hypotheses put forward to explain cycles, perhaps the simplest is that cycles arise directly from predator-prey interactions. Within this conceptual framework, theoretical modeling of cycles has traditionally been developed using deterministic population-level models (PLMs). Such discussions begin with Volterra-like equations, which are coupled differential equations for the predator and prey densities. Equations of this type encapsulate the simplest processes of predator and prey mortality, prey reproduction and competition, and predation. Surprisingly these models do not predict stable cycles: additional biological mechanisms, such as predator satiation, need to be included within the framework of differential equations to give cycles [@may74]. It seems puzzling that cycles, which are so easy to understand intuitively, can only be described mathematically in models which include these more subtle mechanisms. In order to probe this issue, we shall take a different approach here, and describe the predator-prey system using an individual level model (ILM). The individuals, which are either predators or prey, are acted upon by simple stochastic processes of mortality, reproduction, and predation. We are able to derive an exact description of this model when the number of individuals is large and finite. We find that the predator and prey numbers undergo large cycles, just as one would expect intuitively. The cycles, which arise from a novel resonance effect, disappear only when the number of individuals is taken to be strictly infinite, that is, when the PLM is recovered. From a statistical physics viewpoint, we would term the PLM a mean field theory of the underlying “microscopic” ILM which includes statistical fluctuations. Predator-prey cycles observed in nature will have a stochastic component — this will affect both their amplitude and phase. Therefore care must be taken in averaging over replicates. A direct average of the population densities from different replicates will result in a constant average density since, in the absence of an external “forcing”, there is a lack of synchrony between the cycles from different replicates. This fact is crucial when modeling predator-prey cycles. A given PLM is written in terms of a population density, which can be thought of as the result of an average of the population numbers from a large number of ILM replicates. If a given ILM shows oscillatory behavior, such cycles will be lost in the modeling transition to a PLM. Thus, it is necessary to study quantities such as the autocorrelation function and power spectrum arising from an ensemble of ILMs, in order to determine the presence and properties of predator-prey cycles. The specific ILM we study in this paper is a non-spatial stochastic model. At a given time, a realization of the ILM consists of $n$ individuals of species $A$ (the predators) and $m$ individuals of species $B$ (the prey). Since we are interested in what is essentially the simplest model of predator-prey interactions, we include only birth processes $BE \stackrel {b}{\rightarrow } BB$, death processes $A \stackrel {d_{1}}{\rightarrow } E$, $B \stackrel {d_{2}}{\rightarrow } E$, and predator-prey interactions $AB \stackrel {p_{1}} {\rightarrow } AA$, $AB \stackrel {p_{2}}{\rightarrow } AE$. Here $(b, d_{1},d_{2}, p_{1}, p_{2})$ are rate constants. The symbol $E$ corresponds to what would be available sites in a spatial model. In this non-spatial model, the $E$’s are $(N-n-m)$ passive constituents of the system, which are required for prey reproduction, and which result in intra-specific prey competition. Note, the overall number of $A, B$ and $E$ constituents is fixed to be $N$. The dynamics of the model consists of choosing constituents at random and implementing the rules given above. The time dynamics of the model can either be numerically simulated or studied analytically using the formalism of master equations [@ren91; @van92]. In the latter case the transition rates $T(n',m'\|n,m)$ from the state $(n,m)$ to the state $(n',m')$ are given by $$\begin{aligned} T(n-1,m\|n,m) &=& d_{1} n\,, \nonumber \\ T(n,m+1\|n,m) &=& 2 b \frac{m}{N} (N - n - m)\,, \nonumber \\ T(n,m-1\|n,m) &=& 2 p_{1} \frac{n m}{N} + d_{2} m\,, \nonumber \\ T(n+1,m-1\|n,m) &=& 2 p_{2} \frac{n m}{N}\,, \label{trans_rates}\end{aligned}$$ where the $b$ and $d_i$ have been scaled by a factor of $(N-1)$ and the $d_i$ by a factor of $N$. We stress that the individuals of a given species in our model are identical, and thus the term ILM should not be confused with “agent based models” which are often designed to study the ecological effects of behavioral and physiological variation among individuals. We have already given an extensive discussion of this approach elsewhere in the context of competition models [@mck04], and we refer the reader to this paper for a fuller discussion of the formalism. For predator-prey models however, including stochastic dynamics leads to more marked effects than in competition models, as we now discuss. The master equation for the probability that the system consists of $n$ predators and $m$ prey at time $t$, $P(n,m,t)$, is $$\begin{aligned} \frac{dP(n,m,t)}{dt} &=& ( {\cal E}_{x} - 1 )\, \left[ T(n-1,m\|n,m)P(n,m,t) \right] \nonumber \\ + ( {\cal E}^{-1}_{y} &-& 1 )\, \left[ T(n,m+1\|n,m)P(n,m,t) \right] \nonumber \\ + ( {\cal E}_{y} &-& 1 )\, \left[ T(n,m-1\|n,m)P(n,m,t) \right] \nonumber \\ + ( {\cal E}^{-1}_{x} {\cal E}_{y} &-& 1 )\, \left[ T(n+1,m-1\|n,m)P(n,m,t) \right]\,, \label{master}\end{aligned}$$ where the step operators $\cal E$ are defined by their actions on functions of $n$ and $m$ by ${\cal E}^{\pm 1}_{x}f(n,m,t) = f(n \pm 1,m,t)$ and ${\cal E}^{\pm 1}_{y}f(n,m,t) = f(n,m \pm 1,t)$. The mean field limit of this ILM may be obtained by multiplying (\[master\]) by $n$ and $m$ in turn, and subsequently summing over all allowed values of $m$ and $n$. This gives equations for the mean values $f_{1}=\langle n \rangle/N$ and $f_{2}=\langle m \rangle/N$ in the limit $N \to \infty$ if we ignore terms which are $1/N$ down on others and make the replacements $\langle m^{2} \rangle \rightarrow \langle m \rangle^{2}$ and $\langle m n \rangle \rightarrow \langle m \rangle \langle n \rangle$. This mean field theory, or PLM, takes the form $$\begin{aligned} \frac{df_1}{dt} &=& n(f_{2})f_{1} - \mu f_{1} \nonumber \\ \frac{df_2}{dt} &=& rf_{2}\left( 1 - \frac{f_{2}}{K} \right) - g(f_{2})f_{1}\,. \label{Volterra}\end{aligned}$$ The eqs. (\[Volterra\]) are frequently referred to as the Volterra equations, to distinguish them from the Lotka-Volterra equations which have no term in $f_{2}/K$ [@ren91]. The constants $\mu$, $r$, and $K$ are simply functions of the rate constants: $$\mu = d_{1}\,, \ \ r = 2b-d_{2}\,, \ \ K = 1 - \frac{d_{2}}{2b}\,, \label{const_rateconst}$$ and the linear numerical and functional responses are given by $n(f_{2})=2p_{1}f_{2}$ and $g(f_{2})=2(p_{1}+p_{2}+b)f_{2}$ respectively. ![Predator and prey densities as a function of time. The upper panel shows the predator density $f_{1}$ for $N=3200$. The blue line is calculated from numerical integration of the mean field Volterra Eqs. (\[Volterra\]). The purple line is the average of the predator density time series from 500 replicates generated from the ILM, and is almost indistinguishable from the mean field solution. The red line is the predator density time series for a single typical replicate. The lower panel is the equivalent plot for the prey density $f_{2}$. Parameter values are $b=0.5$, $d_{1}=0.1$, $d_{2}=0.0$, $p_{1}=0.5$, and $p_{2}=0.1$.[]{data-label="fig1"}](mckane_fig1.eps){width="7.5cm"} As is well known [@ren91], the analysis of this model shows a complete absence of cycles. There is a single fixed point for which the predators and prey have non-zero population sizes. Denoting these stationary values by $f^{(s)}_{1}$ and $f^{(s)}_{2}$, then in terms of the original rate constants they are given by: $$\label{ss} f^{(s)}_{1}= \frac{(2bp_{1}-bd_{1}-p_{1}d_{2})}{2p_{1}(p_{1}+p_{2}+b)},\ f^{(s)}_{2} = \frac{d_{1}}{2p_{1}}\,.$$ The stability of this fixed point may be studied may performing linear stability analysis. This results in a stability matrix which is given by $$A = \left( \begin{array}{cc} 0 & 2p_{1} f^{(s)}_{1} \\ -2 (p_{1}+p_{2}+b) f^{(s)}_{2} & -2b f^{(s)}_{2} \end{array} \right)\,. \label{stability}$$ We have expressed the entries in terms of the fixed point values, since these are manifestly positive, and it is easy to see that the eigenvalues of $A$ both have a negative real part, implying that the fixed point is stable. The entries of $A$ will appear again below in the analysis of the cycling behavior for finite $N$. For now let us simply remark that, while there is no limit cycle in the Volterra system (\[Volterra\]), a limit cycle does exist in the Lotka-Volterra equations (obtained by taking $K \rightarrow \infty$), but it is neutrally stable due to a conserved quantity in the model. This unrealistic behavior disappears with the introduction of a finite carrying capacity, $K$, in (\[Volterra\]), but, as mentioned above, leads to a complete absence of cycling behavior. We will show below that cycles can be found in the ILM, but only when $N$ is finite; the $N \to \infty$ limit which was taken in order to derive the PLM, eliminates the cycles present in the original ILM. To see this, let us first note that the ensemble averaged population density of the ILM, determined from numerical simulations, agrees beautifully with the solution of this deterministic model (Fig. 1, purple and blue lines respectively) showing a decaying oscillatory transient followed by a constant steady-state density, typical of a Volterra system. In marked contrast, individual realizations of the ILM show large persistent cycles (Fig. 1, red line). The amplitude of these oscillations is much larger than the naive estimate based on the law of large numbers. In fact, the oscillations are of order $(1/\sqrt{N})$ as would be expected, but amplified by a very large factor due to a noise-induced resonance effect, as explained below. This cycling behavior can be investigated analytically by extracting an “effective theory” valid for large $N$, through applying a standard method to the master equations, due to van Kampen [@van92]. Essentially the method involves the replacements $n/N = f_{1} + x/\sqrt{N}$ and $m/N = f_{2} + y/\sqrt{N}$ in the transition probabilities that appear in the master equation. By changing from a description based on the (discrete) variables $n$ and $m$ to one based on the (continuous) variables $x$ and $y$, terms of different orders in $1/N$ can be identified in the master equation: the leading order terms gives rise to a deterministic set of equations and the next-to-leading order terms give rise to a linear Fokker-Planck equation. The leading order set of equations (mean field theory) are the PLM and are the Volterra equations (\[Volterra\]) which we have already obtained by a more direct method. At next-to-leading order, rather than write down the Fokker-Planck equation, it is simpler to give the set of Langevin equations to which it is equivalent [@van92]. They take the form $$\begin{aligned} \dot{x} &=& a_{11}x+a_{12}y+\eta_{1}(t) \nonumber \\ \dot{y} &=& a_{21}x+a_{22}y+\eta_{2}(t)\,. \label{Langevin}\end{aligned}$$ These are a pair of differential equations which describe the stochastic behavior of the ILM at large $N$: $x(t)$ and $y(t)$ are stochastic corrections to the deterministic behavior of the predator and prey densities respectively, at large but finite $N$. The constants, $a_{ij}$, appearing in Eq.(\[Langevin\]) are exactly the entries of the matrix $A$, Eq. (\[stability\]) found from a linear stability analysis about the non-trivial fixed point of Eq.(\[Volterra\]). The noise covariance matrix $b_{ij}$, which is responsible for generating the large-scale oscillations, cannot be determined from Eq.(\[Volterra\]) and is derived from the master equation using the van Kampen expansion. Since the noise is white, $b_{ij} = \langle \tilde{\eta}_{i} (\omega) \tilde{\eta}_{j} (-\omega) \rangle$ is independent of the frequency $\omega$. The explicit expressions for these constants are $$\begin{aligned} \nonumber b_{11} & = & 2d_{1}f^{(s)}_{1},\\ b_{12} & = & b_{21}= - d_{1} f^{(s)}_{1}, \nonumber \\ b_{22} & = & 2d_{1}(1+p_{2}/p_{1})f^{(s)}_{1} + 2d_{2} f^{(s)}_{2}\,. \label{bij}\end{aligned}$$ As discussed earlier, it is not the average behavior of replicates that interests us, but rather measures which characterize the oscillations. Examining $x$ and $y$ as functions of frequency allow us to determine the nature of the oscillations. ![A plot of the power spectrum $P(\omega )$ for the predator time series, as a function of frequency $\omega $. The red line corresponds to $P$ calculated from 500 replicate runs of the ILM. The blue line is the prediction from our theory, namely Eq. (\[power\]). The parameter values are the same as those described in the caption to Fig. 1. The inset shows the analogous power spectra (data and theory) for the prey time series.[]{data-label="fig2"}](mckane_fig2.eps){width="7.5cm"} To search for oscillations in noisy data, one of the most useful diagnostic tools is the power spectrum $P(\omega ) = \langle \|\tilde{x}(\omega)\|^{2} \rangle$, where $\tilde{x}(\omega)$ is the Fourier transform of $x (t)$. Taking the Fourier transform of Eq. (\[Langevin\]), solving for ${\tilde x}(\omega )$, and averaging its squared modulus, we find $$P(\omega ) = \frac {\alpha + \beta \omega ^{2}} {[(\omega ^{2}-\Omega_{0}^{2})^{2}+\Gamma ^{2}\omega ^{2}]}\,, \label{power}$$ where $\alpha$ and $\beta$ are functions of the ILM rates: $\alpha = b_{11} a_{22}^{2} + 2 b_{12} a_{12} \|a_{22}\| + b_{22} a_{12}^{2}$ and $\beta = b_{11}$. The constants in the denominator have the especially simple forms: $\Omega^{2}_{0}=a_{12} \|a_{21}\|$ and $\Gamma = \|a_{22}\|$. The spectrum predicted by Eq. (\[power\]) gives the blue line shown in Fig. 2. The agreement with the spectrum obtained from simulation of the ILM (red line) is excellent. Note, the naive $O(1/\sqrt{N})$ estimate of the size of stochastic fluctuations corresponds to the zero frequency value of $P(\omega)$. Fig. 2 clearly illustrates the very large amplification of these fluctuations due to the resonance effect. The spectrum given above is reminiscent of that for a simple mechanical system — namely a linear damped harmonic oscillator, with natural frequency $\Omega _{0}$ and driven at frequency $\omega $. In a mechanical oscillator the driving frequency must be tuned to achieve resonance. In the stochastic predator-prey model described here no tuning is necessary. The system is driven by white noise, as shown in Eq. (\[Langevin\]), which covers all frequencies — thus the resonant frequency of the system is excited without tuning. We stress, the noise which drives the system is [not external]{}, but arises from the demographic stochasticity contained in the individual processes which define the model. We also stress that there is no external or environmental stochasticity in our model, and that the resonance phenomenon we report here is not related to “stochastic resonance”. The damping term, represented by the constant $\Gamma $, limits the amplitude of the oscillations. Predator-prey systems for which $\Gamma $ happens to be small will be at risk of extinction through resonant oscillations, despite having large population sizes. The resonant oscillation occurs in the regime $2 a_{12} \|a_{21}\| > a_{22}^{2}$, where the resonant frequency $\omega_{0} = \sqrt{\Omega^{2}_{0} - \Gamma^{2}/2}$ is real. A similar analysis can be carried out to obtain the spectrum for the prey time series. It again has the form Eq. (\[power\]), but now with $\alpha = b_{11} a_{21}^{2}$ and $\beta = b_{22}$. The positions of the peaks for these two power spectra are only weakly dependent on the $\alpha$’s and $\beta$’s, and so they are almost coincident. Predator-prey systems (and related host-pathogen systems) have been studied theoretically for decades. Most of the previous studies have focused on the role of environmental stochasticity, the relevance of non-linear interactions or of spatial effects, to explain the mechanism of cycling [@nis82; @ren91; @kai96; @apa01; @bjo01; @pas01; @pas03]. Some authors have discussed the role that demographic stochasticity may have on cycles [@bar60; @ren91]. Most of this discussion has been qualitative; the nearest to our own discussion was a prescient analysis by Bartlett nearly fifty years ago [@bar60], in which he postulated equations similar to (\[Langevin\]). However, he did not note the existence of a resonance, and so proposed cycles with an amplitude which were not enhanced by this effect, and which were therefore of limited biological interest. The idea that external perturbations with a dominant frequency can entrain the predator-prey dynamics in a cyclic nature is fairly intuitive; the phenomenon we discuss here is more fundamental and less intuitive. The noise in our system is internal — purely a result of the demographic stochasticity inherent in discrete birth, death, and predation events. The key point is that this internal noise has a flat power spectrum in the frequency domain; in other words, it excites all frequencies of the system simultaneously. These excitations are typically of limited interest in a large population of $N$ individuals, since they give rise to small $O(1/\sqrt{N})$ fluctuations about the mean population densities. Such is the case, for example, in competition models [@mck04]. The predator-prey system, and related ones such as epidemic models, are exceptional, in that the equations describing linear fluctuations about the steady-state are susceptible to resonant amplification in the vicinity of an internal frequency $\Omega $, which is a property of the population itself. The internal noise, in exciting all frequencies, automatically resonates the system giving rise to large oscillations in the population densities. This phenomenon is “emergent” in the truest sense of the word. We expect that this resonance mechanism will occur in other stochastic systems in which the mean field theory shows damped oscillations. We thank D. Alonso, J. Antonovics, M. Chubynsky, M. Pascual and J. Vandermeer for useful discussions. We acknowledge the NSF for partial support, under grant DEB-0328267. A. A. Berryman, Population Cycles. (Oxford University Press, Oxford, 2002). P. Turchin, Complex Population Dynamics. (Princeton University Press, Princeton, 2003). R. M. Anderson and R. M. May, Phil. Trans. Roy. Soc. B 314, 533 (1986). R. M. Anderson and R. M. May, Infectious Diseases of Humans. (Oxford University Press, Oxford, 1991). J. Maynard Smith, Models in Ecology. (Cambridge University Press, Cambridge, 1974). E. Renshaw, Modelling biological populations in space and time. (Cambridge University Press, Cambridge, 1991). N. G. van Kampen, Stochastic Processes in Physics and Chemistry. (Elsevier, Amsterdam, 1992). A. J. McKane and T. J. Newman, Phys. Rev E 70, 041902 (2004). R. Nisbet and W. Gurney, Modelling Fluctuating Populations. (Wiley, New York, 1982). V. Kaitala, E. Ranta and J. Lindstrom, J. Anim. Ecol. 65, 249 (1996). J. P. Aparicio and H. G. Solari, Math. Bioscience 169, 15 (2001). O. N. Bj[ø]{}rnstad and B. T. Grenfell, Science 293, 638 (2001). M. Pascual, P. Mazzaga and S. A. Levin, Ecology 82, 2357 (2001). M. Pascual and P. Mazzega, Theor. Popul. Biol. 64, 385 (2003). M. S. Bartlett, Stochastic Population Models (Methuen, London, 1960), p64.
--- abstract: 'The application of cooperative localization in vehicular networks is attractive to improve accuracy and coverage. Conventional distance measurements between vehicles are limited by the need for synchronization and provide no heading information of the vehicle. To address this, we present a cooperative localization algorithm using utilizing -only measurements. Simulation results show that both directional and positional of vehicles can be decreased significantly and converge to a low value in a few iterations. Furthermore, the influence of parameters for the vehicular network, such as vehicle density, communication radius, prior uncertainty and measurements noise, is analyzed.' author: - \| Yibo Wu, Bile Peng, Henk Wymeersch, Gonzalo Seco-Granados,\ Anastasios Kakkavas, Mario H. Castañeda Garcia, and Richard A. Stirling-Gallacher\ Department of Electrical Engineering, Chalmers University of Technology\ Munich Research Center, Huawei Technologies Duesseldorf GmbH\ Department of Telecommunications and Systems Engineering, Universitat Autonoma de Barcelona\ Department of Electrical and Computer Engineering, Technische Universität München bibliography: - 'reference.bib' title: Cooperative Localization with Angular Measurements and Posterior Linearization --- Introduction ============ Vehicular localization with high precision is of great importance for future autonomous driving. Among different possibilities, e.g., [@gleason2009gnss], cooperative localization [@wymeersch2009cooperative] enables the possibility for between vehicles, which can lead to more accurate positioning and increased positioning coverage. In cooperative localization, vehicles use on-board sensors, including 5G front-end, radar and stereo cameras [@de2017survey], to obtain measurements relative to the positions of nearby vehicles. Vehicles exchange information related to relative positions and own position estimates to obtain an approximation of their own posterior distribution. [@kschischang2001factor] is a well-known framework for Bayesian inference that can be applied for the cooperative localization problem [@wymeersch2009cooperative]. Cooperative localization is particularly advantageous when vehicles have different prior localization accuracy, because vehicles with high-quality sensors can help vehicles with low quality sensors to reduce their localization errors. The last point is practical in the foreseeable future because vehicles with different levels of sensing precision are expected to coexist[@steinmetz2019theoretical]. The performance of any localization system is limited by the underlying measurements. Conventional measurements include distance and angle between vehicles. In terms of distance measurements, radar can provide high accuracy, but does not include identity information of the target, required for . Measurements based on the travel time of radio signals ( and ) can provide such identity information [@mohammadabadi2014cooperative; @catovic2004cramer; @gholami2011hybrid]. However, and are challenged by the synchronization requirements [@catovic2004cramer]. The clocks of two vehicles need to be synchronized such that the delay can be computed. This can lead to significant localization error because of small clock error [@buehrer2018collaborative], or to use two-way with round-trip delay time instead of the one-way delay to avoid synchronization, which doubles the resource requirement. Achieving a ranging accuracy lower than 10 m by / is very challenging in vehicular environments [@alam2013cooperative]. In contrast, is readily available when the receiver is equipped with an antenna array [@sakagami1992vehicle; @fascista2017angle; @kakkavas2018multi; @garcia2019gaussian]: [@kakkavas2018multi] has investigated the performance of relative positioning using measurements from multiple receiving arrays on the vehicle, and the achieved positioning accuracy met requirements of 5G standardization. While measurements are attractive from a practical point of view, the integration in is non-trivial. Due to the nonlinear relation between the and the vehicle state, analytical computation of the messages in is not possible. Approximations include the use of particles [@etzlinger2013cooperative; @savic2013cooperative] or linearization of the measurement model [@wan2000unscented]. While the increasing number of particles gives better approximation performance, it also increases the computation complexity. To address this problem, [@garcia2019gaussian] uses a model for the measurement likelihood and performs [@garcia2015posterior], for a scenario with unknown positions but known orientation. In this paper, we consider a cooperative localization problem where vehicles’ positions and orientations are unknown. We apply Gaussian parametric BP [@yuan2016cooperative] for the MP, which reduces the communication resource overhead and computational complexity compared to a particle approach. To pass those messages through the nonlinear angle measurement model, [@garcia2015posterior] is applied to linearize the model using with respect to the posterior, which can be calculated by the current messages [@garcia2018cooperative]. Based on the linearized model, the BP is then performed to update the new beliefs. This procedure can be iterated so that the posterior of the vehicle position and orientation can converge. Problem Statement {#section:prob_state} ================= We consider a network comprising a set of vehicles $\mathcal{V} = \{1,...,N\}$. A set of communication links $\mathcal{E} \subset \mathcal{V}\times \mathcal{V}$ are considered to connect each vehicle according to a communication radius $r$. The neighbor set of vehicle $i$ is denoted by $\mathcal{N}_i$. Each vehicle $i \in \mathcal{V}$ has a state $\mathbf{x}_i \in \mathbb{R}^3$, comprising the 2D position $[x_i,y_i]^{\mathsf{T}}$ and the heading $\theta_i \in (-\pi,\pi]$. We denote the joint state of vehicles $i$ and $j$ as $\mathbf{x}_{ij}=[\mathbf{x}^{\mathsf{T}}_i \mathbf{x}^{\mathsf{T}}_j]^{\mathsf{T}}$. Each vehicle is assumed to have knowledge of its prior state by some accessible positioning techniques, e.g., GNSS, assumed to be a Gaussian density $$\begin{aligned} p_{i}(\mathbf{x}_{i}) = \mathcal{N}(\mathbf{x}_{i};\bm{\mu}_{i},\mathbf{P}_{i}), \label{eq:data_distribution}\end{aligned}$$ where $\mathcal{N}(\mathbf{x}_{i};\bm{\mu}_{i},\mathbf{P}_{i})$ denotes a Gaussian distribution in variable $\mathbf{x}_{i}$ with mean vector $\bm{\mu}_{i}=[\mu_{x}, \mu_{y}, \mu_{\theta}]^{\intercal}$ and covariance matrix $\mathbf{P}_{i}$. The measurement model between two vehicles is shown in Fig. \[fig:data\_model\]. Each vehicle $i$ is equipped with linear arrays on its two sides, each of which provides a $\varphi _{i}$ with $0 < \varphi _{i}\leq \pi$. Signals with an measurements within the of node can be measured. The measurement vector $\mathbf{z}_{ij}$ between vehicles $i$ and $j$ is defined as a function of $\mathbf{x}_i$ and $\mathbf{x}_{j}$ with additive Gaussian noise $$\begin{aligned} \mathbf{z}_{ij} = \mathbf{h}_{ij}(\mathbf{x}_{ij}) + \bm{\eta}_{ij}, \label{eq:eq_measure_model}\end{aligned}$$ where $\bm{\eta}_{ij}$ represents the measurement noise, modeled as $\bm{\eta}_{ij} \sim \mathcal{N}(\mathbf{0},\mathbf{R}_{ij})$ and $\mathbf{h}_{ij}(\mathbf{x}_{ij})$ is defined as[^1] $$\begin{aligned} \mathbf{h}_{ij}(\mathbf{x}_{ij}) = \left[\begin{array}{c} \operatorname{atan2}\left((y_j-y_i), (x_j-x_i)\right) - \theta_i\\ \operatorname{atan2}((y_i-y_j), (x_i-x_j)) - \theta_j \end{array}\right], \label{eq:True_measure_model}\end{aligned}$$ in which $\operatorname{atan2}(y,x)$ calculate the four-quadrant inverse tangent of $y$ and $x$. However, the $\operatorname{atan2}$ introduces problems because of its discontinuity at the negative semi-axis of $x$, i.e. $(x,0):x<0$. Instead of modeling the angular measurements by distribution, as [@garcia2019gaussian] has done, we adopt a simple ad-hoc correction from [@crouse2015cubature], which is described in Appendix \[appendix: SLR\]. We denote the vector of all measurements by $\mathbf{z}=[\mathbf{z}_{ij}]_{i,j\in \mathcal{N}_i}$ and the vector of all vehicles’ states by $\mathbf{x}$. The goal of the network is to compute $p_i(\mathbf{x}_i \|\mathbf{z})$, for each vehicle. Belief Propagation and Posterior Linearization ============================================== Belief Propagation Formulation ------------------------------ The standard approach to solve the localization problem is to use belief propagation. We first factorize the joint $$\begin{aligned} p(\mathbf{x} ,\mathbf{z}) & =p(\mathbf{x})p(\mathbf{z} \|\mathbf{x})\\ & = \prod_{i=1}^{N}p_i(\mathbf{x}_i) \prod_{j \in \mathcal{N}_i, j>i} p(\mathbf{z}_{ij}\|\mathbf{x}_{ij}).\end{aligned}$$ A factor graph representation of this joint in combination with loopy allows the computation of approximations of the marginal posteriors $p_i(\mathbf{x}_i \|\mathbf{z})$. The message passing rules at iteration $k$ are as follows (assuming $j \in \mathcal{N}_i$)[@kschischang2001factor] $$\begin{aligned} b_j^{(k-1)}(\mathbf{x}_{j})& \propto p_j(\mathbf{x}_{j}) \prod_{i \in \mathcal{N}_j} m^{(k-1)}_{i \to j}(\mathbf{x}_j)\label{eq:MP1} \\ m^{(k)}_{j \to i}(\mathbf{x}_i) & \propto \int p(\mathbf{z}_{ij}\|\mathbf{x}_{ij}) \frac{b_j^{(k-1)}(\mathbf{x}_{j})}{m^{(k-1)}_{i \to j}(\mathbf{x}_j)}\text{d}\mathbf{x}_{j}. \label{eq:MP2} $$ The approximate marginal posterior at iteration $k$ is $p_j(\mathbf{x}_j \|\mathbf{z}) \approx b_j^{(k)}(\mathbf{x}_{j})$. The process is initialized at $k=0$ by $b_j^{(0)}(\mathbf{x}_{j})=p_j(\mathbf{x}_{j})$ and $m^{(0)}_{i \to j}(\mathbf{x}_j)=1$. The joint posterior of $\mathbf{x}_{i},\mathbf{x}_{j}$ can also be approximated by[@kschischang2001factor] $$\begin{aligned} b^{(k)}(\mathbf{x}_{ij}) &\propto p(\mathbf{z}_{ij}\|\mathbf{x}_{ij}) \frac{b_i^{(k)}(\mathbf{x}_{i})b_j^{(k)}(\mathbf{x}_{j})}{m^{(k)}_{i \to j}(\mathbf{x}_j){m^{(k)}_{j \to i}(\mathbf{x}_i)}}. \label{eq:joint_posterior}\end{aligned}$$ However, due to the nonlinear observation model , in general cannot be executed in closed form: neither the integral nor the product can be computed exactly, except when the observation model is linear with Gaussian noise [@garcia2015posterior]. This motivates the following linearization procedure. Linearization {#section:linearization} ------------- Given a belief $b^{(k)}(\mathbf{x}_{ij})$, we approximate the observation model by $$\begin{aligned} \mathbf{h}_{ij}(\mathbf{x}_{ij}) & \approx \mathbf{C}_{ij}\tilde{\mathbf{x}}_{ij}+\mathbf{e}_{ij}, \label{eq:linearizedMeasurements}\end{aligned}$$ where $\mathbf{e}_{ij}\sim \mathcal{N}(\mathbf{0},\bm{\Omega}_{i,j})$, and $\tilde{\mathbf{x}}_{ij}=[{\mathbf{x}}^{\mathsf{T}}_{ij}~ \mathbf{1}^{\mathsf{T}}]^{\mathsf{T}}$. $\mathbf{C}_{ij}$ is selected to minimize the over the given joint belief $b^{(k)}(\mathbf{x}_{ij})$: $$\begin{aligned} \arg \underset{\mathbf{C}_{ij}}{\min} &~ \mathbb{E} \{ \Vert \mathbf{h}_{ij}(\mathbf{x}_{ij})-\mathbf{C}_{ij}\tilde{\mathbf{x}}_{ij}\Vert ^2\}. \label{eq:optimizationProblem}\end{aligned}$$ Once $\mathbf{C}_{ij}$ is determined, we find that $\bm{\Omega}_{i,j} = \Vert \mathbf{h}_{ij}(\mathbf{x}_{ij})-\mathbf{C}_{ij}\tilde{\mathbf{x}}_{ij}\Vert^2$. To solve this optimization problem, the [@garcia2015posterior] with respect to the posterior is performed, where the details are presented in Appendix \[appendix: SLR\]. To visualize the advantage of posterior , Fig. \[fig:linearization-performance\] shows the true measurement model (\[eq:True\_measure\_model\]) and its approximations (\[eq:linearizedMeasurements\]) with respect to posterior and prior. We observe that the linearized model by posterior is more accurate and has less uncertainty than the model linearized by prior . ![The true measurement model $\mathbf{h}_{ij}(\mathbf{x}_{ij})$ and its approximations by with respect to the prior and posterior, as a function of the $x$-dimension of $\mathbf{x}_i$. The length of the red and blue lines represent 2 standard deviations of the prior and posterior linearized models, respectively. []{data-label="fig:linearization-performance"}](10.eps){width="0.9\linewidth"} Belief Propagation with Linearized Measurement Models ----------------------------------------------------- Once a linearization of all measurement models is obtained, is performed as follows. The likelihood function is now of the form $$\begin{aligned} & p(\mathbf{z}_{ij}\|\mathbf{x}_{ij}) \propto \label{eq:linearizedlikelihood} \\ & \exp\left(-\frac{1}{2}(\mathbf{z}_{ij}-\mathbf{C}_{ij}\tilde{\mathbf{x}}_{ij})^{\mathsf{T}} \bm{\Sigma}^{-1}_{ij} (\mathbf{z}_{ij}-\mathbf{C}_{ij}\tilde{\mathbf{x}}_{ij})\right), \nonumber \end{aligned}$$ where $\bm{\Sigma}_{ij}=\bm{\Omega}_{ij}+\mathbf{R}_{ij}$. This formulation now allows closed-form Gaussian message passing according to – and . The details of the implementation are provided in the Appendix \[appendix: BP\]. The overall algorithm thus operates as described in Algorithm \[alg:algorithm 1\]. The algorithm requires a selection of $K$ (the number of linearization iterations) and $M$ (the number of BP iterations per linearization step). The overall complexity per vehicle is approximately $\mathcal{O}(KM\bar{N}D^3)$, where $D$ is the state dimension and $\bar{N}$ is the average number of neighbors. Given the current beliefs $b^{(k-1)}(\mathbf{x}_{ij})$, solve for each $(i,j) \in \mathcal{E}$ to obtain . Run $M$ iterations of BP on the linearized model. Compute joint beliefs $b^{(k)}(\mathbf{x}_{ij})$ at the current BP iteration. Return marginal beliefs. \[alg:algorithm 1\] Simulation Results ================== In this section we simulated a vehicular network scenario and analyzed the performance of the designed Algorithm \[alg:algorithm 1\]. First, the localization and orientation performance of Algorithm \[alg:algorithm 1\] in the vehicle network is evaluated by the positional and directional . Then, based on this scenario, we analyzed the impact of different network parameters. Simulation Scenario ------------------- The vehicular scenario is based on a road map in central New York Manhattan (latitude: $40.71590$ and longitude: $-73.99560$). The map data is generated from Stamen Map [@ManhattanMap] at a zoom level of 18. Within this map, the scenario is shown in Fig. \[fig:scenario\_final\], where 51 vehicles are possibly connected within the communication radius ($r=30~\text{m}$). The priors are set to $\mathbf{P}_{i} = \text{diag}(\sigma^{2}_{x}, \sigma^{2}_{y}, \sigma^{2}_{\theta})$. Among the vehicles, 6 are chosen as anchors (vehicles or road side units with a very concentrated prior density, set to $\text{diag}(\sigma_{x}^{2},\sigma_{y}^{2} , \sigma_{\theta}^{2})=\text{diag}(0.01, 0.01, 0.01)$). The interactive web map is also provided[^2] [@map-simple-1]. The remaining parameters of this scenario are illustrated in Table \[tb:basic\_setup\], where $R$ denotes the constant value of the measurement variance (approximately 18 degrees standard deviation). $r$ \[m\] $\varphi$ \[rad\] $\sigma_{x}$ \[m\] $\sigma_{y}$ \[m\] $\sigma_{\theta}$ \[rad\] $R$ \[rad$^{2}$\] ----------- ------------------- -------------------- -------------------- --------------------------- ------------------- 30 $\pi$ 5 5 0.35 0.10 : Setup parameters for the vehicular scenario. \[tb:basic\_setup\] ![Scenario of the vehicular network. The interactive web map can be found in [@map-simple-1].[]{data-label="fig:scenario_final"}](Initial_map.pdf){width="1\linewidth"} Results and Discussion ---------------------- ### Convergence Speed ![RMS position and direction error against the number of linearization iteration $k$. The initial position and direction RMSE of vehicles are 7.01m and 0.38 rad, respectively.[]{data-label="fig:RMSE_P_D"}](RMSE_D_and_P.eps){width="1\linewidth"} In order to examine the performance of Algorithm \[alg:algorithm 1\], in Fig. \[fig:RMSE\_P\_D\] we plot the RMS position and direction error against the number of linearization iteration $K$. Notice the performance gap between the prior (dotted lines) and the posterior (solid lines). After each belief propagation iteration, the posterior of each vehicle is closer to the true state than the prior, so the belief propagation has a better performance on the posterior linearization measurement model. Both position RMSE and direction RMSE converged for linearization iteration number larger than 4. Meanwhile, increasing $M$ from 1 to 3 provides significant improvements for both position and orientation estimation accuracy as the beliefs are more accurate. The improvement becomes very small for $M$ greater than 3. ### Localization Performance ![CDF of localization and orientation error, $K=10$.[]{data-label="fig:CDF_P_D"}](CDF_test.eps){width="1\linewidth"} While the above results show the average RMSE of the position and direction, Fig. \[fig:CDF\_P\_D\] shows the cumulative distribution functions (CDFs) of the position and direction errors for $K=10$ for different values of $M$. We observe that for $M=3$ the performance is similar to $M=10$ and that nearly all vehicles can be localized with a position error less then 4 meters and an orientation error less than 0.15 radians (8 degrees). The importance of posterior linearization over prior linearization is again clear. ### Impact of Network parameters ![The impact of 4 vehicle network parameters on localization and orientation performance. Lines with square and triangle markers represent the position and orientation RMSE, respectively. $K=10$, $M=10$ and posterior LF are applied.[]{data-label="fig:impact-parameters"}](overall.eps){width="1\linewidth"} Here, we analyze the impact of modifying the scenario parameters in Table \[tb:basic\_setup\] on localization and orientation estimation performance. In Fig. \[fig:impact-parameters\], we evaluate 4 parameters separately, namely communication radius ($r$), measurement noise variance ($R$), prior uncertainty in position ($\sigma_{p} = (\sigma_{x}^{2}+ \sigma_{y}^{2})^{1/2}$) and prior uncertainty in orientation ($\sigma_{\theta}$) in 4 sub-figures, by plotting the position and direction RMSE as functions of one of them, while keeping the rest fixed to the values of Table \[tb:basic\_setup\]. - The top left sub-figure shows the impact of the communication radius $r$. Both RMSEs are reduced rapidly by increasing $r$ from 10 m to 30 m since each vehicle has more neighbors and the network connectivity increases quickly, up to the point where all vehicles are in each others’ communication range. We note that with increased connectivity comes increased computational complexity. - In the top right sub-figure, we vary the measurements noise variance $R$. We note that both direction and position RMSE increase approximately linearly in $\sqrt{R}$. This emphasizes the need for good measurements. - The influence of the prior position uncertainty ($\sqrt{\sigma_{x}, \sigma_{y}}$) is shown in the bottom left sub-figure. The red dashed line describes the prior position RMSE. We notice the increase of $\sigma_p$ from 0 m to 10 m has small effect on both position and direction performance (less than 2 m/0.05 rad), showing the good performance of the proposed method. For position uncertainty over 10 m, Algorithm \[alg:algorithm 1\] is still able to improve performance over the prior RMSE, but leads to progressively larger errors. This is in contrast to range-based cooperative localization [@wymeersch2009cooperative], where no prior information was needed. - The influence of the direction uncertainty ($\sigma_{\theta}$) is shown in the bottom right sub-figure, where we observe a rapid increase in RMSE. This is because the measurements depend on the orientation of the receiving vehicles. For larger prior orientation uncertainty, Algorithm \[alg:algorithm 1\] is less affected. Conclusion ========== We have applied PLBP to cooperative localization (position and orientation estimation) of vehicles with -only measurements. Multiple conditions of the vehicular network, including the vehicle density, communication radius, prior uncertainty and measurement noise variance have been discussed. Numerical results show that the proposed algorithm has good performance in terms of both position and orientation estimation, and only a few iterations are required for convergence. This makes the algorithm attractive for real-time processing. Acknowledgment {#acknowledgment .unnumbered} ============== This research was supported, in part, by the EU Horizon 2020 project 5GCAR (Fifth Generation Communication Automotive Research and innovation) and the Spanish Ministry of Science, Innovation and Universities under Grant TEC2017-89925-R. Steps of the posterior linearization {#appendix: SLR} ==================================== This section illustrates the procedures of SLR on the measurement model and the approximation of the parameters ($\mathbf{C}_{ij},\mathbf{\Omega}_{i,j}$) with respect to the joint posterior $p(\mathbf{x}_{ij}\|\mathbf{z}_{ij})$ = $\mathcal{N}(\mathbf{x}_{ij};\bm{\mu}_{ij};\mathbf{P}_{ij}) $. First, according to the joint posterior of $\mathbf{x}_{i},\mathbf{x}_{j}$, we select $L$ sigma-points $\mathcal{X}_{1},...,\mathcal{X}_{L}$ and weights $\omega_{1},...,\omega_{L}$ using a sigma-point method such as the unscented transform [@julier2004unscented]. Then we calculate the transformed sigma points by $$\begin{aligned} \label{eq:sigma-points} \mathcal{Z}_{l} = \mathbf{h}_{ij}(\mathcal{X}_{l}) \hspace{10pt} l = 1,...,L \end{aligned}$$ However, as mentioned in Section \[section:prob\_state\], the function arctan has discontinuity problem at the negative $x$ semi-axis. The sigma points transformation needs an ad-hoc modification so that the difference between angles $\mathcal{Z}_l - \mathbf{z}_{ij}$ must be bounded in $\pm \pi$. $\mathcal{Z}_{l}$ can be corrected to $\hat{\mathcal{Z}_{l}}$ by the following transformation: $$\begin{aligned} \label{eq:ad-hoc} \hat{\mathcal{Z}_{l}} = \mathbf{z}_{ij} + \pi - \text{modulo}((\mathbf{z}_{ij} -\mathcal{Z}_{l})+\pi)_{2\pi}\end{aligned}$$ where $\hat{\mathcal{Z}_{l}}$ denotes the corrected sigma point, $z_{ij}$ is the measurements and $\text{modulo}(\cdot)_{2 \pi}$ represents the modulo operation. Introducing $\mathbf{C}_{ij}=[\mathbf{A}_{ij}~ \mathbf{b}_{ij}]$, so that $$\begin{aligned} \mathbf{h}_{ij}(\mathbf{x}_{ij}) & \approx \mathbf{A}_{ij} \mathbf{x}_{ij} + \mathbf{b}_{ij}+\mathbf{e}_{ij},\end{aligned}$$ the solution of the approximation of $\mathbf{A}_{ij},\mathbf{b}_{ij},\Omega_{i,j}$ is $$\begin{aligned} \mathbf{A}_{ij} &= \mathbf{C}_{xz}^{\mathsf{T}}\mathbf{P}_{ij}^{-1}\label{eq:sigma-A}\\ \mathbf{b}_{ij} &= \bar{z} - \mathbf{A}_{ij}\bm{\mu}_{ij}\label{eq:sigma-b}\\ \mathbf{\Omega}_{i,j}& = \mathbf{C}_{zz} - \mathbf{A}_{ij}\mathbf{P}_{ij}\mathbf{A}_{ij}^{\mathsf{T}}\label{eq:sigma-Omega}\end{aligned}$$ where $\bar{z}$, $\mathbf{C}_{xz}$ and $\mathbf{C}_{zz}$ are approximated using the sigma-points and weights by $$\begin{aligned} \bar{z} \approx& \sum_{j=1}^{L}\omega_{j}\hat{\mathcal{Z}_{l}} \label{eq:sigma-z} \\ \mathbf{C}_{xz} \approx& \sum_{j=1}^{L}\omega_{j}(\mathcal{X}_{j} - \bm{\mu}_{ij})(\hat{\mathcal{Z}_{l}} - \bar{z})^{\mathsf{T}}\label{eq:sigma-Psi}\\ \mathbf{C}_{zz} \approx& \sum_{j=1}^{L}\omega_{j}(\hat{\mathcal{Z}_{l}} - \bar{z})(\hat{\mathcal{Z}_{l}} - \bar{z})^{\mathsf{T}}.\label{eq:sigma-Phi}\end{aligned}$$ Implementation of BP in the linearized model {#appendix: BP} ============================================ This section illustrates the derivation of equation – and . Once we have the approximated linearization model \[eq:linearizedMeasurements\], we can represent the BP message $m^{(k)}_{i\rightarrow j}$ by the Gaussian format [@garcia2018cooperative] $$\label{eq:msg_a} m^{(k)}_{i\to j}(\mathbf{x}_j) \propto \mathcal{N}(\bm{\alpha}^{(k)}_{ij}; \mathbf{H}^{(k)}_{ij}\mathbf{x}_{j},\bm{\Gamma}^{(k)}_{ij})$$ where $\bm{\alpha}^{(k)}_{ij}$, $\mathbf{H}^{(k)}_{ij}$ and $\bm{\Gamma}^{(k)}_{ij}$ are \[eq:msg\_abc\] $$\label{eq:msg_b} \bm{\alpha}^{(k)}_{ij} = [\mathbf{z}_{ij}]_1 - \mathbf{A}_{i}\bm{\mu}^{(k-1)}_{ij} - b_{ij}$$ $$\label{eq:msg_c} \mathbf{H}^{(k)}_{ij} = \mathbf{A}_{j}$$ $$\label{eq:msg_d} \bm{\Gamma}^{(k)}_{ij} = \mathbf{R}_{ij} + \bm{\Omega}_{ij} + \mathbf{A}_{i}\mathbf{P}^{(k-1)}_{ij}\mathbf{A}_{i}^\intercal$$ where $[\mathbf{z}_{ij}]_1$ is the measurement received by vehicle $i$, $\mathbf{A}_{i},\mathbf{A}_{j}$ are defined at Section \[section:linearization\] and $\bm{\mu}^{(k-1)}_{ij}$ and $\mathbf{P}^{(k-1)}_{ij}$ are found from the relation $$\begin{split} \mathcal{N}(\bm{\mu}^{(k-1)}_{ij},\mathbf{P}^{(k-1)}_{ij}) \propto\mathcal{N}(\mathbf{x}_i;\bm{\mu}_i,\mathbf{P}_i) \prod_{j'\in \mathcal{N}_i\backslash j} m^{(k-1)}_{j'\rightarrow i}(\mathbf{x}_{i}) \end{split} \label{eq:msg_x_ij}$$ where the Kalman update step [@garcia2018cooperative Algorithm 1] is performed to update each message $m^{(k-1)}_{j'\to i}(\mathbf{x}_{i})$ on the prior state $\mathcal{N}(\mathbf{x}_i;\bm{\mu}_i,\mathbf{P}_i)$. To get the local belief (\[eq:MP1\]) at the $k$-th iteration, we can also use Kalman filter update step to update the vehicle prior with all its incoming messages. $$\label{eq:marginal_distribution} b_j^{(k)}(\mathbf{x}_{j}) = \mathcal{N}(\mathbf{x}_{j};\bm{\mu}_{j},\mathbf{P}_{j}) \times \prod_{i \in \mathcal{N}_j} m^{(k)}_{i \to j}(\mathbf{x}_j)$$ The $k$-th iteration joint posterior (\[eq:joint\_posterior\]) is expressed as [@garcia2018cooperative] $$\begin{aligned} & b^{(k)}(\mathbf{x}_{ij})=\mathcal{N}(\mathbf{x}_{i};\bm{\mu}_{i},\mathbf{P}_{i}) \prod_{j'\in \mathcal{N}_i\backslash j} m^{(k)}_{j'\to i}(\mathbf{x}_{i}) \\ & \times \mathcal{N}(\mathbf{x}_{j},\bm{\mu}_{j},\mathbf{P}_{j})\times \prod_{i'\in \mathcal{N}_j\backslash i} m^{(k)}_{i'\rightarrow j}(\mathbf{x}_{j}) p(\mathbf{z}_{ij}\|\mathbf{x}_{i},\mathbf{x}_{j}) \nonumber \end{aligned}$$ where we can also apply Kalman filter update [@garcia2018cooperative Algorithm 1] as in (\[eq:msg\_x\_ij\]). \[reference\] [^1]: For simplicity we consider the center points of the two arrays on each vehicle to coincide. The effect of the relative position and orientation of the antenna arrays is outside the scope of this paper and related work can be found in [@shen2010accuracy]. [^2]: The results of the scenario can be visualized by an interactive web map in [@map-simple-2], where the red, blue, and green dots represent the true, prior and estimated positions, respectively.
--- abstract: 'Investigating for interior regularity of viscosity solutions to the fully nonlinear elliptic equation $$F(x,u,\triangledown u,\triangledown ^2 u)=0,$$ we establish the interior $C^{1+1}$ continuity under the assumptions that $F$ is uniformly elliptic, H$\ddot o$lder continuous and satisfies the natural structure conditions of fractional order, but without the concavity assumption of $F$. These assumptions are weaker and the result is stronger than that of Caffarelli and Wang\[1\], Chen\[2\].' author: - 'G.C.Dong, B.J.Bian and Z.C.Guan[^1]' title: The Second Order Estimate for Fully Nonlinear Uniformly Elliptic Equations without Concavity Assumption --- regularity, viscosity solutions, fully nonlinear elliptic equation, concavity. Introduction ============ The study of solvability problems for the second order fully nonlinear uniformly elliptic equation, i.e., the existence, uniqueness and regularity of solutions with Dirichlet boundary data can be divided into two classes naturally. The first one is about classical solvability, i.e., solution $u \in C^2$. The related results are rich and systematic, See \[3\], \[4\], \[5\] and therein. All of them is obtained under the assumption of concavity for the equation with respect to their arguments. The second is without the concave condition of $F$. In this case, we must seek for solution in generalized sense. The suitable one is viscosity solution\[6\]. But in this case the results are incomplete. Because the regularity is a key stone for existence and uniqueness, we prove the interior $C^{1+1}$ continuity without the concavity assumption of $F$ in this paper and the existence and uniqueness results will be in next one. Our assumptions are $F(x,z,p,X) \in C^\beta (\beta >0)$ of its arguments only, weaker than those in Caffarelli and Wang\[1\] ($\beta =1$) and Chen \[2\]($\beta>1/2$). And the result, $u\in C^{1+1}$ is stronger than theirs ($u\in C^{1+\alpha},0<\alpha $ and small ). This paper is organized in five sections: The second is preliminaries, statements for conditions and the main results. The third is general comparison principle. For solution $u(x)(x\in \Omega, \Omega\subset \mathbb{R}^n )$, we investigate the general conditions for $u(x)-u(y)-\Phi(x,y)$ takes maximum in a $\mathbb{R}^{2n}$ domain $Q\subset \Omega\times \Omega$, where $\Phi(x,y)\in C^2(Q)$. If these conditions are violate, together with the assumption $[u(x)-u(y)-\Phi]_{\partial Q}\leq 0$, we have $u(x)-u(y)-\Phi\leq 0$, and the useful estimation $u(x)-u(y)\leq \Phi$ follows. The fourth is H$\ddot{o}$lder and Lipschitz continuity. The H$\ddot{o}$lder and Lipschitz estimates are obtained by selecting suitable $Q$ and $\Phi$. Although these results are not new, we prove them here for the sake of applications of the results of section 3. The last section is $C^{1+1}$ estimate. Having got the interior Lipschitz estimate for $u(x)$, we can conclude that there exist Caffarelli points such that in small neighborhood of them $u(x)$ can be separated into linear and second order parts. We estimate the lipschitz coefficient of $u(x)$ subtracting linear part in suitable neighborhood of Caffarelli point by selecting suitable comparison function $\Phi(x,y)$. By this way we get a fundamental lemma: the lipschitz coefficient is diminishing in certain constant ratio accompanied with diminishing of radius of spherical neighborhood in suitably constant ratio. The $C^{1+\alpha}$ estimate and $C^{1+1}$ in small part follows from this fundamental lemma. And the $C^{1+1}$ estimate in the whole region follows by putting the estimates in small parts altogether. Preliminaries ============= Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with the boundary $ \partial \Omega \in C$. We consider the regularity problem for solutions of the equation $$\label{eq1} F(x,u,\triangledown u,\triangledown ^2 u)=0 \qquad \qquad in \qquad \Omega$$ where $F(x,u,p,X)$ is a function on $\Gamma=\Omega\times \mathbb{R}\times \mathbb{R}^n\times \mathbb{S}^n $, $\mathbb{S}^n$ the space of $n\times n$ symmetric matrices equipped with usual order. We assume that $F(x,u,p,X)$ is uniformly elliptic in the following sense $$\label{eq2} \lambda Tr(X-Y)\le F(x,z,p,Y)-F(x,z,p,X)\le \Lambda Tr(X-Y)$$ for all $(x,z,p,X)\in \Gamma$, $Y\in \mathbb{S}^n$ and $X\ge Y$, where $\lambda$ and $\Lambda$ are positive constants with $\lambda \le \Lambda$. Assume that $F$ is monotone with respect to z $$\label{eq3} F(x,z,p,X)-F(x,w,p,X)\le 0$$ $\forall z\le w$. Furthermore, we suppose that there exist positive constants $\mu$ and $\beta$ with $\beta \le 1$ such that $$\label{eq4} \begin{aligned} \|F(x,z,p,X)-F(y,w,q,X)\|\qquad\qquad\\ \le \mu [\|x-y\|(1+\|p\|+\|q\|)+\frac {\|p-q\|}{1+\|p\|+\|q\|}]^\beta\\ \cdot (1+\|p\|^2+\|q\|^2+\|\|X\|\|)\qquad \qquad\qquad \end{aligned}$$ $\forall$ $(x,z,p,X)\in \Gamma$ and $(y,w,q,X)\in \Gamma $. (\[eq4\]) is called natural structural condition for $F$ of fractional order (order $\beta$, where $0<\beta\leq 1$). Now we give the definition of viscosity solutions and main theorem. [Definition 1.]{} Let $u$ be an upper semi-continuous (resp. lower semi-continuous) function in $ \Omega $. $u$ is said to be a viscosity subsolution (resp. supersolution) of ($\ref{eq1}$) if for all $\phi \in C^2(\Omega)$ the following inequality $$F(x_0,u(x_0),\triangledown \phi (x_0),\triangledown ^2 \phi (x_0))\le 0$$ $$(resp.\qquad F(x,u(x_0),\triangledown \phi(x_0),\triangledown ^2 \phi(x_0))\ge 0)$$ holds at each local maximum (resp. minimum) point $x_0\in \Omega$ of $u-\phi$. [Definition 2.]{} $u\in C(\Omega)$ is said to be a viscosity solution of ($\ref{eq1}$) if $u$ is both a viscosity subsolution and a supersolution. [Theorem 2.1.]{} Assume $F(x,z,p,X)$ satisfies the conditions $($\[eq2\]$)-($\[eq4\]$)$ and $u$ is a viscosity solution of ($\ref{eq1}$), then $ u \in C^{1+1}(\Omega)$. In this paper, we don’t study any boundary value problem for (\[eq1\]). About the existence and uniqueness for solution of boundary value problems, we shall discuss them in a next paper. General Comparison Principle ============================ For suitably selected regular non-negative function $\Phi (x,y)$ and a bounded domain $Q \subset \mathbb{R}^{2n}$, we suppose that $u(x)-u(y)-\Phi (x,y)$takes a positive maximum value at an interior point $(\bar x,\bar y)\in Q$, then $$u(x)-u(y)-\Phi (x,y) \le u(\bar x)-u(\bar y)-\Phi (\bar x,\bar y)> 0$$ For simplicity, we omit the upper bar on $x,y$ in the following, i.e., write $(x,y)=(\bar x,\bar y)$, thus $$\label{eq5} u(x)-u(y)> \Phi (x,y)\ge 0.$$ In particular we have, $x\neq y$. From \[6\], there exist $X,Y \in \mathbb{S}^n$ such that $$\label{eq6} F(x,u(x),\Phi_x(x,y),X)-F(y,u(y),- \Phi_y(x,y),-Y)\le0$$ and $$\label{eq7} \left( \begin{array}{cc} X&0\\0&Y \end{array} \right) \le \left( \begin{array}{cc} {\Phi _{xx}}&{\Phi _{xy}}\\ {\Phi _{yx}}&{\Phi _{yy}} \end{array} \right)$$ From above inequality, we have $$\label{eq8} \left( \begin{array}{cc} X+Y&X-Y\\X-Y&X+Y \end{array} \right) \le \left( \begin{array}{cc} Z_1&Z\\Z&Z_2 \end{array} \right),$$ where $$\label{eq9} \begin{aligned} Z_1=(\frac{\partial}{\partial x}+\frac{\partial}{\partial y})^2\Phi ,\\ Z_2=(\frac{\partial}{\partial x}-\frac{\partial}{\partial y})^2\Phi ,\\ Z=(\frac{\partial}{\partial x}-\frac{\partial}{\partial y})(\frac {\partial}{\partial x}+\frac{\partial}{\partial y})\Phi . \end{aligned}$$ A consequence of (\[eq8\]) are $$\label{eq10} X+Y-Z_1\le 0,\qquad X+Y-Z_2\le 0.$$ $\forall$ $\sigma\in \mathbb{R}$ and $\xi \in \mathbb{R}^n $, multiplying (\[eq8\]) from left and right by $(\sigma \xi ,\xi )$ and $(\sigma \xi ,\xi )^T $ respectively, we get $$\begin{aligned} \sigma ^2<(X+Y-Z_1)\xi ,\xi >+2\sigma <(X-Y-Z)\xi ,\xi> \\ +<(X+Y-Z_2)\xi ,\xi>\le 0.\qquad \qquad \end{aligned}$$ Hence, for $\xi \in \mathbb{R}^n $ with $ \|\xi \|=1 $, $$\begin{aligned} <(X-Y-Z)\xi,\xi>^2\le \qquad \qquad \\ <(X+Y-Z_1)\xi ,\xi ><(X+Y-Z_2)\xi ,\xi>. \end{aligned}$$ i.e., $$\begin{aligned} \|\|X-Y-Z\|\|^2\le \qquad \qquad \\ \|\|X+Y-Z_1\|\| \|\|X+Y-Z_2\|\|\le \\ \|\|X+Y-Z_1\|\|^2+\|\|Z_2-Z_1\|\|\|\|X+Y-Z_1\|\|.\end{aligned}$$ This inequality implies that $$\|\|X-Y-Z\|\|$$ $$\le \|\|X+Y-Z_1\|\|+\|\|Z_2-Z_1\|\|^{1/2}\|\|X+Y-Z_1\|\|^{1/2}$$ $$\le C_1[\|Tr(X+Y-Z_1)\|+\|\|Z_2-Z_1\|\|^{1/2}\|Tr(X+Y-Z_1)\|^{1/2}].$$ Thus, $$\|\|X\|\|+\|\|Y\|\|$$ $$\le \|\|X+Y\|\|+\|\|X-Y\|\|$$ $$\le \|\|Z_1\|\|+\|\|Z\|\|+\|\|X+Y-Z_1\|\|+\|\|X-Y-Z\|\|$$ $$\le C_1[\|\|Z_1\|\|+\|\|Z\|\|+2\|Tr(X+Y-Z_1)\|$$ $$+\|\|Z_2-Z_1\|\|^{1/2}\|Tr(X+Y-Z_1)\|^{1/2}].$$ Taking a positive parameter $\omega $ to be determined later. From the above inequality, we have $$\label{eq16}\begin{aligned} \omega (\|\|X\|\|+\|\|Y\|\|) \qquad \qquad \qquad \qquad \\ \le C_1\omega[\|\|Z_1\|\|+\|\|Z\|\|+2\|Tr(X+Y-Z_1)\|] \\ +\frac {\lambda}{2}\|Tr(X+Y-Z_1)\|+\frac {2C_1^2\omega ^2}{\lambda}\|\|Z_2-Z_1\|\|. \end{aligned}$$ On the other hand, by ($\ref{eq2}$) and ($\ref{eq10}$), we have $$\lambda \|Tr(X+Y-Z_1)\|=\lambda Tr(Z_1-X-Y)$$ $$\le F(x,u(x),\Phi _x,X)-F(x,u(x),\Phi _x,Z_1-Y).$$ Applying ($\ref{eq6}$) and ($\ref{eq3}$), we have $$F(x,u(x),\Phi _x,X)\le F(y,u(y),-\Phi _y,-Y)$$ $$\le F(y,u(x),-\Phi _y,-Y).$$ Separating $Z_1$ into $Z_1^++Z_1^-$ where $Z_1^+\ge 0$, $Z_1^-\le 0$ and by (\[eq4\]), we obtain $$-F(x,u(x),\Phi _x,-Y+Z_1)$$ $$\le -F(x,u(x),\Phi _x,-Y) +\Lambda TrZ_1^+-\lambda TrZ_1^-$$ Combining the above inequalities and applying (\[eq4\]), we have $$\lambda \|Tr(X+Y-Z_1)\|$$ $$\le C_2\{\|\|Z_1\|\|+ [(1+A)\|x-y\|+\frac {\|Z_0\|}{1+\Lambda}]^\beta$$ $$\label{eq17} \cdot(1+A^2+\|\|X\|\|+\|\|Y\|\|)\},$$ where $$\label{eq18}\begin{aligned} Z_0=\Phi_x+\Phi_y=(\frac {\partial}{\partial x}+\frac {\partial}{\partial y})\Phi ,\\ A=\|Z_0\|+\|(\frac {\partial}{\partial x}-\frac {\partial}{\partial y})\Phi\|. \end{aligned}$$ Taking the parameter $\omega$ to be $$\label{eq19} \omega=C_2[(1+A)\|x-y\|+\frac {\|Z_0\|}{1+\Lambda}]^\beta.$$ Combining (\[eq16\]), (\[eq17\]) and (\[eq19\]), we have $$\label{eq20}\begin{aligned} (\frac \lambda 2 -2C_1\omega)\|Tr(X+Y-Z_1)\|\qquad \qquad \\ \le C[\|\|Z_1\|\|+(1+A^2+\|\|Z\|\|)\omega+\|\|Z_2-Z_1\|\|\omega ^2]. \end{aligned}$$ As $\omega$ is small, $\omega \ll 1$, (\[eq20\]) represent an upper estimate for $\|Tr(X+Y-Z_1)\|$. We still need a lower estimate for $\|Tr(X+Y-Z_1)\|$. Let $P$ be a $n\times n$ diagonal matrix with $0<P\le I$, where $I$ is unit matrix. Since $Tr$ is invariant under coordinate rotation, denote $S$ be coordinate rotation matrix, which is symmetric and satisfies $S^2=I$, then $$-Tr[SPS(Z_2-Z_1)]\qquad \qquad$$ $$=-Tr[S^2PS(Z_2-Z_1)S]=-Tr[PS(Z_2-Z_1)S]$$ $$=Tr[PS(Z_1-X-Y)S]-Tr[PS(Z_2-X-Y)S]$$ Applying (\[eq10\]), we get $$S(Z_1-X-Y)S\ge 0,\ S(Z_2-X-Y)S\ge 0$$ Since $P$ is diagonal and $0<P\le I$, we have $$PS(Z_1-X-Y)S \le S(Z_1-X-Y)S,\ PS(Z_2-X-Y)S\ge 0.$$ Hence $$-Tr[SPS(Z_2-Z_1)]\le Tr[S(Z_1-X-Y)S]$$ $$=Tr(Z_1-X-Y)=\|Tr(X+Y-Z_1)\|$$ Setting $$\Upsilon=\frac {(x-y)\otimes (x-y)}{\|x-y\|^2}$$ it satisfies $0\le \Upsilon\le I$, and selecting $S$ such that $SPS=\frac{1}{1+\varepsilon}(\Upsilon+\varepsilon I)$and let $\varepsilon\to 0$,we have $$\label{eq21} -Tr[\Upsilon(Z_2-Z_1)]\le \|Tr(X+Y-Z_1)\|.$$ (\[eq21\]) is a lower estimate for $Tr(X+Y-Z_1)$ which are needed. Combining (\[eq20\]) and (\[eq21\]), we have $$\label{eq22}\begin{aligned} -(\frac \lambda 2-2C_1\omega)Tr(\Upsilon Z_2) \qquad\qquad \\ \le C[\|\|Z_1\|\|+(1+A^2+\|\|Z\|\|)\omega +\|\|Z_2\|\|\omega^2]. \end{aligned}$$ which is a necessary condition for $u(x)-u(y)-\Phi (x,y)$ takes positive maximum value in $Q$. If (\[eq22\]) is violate, i.e. $$\label{eq23}\begin{aligned} -(\frac \lambda 2-2C_1\omega)Tr(\Upsilon Z_2) \qquad\qquad \\ > C[\|\|Z_1\|\|+(1+A^2+\|\|Z\|\|)\omega +\|\|Z_2\|\|\omega^2], \end{aligned}$$ then $u(x)-u(y)-\Phi (x,y)$ cannot take positive maximum value in $Q$, hence $\forall \ (x,y)\in Q$, we have $$\label{eq24} u(x)-u(y)-\Phi (x,y)\le [u(x)-u(y)-\Phi (x,y)]\|_{\partial Q}$$ The above all are general discuss. H$\ddot o$lder and Lipschitz continuity {#sec:Lipschitzcontinuity} ======================================= [Theorem 4.1]{} Let $u$ be a viscosity solution to (\[eq1\]) in $\Omega$ and suppose that (\[eq2\])-(\[eq4\]) hold, then there exists a constant $\alpha \in (0,1)$, such that $u$ is H$\ddot o$lder continuous with exponent $\alpha$ in $\Omega $ and the following estimate holds $\forall$ $x,y\in \Omega$, $$\label{eq25} \|u(x)-u(y)\|\le \frac C{d^\alpha}\|x-y\|^\alpha,$$ where $d= min[d(x,\partial \Omega),d(y,\partial \Omega)]$ and $C$ is a constant depending only on $n,\lambda,\Lambda,\mu,\beta$ and $sup_{x\in \Omega}\|u(x)\|$. Let $x_0\in \Omega$ and $R>0$ such that $B(x_0,2R)\subset \Omega$. Without loss of generality, suppose $x_0$ is the origin and $sup_{B_{2R}}\|u(x)\|=1$. Let $\alpha<1$ and $K>1$ be two constants to be chosen and taking the domain $$Q=\{(x,y)\in \Omega \|\|x\|^2+\|y\|^2<R^2\}$$ and the function $$\label{eq226}\Phi (x,y)=\frac 2{R^2}(\|x+y\|^2+\|x-y\|^2)+K\frac {\|x-y\|^\alpha}{R^\alpha}$$ and setting $$w=u(x)-u(y)-\Phi (x,y),$$ then we have $w\le0$ on $\partial Q$. We claim that $w\le 0$ in $Q$. If it is not true, there exists a positive maximum value of $w$ in $Q$ at $(x,y)$ and by (\[eq5\]) and (\[eq6\]), we have $$\|x-y\|\le (\frac {\|u(x)-u(y)\|}{K})^{1/\alpha} R\le (\frac 2 K)^{1/\alpha}R,$$ or $$\label{eq26} K\left( \frac{\|x-y\|}{R}\right)^\alpha\le 2.$$ It is easy to see $$\|Z_0\|\le CR^{-1},$$ $$\begin{aligned} \frac {\alpha K}{R^\alpha}\|x-y\|^{\alpha-1}\le A \le C(R^{-1}+\alpha K \frac {\|x-y\|^{\alpha -1}}{R^\alpha }) \end{aligned}$$ where $Z_0$ and $A$ are defined by (\[eq18\]). From (\[eq19\]) and (\[eq26\]), we have $$\begin{aligned} \omega\le C[\frac {\|x-y\|}{R} +\alpha K (\frac {\|x-y\|} R)^\alpha +(\alpha K)^{-1}(\frac {\|x-y\|} R)^{1-\alpha}]^\beta\\ \end{aligned}$$ $$\le C(\alpha +\alpha^{-1}K^{-1/\alpha})^\beta=o(1),$$ as taking $\alpha$ small first and then taking $K$ large. By the definition of (\[eq9\]), we get $$\begin{aligned} \|\|Z_1\|\|\le CR^{-2}, Z=0,\\ \|\|Z_2\|\|\le C R^{-2}+C\alpha K R^{-\alpha}\|x-y\|^{\alpha-2}. \end{aligned}$$ Hence, for our $\Phi$, by applying (\[eq26\]), (\[eq22\]) becomes $$\label{eq30}\begin{aligned} \frac {4\alpha(1-\alpha)K}{R^\alpha}\|x-y\|^{\alpha-2}+O(R^{-2})\\ \le C[R^{-2}+(\alpha K R^{-\alpha}\|x-y\|^{\alpha-1})^2\omega +\alpha KR^{-\alpha}\|x-y\|^{\alpha-2}\omega^2]\\ \le C[R^{-2}+\alpha^2 K R^{-\alpha}\|x-y\|^{\alpha-2}\omega +\alpha KR^{-\alpha}\|x-y\|^{\alpha-2}\omega^2]. \end{aligned}$$ (\[eq30\]) can not hold when we take $\alpha$ small first and then take $K$ large. Thus $u(x)-u(y)-\Phi$ cannot takes positive maximum value in $Q$. Combining estimate for $[u(y)-u(x)-\Phi]_{\partial Q}$, we have $$\|u(x)-u(y)\|\le \Phi=\frac 2{R^2}(\|x+y\|^2+\|x-y\|^2)+K\frac {\|x-y\|^\alpha}{R^\alpha}.$$ Especially, $$\|u(x)-u(0)\|\le \frac 4{R^2}\|x\|^2+K\frac {\|x\|^\alpha}{R^\alpha}\le (4+K)\frac {\|x\|^\alpha}{R^\alpha},$$ By coordinate translation, $\forall$ $y$ such that $\|y\|<R$, we have $$\|u(x+y)-u(y)\|\le (4+K)\frac {\|x\|^\alpha}{R^\alpha}.$$ The theorem follows by substituting $x+y$ for $x$. [Theorem 4.2]{} Let $u$ be a viscosity solution of (\[eq1\]) in $\Omega$ and suppose that $(\ref{eq2})-(\ref{eq4})$ hold, then u is locally Lipschitz continuous and satisfies the following estimate $$\label{eq31} \|u(x)-u(y)\|\le \frac C d \|x-y\|.$$ Let $x_0\in \Omega$ and $R>0$ such that $B(x_0,2R)\subset \Omega$. Without loss of generality, suppose $x_0$ is the origin and $sup_{B_{2R}}\|u(x)\|=1$. Consider the function $$\label{eq32}\begin{aligned} \Phi(x,y)=\frac 2{R^2}(\|x+y\|^2+\|x-y\|^2)+4[1-\frac 1 4(\frac{K\|x-y\|}{R})^\gamma]\frac{K\|x-y\|}R\end{aligned}$$ in the domain $$Q=\{(x,y)\in \Omega\|\|x\|^2+\|y\|^2<R^2, K\|x-y\|<R\}.$$ where $\gamma(\le \alpha\beta)$ and $K(\ge 1)$ are positive constants to be chosen. Obviously, we have $w=u(x)-u(y)-\Phi(x,y)\le 0$ on $\partial Q$. If $w$ takes positive maximum in $Q$ at (x,y), by $(\ref{eq5}),(\ref{eq25})$ and $(\ref{eq32})$, we have $$\frac {2\|x+y\|^2}{R^2}\le \Phi\le u(x)-u(y)\le \frac {C\|x-y\|^\alpha}{R^\alpha},$$ hence $$\|x+y\|\le CR(\frac{\|x-y\|}R)^{\alpha/2}$$ and $$K\frac {\|x-y\|}{R}\le \Phi\le u(x)-u(y)\le \frac {C\|x-y\|^\alpha}{R^\alpha},$$ $$K(\frac {\|x-y\|}{R})^{1-\alpha}\le C,\ K^{\frac 1{1-\alpha}}\frac {\|x-y\|}{R}\le C,\ K\frac {\|x-y\|}R\le CK^\frac {-\alpha}{1-\alpha}.$$ Thus $$K(\frac {\|x-y\|}{R})\le C K^{-\frac \alpha{2(1-\alpha)}}(\frac {\|x-y\|}R)^{\frac \alpha 2},$$ and the corresponding $Z_0$, $A$ and $\omega $ defined by (\[eq18\]) and (\[eq19\]) have the following estimates. $$\|Z_0\|=O(R^{-1}(\frac {\|x-y\|}R)^\frac \alpha 2),\ \frac K R\le A=O(\frac K R),$$ $$\omega=O([(1+A)\|x-y\|+\frac {\|Z_0\|}{1+A}]^\beta)$$ $$=O(K^{-\frac {\alpha\beta}{2(1-\alpha)}}(\frac {\|x-y\|}R)^\frac {\alpha\beta}2)=o(1)$$ as $K$ is large. And $$\|\|Z_1\|\|=O(R^{-2}),\ Z=0,\ \|\|Z_2\|\|=O(\frac K{R\|x-y\|}),$$ $$-Tr(\Upsilon Z_2)=O(R^{-2})+(1+\gamma)\gamma\frac {K^2}{R^2}(\frac {K\|x-y\|}R )^{\gamma-1},$$ where $Z,Z_1,Z_2$ are defined by (\[eq9\]). Substituting the above estimates into (\[eq22\]), we obtain $$\label{eq33}\begin{aligned} [\frac \lambda 2+o(1)][O(R^{-2})+(1+\gamma)\gamma\frac {K^2}{R^2}(\frac {K\|x-y\|}R )^{\gamma-1}]\\ \le C[O(R^{-2})+K^{-\frac {\alpha\beta}{2(1-\alpha)}}(\frac {\|x-y\|}R)^{\frac {\alpha\beta} 2} \frac {K^2}{R^2}\\ +K^{-\frac {\alpha\beta}{1-\alpha}}(\frac {\|x-y\|}R)^{\alpha\beta}\frac K{R\|x-y\|}. \end{aligned}$$ Since $\frac {K\|x-y\|}R\le C$, then $(\frac {K\|x-y\|}R)^{\gamma -1}\ge C$. If we take $\gamma=\alpha\beta$ and $K$ large enough, (\[eq33\]) cannot hold. This means $$\|u(x)-u(y)\|\le \Phi (x,y).$$ In particular, we have $$\|u(x)-u(0)\|\le \Phi (x,0)$$ $$\le \frac {4\|x\|^2}{R^2}+\frac {4K\|x\|}R\le 4(1+K)\frac{\|x\|}R.$$ Applying coordinate translation, $\forall$ $y$ such that $\|y\|<K$ we have $$\|u(x+y)-u(y)\|\le 4(1+K)\frac{\|x\|}R.$$ The theorem is proved by substituting $x+y$ for $x$. $C^{1+1}$ interior estimate =========================== [Definition 3.]{} $u$ is said in $C^{1+1}(\Omega)$, if $Du(x)$ exists $\forall$ $x\in \Omega$ and moreover, $Du(x)$ satisfies Lipschitz condition for all closed subset $\tilde \Omega \subset \subset \Omega$, and, for all line segment $\overline{xy}\in\tilde\Omega$, $$\label{eq34} \|Du(x)-Du(y)\| \le C\|x-y\|,$$ where $C$ depends only on $n,\lambda,\Lambda,\mu,\beta$ and $dist(\tilde \Omega,\partial\Omega)$. If (\[eq34\]) is replaced by H$\ddot o$lder condition and $$\label{eq35} \|Du(x)-Du(y)\| \le C\|x-y\|^\alpha (0<\alpha<1),$$ we call $u\in C^{1+\alpha}(\Omega).$ The following is a covering theorem. [Theorem 5.1]{} For any given sufficiently small positive constant $\theta$ and $\forall\ B(x_0,R_0)\subset \Omega$ with $R_0\le\theta$, $u\in C^{1+1}(B(x_0,\xi R_0))$ where $\xi$ is a constant $\xi\in(0,1)$, then $u\in C^{1+1}(\Omega).$ For any $\tilde \Omega \subset \subset \Omega$ and all $x,y\in \tilde\Omega$ with $\bar{xy}\in \tilde \Omega $, we cover $\bar{xy}$ by a finite set of spheres $\{B(z_i,\xi R_i)\}_{0\le i\le m-1}\subset \tilde \Omega$, with $B(z_i,R_i)\subset\tilde\Omega$, where $R_i\le \theta$ with $x_0=x,x_m=y$ and $\bar{x_ix_{i+1}}\in B(z_i,\xi R_i))(0\le i\le m-1)$. Hence we have $$\|Du(x)-Du(y)\|=\|\sum_{i=0}^{m-1}[Du(x_i)-Du(x_{i+1})]\|$$ $$\le \sum _{i=0}^{m-1}\|Du(x_i)-Du(x_{i+1})\|\le C\sum_{i=0}^{m-1}\|x_i-x_{i+1}\|= C\|x-y\|.$$ The theorem is proved. Fix a $B(x_0,R_0)$ such that $\overline{B(x_0,R_0)}\subset\Omega$ and by section \[sec:Lipschitzcontinuity\], we see that there exists a constant $M_1$ such that $\forall$ $x,y\in B(x_0,R_0)$ $$\label{eq36} \|u(x)-u(y)\| \le M_1\|x-y\|.$$ Without loss of generality we assume $\theta \le M_1$,otherwise substituting $M_1$ by max$(\theta,M_1)$. We consider a function $v(y)$ in $B(0,1)$ as follows $$\begin{aligned} v(y)=\frac {u(x_0+R_0y)-u(x_0)}{\widetilde{M}},\qquad\qquad\\ \widetilde{M}=osc_{x\in B(x_0,R_0)}u(x)\qquad\qquad\\ +R^2_0[1+sup_{\|p\|\le M_1}sup_{x\in B(x_0,R_0)}\|F(x,u(x),p,0)\|] .\end{aligned}$$ By the definition of viscosity solution it is easy to see in $B(0,1)$, $$\label{eq37} \|v\|\le 1,\ v\in S(f),\ \|f\|\le 1,$$ where $S(f)$ denotes the class of viscosity solutions to elliptic equation related to Pucci’s extremal operator(see \[7\]). It is well known that (see\[8\]) the set $Y_E$ of points$\{y_1\}$ in $B(0,1)$ satisfying the following inequality $$\label{eq38} \|v(y)-v(y_1)-<a,y-y_1>\| \le E\|y-y_1\|^2.$$ $\forall$ $y\in B(0,1)$ has density $\frac{\|Y_E\|}{\|B(0,1)\|}$greater than $1-E^{-\Gamma}$, where the constant $E$ is suitably large, i.e., $E$ is bounded below by a constant $E_0$ depending only on $n,\lambda,\Lambda,\mu$ and unbounded above. And $\Gamma$ satisfies $0<\Gamma<1$ and depends only on $n,\lambda,\Lambda,\mu$ as well. All $y_1$ $\in$ $Y_E$ are called Caffarelli points in $B(0,1)$. In the following we always fix a large $E$ for our study. Taking any point $y_1\in Y_E$ and scaling back as $x_1$, we have $x_1\in B(x_0,R_0)$ such that $\forall$ $x\in B(x_0,R_0)$ $$\label{eq39}\begin{aligned} \|u(x)-u(x_1)-<a,x-x_1>\|\qquad\qquad\\ \le \frac E{R_0^2} M\|x-x_1\|^2\le \frac {C_1EM_1}{R_0}\|x-x_1\|^2 ,\end{aligned}$$ where $$\label{eq40}\begin{aligned} C_1=3+sup_{\|p\|\le M_1}sup_{x\in B(x_0,R_0)}\|F(x,u(x),p,0)\|.\end{aligned}$$ The inequality (\[eq39\]) is called Caffarelli expansion of $u(x)$ in $B(x_0,R_0)$ and $x_1$ is called a Caffarelli point in $B(x_0,R_0)$, and (\[eq39\]) is valid under the restriction $\theta\leq M_1$, hence $R_0\le \theta\le M_1$, and the constant vector $a$ can be determined by dividing (\[eq39\]) by $\|x-x_1\|$, and then let $x\rightarrow x_1$ in any fixed direction $l=\frac {x-x_1}{\|x-x_1\|}.$ We have $$<Du(x_1),l>-<a,l>=0,$$ or $a=Du(x_1)$. This means that at any Caffarelli point $x_1$, $Du(x_1)$ exists. Moreover, $$\label{eq41} \|Du(x_1)\|\le \lim_{y\rightarrow x_1}\frac {\|u(x_1)-u(y)\|}{\|x_1-y\|}\le M_1.$$ Since Caffarelli point of $B(x_0,R_0)$ is always a Caffarelli point of $B(y_0,S_0)$, where $B(y_0,S_0)\subset B(x_0,R_0)$. And the inverse is not true. So that talk about Caffarelli point, we must point out its related sphere simultaneously. Let $x_1$ be a Caffarelli point of $B(x_0,R_0)$ such that $B(x_1,\sqrt{2}R_1)\subset B(x_0,R_0)$ and assume that $$\label{eq42} \|u(x)-u(y)-<Du(x_1),x-y>\|\le K\|x-y\|.$$ $\forall x,y \in B(x_1,\sqrt{2}R_1).$ Applying (\[eq36\]) and (\[eq41\]) we have the Lipschitz coefficient $K\le 2M_1$. We estimate the Lipschitz coefficient in the following region of $\mathbf{R}^{2n}$. $$\label{eq43} Q:\{x,y\|J\equiv {(\frac{\|x+y-2x_1\|}{2R_1})}^2+{(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}}+e^{-2m(1+2\delta)\sigma} < 1\}$$ where $\epsilon, \sigma$ are small positive constants and only $\epsilon$ is depending on $E$, $\delta$ is a constant, $\delta\in [\frac 1 2, \frac 3 2], m(>1)$ is a constant independent of $E$. It is easy to show that $$Q\subset\{x,y\|{(\frac{\|x+y-2x_1\|}{2R_1})}^2+{(\frac{\|x-y\|}{2R_1})}^2 < 1\}$$ $$\subset \{x\|\|x-x_1\|<\sqrt{2}R_1\}\times \{y\|\|y-x_1\|<\sqrt{2}R_1\}$$ $$\subset \{x\|\|x-x_1\|<R_0\}\times \{y\|\|y-x_1\|<R_0\}$$ Take the comparison function $\Phi(x,y)$ to be $$\label{eq44} \Phi(x,y)=KJ^{\delta/2}[(1+\varphi)\|x-y\|-\Psi(\|x-y\|)],$$ where $\Psi$ is a $C^2$ function to be determined which satisfies $$\label{eq45} \|\Psi'(\|x-y\|)\| \le \varphi, \qquad \Psi''(\|x-y\|)=O(\frac1{\|x-y\|}).$$ We investigate the conditions for $$\label{eq46} \|u(x)-u(y)-<Du(x_1),x-y>\| \le \Phi(x,y),$$ $\forall x,y \in Q$ On $\partial Q$, applying (\[eq42\]),(\[eq44\]) and (\[eq45\]), we see that (\[eq46\]) is satisfied. In $Q$ we have $$\begin{aligned} \|Z_0\|=\|(\frac{\partial}{\partial x}+\frac{\partial}{\partial y})[\mp(Du(x_1))+\Phi]\|\qquad \qquad\\=4K\delta J^{\frac{\delta}{2}-1}\frac{\|x+y-2x_1\|}{R_1^2}[(1+\varphi)\|x-y\|-\Psi(\|x-y\|)]\\ \le 4(1+\varphi)K\delta J^{\frac{\delta-1}2}\frac{\|x-y\|}{R_1}\qquad \qquad \\ =O(K\epsilon{(\frac{\|x-y\|}{2\epsilon R_1})}^{1-\frac {1}{2} \frac{\sigma}{\ln E}}).\qquad \qquad \\ A=\|Z_0\|+\|(\frac{\partial}{\partial x}-\frac{\partial}{\partial x})[\mp<Du(x_1),x-y>+\Phi]\|=O(1).\\ \omega=C_2{[(1+A)\|x-y\|+\frac{\|Z_0\|}{1+A}]}^\beta\qquad\qquad\\ =O({[(\epsilon R_1+K\epsilon){(\frac{\|x-y\|}{2\epsilon R_1})}^{1-\frac {1}{2} \frac{\sigma}{\ln E}}]}^\beta)\qquad\qquad\\ =O({(K\epsilon)}^\beta {(\frac{\|x-y\|}{2\epsilon R_1})}^{\beta[1-\frac {1}{2} \frac{\sigma}{\ln E}]})=o(1).\qquad \end{aligned}$$ When $\epsilon$ is small and we restrict $R_1$ by $$\label{eq47} R_1 \le K.$$ It is easy to calculate $$\\|Z_1\\|=O(\frac{K}{R_1^2}J^{\delta/2-1}\|x-y\|),$$ $$\\|Z\\|=O(\frac{K}{R_1}J^{\frac{\delta-1}{2}}),$$ $$\\|Z_2\\|=O(KJ^{\delta/2}\frac 1 {\|x-y\|}),$$ by using (\[eq45\]). $$\label{eq48} -Tr(\Upsilon Z_2)\geq 4KJ^{\delta/2}\Psi''(\|x-y\|)+ O(KJ^{\delta/2-1}\frac{\sigma}{\ln E}\frac 1 {2\epsilon R_1} {(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}-1}).$$ The validity of (\[eq48\]) is due to $$\frac\delta 2(1-\frac\delta 2)J^{\frac\delta 2-2}Tr\{\Upsilon[(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}){(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}}]$$ $$\otimes[(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}){(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}}]\}[(1+\varphi)\|x-y\|-\Psi(\|x-y\|)]\geq 0,$$ and $$(\frac{\partial}{\partial x}-\frac{\partial}{\partial y})\Psi=\Psi'\frac{2(x-y)}{\|x-y\|},$$ $${(\frac{\partial}{\partial x}-\frac{\partial}{\partial y})}^2\Psi=2\Psi'(\frac{\partial}{\partial x}-\frac{\partial}{\partial y})\frac{x-y}{\|x-y\|}+ 4\Psi''\frac{(x-y)\otimes(x-y)}{{\|x-y\|}^2}$$ $$\cdot Tr[\Upsilon{(\frac{\partial}{\partial x}-\frac{\partial}{\partial y})}^2\Psi]=4\Psi''(\|x-y\|),$$ We select $\Psi(\|x-y\|)$ such that $$\Psi''(\|x-y\|)=\left\{\begin{array}{l} \frac{\varphi}{3{\|x-y\|}^{1-\frac{2\sigma}{\ln E}}{[2\epsilon R_1 E^{-m\sigma(1+2\delta)}]}^{\frac{2\sigma}{\ln E}}}{[(m+1)\sigma(1+2\delta)\ln E]}^{-1}, \\ \textrm{when $0<\|x-y\|\le 2\epsilon R_1 E^{-m\sigma(1+2\delta)} $},\\ \frac{\varphi}{3\|x-y\|}{[\ln {\frac{2\epsilon R_1}{\|x-y\|}} +\sigma(1+2\delta)\ln E]}^{-\frac 1 2}{[(m+1)\sigma(1+2\delta)\ln E]}^{-\frac 1 2}, \\ \textrm{when $ 2\epsilon R_1 E^{-m\sigma(1+2\delta)} \le \|x-y\| \le 2\epsilon R_1 $}, \end{array} \right.$$ Integrating the above expression we have $$\Psi'(\|x-y\|)=\left\{\begin{array}{l} \frac{\varphi}{3\frac{2\sigma}{\ln E}}{[\frac{\|x-y\|}{2\epsilon R_1 E^{-\sigma(1+2\delta)}}]}^{\frac{2\sigma}{\ln E}}{[(m+1)\sigma(1+2\delta)\ln E]}^{-1},\\ \textrm{when $0<\|x-y\|\le 2\epsilon R_1 E^{-m\sigma(1+2\delta)} $},\\ \frac{\varphi}{3\frac{2\sigma}{\ln E}}{[(m+1)\sigma(1+2\delta)\ln E]}^{-1}+2\\ -2{[\ln {\frac{2\epsilon R_1}{\|x-y\|}}+\sigma(1+2\delta)\ln E]}^{\frac 1 2}{[(m+1)\sigma(1+2\delta)\ln E]}^{-\frac 1 2},\\ \textrm{when $ 2\epsilon R_1 E^{-m\sigma(1+2\delta)} \le \|x-y\| \le 2\epsilon R_1 $}, \end{array} \right.$$ $$\Psi(\|x-y\|)=\int_0^{\|x-y\|}\Psi'(t)dt.$$ Hence (\[eq45\]) is valid by the explicit expression of $\Psi,\Psi'$. We want to prove that $$\label{eq49} -Tr(\Upsilon Z_2)\geq KJ^{\delta/2}\Psi''(\|x-y\|)$$ under suitable conditions. (\[eq49\]) follows from $$J^{\delta/2-1}\frac{\sigma}{\ln E}\frac 1 {2\epsilon R_1}{(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}-1}\ll J^{\delta/2}\Psi''(\|x-y\|),$$ or $$I=\frac 3 \varphi \frac{\sigma}{\ln E}\frac 1 {2\epsilon R_1}{(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}-1}\frac 1 {\Psi''(\|x-y\|)}\ll J,$$ $$={(\frac{\|x+y-2x_1\|}{2R_1})}^2+{(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}}+e^{-2m(1+2\delta)\sigma}.$$ In the interval $$2\epsilon R_1 E^{-m\sigma(1+2\delta)} \le \|x-y\| \le 2\epsilon R_1,$$ $$I=\frac 3 \varphi \frac{\sigma}{\ln E}\frac 1 {2\epsilon R_1}{(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}-1}\|x-y\|{[\ln {\frac{2\epsilon R_1}{\|x-y\|}}+\sigma(1+2\delta)\ln E]}^{\frac 1 2}$$ $$\cdot{[(m+1)\sigma(1+2\delta)\ln E]}^{\frac 1 2}$$ $$\leq \frac 3 \varphi \sigma^2 (m+1)(1+2\delta){(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}}\ll {(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}},$$ if the following condition is true $$\label{eq50} (m+1)\sigma^2\ll 1.$$ In the interval $0<\|x-y\|< 2\epsilon R_1 E^{-m\sigma(1+2\delta)}$, $$I=\frac 3 \varphi \frac{\sigma}{\ln E}\frac 1 {2\epsilon R_1}{(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}-1}{\|x-y\|}^{1-\frac{2\sigma}{\ln E}}$$ $$\cdot{[2\epsilon R_1 E^{-m \sigma (1+2\delta)}]}^{\frac{2\sigma}{\ln E}}[(m+1)\sigma(1+2\delta)\ln E]$$ $$=\frac 6 \varphi \sigma^2 (1+2\delta)E^{-m(1+2\delta)\frac{2\sigma}{\ln E}}(m+1)$$ $$=\frac 6 \varphi (m+1) \sigma^2 (1+2\delta)e^{-2m(1+2\delta)\sigma}\ll e^{-2m(1+2\delta)\sigma},$$ if the condition (\[eq50\]) is true. Hence, (\[eq49\]) is valid under the restriction (\[eq50\]). It is easy to prove that $$\\|Z_1\\|\ll -Tr(\Upsilon Z_2)$$ when $$\sigma \epsilon^2 \ln E \ll 1 ;$$ $$\\|Z_2\\| \omega^2 \ll -Tr(\Upsilon Z_2)$$ when $$\epsilon^{2\beta} \sigma \ln E {(\frac{\|x-y\|}{2\epsilon R_1})}^{2\beta [1-\frac 1 2 \frac{\sigma}{\ln E}]-\frac{2\sigma}{\ln E}}$$ $$\ll \epsilon^{2\beta} \sigma \ln E \ll 1 ;$$ $$\\|Z\\|\ll -Tr(\Upsilon Z_2)$$ when $$\epsilon^{\beta} \sigma \ln E \ll 1 ;$$ $$\omega \ll -Tr(\Upsilon Z_2)$$ when $$\epsilon^{1+\beta} R_1 \sigma \ln E \ll 1 .$$ Taking $\epsilon = {(\ln E)}^{-\frac 2 \beta}$, then the estimates for $\\|Z_1\\|,\\|Z_2\\|,\\|Z\\|,\omega$ are all true. Hence we have the following lemma. [Lemma 5.2]{} The estimates (\[eq46\]) is valid when (\[eq47\]) and (\[eq50\]) are true. The lemma is proved since (\[eq23\]) is true by the above estimates. (\[eq46\]) implies that $$\label{eq51} \begin{aligned} \|u(x)-u(y)-<Du(x_1),x-y>\|\leq \qquad \qquad \qquad \\ (1+\varphi)K{[{(\frac{\|x+y-2x_1\|}{2R_1})}^2 +{(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}}+e^{-2m(1+2\delta)\sigma}]}^{\frac \delta 2}\|x-y\|, \end{aligned}$$ $\forall x,y \in Q.$ Taking constants $m,\sigma$ such that $m\sigma$ is large and $m\sigma^2$ is small, hence (\[eq50\]) is valid. Taking $\|x-y\|$ small such that $$\label{eq52} {(\frac{\|x-y\|}{2\epsilon R_1})}^{\frac{2\sigma}{\ln E}}\leq e^{-2m(1+2\delta)\sigma}$$ then (\[eq51\]) implies that $$\label{eq53} \begin{aligned} \|u(x)-u(y)-<Du(x_1),x-y>\|\leq \qquad \qquad \\ (1+\varphi)K{[{(\frac{\|x+y-2x_1\|}{2R_1})}^2 +2e^{-2m(1+2\delta)\sigma}]}^{\frac \delta 2}\|x-y\|, \end{aligned}$$ $\forall x,y \in Q $ satisfies (\[eq52\]). We relax the restriction (\[eq52\]) for $x,y$ and assume $x,y$ satisfying $$\forall x,y \in \check{Q}:\{x,y\|{(\frac{\|x+y-2x_1\|}{2R_1})}^2 +{(\frac{\|x-y\|}{2 R_1})}^{2} < 1-e^{-2m(1+2\sigma)\delta}\},$$ interpolating line segment $\overline{xy}$ by $z_i=\frac{x+y} 2 + \frac{i}{2M}(x-y),$ where $i=-M, -M+1, \cdots, 0, \cdots, M-1, M$. By applying (\[eq53\]), we have $$\|u(z_{-i})-u(z_{-i-1})-<Du(x_1),z_{-i}-z_{-i-1}>\|$$ $$+\|u(z_{i+1})-u(z_i)-<Du(x_1),z_{i+1}-z_i>\|$$ $$\leq (1+\varphi)K[(U+V)^{\frac{\delta}{2}}+(U-V)^{\frac{\delta}{2}}]\frac{\|x-y\|}{2M},$$ where $$U={(\frac{\|x+y-2x_1\|}{2R_1})}^2 +{(\frac{2i+1}{2M})}^2{(\frac{\|x-y\|}{2R_1})}^2 +2e^{-2m(1+2\delta)\sigma},$$ $$V=\frac{2i+1}{M}<\frac{x+y-2x_1}{2R_1}, \frac{x-y}{2R_1}>.$$ Since $0<\frac{\delta}{2}\leq \frac{3}{4}\leq 1$, we have $${(U+V)}^{\frac{\delta}{2}}+{(U-V)}^{\frac{\delta}{2}}\leq 2U^{\frac{\delta}{2}}$$ $$\leq 2{[{(\frac{\|x+y-2x_1\|}{2R_1})}^2 +{(\frac{\|x-y\|}{2 R_1})}^{2} + 2e^{-2m(1+2\delta)}]}^{\frac{\delta}{2}}$$ Summing for $i=0,1,\cdots,M-1$, we have $$\label{eq54}\begin{aligned} \|u(x)-u(y)-<Du(x_1),x-y>\|\leq \qquad \qquad \\ (1+\varphi)K{[{(\frac{\|x+y-2x_1\|}{2R_1})}^2 +{(\frac{\|x-y\|}{2R_1})}^2+\xi^2]}^{\frac \delta 2}\|x-y\|, \end{aligned}$$ $\forall x,y \in \tilde{Q}:\{x,y\|{(\frac{\|x+y-2x_1\|}{2R_1})}^2 +{(\frac{\|x-y\|}{2R_1})}^2<1-\xi^2\}$ where $$\label{eq55} 2e^{-2m(1+2\delta)\sigma}\leq 2 e^{-3m\sigma}\equiv {\xi}^2 \ll 1.$$ $\forall \delta \in [\frac 1 2, \frac 3 2].$ Now we state and prove the fundamental lemma on decreasing of Lipschitz coefficient in certain constant ratio accompanied with decreasing of radius of spherical region in suitable constant ratio. [Remark 5.1]{} Let $y\to x$ in (5.22), approximately we have $$\label{remark1} \|Du(x)-Du(x_1)\|\leq (1+\varphi+\varepsilon)(\frac{\|x-x_1\|}{R_1})^\delta), (\varepsilon=\frac{\xi R_1}{\|x-x_1\|}),$$ $\delta=\frac{3}{2}$ is a sharp estimate. But Eq.(\[remark1\]) is valid only in the spherical shell $\xi R_1\leq\|x-x_1\|\leq R_1$, and not for the sphere $\|x-x_1\|\leq R_1$, so that, Eq.(\[remark1\]) and (5.22) are reasonable. [Lemma 5.3]{} Let $x_1$ be a Caffarelli point of $B(x_0,R_0)$ satisfying the Lipschitz condition (\[eq42\]), $\forall x,y \in B(x_1,\sqrt{2}R_1) \subset B(x_0,R_0)$.Under the restriction (\[eq47\]), $ \forall $ points $x_2 \in B(x_1,\xi R_1)$, $\exists$ Caffarelli point $y_1$ of $B(x_1,R_1)$ such that $y_1 \in B(x_1,\xi R_1)$, and $\|y_1-x_2\|\leq \frac{\xi}{3} R_1$, we have $$\label{eq56} \|Du(y_1)-Du(x_1)\|\leq \eta K,$$ $$\label{eq57} \|u(x)-u(y)-<Du(y_1),x-y>\|\leq \zeta K\|x-y\|,$$ $\forall x,y$ such that $\{x,y\|{\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2\leq \frac {R_1^2} 9 \}$, where $\eta$ is a small constant and $\zeta$ is a constant ,$\zeta <1$. Both $\eta,\zeta$ depend on $\xi$ only when we fixed $\delta$ to be $\delta=\frac 1 2$. Since $x_2\in B(x_1,\xi R_1)$, in $B(x_2,\frac{\xi}{3} R_1)\cap B(x_1,\xi R_1)$, $\exists$ Caffarelli point of $B(x_1,R_1)$, because of $B(x_2,\frac{\xi}{3} R_1)\cap B(x_1,\xi R_1) \supset B(x_2+\frac{x_1-x_2}{\|x_1-x_2\|}\frac{\xi}{6}, \frac{\xi}{6}R_1)$ and the density of sphere $B(x_2+\frac{x_1-x_2}{\|x_1-x_2\|}\frac{\xi}{6}, \frac{\xi}{6}R_1)$ with respect to $B(x_1,R_1)$ satisfying ${(\frac{\xi}{6})}^n \gg E^{-\Gamma}$ (If this condition is not satisfied, since $\xi$ and $\Gamma$ are independent of $E$, we substitute $E$ by a large one, such that this condition is satisfied.) We denote one of Caffarelli point by $y_1$. Hence $y_1 \in B(x_1,\xi R_1), \|y_1 -x_2\|\leq \frac{\xi}{3} R_1$ and moreover $Du(y_1)$ exists. Take $y=y_1$ in (\[eq54\]) and then divide it by $\|x-y_1\|$ and let $x\rightarrow y_1$, we have $$\|Du(y_1)-Du(x_1)\|\leq (1+\varphi){(2\xi^2)}^{\frac \delta 2} K$$ $$=(1+\varphi){(2\xi^2)}^{\frac 1 4} K\equiv \eta K ,$$ this is (\[eq56\]). Since $$\{x,y\|{\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2\leq \frac {R_1^2} 9 \}$$ $$\subset \{x,y\|{\|\frac {x+y} 2 -x_1\|}^2 +{(\frac {\|x-y\|} 2)}^2\leq \frac {R_1^2} 4 \} \subset \tilde{Q}.$$ applying (\[eq54\]) and (\[eq56\]) we have $$\|u(x)-u(y)-<Du(y_1),x-y>\|\leq [\eta + (1+\varphi){(\frac 1 4 +\xi^2)}^{\frac 1 4}]K\|x-y\|\leq \zeta K\|x-y\|.$$ $$\forall \{x,y\|{\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2\leq \frac {R_1^2} 9 \} ,$$ where we denote $$\label{eq58} \zeta = \eta +(1+\varphi)[{(\frac 1 4+ 2\xi^2)}^{\frac 1 4}{(1+9\xi^2)}^{\frac 1 2}+\eta {(\frac 1 9 +2\xi^2)}^{-\frac 1 2}].$$ We take a little larger $\zeta$ over our necessary for the sake of next lemma. Since $\xi, \eta$ are small, we have $$\zeta \thickapprox (1+\varphi){(\frac 1 4)}^{\frac 1 4}=[1+\frac 1 2 (\sqrt{2}-1)]\frac{ 1 }{\sqrt{2}}=\frac 1 2 (1+\frac {1} {\sqrt{2}})<1,$$ hence we can take $\xi$ small such that $\zeta<1$. The lemma is proved completely. [Remark 5.2]{} The only reason for approximating the general point $x_2$ by Caffarelli point $x_1,y_1,\ldots$ is that there exists first derivative on Caffarelli points. [Lemma 5.4]{} We have $u \in C^{1+\alpha}(B(x_0,\frac{\xi}{\sqrt{2}} R_0))$, where $$\label{eq59} \alpha=\frac {\ln \zeta}{\ln \frac 1 3}$$ Fixed a $x_2 \in (B(x_0,\frac{\xi}{\sqrt{2}}R_0))$. Take a Caffarelli point $x_1$ of $(B(x_0,\frac{\xi}{\sqrt{2}} R_0))$ such that $\|x_1-x_2\|\leq \frac{\xi}{\sqrt{2}} R_0$. Take $R_1=(1-\xi)\frac{1}{\sqrt{2}} R_0$ and $K=2M_1$. Denote $R^{(0)}=R_1, R^{(k)}=\frac {R_1}{3^k}, K^{(0)}=K, K^{(k)}=\zeta^k K (k=0,1,2,\cdots)$. Denote $x_1\equiv y_0$, take Caffarelli point $y_1$ of $B(y_0,R^{(0)})$ such that $y_1 \in B(y_0,\xi R^{(0)})\cap B(x_2,\frac{\xi}{3} R^{(0)})$, hence $\|y_1-x_2\|< \frac{\xi}{3} R^{(0)}=\xi R^{(1)},$ i.e. $y_1\in B(x_2,\xi R^{(1)})$, take Caffarelli point $y_2$ of $B(y_1,R^{(1)})$ such that $y_2 \in B(y_1,\xi R^{(1)})\cap B(x_2,\frac{\xi}{3} R^{(1)}) \cdots $ In general, take Caffarelli point $y_k$ of $B(y_{k-1},R^{(k-1)})$ such that $y_k \in B(y_{k-1},\xi R^{(k-1)})\cap B(x_2,\frac{\xi}{3} R^{(k-1)})$, $\forall k=1,2,\cdots.$ Since we restrict $R_0$ to be $\leq M_1$, we have $$R^{(k)}=\frac {R_1}{3^k}\leq \frac {R_0}{3^k \sqrt{2}}\leq \frac {M_1}{3^k}\leq K \zeta^k = K^{(k)} (k=0,1,2,\cdots),$$ i.e. (\[eq47\]) is valid for all $k=0,1,2,\cdots$. We prove by induction that $$\label{eq60} \|Du(y_k)-Du(y_{k-1})\|\leq K^{(k)},$$ $$\label{eq61} \|u(x)-u(y)-<Du(y_k),x-y>\|\leq K^{(k)}\|x-y\|,$$ $\forall x,y$ such that $\{ {\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2\leq {(R^{(k)})}^2 \}$, $\forall k=1,2,\cdots$ When $k=1$, (\[eq60\]) follows from (\[eq56\]) and $\eta K\leq \zeta K=K^{(1)}$, (\[eq61\]) follows from (\[eq57\]). When (\[eq60\]) and (\[eq61\]) are valid for $k-1$, substituting $y_0=x_1, R^{(0)}=R_1, K^{(0)}=K=2M_1$ by $y_{k-1}, R^{(k-1)}, K^{(k-1)}$ in lemma 5.2, 5.3. Since $R^{(k)}\leq K^{(k)}$, hence lemma 5.2 is valid and lemma 5.3 is valid by using $y_k, R^{(k)}$ and $K^{(k)}$ instead of $y_1, R^{(1)}=\frac {R_1} 3, K^{(1)}=\zeta K$. (\[eq60\]) and (\[eq61\]) follow for $k$ by applying lemma 5.3 and $\eta K^{(k-1)}\leq \zeta K^{(k-1)}=K^{(k)}$. i.e. (\[eq60\]) and (\[eq61\]) are valid $\forall k=1,2,\cdots$. Applying (\[eq60\]) we have $$\sum_{k=1}^\infty \|Du(y_k)-Du(y_{k-1})\|\leq \sum_{k=1}^\infty K \zeta ^k <\infty.$$ Hence the series $$\sum_{k=1}^\infty [Du(y_k)-Du(y_{k-1})]$$ converges, denote its limit by $\tilde{a}-Du(x_1)$. $\forall x,y$ satisfies $\{x,y {\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2\leq {(R^{(k)})}^2 \}$, applying (\[eq61\]) we have $$\label{eq62} \begin{aligned} \|u(x)-u(y)-<\tilde{a},x-y>\| \leq \|u(x)-u(y)-<Du(y^k),x-y>\| \\ + \|<Du(y^k)-\tilde{a},x-y>\| \qquad\qquad\qquad\\ \leq K^{(k)}\|x-y\|+(\sum_{l=k+1}^\infty K\zeta^l)\|x-y\|=\frac {1+\zeta}{1-\zeta}K^{(k)}\|x-y\| \\ =\frac {1+\zeta}{1-\zeta}K \zeta^k \|x-y\|=\frac {1+\zeta}{1-\zeta}K {(\frac{R^{(k)}}{R^{(0)}})}^\alpha \|x-y\|\qquad\\ \leq \frac {1+\zeta}{1-\zeta}K {(\frac{3R^{(k+1)}}{(1-\xi)R^{(0)}})}^\alpha \|x-y\| \leq CR^\alpha \|x-y\|,\qquad \end{aligned}$$ when we denote $$R= {({\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2)}^{\frac 1 2},$$ $$\label{eq63} C=\frac {1+\zeta}{1-\zeta} 2 M_1 {(\frac {3}{1-\xi})}^\alpha {R_0}^{-\alpha},$$ and assume $$R^{(k+1)}\leq R\leq R^{(k)} (k=0,1,2,\cdots)$$ Put the case $k=0,1,2,\cdots$ together, we obtain that (\[eq62\]) is true $$\forall 0\leq {({\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2)}^{\frac 1 2}\leq R\leq R^{(1)}=\frac {R_1} 3= \frac{1-\xi}{3\sqrt{2}} R_0.$$ Putting $y=x_2$ and dividing (\[eq62\]) by $\|x-x_2\|$, then let $x\rightarrow x_2$, we have $Du(x_2)$ exists and $\tilde{a}=Du(x_2)$. Hence we have $$\label{eq64} \|u(x)-u(y)-<Du(x_2),x-y>\|\leq CR^\alpha \|x-y\|,$$ $\forall x,y$ satisfies $\{x,y\|{({\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2)}^{\frac 1 2}\leq R\leq R_1 =\frac{1-\xi}{3\sqrt{2}} R_0\}$ (\[eq64\]) implies that $$\|Du(x)-Du(x_2)\|\leq C{\|x-x_2\|}^\alpha, \forall x,x_2 \in B(x_0,\frac{\xi}{\sqrt{2}} R_0)$$ Hence $$u \in C^{1+\alpha}(B(x_0,\frac{\xi}{\sqrt{2}} R_0))$$ The lemma is proved. Now we study the case that Lipschitz coefficient contains a factor $R^\gamma$, where $\gamma \in (0,1)$. [Lemma 5.5]{} $\forall x_2 \in B(x_0,\frac{\xi}{\sqrt{2}} R_0)$, if constant $\gamma \in (0,1)$ and positive constant $H,S\leq \frac{(1-\xi)}{3\sqrt{2}}R_0$ exist such that we have $$\label{eq65} \|u(x)-u(y)-<Du(x_2),x-y>\|\leq H R^\gamma \|x-y\|,$$ $\forall x,y$ satisfies $\{x,y\|{({\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2)}^{\frac 1 2}\leq R\leq S\}$ Then in case $$\label{eq66} \gamma \leq 1-\alpha,$$ where $\alpha$ is defined by (\[eq59\]). $\exists$ constant $\tilde{H}(>H)$ and $\tilde{S}(<S)$ such that we have $$\label{eq67} \|u(x)-u(y)-<Du(x_2),x-y>\|\leq H R^{\gamma+\alpha} \|x-y\|,$$ $\forall x,y$ satisfies $\{x,y\|{({\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2)}^{\frac 1 2}\leq R\leq \tilde{S}\}.$ Denote $R^{(0)}=min\{S,H^{\frac 1 {1-\gamma}}\}, K^{(0)}=H{R^{(0)}}^\gamma,$ where the meaning of $R^{(0)}$ is different from that in lemma 5.4. Let $y_0 \in B(x_2,\xi R^{(0)})$ be a Caffarelli point of $u(x)$ in $B(x_2,R^{(0)})$. In the region $B(y_0,(1-\xi )R^{(0)})$, the lemma 5.2 for estimate $\|u(x)-u(y)-<Du(y_0),x-y>\|$ is also valid in the present case, this is because of (\[eq47\]) $$R^{(0)} \leq H{(R^{(0)})}^\gamma =K^{(0)}$$ is true. Take $K=H{(R^{(0)})}^\gamma, \delta=\frac 1 2 +\gamma $. From (\[eq54\]), $\forall x,y \in \tilde{Q} :\{x,y\|{(\frac {\|x+y-2y_0\|} {2R^{(0)}})}^2 +{(\frac {\|x-y\|} {2R^{(0)}})}^2 \leq 1-\xi^2 \},$ we have $$\|u(x)-u(y)-<Du(y_0),x-y>\|$$ $$\leq (1+\varphi) H{(R^{(0)})}^\gamma {[{(\frac {\|x+y-2y_0\|} {2R^{(0)}})}^2 +{(\frac {\|x-y\|} {2R^{(0)}})}^2+2\xi^2]}^{\frac 1 4 +\frac \gamma 2}\|x-y\|.$$ Hence we have for all Caffarelli point $y_1$ of $B(y_0,R^{(0)})$ satisfying $y_1 \in B(y_0,\xi R^{(0)})\cap B(x_2,\frac{\xi}{3} R^{(0)})$, we have $$\label{eq68} \|Du(y_1)-Du(y_0)\|\leq (1+\varphi) H{(R^{(0)})}^\gamma {(2\xi^2)}^{\frac 1 4 +\frac \gamma 2} \leq (1+\varphi) K^{(0)} {(2\xi^2)}^{\frac 1 4 +\frac \gamma 2}=\eta K^{(0)},$$ $\forall x,y$ satisfying $\{x,y\|{\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2 <{(\frac {R^{(0)}} 3)}^2 \}$, we have $$\label{eq69}\begin{aligned} \|u(x)-u(y)-<Du(y_1),x-y>\| \qquad\qquad \\ \leq [(1+\varphi)H{(R^{(0)})}^\gamma {(\frac 1 9 +2\xi^2)}^{\frac \gamma 2} {(\frac 1 4 +2\xi^2)}^{\frac 1 4}+\eta K^{(0)}]\|x-y\|\\ \leq \zeta \frac {K^{(0)}}{3^\gamma} \|x-y\|, \qquad\qquad\qquad \end{aligned}$$ where $\zeta$ is defined by (\[eq58\]). (\[eq68\]) and (\[eq69\]) are the similar relation of lemma 5.3 in the present case. Define $ R^{(k)}=\frac{R^{(0)}}{3^k}, K^{(k)}=H{(R^{(k)})}^\gamma \zeta^k (k=1,2,3,\cdots)$ Take Caffarelli point $y_k$ of $B(y_{k-1},R^{(k-1)})$ such that $y_k \in B(y_{k-1},\xi R^{(k-1)})\cap B(x_2,\frac{\xi}{3} R^{(k-1)})$ . First we verify (\[eq47\]) in the present case. Applying (\[eq66\]) we have $$3^{1-\gamma}\zeta \geq 3^\alpha \zeta=3^{\frac{\ln{\frac 1 \zeta}}{\ln 3}}\zeta =\frac{1}{\zeta} \zeta =1 ,$$ or it is $$\frac 1 {3^k} \leq \frac{\zeta^k}{{(3^k)}^\gamma}.$$ Hence $$R^{(k)}=\frac{R^{(0)}}{3^k}\leq \frac{K^{(0)}}{3^k}=\frac{H{(R^{(0)})}^\gamma}{3^k}$$ $$\leq H{(\frac{R^{(0)}}{3^k})}^\gamma \zeta^k=H{(R^{(k)})}^\gamma\zeta^k=K^{(k)},$$ i.e. (\[eq47\]) is valid in the present case. Then we prove (\[eq60\]) and (\[eq61\]) by induction. When $k=1$, applying (\[eq68\]) we have $$\|Du(y_1)-Du(y_0)\|\leq \eta K^{(0)} =\frac{\eta}{\zeta}3^\gamma K^{(1)} \leq \frac{3\eta}{\zeta}K^{(1)}\leq K^{(1)}.$$ Applying (\[eq69\]) we have $$\|u(x)-u(y)-<Du(y_1),x-y>\|\leq \zeta \frac {K^{(0)}}{3^\gamma} \|x-y\| =K^{(1)}\|x-y\|,$$ $\forall x,y$ satisfies $\{x,y\|{\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2 <{(\frac {R^{(0)}} 3)}^2 ={(R^{(1)})}^2 \}$ i.e. (\[eq60\]) and (\[eq61\]) are valid for $k=1$. Suppose (\[eq60\]) and (\[eq61\]) are valid for $k-1$. Since $K^{(k)}=H{(R^{(k)})}^\gamma \zeta^k$, take $\delta=\frac 1 2 +\gamma$ and apply (\[eq54\]), $$\label{eq70}\begin{aligned} \|u(x)-u(y)-<Du(y_k),x-y>\| \qquad\qquad \\ \leq (1+\varphi)K^{(k)} {[{(\frac {\|x+y-2y_k\|} {2R^{(k)}})}^2 +{(\frac {\|x-y\|} {2R^{(k)}})}^2+2\xi^2]}^{\frac 1 4 +\frac \gamma 2}\|x-y\|, \end{aligned}$$ $\forall x,y$ satisfies $$\{x,y\|{(\frac {\|x+y-2y_k\|}{2R^{(k)}})}^2 +{(\frac {\|x-y\|} {2R^{(k)}})}^2 < 1-\xi^2 \}.$$ Hence applying (\[eq70\]) we have $$\|Du(y_{k+1})-Du(y_k)\|\leq (1+\varphi)K^{(k)}{(2\xi^2)}^{\frac 1 4 +\gamma}\leq (1+\varphi)K^{(k)}{(2\xi^2)}^{\frac 1 4}$$ $$=\eta K^{(k)} =\frac{\eta}{\zeta}3^\gamma K^{(k+1)} \leq \frac{3\eta}{\zeta}K^{(k+1)}\leq K^{(k+1)}.$$ And $$\|u(x)-u(y)-<Du(y_{k+1}),x-y>\|$$ $$\leq (1+\varphi) [K^{(k)}{(\frac 1 9 +2\xi^2)}^{\frac \gamma 2} {(\frac 1 4 +2\xi^2)}^{\frac 1 4}+\eta K^{(k)}]\|x-y\|$$ $$=\frac{\zeta}{3^\gamma}K^{(k)} \|x-y\|=K^{(k+1)}\|x-y\|,$$ $\forall x,y$ satisfies $\{x,y\|{\|\frac {x+y} 2 -x_2\|}^2 +{(\frac {\|x-y\|} 2)}^2 <{(\frac {R^{(k)}} 3)}^2 ={(R^{(k+1)})}^2 \}.$ Hence (\[eq60\]) and (\[eq61\]) are valid for all $k=1,2,\cdots$. Since $Du(x) \in C, \forall x \in B(x_0,\frac{\xi}{\sqrt{2}} R_0)$. Applying (\[eq60\]) and (\[eq61\]) we have $$\|u(x)-u(y)-<Du(x_2),x-y>\|$$ $$\leq \|u(x)-u(y)-<Du(y_k),x-y>\|+[\sum_{l=k+1}^\infty K^{(l)}]\|x-y\|$$ $$\leq K^{(k)}\|x-y\| +[\sum_{l=k+1}^\infty K^{(l)}]\|x-y\|$$ $$=H[{(R^{(k)})}^\gamma \zeta^k+\sum_{l=k+1}^\infty {(R^{(l)})}^\gamma \zeta^l]\|x-y\|$$ $$=\frac{1+\frac{\zeta}{3^\gamma}}{1-\frac{\zeta}{3^\gamma}}H {(R^{(k)})}^\gamma \zeta^k \|x-y\|$$ $$=\frac{1+\frac{\zeta}{3^\gamma}}{1-\frac{\zeta}{3^\gamma}}H {(R^{(k)})}^\gamma {(\frac{R^{(k)}}{R^{(0)}})}^\alpha \|x-y\|.$$ In the region $R^{(k+1)}\leq {({\|x-x_2\|}^2+{\|y-x_2\|}^2)}^{\frac 1 2}=R\leq R^{(k)}$, we have $$R^{(k)}=3 R^{(k+1)}\leq 3 R.$$ Hence $$\|u(x)-u(y)-<Du(x_2),x-y>\|\leq \tilde{H} R^{\gamma+\alpha}\|x-y\|,$$ where $$\label{eq71} \tilde{H}=\frac{1+\frac{\zeta}{3^\gamma}}{1-\frac{\zeta}{3^\gamma}}3^{\gamma+\alpha} H{(R^{(0)})}^{-\alpha}.$$ Putting $k=1,2,3,\cdots$ together we have $$\|u(x)-u(y)-<Du(x_2),x-y>\|\leq \tilde{H} R^{\gamma+\alpha}\|x-y\|,$$ $\forall x,y$ satisfies $\{x,y\|{\|x-x_2\|}^2+{\|y-x_2\|}^2\leq R\leq \check{S}\},$ $$\label{eq72} \check{S}=\frac{R^{(0)}}{3} \leq \frac{1-\xi}{3} \min\{S,H^{\frac{1}{1-\gamma}}\}.$$ The lemma is proved. [Lemma 5.6]{} $\forall (x_0,R_0)\subset \Omega$, we have $$u(x) \in C^{1+1}(B(x_0,\frac{\xi}{\sqrt{2}} R_0)),$$ where $\xi$ is defined by (\[eq55\]). Since $\alpha$ and $\zeta$ are defined by (\[eq59\]) and (\[eq58\]), substituting $\zeta$ by a little large one such that $\zeta<1$ still valid and $\frac 1 \alpha $ is a positive integer $M$. Applying lemma 5.4 once we obtain (\[eq64\]). Then applying lemma 5.5 successively. Denote $$\label{eq73} R_j^{(0)}=\min \{S_j,H_j^{\frac 1 {1-j\alpha}}\}, \qquad j=1,2,\cdots,M.$$ where $S_j,H_j$ are the value of $S,H$ in lemma 5.5 corresponding to $\gamma=j \alpha$. Applying lemma 5.4 we have $$S_1=\frac{1-\xi}{3\sqrt{2}} R_0,$$ $$H_1=\frac{1+\zeta}{1-\zeta} 2 M_1 {(\frac 3 {1-\xi})}^\alpha {(\frac{1-\xi}{\sqrt{2}}R_0)}^{-\alpha}.$$ Since $M_1\geq R_0$, it is easy to obtain that $$\label{eq74} H_1\geq S_1^{1-\alpha}.$$ Hence applying (\[eq73\]) we have $$R_1^{(0)}= S_1.$$ Applying lemma 5.5 we have $$\label{eq75} S_{j+1}=\frac{1-\xi}{3} \min\{S_j,H_j^{\frac 1 {1-j \alpha}}\}.$$ $$\label{eq76} H_{j+1}=\frac{1+\frac{\zeta}{3^{j \alpha}}}{1-\frac{\zeta}{3^{j \alpha}}} 3^{(j+1)\alpha}H_j{(R_j^{(0)})}^{-\alpha} \geq H_j{(R_j^{(0)})}^{-\alpha}.$$ We prove by induction that $$\label{eq77} H_j\geq S_j^{1-j \alpha}$$ is true, then by (\[eq73\]) we have $R_j^{(0)}=S_j$. When $j=1$, (\[eq77\]) is true by (\[eq74\]). If (\[eq77\]) is true for $j$, applying (\[eq76\]) we have $$H_{j+1}\geq H_j{(R_j^{(0)})}^{-\alpha}\geq S_j^{1-j \alpha} S_j^{-\alpha} = S_j^{1-(j+1)\alpha},$$ i.e. (\[eq77\]) is valid for $j$ substitute by $j+1$. Hence (\[eq77\]) is true $\forall j=1,2,\cdots, M$. Applying (\[eq75\]) we have $$S_{j+1} =\frac{1-\xi}{3} S_j (j=1,2,\cdots,M-1).$$ Hence $$S_{j+1}={(\frac{1-\xi}{3})}^j S_1={(\frac{1-\xi}{3})}^{j+1}\frac{R_0}{\sqrt{2}}.$$ Especially we have $$S_M=R_M^{(0)}={(\frac{1-\xi} 3)}^M \frac{R_0}{\sqrt{2}}=[1+o(1)]e^{-\frac{\ln 3}{\alpha}}\frac{R_0}{\sqrt{2}}$$ $$\thickapprox e^{ \frac{{(\ln 3)}^2}{\ln [{\frac 1 2}(1+\frac 1 {\sqrt{2}})]}}\frac{R_0}{\sqrt{2}},$$ when $\xi$ is small. $e^{ \frac{{(\ln 3)}^2}{\ln [{\frac 1 2}(1+\frac 1 {\sqrt{2}})]}}$ is a constant $<1$ and is independent of $\xi$. Hence $$\|u(x)-u(y)-<Du(x_2),x-y>\|\leq \tilde{C} \frac{R}{R_0}\|x-y\|$$ $$\forall x,y \in B(x_0,R_M^{(0)}).$$ Since $R_M^{(0)} > \xi \frac{R_0}{\sqrt{2}},$ hence we have $\forall x_2,x_3 \in B(x_0,\frac{\xi}{\sqrt{2}} R_0),$ $$\|Du(x_3)-Du(x_2)\|\leq \tilde{C} \frac{\|x_3-x_2\|}{R_0}$$ The lemma is proved. Our main theorem 2.1. Theorem 2.1 follows by applying theorem 5.1 and lemma 5.7. [A final remark]{} After we proved the regularity result $u\in C^{1+1}(\Omega)$, a natural question has arisen. Is the regularity result the best possible or not? If it is yes, we must give an example to show that the solution of (\[eq1\]) under the conditions (\[eq2\])-(\[eq4\])) cannot belong to $C^2(\Omega)$. More exactly, the solution cannot possesses more regularity when it is lack of concavity assumption, so that more regularity assumptions on $F$ has no effect. Hence we present the following problem. [An open problem]{} Let the equation (\[eq1\]) satisfying the following assumptions: \(a) $F(x,z,p,X)$ is sufficiently smooth with respect to its arguments $x,z,p,X$. \(b) Assumption (\[eq2\]). \(c) $F$ is strict monotone with respect to $z$ in the following meaning $$F(x,z,p,X)+K(w-z)\leq F(x,w,p,X),$$ \(d) $F$ satisfies the natural structure condition $$\|F\|+(1+\|p\|)\|F_p\|+\|F_z\|+\frac 1{1+\|p\|}\|F_x\|$$ $$\leq \mu (1+\|p\|^2+\|X\|),$$ where $\lambda , \Lambda, K, \mu$ are positive constants. Can you construct an example to show that the solution $u(x)$ of (\[eq1\]) under the assumptions (a)-(d) does not belong to $C^2(\Omega)$? \[ [Department of Mathematics, Zhejiang University, Hangzhou, China(310027)]{} [99]{} =0.2ex L.A.Caffarelli and L.Wang, “A Harnack inequality approach to the interier regularity of elliptic equations", Indiana Univ. Math. J.,42(1993) 145-158. Y.Z.Chen, “$C^{1,\alpha}$ regularity of viscosity solutions of fully nonlinear elliptic PDE under natural structure conditions", J. Partial Diff. Equa., 6(1993) 193-216. L.C.Evens, “Classical solutions of fully nonlinear convex, second order elliptic equations”, Comm. Pure Appl. Math., 35(1982) 333-363. N.V.Krylov, “ Nonlinear elliptic and parabolic equations of second order, ” Reidel, 1987. G. C. Dong, “Nonlinear partial differential equations of second order,” Trans. Math. Monograph 95, A.M.S.,Providence, Rhode Island, 1993. M. G. Crandall, H. Ishii and P. L. Lions, “User’s guide to viscosity solutions of second order partial differential equations," Bull Amer. Math. Soc., 29(1992) 1-67. C. Pucci, “Operatori estremani," Ann. Math. Pure Appl., 72(1966) 141-170. L. A. CAffarelli, “Interior estinates for solutions of fully nonlinear equations, Annal of Math., 130(1980) 189-213.******** [^1]: This work was supported by NNSF of China
--- abstract: 'We show that a kilometer-scale neutrino observatory, though optimized for TeV to PeV energy, is sensitive to the neutrinos associated with super-EeV sources. These include super-heavy relics, neutrinos associated with the Greisen cutoff, and topological defects which are remnant cosmic structures associated with phase transitions in grand unified gauge theories. It is a misconception that new instruments optimized to EeV energy are required to do this important science, although this is not their primary goal. Because kilometer-scale neutrino telescopes can reject atmospheric backgrounds by establishing the very high energy of the signal events, they have sensitivity over the full solid angle, including the horizon where most of the signal is concentrated. This is important because up-going neutrino-induced muons, routinely considered in previous calculations, are absorbed by the Earth.' address: 'Univ. of Wisconsin, Dept. of Physics, 1150 University Avenue, Madison, Wisconsin 53706, USA.' author: - 'J. Alvarez-Muñiz and F. Halzen' title: ' $10^{20}$eV cosmic-ray and particle physics with kilometer-scale neutrino telelscopes' --- 0.5cm PACS number(s): 95.55.Vj, 96.40.Tv, 98.70.Sa, 98.80.Cq Introduction ============ It has been realized for some time that topological defects are unlikely to be the origin of the structure in the present Universe [@TDstructure]. Therefore the observation of their decay products, in the form of cosmic rays or high energy neutrinos, becomes the most straightforward way to search for these remnant structures from grand unified phase transitions [@TD]. Such search represents an example of fundamental particle physics that can only be done with cosmic beams. We here point out that a kilometer-scale neutrino observatory [@physrep], such as IceCube, has excellent discovery potential for topological defects. The instrument can identify the characteristic signatures in the energy and zenith angle distribution of the signal events. It is a common misconception that different instruments[@auger; @owl], optimized to EeV signals, are required to do this important science, although this is not their primary motivation. Our conclusions for topological defects extend to other physics associated with $10^{20} - 10^{24}$eV energies. We will illustrate our claims by demonstrating IceCube sensitivity to: - generic topological defects with grand-unified mass scale $M_X$ of order $10^{14}-10^{15}$GeV and a particle decay spectrum consistent with all present observational constraints[@protheroe; @sigl; @pillado], - superheavy relics, normalized to the Z-burst scenario[@weiler] where the observed ultra high energy cosmic rays (UHECR) of $\sim 10^{20}$eV energy and above are locally produced by the interaction of superheavy relic neutrinos with the cosmic neutrino background radiation [@gelmini], - neutrinos produced by superheavy relics which themselves decay into the UHECRs [@berez; @sarkar], and - the flux of neutrinos produced in the interactions of UHECR cosmic rays with the microwave background [@steckerCMB], the so called Greisen neutrinos. This flux, which originally inspired the concept of a kilometer-scale neutrino detector, is mostly shown for comparison. The basic reasons for our more optimistic conclusions about the sensitivity of a detector such as IceCube are simple. Unlike first-generation neutrino telescopes, IceCube can measure energy and can therefore separate very high energy signals from the low energy atmospheric neutrino background by energy measurement [@icecube] (see below). The instrument can therefore isolate high energy events over $4\pi$ solid angle, and not just in the hemisphere where the neutrinos are identified by their penetration of the Earth. This is of primary importance here because neutrinos from topological defects have energies high enough so that they are efficiently absorbed by the Earth [@gandhi]. The signal from above and near the horizon typically dominates the up-going neutrino fluxes by an order of magnitude. We will show that the zenith angle distribution of neutrinos associated with topological defects form a characteristic signature for their extremely high energy origin. Neutrino events =============== We calculate the neutrino event rates by convoluting the $\nu_\mu+\bar\nu_\mu$ flux from the different sources considered in this paper, with the probability of detecting a muon produced in a muon-neutrino interaction in the Earth or atmosphere: $$N_{\rm events}=2\pi~A_{\rm eff}~T~\int\int~{dN_\nu\over dE_\nu}(E_\nu) P_{\nu\rightarrow\mu}(E_\nu,E_\mu {\rm (thresh)},\cos\theta_{\rm zenith}) ~dE_\nu~d\cos\theta_{\rm zenith}$$ where $T$ is the observation time and $\theta_{\rm zenith}$ the zenith angle. We assume an effective telescope area of $A_{\rm eff}=1~{\rm km^2}$, a conservative assumption for the very high energy neutrinos considered here. It is important to notice that the probability ($P_{\nu\rightarrow\mu}$) of detecting a muon with energy above a certain energy threshold $E_\mu$(threshold), produced in a muon-neutrino interaction, depends on the angle of incidence of the neutrinos. This is because the distance traveled by a muon cannot exceed the column density of matter available for neutrino interaction, a condition not satisfied by very high energy neutrinos produced in the atmosphere. They are absorbed by the Earth and only produce neutrinos in the ice above, or in the atmosphere or Earth near the horizon. The event rates in which the muon arrives at the detector with an energy above $E_\mu$(threshold)=1 PeV, where the atmospheric neutrino background is negligible, are shown in Table I. Fig.1 shows the $\nu_\mu+\bar\nu_\mu$ fluxes used in the calculations. We first calculate the event rates corresponding to the largest flux from topological defects [@protheroe] allowed by constraints imposed by the measured diffuse $\gamma$-ray background in the vicinity of 100 MeV. The corresponding proton flux has been normalized to the observed cosmic ray spectrum at $3\times 10^{20}$eV; see Fig.2 of reference [@protheroe]. Models with p$<$1 appear to be ruled out [@sigl] and hence they are not considered in the calculation. As an example of neutrino production by superheavy relic particles, we consider the model of Gelmini and Kusenko [@gelmini]. In Figs.2 and 3 we show the event rates as a function of neutrino energy. We assume a muon energy threshold of 1 PeV. We also show in both plots the event rate due to the Waxman and Bahcall bound [@wblimit]. This bound represents the maximal flux from astrophysical, optically thin sources, in which neutrinos are produced in p-p or p-$\gamma$ collisions. The atmospheric neutrino events are not shown since they are negligible above the muon energy threshold we are using. The area under the curves in both Figs. is equal to the number of events for each source. In Fig.4 we plot the event rates in which the produced muon arrives at the detector with an energy greater than $E_\mu$(threshold). In Fig.5 we finally present the angular distribution of the neutrino events for the different very high energy neutrino sources. The characteristic shape of the distribution reflects the opacity of the Earth to high energy neutrinos, typically above $\sim$100 TeV. The limited column density of matter in the atmosphere essentially reduces the rate of downgoing neutrinos to interactions in the 1.5km of ice above the detector. The events are therefore concentrated near the horizontal direction corresponding to zenith angles close to $90^{\rm o}$. The neutrinos predicted by the model of Gelmini and Kusenko are so energetic that they are even absorbed in the horizontal direction as can be seen in Fig.5. 0.3cm Model $N_{\nu_\mu+\bar\nu_\mu}$ (downgoing) $N_{\nu_\mu+\bar\nu_\mu}$ (upgoing) --------------------------------------------------------------- --------------------------------------- ------------------------------------- TD, $M_X=10^{14}$ GeV, $Q_0=6.31\times 10^{-35}$, p=1 11 1 TD, $M_X=10^{14}$ GeV, $Q_0=6.31\times 10^{-35}$, p=2 3 0.3 TD, $M_X=10^{15}$ GeV, $Q_0=1.58\times 10^{-34}$, p=1 9 1 TD, $M_X=10^{15}$ GeV, $Q_0=1.12\times 10^{-34}$, p=2 2 0.2 Superheavy Relics Gelmini [et al.]{} [@gelmini] 30 $1.5\times 10^{-7}$ Superheavy Relics Berezinsky [et al.]{} [@berez] 2 0.2 Superheavy Relics Birkel [et al.]{} [@sarkar] 1.5 0.3 p-$\gamma_{\rm CMB}$ $(z_{\rm max}=2.2)$ [@steckerCMB] 1.5 $1.2\times 10^{-2}$ W-B limit $2\times 10^{-8}~E^{-2}~{\rm (cm^2~s~sr~GeV)^{-1}}$ 8.5 2 Atmospheric background $2.4\times 10^{-2}$ $1.3\times 10^{-2}$ [Table I:]{} Neutrino event rates (per year per ${\rm km^2}$ in $2\pi$ sr) in which the produced muon arrives at the detector with an energy above $E_\mu$(threshold)=1 PeV. Different neutrino sources have been considered. The topological defect models (TD) correspond to highest injection rates $Q_0~({\rm ergs~cm^{-3}~s^{-1}})$ allowed in Fig.2 of [@protheroe]. Also shown is the number of events from p-$\gamma_{\rm CMB}$ interactions in which protons are propagated up to a maximum redshift $z_{\rm max}=2.2$ [@steckerCMB] and the number of neutrinos from the Waxman and Bahcall limit on the diffuse flux from optically thin sources [@wblimit]. The number of atmospheric background events above 1 PeV is also shown. The second column corresponds to downward going neutrinos (in $2\pi$ sr). The third column gives the number of upward going events (in $2\pi$ sr). We have taken into account absorption in the Earth according to reference [@gandhi]. IceCube will detect the sum of the event rates given in the last two columns. 0.5cm Energy measurement is critical for achieving the sensitivity of the detector claimed. For muons, the energy resolution of IceCube is anticipated to be $25\%$ in the logarithm of the energy, possibly better. The detector is able to determine energy to better than an order of magnitude, sufficient for the separation of EeV signals from atmospheric neutrinos with energies below 100 TeV. Notice that one should also be able to identify electromagnetic showers initiated by electron and tau-neutrinos. Their energy measurement is linear and expected to be better than $20\%$. Such EeV events will be gold-plated, unfortunately their fluxes are expected to be even lower. For instance for the first TD model in Table I (p=1, $M_X=10^{14}$ GeV and $Q_0=6.31 \times 10^{-35}~{\rm ergs~cm^{-3}~s^{-1}}$), we expect $\sim 1$ contained shower per year per ${\rm km^2}$ above 1 PeV initiated in charged current interactions of $\nu_e+\bar\nu_e$. The corresponding number for the Gelmini and Kusenko flux is $\sim 4~{\rm yr^{-1}~km^{-2}}$. One should also worry about the fact that a very high energy muon may enter the detector with reduced energy because of energy losses. It could become indistinguishable from atmospheric background [@gaisser]. We have accounted for the ionization as well as catastrophic muon energy losses which are incorporated in the calculation of the range of the muon. In the PeV regime region this energy reduction is roughly one order of magnitude, it should be less for the higher energies considered here. In conclusion, if the fluxes predicted by our sample of models for neutrino production in the super-EeV region are representative, they should be revealed by the IceCube observatory operated over several years. Acknowledgements {#acknowledgements .unnumbered} ================ We thank J.J. Blanco-Pillado for making available to us his code to obtain the neutrino fluxes from topological defects and E. Zas for helpful discussions. This research was supported in part by the US Department of Energy under grant DE-FG02-95ER40896 and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. J.A. thanks the Department of Physics, University of Wisconsin, Madison and the Fundación Caixa Galicia for financial support. [999]{} See Proceedings of the First International Workshop on Particle Physics and the Early Universe, Ambleside, England, 1997, Ed. L. Roszkowski, pp. 403-432. P. Bhattacharjee, C.T. Hill and D.N. Schramm, Phys. Rev. Lett. [69]{}, 567 (1992). T.K. Gaisser, F. Halzen and T. Stanev, Phys. Rep. [258]{}, 173 (1995), and references therein. K.S. Capelle, J.W. Cronin, G. Parente and E. Zas, Astropart. Phys. [8]{}, 321 (1998). D. Cline and F.W. Stecker, contributed to OWL/AW Neutrino Workshop on Observing Ultrahigh Energy Neutrinos, Los Angeles, California, Nov. 1999, astro-ph/0003459. R.J. Protheroe and T. Stanev, Phys. Rev. Lett. [77]{}, 3708 (1996) and Erratum [78]{}, 3420 (1997). G. Sigl, S. Lee, P. Bhattacharjee and S. Yoshida, Phys. Rev. D [59]{}, 043504 (1998). J.J. Blanco-Pillado, R. A. Vázquez and E. Zas, Phys. Rev. Lett. [19]{}, 3614 (1997). T. Weiler, Astropart. Phys. [11]{}, 303 (1999); D. Fargion, B. Mele and A. Salis, Astrophys. J. [517]{}, 725 (1999). G. Gelmini and A. Kusenko, Phys. Rev. Lett. [84]{}, 1378 (2000). V. Berezinsky, M. Kachelrei[ß]{} and A. Vilenkin, Phys. Rev. Lett. [79]{}, 4302 (1997). M. Birkel and S. Sarkar, Astropart. Phys. [9]{}, 297 (1998). F.W. Stecker et al., Phys. Rev. Lett., [66]{}, 2697 (1991). The IceCube NSF proposal [http://pheno.physics.wisc.edu/icecube/]{} R. Gandhi, C. Quigg, M.H. Reno, I. Sarcevic, Phys. Rev. D [58]{}, 093009 (1998). E. Waxman and J. Bahcall, Phys. Rev. D [59]{}, 023002 (1998); E. Waxman and J. Bahcall, hep-ph/9902383. T.K. Gaisser, talk given at OECD Megascience Forum Workshop, Taormina, Italy, May 1997, astro-ph/9704061.
--- abstract: 'The thermal fluctuation of mirror surfaces is the fundamental limitation for interferometric gravitational wave (GW) detectors. Here, we experimentally demonstrate for the first time a reduction in a mirror’s thermal fluctuation in a GW detector with sapphire mirrors from the Cryogenic Laser Interferometer Observatory at 17K and 18K. The detector sensitivity, which was limited by the mirror’s thermal fluctuation at room temperature, was improved in the frequency range of 90Hz to 240Hz by cooling the mirrors. The improved sensitivity reached a maximum of $2.2 \times 10^{-19}\,\textrm{m}/\sqrt{\textrm{Hz}}$ at 165Hz.' author: - Takashi Uchiyama - Shinji Miyoki - Souichi Telada - Kazuhiro Yamamoto - Masatake Ohashi - Kazuhiro Agatsuma - Koji Arai - 'Masa-Katsu Fujimoto' - Tomiyoshi Haruyama - Seiji Kawamura - Osamu Miyakawa - Naoko Ohishi - Takanori Saito - Takakazu Shintomi - Toshikazu Suzuki - Ryutaro Takahashi - Daisuke Tatsumi bibliography: - 'clio\_prl.bib' title: Reduction of thermal fluctuations in a cryogenic laser interferometric gravitational wave detector --- #### Introduction Two hundred years ago, Robert Brown investigated the random motion of small particles in water [@brownian]. This random motion is now understood to be an irreducible natural phenomenon, and it creates a fundamental limit on the precision of measurements, including the measurement of fundamental constants, high-resolution spectroscopy and fundamental physics experiments using a frequency-stabilized laser [@freq_standard] and gravitational wave (GW) detection by a laser interferometer with suspended mirrors [@salson_prd]. In the case of interferometric GW detectors, thermal noise in the mirror typically limits the detector sensitivity to approximately a few hundred Hz, which lies in the important frequency region for the detection of GWs from the binary coalescence of neutron stars. Although kilometer-scale first-generation laser interferometric GW detectors, such as Laser Interferometer Gravitational-Wave Observatory (LIGO) [@ligo] and VIRGO [@virgo], have already performed several long-term observation runs, no GW signal has yet been observed. The sensitivity must be improved by one order of magnitude to be able to detect GWs within a single year of observation. According to the fluctuation-dissipation theorem [@fdt], the power of thermal fluctuations is proportional to both temperature and mechanical loss. Thus, a mirror constructed of a low-loss material operating at a cryogenic temperature (a cryogenic mirror) is a good candidate for creating a low-thermal-fluctuation mirror. Because of various difficulties associated with implementing cryogenic mirrors in interferometers, major efforts have thus far been devoted to finding low-loss materials at room temperature. We originally proposed using a suspended sapphire mirror cooled to less than 20K for a GW detector [@sapphire_cooling], and we have researched and developed this technique [@sapphire_subQ; @sapphire_fiberQ; @coating_Q; @sapphire_thermalCond; @sapphire_optabsorb; @cryogenic_contami; @clik_lism; @thermal_lens]. Here, we demonstrate for the first time a reduction in mirror thermal fluctuations using this cryogenic mirror in a working GW detector. This reduction is the primary purpose of the Cryogenic Laser Interferometer Observatory (CLIO) [@clio], which is the first GW detector to use a cryogenic mirror. #### Experiment CLIO was built at an underground site in the Kamioka mine, which is located 220 km northwest of Tokyo in Japan. CLIO is a Michelson interferometer with 100-m Fabry-Perot (FP) arm cavities, each consisting of front and end mirrors [@clio; @yamamoto_clio]. The front mirrors are suspended closest to the beam splitter and were cooled to 17K and 18K for the sensitivity measurement reported in this study. The end mirrors remained at 299K. The cavity mirror substrate material is sapphire. The cylindrical substrate has a diameter of 100mm, a thickness of 60mm and a mass of 1.8kg. One surface of the substrate has a highly reflective multilayered film coating of SiO$_{2}$ and Ta$_{2}$O$_{5}$. The mirror is suspended at the final stage of a six-stage suspension system [@suspension_clio], and the length of the wire suspending the mirror is 400mm. The suspension system is installed in a cryostat with two layers of radiation shielding (an outer and an inner shield) [@suspension_clio]. A two-stage pulse-tube cryo-cooler [@cryocooler] cools the outer and inner shields to approximately 70K and 10K, respectively. It took approximately 250 hours to cool the mirrors, and the vacuum pressure was less than $10^{-4}$Pa. To cool the front mirrors, we changed the mirror suspension wires from Bolfur to 99.999% purity aluminum to provide higher thermal conductivity, and we added three heat links to each suspension system [@yamamoto_clio]. Bolfur wire is an amorphous metal wire made by Unitika, Ltd. with a diameter of 50$\mu$m. The aluminum wire used has a diameter of 0.5mm. The same aluminum wire was used for the heat links, which provide thermal conduction between the suspended masses (Damping Stage, Cryo-base and Upper Mass) and the inner shield in the cryostat [@suspension_clio]. The lengths of the heat links between the inner shield and the Cryo-base, between the Cryo-base and the Upper Mass and between the Damping Stage and the inner shield were 315mm, 115mm and150 mm, respectively; each had one heat link. Thermometers were attached to the Cryo-base and the Upper Mass. We estimated the temperature of the mirror from the temperatures of the Cryo-base and the Upper Mass and from the thermal conductivity of the mirror suspension wires and the heat link. One 100-m FP cavity serves as a reference for the laser frequency stabilization, and the length of the other 100-m FP cavity is controlled to maintain the optical resonance [@clio]. Coil-magnet actuators, consisting of magnets glued to the mirror and coils facing toward the magnets, are used to control the cavity length. The GW signal is included in the feedback signal for this length control. In this study, the sensitivity was characterized by the power spectrum density of the displacement in units of m/$\sqrt{\textrm{Hz}}$ and is calculated using the following three measurements: the feedback signal, V/$\sqrt{\textrm{Hz}}$, the loop gain of the length control system, and the response function of the coil-magnet actuators, m/V. The thermal fluctuation of a mirror surface is caused by several different loss mechanisms. The thermoelastic damping and the internal frictional loss of the sapphire substrate and of the reflective coating films were considered in this study. In the case of the sapphire mirror at room temperature, the largest loss mechanism is the thermoelastic damping of the sapphire substrate (thermoelastic noise). At temperature below 20K, the internal frictional loss of the reflective coating films is expected to be the largest loss mechanism. The theory of thermoelastic noise [@braginsky; @cerdonio] has previously been validated experimentally [@black]. The power spectrum density ($\textrm{m}^2/\textrm{Hz}$) of the thermoelastic noise in a mirror was shown in Black et al. [@black] and described as follows: $$S_{\mathrm{TE}}(\omega) = \frac{4}{\sqrt{\pi}}\frac{\alpha^2(1+\sigma)^2}{\kappa}k_{\mathrm{B}}T^2WJ(\Omega),$$ where $J(\Omega)$ is $$J(\Omega)=\frac{\sqrt{2}}{\pi ^{3/2}} \int_{0}^{\infty}du \int_{-\infty}^{+\infty}dv\frac{u^3e^{-u^2/2}}{(u^2+v^2)[(u^2+v^2)^2+\Omega^2]},$$ and $\Omega$ is $$\Omega = \frac{\omega}{\omega_{\mathrm{c}}},$$ where $\omega_{\mathrm{c}}$ is $$\omega_{\mathrm{c}}=\frac{2\kappa}{\rho CW^2}.$$ In these equations, $\omega$ is the angular frequency, $\alpha$ is the thermal expansion coefficient, $\sigma$ is the Poisson’s ratio, $\kappa$ is the thermal conductivity, $\rho C$ is the specific heat per unit volume, $k_\mathrm{B}$ is the Boltzmann constant, $T$ is the temperature of the mirror, and $W$ is the beam spot radius on the mirror. In the case of the CLIO mirror at room temperature, $\Omega \gg 1$ is satisfied near 100Hz. In this case, the power spectrum density ($\textrm{m}^2/\textrm{Hz}$) is simplified as follows: $$S_{\mathrm{TE}}^{\Omega \gg 1}(\omega) = \frac{16}{\sqrt{\pi}}\alpha^2(1+\sigma)^2\frac{k_{\mathrm{B}}T^2\kappa}{(\omega \rho C)^2}\frac{1}{W^3}.$$ The power spectrum density ($\textrm{m}^2/\textrm{Hz}$) of the thermal noise in a mirror caused by the internal frictional loss of the substrate and of the coating films was shown in Nakagawa et al. [@nakagawa] as follows: $$S(f) = \frac{2k_BT(1-\sigma^2)}{\pi^{3/2}fWE}\phi_{\mathrm{substr}}\{1+\frac{2}{\sqrt{\pi}}\frac{(1-2\sigma)}{(1-\sigma)}\frac{\phi_{\mathrm{coat}}}{\phi_{\mathrm{substr}}}(\frac{d}{W})\}.$$ In this equation, $E$ is the Young’s modulus, $d$ is the thickness of the coating films and $\phi_{\mathrm{substr}}$ and $\phi_{\mathrm{coat}}$ are the internal frictional loss of the substrate and of the coating films, respectively. #### Results ![\[fig1\]Comparison of CLIO displacement sensitivity curves. Cryogenic sensitivity (CryoSens) was measured with the front mirrors at 18K and 17K and with the end mirrors at room temperature (299K). Room-temperature sensitivity (RoomSens) was measured with all of the mirrors at 299K. CryoSens and RoomSens were measured on March 20, 2010 and November 5, 2008, respectively. The magnified sensitivity curves are shown in Fig. \[fig2\]. Mirror thermal noise estimate curves corresponding to each of the sensitivity measurements are also shown (CryoMT and RoomMT). The CryoMT and the RoomMT were estimated to be $1.5\times10^{-19}\,\textrm{m}/ \sqrt{\textrm{Hz}}\times(100\,\textrm{Hz}/f )$ and $3.7\times10^{-19}\,\textrm{m}/\sqrt{\textrm{Hz}}\times(100\,\textrm{Hz}/f )$, respectively. The low-frequency noise model (LFNM) consists of the fitted lines for the noise floor level of CryoSens from 40Hz to 70Hz and for RoomSens from 20Hz to 70Hz. The LFNM was estimated to be $3.7\times 10^{-19}\,\textrm{m}/\sqrt{\textrm{Hz}}\times (100\,\textrm{Hz}/f)^{2.5}$. The high-frequency noise model (HFNM) consists of the fitted lines for the noise floor levels of both sensitivity curves above 400Hz. Based on the characteristics of the 100-m Fabry-Perot cavities of CLIO, the HFNM was found to be $1.4\times10^{-19}\,\textrm{m}/\sqrt{\textrm{Hz}}\times \sqrt{1+(f/250\,\textrm{Hz})^2}$.](uchiyama_fig1.eps){width="0.85\columnwidth"} For comparison, Fig. \[fig1\] presents both the displacement sensitivity curve measured with the front CLIO mirrors cooled to 17K and 18K (cryogenic sensitivity; CryoSens) as well as the curve without cooled mirrors (room temperature sensitivity; RoomSens). CryoSens and RoomSens were measured on March 20, 2010 and November 5, 2008, respectively. The loop gain of the length control was measured immediately after each feedback signal measurement, and the response function of the coil-magnet actuators was calibrated for each experimental configuration. A sensitivity curve consists of frequency-dependent noise floors and multiple line noises. The noise floor level of CryoSens from 90Hz to 240Hz is below the noise floor level of RoomSens. By reducing this noise floor level, the detection range for GW signals from the binary coalescence of neutron stars in the optimal direction was improved from 150kpc to 160kpc. The noise floor at 165Hz was reduced from $3.1\times10^{-19}\,\textrm{m}/\sqrt{\textrm{Hz}}$ to $2.2\times10^{-19}\,\textrm{m}/\sqrt{\textrm{Hz}}$ by cooling the front mirrors. #### Discussion The noise floors of the two sensitivities shown in Fig. \[fig1\] are very similar from 40Hz to90 Hz and from 400Hz to 5kHz, although additional line noises appeared in CryoSens. The line noise near 30Hz results from the mechanical resonance of the cooled front mirror suspension systems. The line noise near 120Hz and its higher-order harmonics are also caused by mechanical resonance in the suspension wires of the cooled front mirrors. The line noise at 60Hz and its higher harmonics are similar to electric power line noise. Objects Unit Room temperature Cryogenic --------------------------- ---------------------------------------------------------------- ------------------ ----------------------- -------------------------- Mirror temperature K $299$ $17\, \textrm{and}\, 18$ $W _{\mathrm{front}}$ Beam spot radius on front mirrors mm $4.9$ $4.9$ $W _{\mathrm{end}}$ Beam spot radius on end mirrors mm $8.5$ $8.5$ Material properties of Sapphire $\alpha$ Thermal expansion coefficient [@data_book; @th_expansion_cryo] 1/K $5.4 \times 10^{-6}$ $5.6 \times 10^{-9}$ $\rho C$ Specific heat per unit volume [@data_book] J/K/m$^{3}$ $3.1 \times 10^{6}$ $2.8 \times 10^{3}$ $\kappa$ Thermal conductivity [@data_book] W/m/K $46$ $1.6 \times 10^{4}$ $\sigma$ Poisson’s ratio[^1] $0.27$ $0.27$ $E$ Young’s modulus [@coating_Q] Pa $40 \times 10^{10}$ $40\times 10^{10}$ $\phi _{\mathrm{substr}}$ Mechanical loss [@sapphire_subQ] $1/4.6 \times 10^{6}$ $1/1.5 \times 10^{8}$ $\phi _{\mathrm{coat}}$ Mechanical loss in coating films [@coating_Q] $4.0 \times 10^{-4}$ $4.0 \times 10^{-4}$ $d$ Thickness of coating films $\mathrm{\mu}$ m $3.9$ $3.9$ Figure \[fig1\] also shows the mirror thermal noise estimates corresponding to each sensitivity measurement (CryoMT and RoomMT), with the fitted lines for the noise floor in low-frequency region (the low-frequency noise model; LFNM) and high-frequency region (the high-frequency noise model; HFNM). The parameters for the thermal noise calculation are summarized in Table \[table:parameters\]. We did not perform any fits to the parameters. The RoomMT was calculated to be $3.7\times10^{-19}\,\textrm{m}/\sqrt{\textrm{Hz}}\times(100\,\textrm{Hz}/f )$. Because the mirror thermal noise is larger with a smaller laser beam, the thermal noise of the front mirror is approximately twice as large as the thermal noise of the end mirror in the RoomMT. The CryoMT was greatly reduced to $1.5\times10^{-19}\,\textrm{m}/ \sqrt{\textrm{Hz}}\times(100\,\textrm{Hz}/f )$ by the decrease in the thermal fluctuation of the cooled front mirrors. If all CLIO mirrors are cooled to 20K, the mirror thermal noise near 100Hz should be reduced to $1.7\times10^{-20}\,\textrm{m}/ \sqrt{\textrm{Hz}}\times(100\,\textrm{Hz}/f )^{1/2}$. The LFNM was estimated from the noise floor of CryoSens between 40Hz and 70Hz and the noise floor of RoomSens between 20Hz and 70Hz, with a value of $3.7\times10^{-19}\,\textrm{m}/\sqrt{\textrm{Hz}}\times(100\,\textrm{Hz}/f )^{2.5}$. A discussion of the origin of the LFNM is outside the scope of the current work. The HFNM was estimated from the noise floors of both sensitivity curves above 400Hz by assuming a simple pole frequency dependence [@salson_book]. The HFNM consists of photocurrent shot noise and laser intensity noise at a frequency of 15.8MHz and was estimated to have a value of $1.4\times10^{-19}\,\textrm{m}/\sqrt{\textrm{Hz}}\times\sqrt{1+(f/250\,\textrm{Hz})^2}$; the HFNM depends on the length and Finesse of the 100-m FP cavity. Cavity pole frequencies of 262 Hz and 246 Hz were obtained for the measurements of CryoSens and RoomSens, respectively. The systematic error in the noise floor amplitude due to the discrepancy in the cavity pole frequencies was less than 2% at frequency below 200Hz. ![\[fig2\]Comparison of CLIO displacement sensitivity curves and noise models. The cryogenic sensitivity (CryoSens) and the room-temperature sensitivity (RoomSens) show similar curves, as observed in Fig. \[fig1\]. The cryogenic noise model (CryoNM) is the noise-floor model curve for CryoSens and is calculated as the quadrature sum of the low-frequency noise model (LFNM), the high-frequency noise model (HFNM) and the mirror thermal noise estimate for CryoSens (CryoMT). The room-temperature noise model (RoomNM) is the noise-floor model curve for RoomSens and is calculated as the quadrature sum of the LFNM, the HFNM and the mirror thermal noise estimate for RoomSens (RoomMT). Error bars are shown for both CryoNM and RoomNM, representing the quadrature sums of the calibration error and the noise estimation errors. We estimated a systematic calibration error of $\pm$2.5%, a $\pm$5% estimation error for CryoMT and RoomMT and a $\pm$10% statistical error for LFNM and HFNM. The noise estimation error is the quadrature sum of the error in the mirror thermal noise estimate (CryoMT or RoomMT) and the errors in the LFNM and HFNM.](uchiyama_fig2.eps){width="0.85\columnwidth"} Figure \[fig2\] compares CryoSens, RoomSens, the cryogenic noise model (CryoNM) and the room-temperature noise model (RoomNM). CryoNM is the quadrature sum of the LFNM, the HFNM and CryoMT. RoomNM is the quadrature sum of the LFNM, the HFNM and RoomMT. Error bars are shown for both CryoNM and RoomNM, representing the quadrature sum of the calibration error and the noise estimation error. We estimated a $\pm$2.5% systematic error for the calibration, a $\pm$5% estimation error for CryoMT and RoomMT and a $\pm$10% statistical error for LFNM and HFNM. The noise estimation error is the quadrature sum of the error in the mirror thermal noise estimate (CryoMT or RoomMT) and the errors in LFNM and HFNM. The noise floor of RoomSens agrees with RoomNM. The noise floor of CryoSens is below RoomNM and agrees instead with CryoNM. The difference between CryoNM and RoomNM is solely based on the mirror thermal noise estimate. Thus, we conclude that the sensitivity limitation due to thermal noise in the mirror at room temperature was improved by the noise reduction because of the decreased thermal fluctuation in the cooled front mirrors. This observation is the first demonstration that mirrors display less thermal fluctuation at cryogenic temperatures than at room temperature. #### Conclusion The thermal fluctuation of mirror surfaces represents the fundamental limitation on experiments that require high-precision measurements, such as experiments using a frequency-stabilized laser and GW detection using a laser interferometer. The cryogenic mirror technique uses a low-mechanical-loss mirror at a low temperature to reduce such fluctuations. Our demonstration using an actual GW detector directly proves that the cryogenic mirror technique effectively improves the sensitivity of the GW detector. This cryogenic mirror technique will be used in advanced GW detectors, such as KAGRA formerly called the Large-Scale Cryogenic Gravitational Wave Telescope [@lcgt], on which construction began in the middle of 2010 in Japan, and the Einstein Telescope project in Europe [@et]. The mirror thermal noise of KAGRA near 100Hz, which limits the sensitivity to $4\times10^{-20}\,\textrm{m}/ \sqrt{\textrm{Hz}}\times(100\,\textrm{Hz}/f )$ at 299K, will be reduced to $5\times10^{-21}\,\textrm{m}/ \sqrt{\textrm{Hz}}\times(100\,\textrm{Hz}/f )^{1/2}$ at 20K due to the cryogenic mirror technique. The KAGRA sensitivity will make it possible to detect a GW signal from the binary coalescence of neutron stars in the optimal direction up to 250Mpc with a signal-to-noise ratio of 10 and the event rate is expected to be approximately 6 events per year [@lcgt]. We believe that our achievement represents a breakthrough in the study of thermal fluctuations, laser frequency stabilization and GW detection. We wish to thank our CLIO, TAMA and KAGRA collaborators, the Kamioka Observatory, Institute for Cosmic Ray Research, the University of Tokyo and Dr. Hiroaki Yamamoto of the LIGO laboratory for his advice on the manuscript. This work was supported in part by a Grant-in-Aid for Scientific Research on Priority Areas (No. 415) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT). This work was also supported in part by a Grant-in-Aid for Scientific Research (A, No. 18204021) by the Japan Society for the Promotion of Science (JSPS). [^1]: We found various values for the Poisson’s ratio of sapphire between 0.23 and 0.30. We used the averaged value.
--- abstract: 'The detections of some long gamma-ray bursts (LGRBs) relevant to mergers of neutron star (NS)-NS or black hole (BH)-NS, as well as some short gamma-ray bursts (SGRBs) probably produced by collapsars, muddle the boundary of two categories of gamma-ray bursts (GRBs). In both cases, a plausible candidate of central engine is a BH surrounded by a hyperaccretion disc with strong outflows, launching relativistic jets driven by Blandford-Znajek mechanism. In the framework of compact binary mergers, we test the applicability of the BH hyperaccretion inflow-outflow model on powering observed GRBs. We find that, for a low outflow ratio, $\sim 50\%$, postmerger hyperaccretion processes could power not only all SGRBs but also most of LGRBs. Some LGRBs might do originate from merger events in the BH hyperaccretion scenario, at least on the energy requirement. Moreover, kilonovae might be produced by neutron-rich outflows, and their luminosities and timescales significantly depend on the outflow strengths. GRBs and their associated kilonovae are competitive with each other on the disc mass and total energy budgets. The stronger the outflow, the more similar the characteristics of kilonovae to supernovae (SNe). This kind of ‘nova’ might be called ‘quasi-SN’.' author: - \| Cui-Ying Song, Tong Liu[^1], and Ang Li\ Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China date: 'Accepted XXX. Received YYY; in original form ZZZ' title: 'Outflows from black hole hyperaccretion systems: short and long-short gamma-ray bursts and “quasi-supernovae”' --- \[firstpage\] accretion, accretion discs - black hole physics - gamma-ray burst: general - magnetic fields - stars: neutron Introduction {#sec:intro} ============ Gamma-ray bursts (GRBs) are classified into two categories divided by $T_{90}\sim 2 ~\rm s$, i.e., long- and short-duration GRBs [LGRBs and SGRBs, see e.g., @Kouveliotou1993]. It is well known that SGRBs are possibly originated from the mergers of neutron star (NS)-NS or black hole (BH)-NS [e.g., @Paczynski1986; @Eichler1989; @Paczynski1991; @Popham1999; @Narayan1992; @Berger2014], and LGRBs associated with type Ib/c supernovae (SNe) could be powered by collapsars [e.g., @Tutukov1992; @Galama1998; @Hjorth2003; @Stanek2003; @Woosley2006; @Campana2006; @Fruchter2006; @Kumar2015]. With the accumulation of observational data, some LGRBs, such as GRB 060614 [e.g., @Gehrels2006; @Zhang2007b; @Zhang2009], were found to be relevant to mergers, while some SGRBs such as GRB 090426 [e.g., @Antonelli2009; @Levesque2010; @Xin2011] were produced probably by collapsars. Those bursts muddle the LGRB-SGRB boundary. @Zhang2009 summarized the nature of GRBs and proposed a new classification approach mainly based on their progenitors. Two types of GRB central engines have been widely discussed: hyper-accreting stellar mass BHs [e.g., @Woosley1993; @Lei2009; @Lei2013; @Liu2017a] and rapidly spinning and highly magnetized NSs [magnetars, see e.g., @Usov1992; @Thompson1994; @Dai1998; @Zhang2001; @Dai2006]. The detection of gravitational waves (GWs) from close compact binary mergers [e.g., @Eichler1989; @Schutz1989; @Cutler1994; @Lipunov1997; @Abadie2010] provides a direct way to verify the progenitors of SGRBs if the association of GWs with SGRBs can be confirmed. After merger events, optical/near-infrared (NIR) emission from the radioactive decay of heavy r-process elements are produced by the merger remains. These transient events are named ‘kilonovae’ [@Li1998]. Recently, the NS-NS merger GW event (GW170817) was detected by the advanced LIGO/Virgo collaboration [e.g., @Abbott2017a; @Abbott2017b; @Alexander2017; @Blanchard2017; @Coulter2017; @Hallinan2017; @Troja2017; @Shappee2017]. Its electromagnetic counterpart, GRB 170817A accompanied by a kilonova AT 2017gfo, was also discovered [e.g., @Abbott2017c; @Chornock2017; @Cowperthwaite2017; @Evans2017; @Kilpatrick2017; @Margutti2017; @Nicholl2017; @Smartt2017]. This event provides the first direct evidence for the progenitor hypothesis of SGRBs. Outflows may present in accretion processes, especially for super-Eddington accretion discs [e.g., @Shakura1973]. They were widely studied by theoretical analyses [e.g., @Blandford1999; @Liu2008; @Gu2015], numerical simulations [e.g., @Eggum1988; @Okuda2002; @Ohsuga2005; @Ohsuga2011; @Jiang2014; @Jiang2017; @McKinney2014; @Sadowski2014; @Sadowski2015], as well as observations [e.g., @Wang2013; @Cheung2016; @Parker2017a]. @Narayan1994 proposed that the advection-dominated accretion flows (ADAFs) with the positive Bernoulli parameter might generate outflows and, by extension, jets [e.g., @Narayan1995; @Abramowicz1995]. @Blandford1999 emphasized the roles of the outflows in ADAFs and developed a variant named advection-dominated inflow-outflow solution (ADIOS). They also constructed 1D and 2D self-similar solutions and found that the structure and radiation of flows were subject to the outflows [@Blandford2004]. As to the study of the vertical structures of the discs, strong outflows were required in both optically-thick and optically-thin flows, resulted from energy equilibrium [e.g., @Gu2007; @Gu2015]. Outflows can also exist in neutrino-dominated accretion flows [NDAFs, see reviews by @Liu2017a]. @Liu2012a visited the vertical structures and luminosities of NDAFs. They found that outflows might be present in the outer region of the discs, depending on the vertical distributions of the Bernoulli parameter. A wide variety of mechanisms could lead to the generation of outflows. In optically-thick accretion flows, photons could exert radiation pressure upon materials to blow them away. For the optically-thin cases, such as ADAFs, they might possess a positive Bernoulli parameter due to their high internal energy [@Narayan1994; @Gu2015]. Moreover, the Blandford-Payne process [@Blandford1982] could produce outflows for both optically-thin and optically-thick discs [e.g., @Ma2018]. Apart from theoretical studies, several simulation results also implied that outflows play essential roles in accretion systems. It was first pointed out by @Stone1999, where 2D hydrodynamic numerical simulations were carried out. Many later simulations confirmed their conclusion, for example in @Hawley2001, @Machida2001, @Igumenshchev2003, @Pang2011, and @Yuan2014. Furthermore, both the inflow and outflow mass accretion rates decreased inward, following a power-law $\dot{M}\propto r^s$ ($0\leq s\leq 1$). For the outflows, $s\approx 1$ was possible which means that more than $90\%$ of the materials could be pushed into powerful outflows from the accretion disc [e.g., @Yuan2012a; @Yuan2012b; @Begelman2012]. In the global 3D radiation magneto-hydrodynamical simulation, @Jiang2014 found that the radiation-driven outflows were formed along the rotation axis and about $20\%$ of the radiative energy were carried by outflows. In a recent study of super-Eddington accretion flows onto supermassive BHs (SMBHs), the outflow speed could approach $\sim 0.1-0.4~c$, and the mass flux lost could reach $15\%-50\%$ of the net mass accretion rates [@Jiang2017]. The 3D general-relativistic magnetohydrodynamic simulations of NDAFs indicated that the velocities of powerful outflows likely approached $\sim 0.03-0.1~c$ [e.g., @Siegel2017]. Recent observations also showed the importance of outflows in accretion systems. For Galactic SMBH accretion, more than $99\%$ original gas escaped from the disc [@Wang2013]. In quiescent galaxies, @Cheung2016 observed that centrally-driven winds could suppress the star formation. The ultra-fast outflow of the Seyfert I galaxy IRAS 13224-3809 was discovered by @Parker2017a. They also proposed that the outflows could be detected from the long-term X-ray variability [@Parker2017b]. Furthermore, the observed kilonova following GRB 130603B [@Berger2013a; @Tanvir2013] might also related to disc winds [e.g., @Metzger2014a]. Outflows play important roles in the BH accretion processes, however, few studies have been done on the BH hyperaccretion system. To investigate the nature of the GRB central engine, we necessarily confront the BH hyperaccretion inflow-outflow model with observational data. In paper I, we constrained the characteristics of the progenitor stars of LGRBs and Ultra-LGRBs with the BH hyperaccretion inflow-outflow model in the collapsar scenario [@Liu2018]. In the present work, the applicability of the model is tested to power both SGRBs and LGRBs in the compact binary merger scenario. We further examine the properties of kilonovae triggered by strong neutron-rich outflows. The paper is organized as follows. In Section 2, we describe our BH hyperaccretion inflow-outflow model. The GRB data are presented in Section 3. The properties of kilonovae are shown in Section 4. Section 5 is a brief summary. Model ===== There are two types of GRB central engine candidates widely discussed: Magnetar and hyperaccreting BH with stellar mass. After the mergers of NS-NS or BH-NS, an NS with high spin (period $\sim1~ \rm ms$) and high surface magnetic field ($\sim10^{15}~\rm G$) might be formed, known as a millisecond magnetar. The released spin-down energy could power GRBs. Alternatively, in the BH accretion disc models, the GRB jet can be produced either through the neutrino-antineutrino annihilation process [see e.g., @Popham1999; @Di2002; @Narayan2001; @Liu2007], or via the electromagnetic processes. Some general-relativistic magnetohydrodynamics simulations have showed the evidences for the Blandford-Znajek (BZ) mechanism [@Blandford1977; @Blandford1977a] in GRB central engines [e.g ., @Nagataki2009; @Tchekhovskoy2012]. @Barkov2010 has confirmed the possibility of the magnetically driven stellar explosions, and has pointed out the required magnetic flux in excess of $10^{28}~\rm G~cm^{2}$ in the close compact binary scenario for LGRBs. Moreover, some studies have shown that the BZ mechanism is more efficient than the neutrino annihilation if the magnetic fields are strong enough or the accretion rates are lower than the ignition accretion rates of NDAFs [e.g., @Kawanaka2013; @Liu2015a; @Lei2017; @Liu2017a]. The BH spin energy might be extracted via the BZ mechanism when a strong magnetic field ($\sim10^{13}-10^{15} ~\rm G$) threads the spinning BH and is connected with a distant astrophysical load. Essentially all central engine models require a strong, large-scale magnetic field to launch GRBs. It may inherit and redistribute the large-scale magnetic field of the merging components following the magnetic flux conservation [e.g., @Liu2016; @Punsly2016]. In the BH hyperaccretion scenario, introducing the effects of the outflows and the relativistic jets driven by the BZ mechanism, the disc model is called BH hyperaccretion inflow-outflow model. Two parameters are required for describing a hyperaccreting stellar mass BH: the dimensionless mass $m_{\rm BH}=M_{\rm BH}/M_{\rm \sun}$ and spin $a_* \equiv cJ_{\rm BH}/GM_{\rm BH}^{2}$. Then the BZ jet power can be estimated as [e.g ., @Lee2000a; @Lee2000b; @Li2000; @McKinney2005; @Barkov2008; @Komissarov2009; @Barkov2010; @Lei2013] L\_[BZ]{}=1.710\^[50]{}a\_[\]{}\^[2]{}m\_[BH]{}\^[2]{}B\_[BH,15]{}\^[2]{}F(a\_\)[ erg s\^[-1]{}]{}, where $B_{\rm BH,15}=B_{\rm BH}/10^{15} {\rm G}$ and $F(a_)=[(1+q^{2})/q^{2}][(q+1/q)\arctan(q)-1]$ with $q=a_{}/(1+\sqrt{1-a_{}^{2}})$. We can evaluate the magnetic field strength when the magnetic pressure on the BH horizon balances the ram pressure of the innermost part of the disc [e.g., @Moderski1997] =P\_[ram]{}\~c\^2 \~, where $r_{\rm BH}=GM_{BH}(1+\sqrt{1-a_^{2}})/c^2$ denotes the event horizon of the BH, and $\dot{M}_{\rm in}$ is the accretion rate at the inner boundary. Then the magnetic field strength can be written as B\_[BH]{} 7.4 10\^[16]{}\_[in]{}\^[1/2]{}m\_[BH]{}\^[-1]{}(1+)\^[-1]{} [G]{}. Inserting above equation into Equation (1), we obtain the BZ jet power as a function of mass accretion rate at and spin of BH, L\_[BZ]{}=9.310\^[53]{}a\_\\^[2]{}\_[in]{} X(a\_\)[ erg s\^[-1]{}]{}, and X(a\_\)=F(a\_\)/(1+)\^[2]{}. The dimensionless BH mass accretion rate at the innermost stable orbit $\dot{m}_{\rm in}$ is defined as \_[in]{}=, where $T_{\rm 90,\rm rest} = T_{90}/(1+z)$ is the duration of the prompt emission in the rest frame, and $z$ stands for the redshift. $m_{\rm disc}=M_{\rm disc}/M_{\odot}$ stands for the dimensionless accretion disc mass, and $f$ represents the fraction of the outflow mass to the disc mass. As mentioned in the Introduction, outflows have been found to be very strong in many studies, therefore we take $f = 99\%$ for the strongest outflow case, and $f=50\%$ as a typical value for comparison. Recently, the simulations of NS-NS mergers [e.g., @Dietrich2015] and BH-NS mergers [e.g., @Foucart2014; @Just2015; @Kyutoku2015; @Kiuchi2015; @Siegel2017] showed that the remnant disc mass $m_{\rm disc}$ likely possessed an upper limit $\sim 0.3~M_{\rm\sun}$, which depends on the equation of state of the NS, the mass ratio, the total mass and the period of the binary [e.g., @Oechslin2006; @Dietrich2015]. We therefore take $m_{\rm disc}=0.01, ~0.1, ~0.3$ in our calculations. On the other hand, the jet power can be obtained from the observational data of GRBs [e.g., @Fan2011; @Liu2015b; @Song2016], i.e., L\_[j]{}, where $E_{\rm \gamma,\rm iso}$, $E_{\rm k,\rm iso}$, and $\theta_{\rm j}$ denote the isotropic radiated energy, the isotropic kinetic energy of afterglows, and the jet opening angle, respectively. ![Luminosities and timescales of BZ jets originated from compact binary mergers in five solid lines, different colors corresponding to different parameters as labelled. The gray filled stars and circles denote SGRBs and LGRBs data, respectively. The magenta star stands for SGRB that may be related to the collapse of massive stars. The colorful circles are for LGRBs possibly from NS-NS or BH-NS mergers.](f1.eps){width="\columnwidth"} GRB Data ======== As shown in Table 1, we collected the data of $T_{90}$, $z$, $E_{\rm \gamma,\rm iso}$, $E_{\rm k,\rm iso}$, $\theta_{\rm j}$ and the peak energy in the rest frame $E_{\rm p, rest}$ of 30 SGRBs and 89 LGRBs. It is worth noting that the measurements of $E_{\rm \gamma,\rm iso}$, $E_{\rm k,\rm iso}$, and $\theta_{\rm j}$ are model dependent. Considerable debates surround the origins of some GRBs due to their perplexing observational phenomena [e.g., @Zhang2009; @Xin2011; @Kann2011; @Li2016], some unusual GRBs are labelled by the superscript $$ in the table, which are shown as following: GRB 060505* This burst has a duration of $4\pm1\rm~s$, the low energy $\sim 10^{49} \rm~erg$ and a very low redshift $z=0.0894$. The presence of a SN is ruled out down to limit of hundreds times fainter than SN 1998bw [@Ofek2007; @Fynbo2006]. The star formation rate, metallicity, ionization state of the host environment are more similar to SGRBs than to LGRBs [@Levesque2007]. GRB 060614 Its $T_{90}\sim100 \rm~s$ in the BAT (15-150 keV) band groups it with LGRBs, while its peak luminosity and temporal lag completely satisfy the SGRB subclass [@Gehrels2006]. The low-star-formation-rate host galaxy [@Savaglio2009], and its irrelevant to any known SN [@GalYam2006; @Fynbo2006; @DellaValle2006] suggest that it is related to a compact binary merger rather than a collapsar [e.g., @Zhang2007a]. Furthermore, the discovery of a NIR bump in afterglow denotes the strong connection between GRB 060614 and a kilonova, and provides tangible evidence to support the merger origin [@Yang2015; @Jin2015; @Horesh2016]. GRB 080913 The burst duration $T_{90}$ in the BAT band is $8\pm1\rm~s$. Considering its high redshift $z=6.7$, the rest-frame duration of this burst is $T_{90, \rm rest}\sim1 \rm~s$. @Palshin2008 fitted the Konus-Wind and the Swift/BAT joint spectral analysis, and derived that the best fit peak energies are $E_{\rm peak}=131_{-48}^{+225} \rm~keV$ and $E_{\rm peak}=121_{-39}^{+232} \rm~keV$ for the cutoff power law and Band-function spectra, respectively. Placing this GRB at $z=1$, it can be classified as SGRBs due to intrinsically short duration and hard spectrum. @PerezRamirez2010 presented the X-ray, NIR and millimetre observations, and proposed that the progenitor of this burst was likely from a BH-NS merger. A maximally-rotating BH might form in the center and power this GRB by the BZ mechanism. However, this GRB is consistent with the lag-luminosity correlation and the Amati relation of LGRBs, which makes the collapsar origin cannot be ruled out [@Greiner2009; @Zhang2009]. GRB 090423 Similar to GRB 080913, this burst was measured with a high redshift $z=8.6$ and a BAT band duration $T_{90}=10.3\pm1.1 \rm~s$. In rest frame, the peak energy and duration are $491\pm200 \rm~keV$ [@Amati2009] and $\sim1.1 \rm~s$, respectively. GRB 090426 It is a SGRB with an observed duration of $T_{90}\sim1.28 \rm~s$ at $z=2.609$ [@Antonelli2009]. On the other hand, the soft spectrum $E_{\rm p, rest}=177_{-65}^{+90} \rm~keV$ [@Amati2009] and burst environment are similar to those of LGRBs. The number density of the medium ($>11.2 \rm~cm^{-3}$) is not consistent with the condition of the compact-binary-merger progenitors, which often occur in low density medium [e.g., @Xin2011]. Above all, the observed LGRBs and SGRBs are contaminated by each other. Some studies also found that the duration distributions of SGRBs and LGRBs overlaped each other [e.g., @Horvath2002]. Certainly $T_{90}$ is not a good criterion to manifest the nature of GRBs, which resulted in the old dichotomy between the LGRBs related to collapsars and the SGRBs originated from compact binary mergers. Figure 1 shows the luminosities and timescales of the BZ jets originated from the compact binary mergers. The solid lines in different colors correspond the BZ jets with the different values of $m_{\rm disc}$, $a_{}$, and $f$. The gray filled stars and circles denote SGRBs and LGRBs data, respectively. The magenta star stands for the SGRB which may related to the collapse of massive stars. The colorful circles are for LGRBs possibly from NS-NS or BH-NS mergers. By comparing the red and black lines, one can find that the luminosities of SGRBs decrease when the outflow increases. For $f = 50\%$, most of LGRBs find their place under our predicted lines of the BZ jets. For $f = 99\%$, at least half of those LGRBs cannot be explained by the model. Therefore, our model can explain not only all SGRBs but also most of LGRBs (with a low outflow ratio). Actually, assuming that $T_{90}$ is near or proportional to the duration of the central engine activity, then the duration of GRBs may be closely related to the properties of progenitors. The collapsar scenario is suggested to produce LGRBs through accretion because of the typical envelope fallback timescale is $10\rm~s$ [e.g., @MacFadyen1999]. The BH accretion disc systems after NS-NS/BH-NS mergers have a typical accretion timescale $\sim 0.01-0.1 \rm~s$, which were raised to account for SGRBs in many models [e.g., @Narayan2001; @Aloy2005]. However, the discover of X-ray flares [e.g., @Burrows2005; @Nousek2006; @Wu2007; @Chincarini2010; @Margutti2010; @Mu2016a; @Mu2016b], extended emission [e.g., @Lazzati2001; @Connaughton2002; @Norris2010; @Norris2011; @Liu2012b] and plateaus [e.g., @Troja2007; @Rosswog2007; @Rowlinson2013] in some GRBs denote that the duration of the GRB central engine activity is much longer than $T_{90}$ in both LGRBs and SGRBs. The gamma-ray duration $T_{90}$ may much shorter than the central engine activity duration, named ‘tip-of-iceberg’ effect [@Lv2014; @Zhang2014; @Li2016; @Gao2017a; @Liu2018]. A longer accretion timescale is needed to explain these observations by the BH hyperaccretion systems. The durations of the compact binary mergers are not necessarily to be “short", and in principle the collapsar model can also bring forth SGRBs [e.g., @Janiuk2008; @Zhang2009]. If it is the case, the values of the jet luminosities and timescales of GRBs are larger than these in Table 1. Thus the admission of the model limitations is stricter than in the current situations. =6.6pt -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Name $T_{90}$ $z$ $E_{\rm \gamma,\rm iso}$ $E_{\rm k,\rm iso}$ $E_{\rm p,\rm rest}$ $\theta_{\rm j}$ $L_{\rm j}$ Ref ------ ---------- ----- -------------------------------------------------------------------------------------------------------------------------------------- --------------------- ---------------------- ------------------ ------------- ----- (s) z ($10^{52}$erg) & ($10^{52}$erg) &(keV) & (rad) & ($10^{50} \rm erg\rm~s^{-1}$) &\ SGRBs\ 050509B & 0.04 & 0.225 & $0.00024_{-0.0001}^{+0.00044}$ & 0.0055 & $100.45_{-98}^{+748.475}$ & $ >0.05 $ & 0.022 & 1, 2\ 050709 & 0.07 & 0.161 & $0.0027\pm0.0011$ & 0.0016 & $96.363_{-13.932}^{+20.898}$ & $ >0.26 $ & 0.2397 & 1, 3\ 050724A & 3 & 0.257 & $0.009_{-0.002}^{+0.011}$ & 0.027 & $138.27_{-56.565}^{+502.8}$ & $ >0.35 $ & 0.0915 & 1, 2\ 051210 & 1.3 & 1.3 & $0.4_{-0.2}^{+0.5}$ & 0.238 & $943_{-598}^{+1495}$ & $>0.05$ & 0.1411 & 1, 2\ 051221A & 1.4 & 0.5465 & $0.28_{-0.1}^{+0.21}$ & 1.26 & $603.135_{-293.835}^{+1020.69}$ & 0.12 & 1.2234 & 1, 2\ 060502B & 0.09 & 0.287 & $0.003_{-0.002}^{+0.005}$ & 0.012 & $437.58_{-244.53}^{+926.64}$ & $>0.05$ & 0.0268 & 1, 2\ 060801 & 0.5 & 1.13 & $0.7_{-0.5}^{+1.5}$ & 0.071 & $1320.6_{-724.2}^{+2279.1}$ & $0.0561_{-0.0063}^{+0.0056}$ & 0.5167 & 1, 2, 4\ 061006 & 0.4 & 0.438 & $3_{-1}^{+4}$ & 0.314 & $819.66_{-402.64}^{+1308.58}$ & $0.407_{-0.173}^{+0.068}$ & 97.3211 & 1, 2, 4\ 061201 & 0.8 & 0.111 & $3_{-2}^{+4}$ & 0.007 & $666.6_{-388.85}^{+888.8}$ & 0.017 & 0.0603 & 1, 2\ 061210 & 0.2 & 0.409 & $0.09_{-0.05}^{+0.16}$ & 0.086 & $760.86_{-436.79}^{+1070.84}$ & $>0.37$ & 8.3909 & 1, 2, 5\ 070429B & 0.5 & 0.902 & $0.07_{-0.02}^{+0.11}$ & 0.451 & $228.24_{-125.532}^{+1418.89}$ & $>0.05$ & 0.2477 & 1, 2\ 070714B & 2.0 & 0.923 & $1.16_{-0.22}^{+0.41}$ & 0.232 & $2153.76_{-730.74}^{+1499.94}$ & $0.33_{-0.11}^{+0.11}$ & 7.2217 & 1, 3, 4\ 070724A & 0.4 & 0.457 & $0.003\pm0.001$ & 0.099 & $~99$ & $0.27_{-0.16}^{+0.16}$ & 1.346 & 1, 3, 4\ 070729 & 0.9 & 0.8 & 0.017 & 0.132 & $840.6_{-351}^{+1526}$ & $>0.05$ & 0.0372 & 1, 6\ 070809 & 1.3 & 0.473 & 0.0056 & 0.391 & $75.6_{-13.9}^{+12.6}$ & $0.4_{-0.333}^{+0.08}$ & 3.5473 & 1, 4, 6\ 071227 & 1.8 & 0.381 & $0.1\pm 0.02$ & 0.025 & $1384 \pm 277$ & $>0.0262$ & 0.0033 & 1, 5, 7\ 080905A & 1.0 & 0.122 & 0.0005 & 0.0024 & $502.8_{-280.5}^{+950.6}$ & $0.28_{-0.16}^{+0.15}$ & 0.0127 & 1, 4, 6\ $090426^{}$ & 1.2 & 2.609 & $0.5\pm0.1$ & 13.5 & $177 _{-65}^{+90}$ & 0.07 & 10.3115 & 1, 8\ 090510 & 0.3 & 0.903 & $4.47_{-3.77}^{+4.06}$ & 0.307 & $7490_{-494.8}^{++532.8}$ & 0.017 & 0.4379 & 1, 6\ 090515 & 0.04 & 0.403 & 0.0008 & 0.062 & $90.1_{-16.8}^{+47.4}$ & $>0.05$ & 0.2753 & 1, 6\ 100117A & 0.3 & 0.92 & $0.09 \pm 0.01$ & 0.11 & $551_{-96}^{+142}$ & $0.27_{-0.15}^{+0.15}$ & 4.6373 & 1, 4, 7\ 100206A & 0.1 & 0.408 & $0.0763_{-0.0229}^{+0.0789}$ & 0.0073 & $638.98_{-131.21}^{+131.21}$ & $>0.05$ & 0.1471 & 1, 9\ 100625A & 0.3 & 0.453 & $0.075\pm0.003$ & 0.0093 & $701.32 \pm 114.71$ & $>0.05$ & 0.051 & 1, 10\ 101219A & 0.6 & 0.718 & $0.49 \pm 0.07$ & 0.045 & $842_{-136}^{+177}$ & $0.29_{-0.14}^{+0.14}$ & 6.3966 & 1, 4, 7\ 111117A & 0.5 & 1.3 & $0.338 \pm 0.106$ & 0.377 & $966 \pm 322$ & 0.105 & 1.8114 & 1, 10\ 120804A & 0.81 & 1.3 & 0.7 & 0.7 & $310.5_{-66.7}^{+151.8}$ & $>0.19$ & 7.1539 & 11\ 130603B & 0.18 & 0.356 & $0.212 \pm 0.023$ & 0.28 & $900 \pm 140$ & 0.07 & 0.9077 & 1, 10\ 131001A & 1.54 & 0.717 & 0.037 & 0.541 & $94.44\pm 24.04$ & $>0.05$ & 0.0805 & 1, 12\ 140622A & 0.13 & 0.959 & 0.0065 & 0.977 & $86.2\pm 15.67$ & $>0.05$ & 1.8522 & 1, 13\ 160821B & 0.48 & 0.16 & $0.021\pm0.002$ & $~8$ & $97.44\pm 22.04 $ & $~0.063$ & 3.8455 & 14, 15\ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \[tab:continued1\] ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Name $T_{90}$ $z$ $E_{\rm \gamma,\rm iso}$ $E_{\rm k,\rm iso}$ $E_{\rm p,\rm rest}$ $\theta_{\rm j}$ $L_{\rm j}$ Ref ------ ---------- ----- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------- ---------------------- ------------------ ------------- ----- (s) z ($10^{52}$erg) & ($10^{52}$erg) &(keV) & (rad) & ($\rm 10^{50} erg\rm~s^{-1}$) &\ LGRBs\ 970508 & $14.0 \pm 3.6$ & 0.8349 & $0.61 \pm 0.13$ & $0.99\pm0.14$ & $145 \pm 43 $ & $0.3775\pm0.0291$ & 1.4765 & 16, 17\ 970828 & 160.0 & 0.96 & $29 \pm 3$ & 37.154 & $586 \pm 117$ & 0.1239 & 0.6212 & 16, 18\ 971214 & $31.23\pm 1.18$ & 3.418 & $21 \pm 3 $ & $8.48\pm 0.97$ & $685 \pm 133$ & $>0.0967\pm 0.0040$ & 1.9483 & 16, 17\ 980613 & $42.0 \pm 22.1$ & 1.0964 & $0.59 \pm 0.09 $ & $1.22\pm0.38$ & $194\pm89$ & $ >0.2194\pm 0.0101$ & 0.2166 & 16, 17\ 980703 & $76.0 \pm 10.2$ & 0.966 & $7.2 \pm 0.7 $ & $2.41\pm0.63$ & $503 \pm 64$ & $0.1957\pm 0.0141$ & 0.4745 & 16, 17\ 990123 & $63.3 \pm 0.3$ & 1.61 & $229 \pm 37 $ & 534 & $1724 \pm 446$ & $0.064 \pm 0.005$ & 6.4408 & 16, 19\ 990510 & $67.58 \pm1.86$ & 1.619 & $17 \pm 3 $ & $13.16\pm1.12$ & $423 \pm 42$ & $0.0586\pm 0.0037$ & 0.2006 & 16, 17\ 990705 & $32.0 \pm 1.4$ & 0.84 & $18 \pm 3 $ & $0.34\pm0.12$ & $459 \pm 139$ & $0.0930\pm 0.0072$ & 0.4557 & 16, 17\ 991216 & $15.17\pm0.091$ & 1.02 & $67 \pm 7 $ & $ 36.64\pm1.79$ & $648\pm134$ & $0.0798\pm0.0126$ & 4.3918 & 16, 17\ 000210 & $9.0 \pm1.4$ & 0.846 & $14.9 \pm 1.6 $ & $0.50 \pm 0.12$ & $753 \pm 26$ & $> 0.1194\pm0.0049$ & 2.2489 & 16, 17\ 000926 & $1.30 \pm 0.59$ & 2.0387 & $27.1\pm5.9 $ & $9.97 \pm 3.75$ & $310\pm20$ & $0.1075\pm0.0054$ & 50.0191 & 16, 17\ 010222 & $74.0 \pm4.1$ & 1.4769 & $81 \pm 9 $ & $22.79\pm2.48$ & $766 \pm 30$ & $0.0559\pm0.0023$ & 0.5426 & 16, 17\ 011211 & $51.0 \pm7.6$ & 2.14 & $5.4\pm0.6 $ & $71.32\pm0.22$ & $186\pm24$ & $0.1114\pm0.0070$ & 2.9279 & 16, 17\ 020813 & 89 & 1.25 & $66 \pm 16 $ & $204.174$ & $590 \pm 151$ & 0.0541 & 0.9993 & 16, 18\ 021004 & $77.1 \pm2.6$ & 2.3304 & $3.3 \pm 0.4 $ & $8.35\pm 1.45$ & $266 \pm 117$ & $0.2211\pm0.0787$ & 1.225 & 16, 17\ 050126 & 30 & 1.29 & $0.8_{-0.2}^{+1.0} $ & $39.8\pm80.4$ & $387.01_{-144.27}^{+1135.84}$ & $0.365_{-0.125}^{+0.095}$ & 20.4159 & 2, 4, 20\ 050315 & $96 \pm10$ & 1.9500 & $5.7_{-0.1}^{+6.2} $ & $512.403_{-65.577}^{+45.299}$ & $126.85_{-123.9}^{+32.45}$ &$0.0759_{-0.0091}^{+0.0080}$ & 4.5837 & 2, 17\ 050318 & $32\pm2$ & 1.4436 & $2.2 \pm 0.16 $ & $11.259_{-0.685}^{+0.867}$ & $115\pm25$ & $0.0380\pm0.0070$ & 0.0742 & 16, 17\ 050319 &$139.4 \pm 8.2$ &3.2425 &$4.6_{-0.6}^{+6.5}$ & $77.896_{-28.695}^{+20.496}$ &$190.912_{-182.428}^{+114.548}$ &$0.0380_{-0.0070}^{+0.0051}$ & 0.1812 & 2, 17\ 050401 & 38 & 2.9 & $35 \pm 7$ & $4570.9\pm1317.6$ & $467\pm110$ & $0.472_{-0.044}^{+0.02}$ & 5168.5851 & 4, 16, 20\ 050416A & 5.4 & 0.654 & $0.1 \pm 0.01$ & $15.1\pm 3.9$ & $25.1\pm4.2$ & $0.237_{-0.059}^{+0.114}$ & 13.0142 & 4, 16, 20\ 050505 & $63\pm2$ & 4.27 & $16_{-3}^{+13}$ &$237.829_{-49.203}^{+98.405}$ & $737.8_{-226.61}^{+1807.61}$ & $0.0290_{-0.0030}^{+0.0059}$ & 0.8928 &2, 17\ 050525 & 11.5 & 0.606 & $2.5 \pm 0.43$ & $28.2\pm 8.1$ & $127\pm10$ & $0.0551_{-0.0062}^{+0.0069}$ & 0.6507 & 4, 16, 20\ 050730 & $155\pm20$ & 3.97 & $9_{-3}^{+8}$ & 86.1223 & $974.12_{-432.39}^{+2798.11}$ & $>0.023$ & 0.0807 & 2, 19\ 050802 & 20 & 1.71 & $1.8197_{-0.30614}^{+1.6477}$ & $616.6\pm 295.4$ & $268.3_{-75.9}^{+623.3}$ & $0.29_{-0.15}^{+0.15}$ &349.8991 & 4, 20,21\ 050814 & 48 & 5.3 & $6_{-1}^{+3}$ & 831.764 & $403.2_{-138.6}^{+378}$ & 0.0419 & 9.6506 & 2, 18\ 050820A & 600 & 2.615 & $97.4 \pm 7.8$ & 53.7145 & $1325\pm277$ & 0.184 & 1.5369 & 16, 19\ 050904 & $183.6\pm13.2$ & 6.295 & $124 \pm 13$ & $88.37_{-44.2}^{+86.3}$ & $3178 \pm 1094$ & $0.0340\pm0.0051$ & 0.4877 & 16, 17\ 050922C & 4.5 & 2.198 & $5.3\pm1.7$ & 47.725 & $415\pm111$ & 0.026 & 1.2736 & 16, 19\ 051109A & 360 & 2.35 & $6.5 \pm 0.7$ & 169.824 & $539\pm200$ & 0.0593 & 0.2884 & 16, 18\ 060124 & $298 \pm2$ & 2.297 & $41 \pm 6$ & $578.87_{-12.66}^{+110.79}$ & $784\pm285$ & $0.0531_{-0.0040}^{+0.0091}$ & 0.9666 & 16, 17\ 060206 & $5.0\pm0.7$ & 4.05 & $4.3 \pm 0.9$ & $386.76 \pm 93.02$ & $394 \pm 46$ & $0.0351\pm 0.0010$ & 24.3279 & 16, 17\ 060210 & $220 \pm 70$ & 3.91 & $42_{-8}^{+35}$ & 1313.2261 & $667.76_{-191.49}^{+1703.77}$ & $0.024 \pm 0.002$ & 0.871 & 2, 19\ 060418 & $52 \pm 1$ & 1.49 & $13 \pm 3$ & 7.5307 & $572 \pm 143$ & $0.029 \pm 0.006$ & 0.0413 & 16, 19\ $060505^{}$& $4\pm1$ &0.089 & $0.0012\pm0.0002$ & 0.028& $482.4_{-167.7}^{+524.8}$ & $\sim0.4$& 0.06275 & 6, 22, 23\ 060526 &$258.8\pm5.4$ & 3.21 & $2.6 \pm 0.3$ & $15.58_{-0.21}^{+0.24}$ & $105 \pm 21$ & $0.0630\pm0.0010$ & 0.0587 & 16, 17\ 060605 & $19 \pm 1$ & 3.8 & $2.5_{-0.6}^{+3.1}$ & 115 & $681.6_{-240}^{+1723.2}$ & $>0.046$ & 3.14 & 2, 19\ 060607A & $100 \pm 5$ & 3.082 & $9_{-2}^{+7}$ & 0.822 & $567.398_{-167.362}^{+889.876}$ & $>0.095$ & 0.1808 & 2, 19\ $060614^{}$ & 6.9 & 0.12 & $0.21 \pm 0.09$ & 1.698 & $55 \pm 45$ & 0.2025 & 0.6328 & 16, 18\ 060707 & 210.0 & 3.42 & $5.4 \pm 1$ & 102.329 & $279 \pm 28$ & $0.1379$ & 2.1525 & 16, 18\ 060714 & $108.2\pm6.4$ & 2.71 & $7.7_{-0.9}^{+7.5}$ & $250.46 \pm 248.11$ & $196.63_{-181.79}^{+348.74}$ & $0.0201\pm0.0010$ & 0.1788 & 2, 17\ 060908 & $18.0\pm0.8$ & 1.8836 & $9.8 \pm 0.9$ & $2017.68_{-504.42}^{+2522.09}$ & $514 \pm 102$ & $0.0080_{-0.0010}^{+0.0051}$ & 1.0394 & 16, 17\ 061007 & $75\pm5$ & 1.262 & $86 \pm 9$ & 29.9425 & $890 \pm 124$ & $>0.138$ & 3.3244 & 16, 19\ 061021 & 79.0 & 0.35 & $10_{-4}^{+8}$ & 6.166 & $661.5_{-337.5}^{+985.5}$ & 0.1501 & 0.3106 & 2, 18\ 061121 & $81\pm 5$ & 1.314 & $22.5 \pm 2.6$ & 20.5215 & $1289 \pm 153$ & 0.099 & 0.6018 & 16, 19\ 061222A & 16.0 & 2.09 & $21_{-4}^{+11}$ & 2290.868 & $710.7_{-210.12}^{+747.78}$ & 0.0471 & 49.5146 & 2, 18\ 070110 & $89 \pm 7$ & 2.352 & $3.0_{-0.5}^{+2.5}$ & 0.687 & $372.072_{-90.504}^{+1035.77}$ & $>0.274$ & 0.518 & 2, 19\ 070125 & $63.0\pm1.7$ & 1.5477 & $80.2 \pm 8$ & $6.43_{0.17 }^{0.9}$ & $934 \pm 148$ & $0.2304\pm0.0105$ & 9.2574 & 16, 17\ 070306 & 3.0 & 1.5 & $6_{-1}^{+5}$ & 67.608 & $300_{-97.5}^{+1340}$ & 0.0768 & 18.081 & 2, 18\ 070318 & $63\pm3$ & 0.84 & $0.9_{-0.2}^{+0.9}$ & 47.2719 & $360.64_{-143.52}^{+818.8}$ & $0.127 \pm 0.008$ & 1.1331 & 2, 19\ 070411 & $101\pm5$ & 2.95 & $10_{-2}^{+8}$ & 83.6596 & $474_{-154.05}^{+2196.2}$ & $0.032 \pm 0.005$ & 0.1875 & 2, 19\ 070508 & 23.4 & 0.82 & $8_{-1}^{+2}$ & 10.715 & $378.56_{-74.62}^{+138.32}$ & 0.0611 & 0.2716 & 2, 18\ 071010A & $6\pm1$ & 0.98 & $0.13\pm0.01$ & 7.2164 & $73_{-69.9}^{+97.7}$ & $0.090 \pm 0.008$ & 0.9812 & 19, 21\ 071010B & $35.7\pm0.5$ & 0.947 & $1.7 \pm 0.9$ & 7.2713 & $101 \pm 20$ & $0.150 \pm 0.006$ & 0.5494 & 16, 19\ 071031 & 150.5 & 2.692 & $3.9\pm0.6$ & 1.554 & $45.23_{-41.54}^{+22.85}$ & $0.070 \pm 0.013$ & 0.0328 & 19, 21\ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- \[tab:continued2\] ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Name $T_{90}$ $z$ $E_{\rm \gamma,\rm iso}$ $E_{\rm k,\rm iso}$ $E_{\rm p,\rm rest}$ $\theta_{\rm j}$ $L_{\rm j}$ Ref ------ ---------- ----- ------------------------------------------------------------------------------------------------------------------------------------------------ --------------------- ---------------------- ------------------ ------------- ----- (s) z ($10^{52}$erg) & ($10^{52}$erg) &(keV) & (rad) & ($10^{50} \rm erg\rm~s^{-1}$) &\ 080310 & 32.0 & 2.43 & 6.0256 & 29.512 & $75.4_{-30.8}^{+72}$ & 0.0628 & 0.7509 & 18, 21\ 080319C & 29.55 & 1.95 & $14.1 \pm 2.8$ & 74.4078 & $906 \pm 272$ & $>0.102$ & 4.5924 & 16, 19\ 080330 & $61 \pm 9$ & 1.51 & $0.21\pm0.05$ & 21.0923 & $<88$ & $>0.087$ & 0.3315 & 19, 24\ 080413B & 8.0 & 1.1 & $2.4 \pm 0.3$ & 138.038 & $150 \pm 30$ & 0.1047 & 20.1874 & 18, 25\ 080603A & 150 & 1.688 & $2.2\pm0.8$ & 52.5129 & $160_{-130}^{+920}$ & $0.071 \pm 0.011$ & 0.247 & 19, 26\ 080710 & $120\pm17$ & 0.845 & $0.8\pm0.4$ & 2.6451 & 200 & $>0.062$ & 0.0102 & 19, 27\ 080810 & $108 \pm5$ & 3.35 & $45 \pm 5$ & 41.8519 & $1470 \pm 180$ & $>0.105$ & 1.9266 & 19, 25\ $080913^{}$ & $8\pm1$ & 6.695 & $8.6\pm2.5$ & $\leqslant 10$ & $710\pm350$ & $0.359_{+0.099}^{-0.125}$ & 114.0568 & 4, 25\ 081008 & $162.2\pm25.0$ & 1.967 & $9.98_{-2.31}^{+2.34}$ & $134.7_{-17.3}^{+18.3}$ & $255.9\pm57.47$ & $0.0227\pm0.0070$ & 0.0682 & 17, 28\ 081203A & 223 & 2.1 & $35 \pm 3$ & 11.2261 & $1541 \pm 757$ & $>0.116$ & 0.4319 & 19, 29\ 081222 & 5.8 & 2.77 & $30 \pm 3$ & 131.826 & $505 \pm 34$ & 0.0489 & 12.5737 & 18, 25\ 090313 & $78 \pm 19$ & 3.375 & 3.2 & 276.8523 & $240.1_{-223.5}^{+885.4}$ & $>0.093$ & 6.7881 & 19, 21\ 090323 & $133.1\pm1.4$ & 3.568 & $410\pm50$ & $116_{-9}^{+13}$ & $1901 \pm 343$ & $0.0489_{-0.0017}^{+0.0070}$ & 2.1579 & 17, 25\ 090328 & $57\pm 3$ & 0.7354 & $13 \pm 3$ & $82_{-18}^{+28}$ & $1028 \pm 312$ & $0.0733_{-0.0140}^{+ 0.0227}$ & 0.7767 & 17, 25\ $090423^{}$ & $10.3\pm 1.1$ & 8.23 &$11 \pm 3$ & $340_{-140}^{+110}$ &$491 \pm 200$ & $0.0262_{-0.0052}^{+0.0122}$ & 10.7949 & 17, 25\ 090424 & $49.47 \pm 0.9$ & 0.544 & $4.6 \pm 0.9$ & 53.1215 & $273 \pm 50$ & $>0.378$ & 12.718 & 19, 25\ 090812 & 75.9 & 2.452 & $40.3 \pm 4$ & 148.827 & $2023 \pm 663$ & $>0.071$ & 2.1671 & 19, 29\ 090902B & $19.328 \pm 0.286$ & 1.8829 & $1.77\pm0.01$ & $56_{-7}^{+3}$ &$596.76\pm17.2974$ & $0.0681\pm0.0035$ & 1.9973 & 3, 17\ 090926A & $20\pm 2$ & 2.1062 & $210_{-8}^{+9}$ & $6.8 \pm 0.2$ & $1279.75\pm62.124$ & $0.1571_{-0.0349}^{+0.0698}$ & 41.4656 & 3, 17\ 091018 & 106.5 & 0.97 & 0.5888 & 12.023 & $51.29\pm23.7$ & 0.0820 & 0.0784 & 18, 28\ 091020 & 65 & 1.71 & $12.2 \pm 2.4$ & 51.286 & $129.809 \pm 19.241$ & 0.1204 & 1.9162 & 7, 18\ 091024 & 1020 & 1.092 & $28 \pm 3$ & 37.2529 & $794 \pm 231$ & $>0.071$ & 0.0337 & 19, 29\ 091029 & 39.2 & 2.752 & $7.4 \pm 0.74$ & 40.303 & $230 \pm 66$ & $>0.192$ & 8.39 & 19, 29\ 091208B & 71 & 1.063 & $2.01 \pm 0.07$ & 50.119 & $297.4846_{-28.6757}^{+37.13}$ & 0.1274 & 1.2276 & 7, 18\ 100418A & $8.0 \pm 2.0$ & 0.6235 & $0.99_{-0.34}^{+0.63}$ & 3.36 & $47.08_{-43.83}^{+3.247} $ & 0.3560 & 5.5352 & 30, 31\ 100621A & $63.6 \pm 1.7$ & 0.542 & $4.37 \pm 0.5$ & 111.7596 & $146 \pm 23.1$ & $>0.234$ & 7.6734 & 19, 29\ 100728B & $12.1 \pm 2.4$ & 2.106 & $2.66 \pm 0.11$ & 95.665 & $406.886 \pm 46.59$ & $>0.063$ & 5.0071 & 7, 19\ 100901A & 439 & 1.408 & 6.3 & 167.3233 & 230 &0.152 & 1.098 & 19, 32\ 100906A & $114.4\pm1.6$ & 1.727 & $28.9 \pm 0.3$ & 23.8173 & $289.062_{-55.0854}^{+47.72}$ & $0.055 \pm 0.002$ & 0.19 & 7, 19\ 110205A & $257 \pm 25$ & 2.22 & $56 \pm 6$ & 31.2172 & $715 \pm 239$ & 0.064 & 0.2237 & 19, 29\ 110213A & $48 \pm 6$ & 1.46 & $6.9 \pm 0.2$ & 25.7527 & $242.064_{-16.974}^{+20.91}$ & $>0.142$ & 1.6843 & 7, 19\ 120119A & $70 \pm 4$ & 1.728 & 36 & 4.17 & $498.9 \pm 22.31$ & $0.032 \pm 0.002$ & 0.0801 & 19, 28\ 120326A & $11.8 \pm 1.8$ & 1.798 & $3.2\pm0.1$ & $14.0 \pm 0.07$ & $152\pm14$ & $0.0803\pm0.0035$ & 1.3142 & 17, 33\ 120521C & $26.7\pm0.4$ & 6 & $8.25_{-1.96}^{+2.24}$ & $22_{-14}^{+37}$ & $682_{-207}^{+845}$ & $0.0524_{-0.0192}^{+0.0401}$ & 1.0885 & 17, 34\ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Notes:\ $^{\star}$ GRBs have unusual characteristics on observations [@Xin2011; @Zhang2009]. References:\ (1) @Liu2015a; (2) @Butler2007; (3) @Zhang2011; (4) @Ryan2015; (5) @Racusin2009; (6) @Kann2011 ; (7) @Zhang2012; (8) @Antonelli2009; (9) @Tsutsui2013; (10) @Zaninoni2016; (11) @Berger2013b; (12) @Cummings2013; (13) @Sakamoto2014; (14) @Stanbro2016 ;(15) @Lv2017; (16) @Amati2008; (17) @Song2016; (18) @Nemmen2012; (19) @Yi2016; (20) @Zhang2007b; (21) @Kann2010; (22) @Xu2009; (23) @Ofek2007; (24) @Guidorzi2009; (25) @Amati2009; (26) @Guidorzi2011; (27) @Kruhler2009; (28) @Dichiara2016; (29) @Ghirlanda2012; (30) @Marshall2011; (31) @Laskar2015; (32) @Gorbovskoy2012; (33) @Demianski2017; (34) @Yasuda2017.\ Quasi-SNe ========= As strong GW sources in the nearby galaxies, the NS-NS/BH-NS mergers are expected to have electromagnetic counterparts such as SGRBs [e.g., @Eichler1989; @Nakar2007; @Berger2014; @Kumar2015; @Levan2016], off-axis emission of SGRB jets [e.g., @Rhoads1999; @Lazzati2017; @Xiao2017], optical/NIR signals powered by the decay of heavy radioactive elements in the ejection matter [e.g., @Li1998; @Metzger2017], radio flares [e.g., @Nakar2011; @Gao2013; @Piran2013], or X-ray emission from GRB central engine [e.g., @Nakamura2014; @Kisaka2015]. Furthermore, @Zhang2013 proposed that potentially an early X-ray afterglow would continue for thousands of seconds followed GW bursts once the NS-NS merger produced a magnetar rather than a BH. So the existence of an X-ray transient might be used as a criterion to judge if it is a remnant magnetar. The BH scenario will not fit the data in this case [@Sun2017]. As mentioned above, the compact objects merger model is accompanied by the ejection of neutron-rich matter. The dynamical ejecta, in a typical timescale of milliseconds, constitute contact-interface materials which are squeezed out by the hydrodaynamic force [e.g., @Oechslin2007; @Bauswein2013; @Hotokezaka2013] or the tidal force [e.g., @Kawaguchi2015]. For NS-NS mergers, the typical ejecta velocity and mass are in the range of $\sim 0.1-0.3~c$ and $\sim 10^{-4}-10^{-2}~M_{\rm \sun}$, respectively [e.g., @Hotokezaka2013]. Recent BH-NS merger simulations revealed that the ejecta mass could reach $0.1~M_{\rm \sun}$ with a similar velocity as in the NS-NS cases [@Kawaguchi2015; @Kawaguchi2016]. Then heavy radioactive elements will form via the r-process of neutron-rich matter. The radioactive decay of these elements provides a source for powering transient optical/NIR emission [@Eichler1989; @Li1998], named ‘kilonova’ [@Kulkarni2005] \[also called ‘macronova’ [@Metzger2010]\]. In addition to the radioactivity of the merger ejecta, the remnant materials’ fall-back accretion [e.g., @Rosswog2007; @Rossi2009; @Chawla2010; @Kyutoku2015], the ejecta from the disc, like winds [e.g., @Metzger2012; @Ma2018] or outflows, and magnetars [e.g., @Zhang2013; @Gao2013; @Yu2013; @Metzger2014; @Gao2015; @Gao2017b; @Yi2017; @Yi2018] can also power kilonovae. For some SGRBs with extended emission or internal X-ray plateaus, the magnetars might form after NS-NS mergers and provide the additional energy injection into the ejecta to power ‘mergernovae’ [e.g., @Yu2013]. In this paper, the nature of the outflows represents the heavy-nuclei-dominated injections into kilonovae. The kilonovae are claimed to be detected in the optical/NIR band, associated with some GRBs, i.e., GRBs 050709 [@Jin2016], 060614 [@Yang2015; @Jin2015; @Horesh2016], 130603B [@Tanvir2013; @Berger2013a; @Fan2013], and 160821B [@Kasliwal2017]. The excess optical emission was also discovered in GRB 080503 with a lack of redshift [@Perley2009]. @Gao2017b revisited the Swift SGRB samples and found three ‘magneter-powered mergernova’ candidates, i.e., GRBs 050724, 070714B, and 061006. The luminosities of these sources are ten times or a hundred times higher than those of typical kilonovae. For the recent GW event, the luminosity of GRB 170817A is one order of magnitude lower than that of associated AT 2017gfo [e.g., @Smartt2017]. The outflow matter is much massive than the dynamical ejecta after mergers, so we just calculated the effects of the outflows on kilonovae. Following @Li1998, we adopt a power law decay model here, and assume the material envelope expanding uniformly with the fixed velocity $V$, constant outflow mass $M_{\rm outflow}=fM_{\rm disc}$, surface radius $R$, and density $\rho$. The critical time $t_{c}$, when the optical depth of the expanding sphere satisfies $\kappa \rho R = 1$, can be calculated as t\_[c]{}&=&()\^[1/2]{}\ &=&1.13 [day]{} ()\^[1/2]{}()\^[-1]{}()\^[1/2]{}, where $\kappa \sim \kappa_{e}$ and $\kappa_{e} \sim 0.1~\rm cm^{2}~ g^{-1}$ [e.g., @Metzger2010] represent the average opacity and the electron scattering, respectively, and $V = 0.1~c$ is adopted. As shown in Figures 1 and 2, we take $m_{\rm disc}$=0.3, 0.1, 0.01, and $f=99\%, ~50\%$ to demonstrate the budget on the outflow strength and remnant inflow mass to kilonovae and GRBs. The luminosity of a kilonova powered by the radioactive decay of nuclei can be estimated as L\_[kilo]{}=L\_[0]{}Y(), where $L_{0}=3\eta fM_{\rm disc}c^{2}/(4\beta t_{\rm c})$, $\tau=t/t_{\rm c}$, and $\beta=V/c$. $\eta =3\times 10^{-6}$ denotes the fraction of rest-mass energy released in radioactive decay [@Metzger2010], and $Y(x)=e^{-x^2}\int_0^x e^{k^2}dk$ is the Dawson’s integral. By assuming blackbody emission, the effective temperature of the thermal emission is given by T\_[eff]{}=()\^[1/4]{}, where $\sigma$ is the Stephan-Boltzmann constant, and $R_{\rm ph}$ is the photosphere radius corresponding the radius of mass shell when the optical depth is equal to 1. The flux density of the source at photon frequency $\nu$ can be described as follows, F\_=. Then we can get the luminosity of an observational frequency $\nu$, i.e., L\_=4D\^2 F\_=. ![(a) Total luminosity of kilonovae. (b) Comparisons of the optical ($\sim 1~\rm~eV$) light curves with SNe and mergernovae. Lines in different colors (black, red, blue, green) correspond to different values of $m_{\rm disc}$ and $f$. Thick solid parts and thin dotted parts indicate the expanding spheres in the optically-thick and optically-thin, respectively. The gray filled circles, triangles, squares, and stars represent SN 1994I, SN 2002bj, SN 2010X, and SN 2013dx data, respectively. The grey solid and dashed lines depict optical light curves of the millisecond-magnetar-powered mergernovae@Yu2013](f2a.eps "fig:") ![(a) Total luminosity of kilonovae. (b) Comparisons of the optical ($\sim 1~\rm~eV$) light curves with SNe and mergernovae. Lines in different colors (black, red, blue, green) correspond to different values of $m_{\rm disc}$ and $f$. Thick solid parts and thin dotted parts indicate the expanding spheres in the optically-thick and optically-thin, respectively. The gray filled circles, triangles, squares, and stars represent SN 1994I, SN 2002bj, SN 2010X, and SN 2013dx data, respectively. The grey solid and dashed lines depict optical light curves of the millisecond-magnetar-powered mergernovae@Yu2013](f2b.eps "fig:") The total luminosities of kilonovae are shown in Figure 2(a). Lines in different color (black, red, blue, green) correspond to different values of $m_{\rm disc}$, and $f$. Thick solid parts and thin dotted parts indicate the expanding spheres in the optically-thick and optically-thin, respectively. We notice that the luminosities of kilonovae increase with the increase of the outflow ratios and residual masses. Figure 2(b) displays the optical ($\sim 1~\rm eV$) light curves of kilonovae, in comparison with SNe and mergernovae. The gray filled stars and circles respectively represent the data of SN 2013dx and SN 1994I. The grey solid and dashed lines depict the optical light curves of the millisecond-magnetar-powered mergernovae, which are adapted from Figure 3 in @Yu2013. From Figure 1 and Figure 2, one can conclude that the energies of a GRB and its associated kilonova appear to be complementary to each other, mainly depending on the neutron-rich outflow ratio. Comparing those light curves, we find that for strong outflows and massive remnants, the durations of the kilonovae powered by the outflows are much longer than those of mergernovae, even approaching those of SNe. That is, the more massive accretion materials become the outflows, the more similar the behaviours of kilonovae become faint SNe, especially the SNe with the steep decay such as SN 2002bj and SN 2010X. Therefore, we prefer the name ‘quasi-SNe’ for these phenomena, and we expect that a new type of ‘nova’ like the faint SNe may be detected after merger events. In addition, the vertical distribution of the outflows might effect the luminosity of the kilonovae for different view angles. Summary {#Summary} ======= The progenitors of GRBs remain mysteries after about fifty years’ discussions. It is still difficult to identify the physical origin of a GRB with multi-wavelength observational data available. The traditional definition of SGRBs and LGRBs by $T_{90}$ might not shed light on their progenitors. After the mergers of NS-NS or BH-NS, a BH might may be born surrounded by a hyperaccretion disc. In the present work, we test the applicability of the BH hyperaccretion inflow-outflow model on powering both LGRBs and SGRBs in the compact binary merger scenario. If about half of the disc materials become outflows, the luminosity and duration of the hyperaccretion processes might satisfy the requirements of not only all SGRBs but also account for most of LGRBs. We also point out that, to verify the origin of GRBs one may need various information, including the characteristics of host galaxies, the SN associations, and the spectral lags, etc. The optical/NIR emission observed in GRB afterglows are possibly powered by the numerous energy sources [for reviews, see e.g., @Metzger2017]. Here we propose a new mechanism of a BH hyperaccretion disc with extreme strong neutron-rich outflows, named quasi-SNe. The luminosities and timescales of quasi-SNe depend significantly on the outflow strengths. Consequently, there is a severe competition between GRBs and the associated quasi-SNe on the disc mass and energy budgets. In contrast, the luminosity of a mergernova depend on the energy injection from a magnetar, and there is no obvious correlation between it and the GRB luminosity, since most of the spin-down energy would be dissipated via GW emission [e.g., @Liu2017b]. In the particular case of GRB 170817A and AT 2017gfo, our model can naturally explain a weak GRB associated by a bright kilonova by considering the vertical distribution of the outflows, even without an off-axis jet for observers [e.g., @Lazzati2017; @Xiao2017; @Zhang2017]. We will investigate further this point in our future work. Very possibly in the near future, more and more GWs’ electromagnetic counterparts could be confirmed, produced by the compact binary mergers (especially for NS-NS and NS-BH). Noted that their optical/NIR emission might be originated from three mechanism: the ejecta or winds (kilonovae), the injection energy of magnetars (mergernovae), or the strong outflows from the hyperaccretion discs (quasi-SNe). Acknowledgements {#acknowledgements .unnumbered} ================ We thank Zi-Gao Dai, Bing Zhang, and Tuan Yi for beneficial discussion and the anonymous referee for very useful suggestions and comments. This work was supported by the National Basic Research Program of China (973 Program) under grant 2014CB845800, the National Natural Science Foundation of China under grants 11473022 and U1431107, and the Fundamental Research Funds for the Central Universities (grant 20720160024). [207]{} Abadie J., et al., 2010, CQGra, 27, 173001 Abbott B. P., et al., 2017, PhRvL, 119, 161101 Abbott B. P., et al., 2017, ApJ, 850, L40 Abbott B. P., et al., 2017, ApJ, 850, L39 Abramowicz M. A., Chen X., Kato S., Lasota J.-P., Regev O., 1995, ApJ, 438, L37 Alexander K. D., et al., 2017, ApJ, 848, L21 Aloy M. 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--- abstract: 'The asymptotic shape theorem for the contact process in random environment gives the existence of a norm $\mu$ on ${\mathbb{R}^d}$ such that the hitting time $t(x)$ is asymptotically equivalent to $\mu(x)$ when the contact process survives. We provide here exponential upper bounds for the probability of the event $\{\frac{t(x)}{\mu(x)}\not\in [1-{\varepsilon},1+{\varepsilon}]\}$; these bounds are optimal for independent random environment. As a special case, this gives the large deviation inequality for the contact process in a deterministic environment, which, as far as we know, has not been established yet.' address: \| Institut Élie Cartan Nancy (math[é]{}matiques)\ Universit[é]{} de Lorraine\ Campus Scientifique, BP 239\ 54506 Vandoeuvre-l[è]{}s-Nancy Cedex France\ author: - Olivier Garet - 'R[é]{}gine Marchand' title: Large deviations for the contact process in random environment --- \[section\] \[theorem\][Conjecture]{} \[theorem\][Lemma]{} \[theorem\][Definition]{} \[theorem\][Corollary]{} \[theorem\][Remark]{} \[theorem\][Proposition]{} Introduction ============ Durrett and Griffeath [@MR656515] proved that when the contact process on ${\mathbb{Z}^d}$ starting from the origin survives, the set of sites occupied before time $t$ satisfies an asymptotic shape theorem, as in first-passage percolation. In [@GM-contact], we extended this result to the case of the contact process in a random environment. The random environment is given by a collection $(\lambda_e)_{e \in {\mathbb{E}^d}}$ of positive random variables indexed by the set of edges of the grid ${\mathbb{Z}^d}$. Given a realization $\lambda$ of this environment, the contact process $(\xi^0_t)_{t\ge 0}$ in the environment $\lambda$ is a homogeneous Markov process taking its values in the set $\mathcal{P}({\mathbb{Z}^d})$ of subsets of ${\mathbb{Z}^d}$. If $\xi_t^0(z)=1$, we say that $z$ is occupied at time $t$, while if $\xi_t^0(z)=0$, we say that $z$ is empty at time $t$. The initial value of the process is $\{0\}$ and the process evolves as follows: - an occupied site becomes empty at rate $1$, - an empty site $z$ becomes occupied at rate: $\displaystyle \sum_{\\|z-z'\\|_1=1} \xi_t^0(z')\lambda_{\{z,z'\}},$ all these evolutions being independent. We study then the hitting time $t(x)$ of a site $x$: $$t(x)=\inf\{ t\ge 0: \; x \in \xi^0_t \}.$$ In [@GM-contact], we proved that under good assumptions on the random environment, there exists a norm $\mu$ on ${\mathbb{R}^d}$ such that for almost every environment, the family $(t(x))_{x\in{\mathbb{Z}^d}}$ satisfies, when $\\|x\\|$ goes to $+\infty$, $$t(x)\sim \mu(x) \quad \text{on the event ``the process survives''}.$$ We focus here on the large deviations of the hitting time $t(x)$ for the contact process in random environment. As far as we know, such inequalities for the classical contact process have not been studied yet, they will be contained in our results. The assumptions we will require on the random environment are the ones we already needed in [@GM-contact]. We denote by $\lambda_c({\mathbb{Z}^d})$ the critical intensity of the classical contact process on ${\mathbb{Z}^d}$, we fix $\lambda_{\min}$ and $\lambda_{\max}$ such that $\lambda_c({\mathbb{Z}^d})<\lambda_{\min}\le \lambda_{\max}$ and we set $\Lambda=[\lambda_{\min},\lambda_{\max}]^{{\mathbb{E}^d}}$. The support of the law $\nu$ of the random environment is included in $\Lambda=[\lambda_{\min},\lambda_{\max}]^{{\mathbb{E}^d}}$; the law $\nu$ is stationary, and if $\operatorname{Erg}(\nu)$ denotes the set of $x\in{\mathbb{Z}^d}\backslash \{0\}$ such that the translation along vector $x$ is ergodic for $\nu$, then the cone generated by $\operatorname{Erg}(\nu)$ is dense in ${\mathbb{R}^d}$. This last condition is obviously fulfilled if $\operatorname{Erg}(\nu)={\mathbb{Z}^d}\backslash\{0\}$. We will sometimes require the stronger following assumptions: The law $\nu$ of the random environment is a product measure: $\nu=\nu_0^{\otimes{\mathbb{E}^d}}$, where $\nu_0$ is some probability measure on $[\lambda_{\min},\lambda_{\max}]$. By taking for $\nu$ the Dirac mass $(\delta_{\lambda})^{\otimes {\mathbb{E}^d}}$, with $\lambda>\lambda_c({\mathbb{Z}^d})$, which clearly fullfills these assumptions, we recover the case of the classical contact process in a deterministic environment. For $\lambda \in \Lambda$, we denote by ${\mathbb{P}}_\lambda$ the (quenched) law of the contact process in environment $\lambda$, and by ${{\overline{\mathbb{P}}}}_\lambda$ the (quenched) law of the contact process in environment $\lambda$ conditioned to survive. We define then the annealed probability measures ${\overline{\mathbb{P}}}$ and ${\mathbb{P}}$: $${{\overline{\mathbb{P}}}}(.)=\int_\Lambda {{\overline{\mathbb{P}}}}_\lambda(.)\ d\nu(\lambda)\quad \text{ and }\quad {{\mathbb{P}}}(.)=\int_\Lambda {{\mathbb{P}}}_\lambda(.)\ d\nu(\lambda).$$ We will study separately the probabilities of the “upper large deviations” and the “lower large deviations”, [i.e. ]{}respectively of the events $\{t(x)\ge (1+{\varepsilon})\mu(x)\}$ and $\{t(x)\le (1-{\varepsilon})\mu(x)\}$. The most general result concerns the quenched “upper large deviations” for the hitting time $t(x)$ and the coupling time $$t'(x)=\inf\{ T\ge 0:\;\forall t\ge T\quad \xi^0_t(x)= \xi^{{\mathbb{Z}^d}}_t(x)\},$$ where $(\xi^{{\mathbb{Z}^d}}_t)_{t \ge 0}$ is the contact process starting from ${\mathbb{Z}^d}$, and for the set of hit points $H_t$ and the coupled region $K'_t$: $$\begin{aligned} H_t = \{x \in {\mathbb{Z}^d}: \; t(x) \le t\}, &&\tilde{H}_t=H_t+[0,1]^d\\ K'_t = \{x\in{\mathbb{Z}^d}: \; t'(x)\le t\}, &&\tilde{K}'_t=K'_t+[0,1]^d.\end{aligned}$$ We only require here Assumptions $(E)$. \[theoGDUQ\] Let $\nu$ be an environment law satisfying Assumptions $(E)$.\ For every ${\varepsilon}>0$, there exist $B>0$ and a random variable $A(\lambda)$ such that for $\nu$ almost every environment $\lambda$, for every $x \in {\mathbb{Z}^d}$, $$\begin{aligned} {\overline{\mathbb{P}}}_{\lambda}\left(t(x)\ge\mu(x)(1+{\varepsilon})\right) & \le & A(\lambda)e^{-B\\|x\\|}, \label{venus} \\ {\overline{\mathbb{P}}}_{\lambda}\left(t'(x)\ge\mu(x)(1+{\varepsilon})\right) & \le & A(\lambda)e^{-B\\|x\\|}, \label{tcouple} \\ {\overline{\mathbb{P}}}_{\lambda}\left(\forall t \ge T \quad (1-{\varepsilon})t A_\mu \subset \tilde{K'_t} \cap \tilde{H_t} \right) & \ge & 1-A(\lambda)e^{-BT}. \label{audessousforme} \end{aligned}$$ We can note that the random variable $A(\lambda)$ is almost surely finite, but that it could often be large. This question will be studied in a forecoming paper about annealed upper large deviations [@GM-contact-gd-annealed]. The key point of the proof of Theorem \[theoGDUQ\], interesting on its own, is to control the times $s$ when a site $x$ is occupied and has infinite progeny. We will denote this event by $\{(0,0) \to (x,s) \to \infty\}$ by analogy with percolation. \[lemme-pointssourcescontact\] There exist $C, \theta,A,B>0$ such that $ \forall\lambda\in\Lambda \quad \forall x\in{\mathbb{Z}^d}$ $$\forall t\ge C\\|x\\|\quad {\overline{\mathbb{P}}}_{\lambda}\left(\operatorname{Leb}\{s \in [0,t]: (0,0)\to (x,s) \to \infty\} \le \theta t \right)\le A\exp(-Bt).$$ For the “lower large deviations”, the subadditivity gives a nice setting and allows to state a large deviations principle in the spirit of Hammersley [@MR0370721]. \[LDP\] Let $\nu$ be an environment law satisfying Assumptions $(E)$.\ Let $x\in{\mathbb{Z}^d}$. There exist a convex function $\Psi_x$ and a concave function $K_x$ taking their values in ${\mathbb{R}}_+$ such that for $\nu$ almost every $\lambda$, $$\begin{aligned} \forall u>0 \quad \lim_{n\to +\infty} -\frac1{n}\log {\overline{\mathbb{P}}}_{\lambda}(t(nx)\le nu) & = & \Psi_x(u); \\ \forall\theta\ge 0\quad\lim_{n\to +\infty} -\frac1{n}\log {\overline{\mathbb{E}}}_{\lambda}[e^{-\theta t(nx)} ]& = & K_x(\theta).\end{aligned}$$ The functions $\Psi_x$ and $K_x$ moreover satisfy the reciprocity relations: $$\forall u>0 \quad \forall \theta\ge 0\quad \Psi_x(u)=\sup_{\theta\ge 0} \{K_x(\theta)-\theta u\}\text{ and }K_x(\theta)=\inf_{u>0} \{\Psi_x(u)+\theta u\}.$$ To obtain effective large deviation inequalities, we moreover have to prove that $\Psi_x(u)>0$ if $u<\mu(x)$. More precisely, \[dessouscchouette\] Let $\nu$ be an environment law satisfying Assumptions $(E')$.\ For every ${\varepsilon}>0$, there exist $A,B>0$ such that for every $x \in {\mathbb{Z}^d}$, for every $t \ge 0$, $$\begin{aligned} {\mathbb{P}}(t(x)\le (1-{\varepsilon})\mu(x)) & \le & A\exp(-B\\|x\\|), \label{defontenay}\\ {\mathbb{P}}(\forall s \ge t \quad H_s \subset (1+{\varepsilon})s A_\mu) & \ge & 1-A\exp(-Bt)\label{decadix}.\end{aligned}$$ The annealed large deviations inequalities imply the quenched ones: setting $$A(\lambda)=\sum_{x\in{\mathbb{Z}^d}}\exp(B\\|x\\|/2){\mathbb{P}}_{\lambda}\left(t(x)\le (1-{\varepsilon})\mu(x)\right),$$ we see that $A(\lambda)$ is integrable with respect to $\nu$, and thus is $\nu$-almost surely finite. So $$\forall x\in{\mathbb{Z}^d}\quad {\mathbb{P}}_{\lambda}(t(x)\le (1-{\varepsilon})\mu(x))\le A(\lambda)\exp(-B/2\\|x\\|).$$ Unfortunately, we do not have a complete large deviation principle as Theorem \[LDP\] for the upper large deviations. However, we will see in Section \[bonnevitesse\] that when the environment is i.i.d, the exponential order given by these inequalities is optimal. Asymptotic shape results for growth models are generally proved thanks to the subadditive processes theory initiated by Hammersley and Welsh [@MR0198576], and especially with Kingman’s subadditive ergodic theorem [@MR0356192] and its extensions. Since Hammersley [@MR0370721], we know that subadditive properties offer a proper setting to study the large deviation inequalities. See also the survey by Grimmett [@MR814710] and the Saint-Flour course by Kingman [@MR0438477]. However, as noted by Sepp[ä]{}l[ä]{}inen and Yukich [@MR1843178], the general theory of large deviations for subadditive processes is patchy. The best known case is first-passage percolation, studied by Grimmett and Kesten in 1984 [@grimmett-kesten]. This paper introduced some lines of proof for the large deviations of growth processes, that have been reused later, for instance in the study of the large deviations for the chemical distance in Bernoulli percolation [@GM-large]. For more recent results concerning first-passage percolation, see Chow–Zhang [@chow-zhang], Cranston–Gauthier–Mountford [@MR2521889], and Théret et al [@MR2464099; @MR2343936; @MR2610330; @RT-IHP; @CT-PTRF; @CT-AAP; @CT-TAM; @RT-ESAIM]. The renormalization techniques used by Grimmett and Kesten are well-known now: static renormalization for “upper large deviations” (control of a too slow growth), dynamic renormalization for “lower large deviations” (control of a too fast growth). However, the possibility for the contact process to die gives rise to extra difficulties that do not appear in the case of first-passage percolation or even of Bernoulli percolation. To our knowledge, the only growth process with possible extinction for which large deviations inequalities have been established is oriented percolation in dimension 2 (see Durrett [@MR757768]). Note also that Proposition 20.1 in the PhD thesis of Couronné [@Couronne] rules out the possibility of a too fast growth for oriented percolation in dimension $d$. In Section 2, we construct the model, give the notation and state previous results, mainly from [@GM-contact]. Section 3 is devoted to the proof of the upper large deviation inequalities, Theorem \[theoGDUQ\], while lower large deviations – Theorems \[LDP\] and \[dessouscchouette\] – are proved in Section 4. Finally, the optimality of the exponential decrease given by these results is briefly discussed in Section 5. Preliminaries ============= Definition of the model ----------------------- Let $\lambda_{\min}$ and $\lambda_{\max}$ be fixed such that $\lambda_c({\mathbb{Z}^d})<\lambda_{\min}\le\lambda_{\max}$, where $\lambda_c({\mathbb{Z}^d})$ is the critical parameter for the survival of the classical contact process on ${\mathbb{Z}^d}$. In the following, we restrict ourselves to the study of the contact process in random environment with birth rates $\lambda=(\lambda_e)_{e \in {\mathbb{E}^d}}$ in $\Lambda=[\lambda_{\min},\lambda_{\max}]^{{\mathbb{E}^d}}$. An environment is thus a collection $\lambda=(\lambda_e)_{e \in {\mathbb{E}^d}} \in \Lambda$. Let $\lambda \in \Lambda$ be fixed. The contact process $(\xi_t)_{t\ge 0}$ in the environment $\lambda$ is a homogeneous Markov process taking its values in the set $\mathcal{P}({\mathbb{Z}^d})$ of subsets of ${\mathbb{Z}^d}$, that we sometimes identify with $\{0,1\}^{{\mathbb{Z}^d}}$: for $z \in {\mathbb{Z}^d}$ we also use the random variable $\xi_t(z)={1\hspace{-1.3mm}1}_{\{z \in \xi_t\}}$. If $\xi_t(z)=1$, we say that $z$ is occupied or infected, while if $\xi_t(z)=0$, we say that $z$ is empty or healthy. The evolution of the process is as follows: - an occupied site becomes empty at rate $1$, - an empty site $z$ becomes occupied at rate $\displaystyle \sum_{\\|z-z'\\|_1=1} \xi_t(z')\lambda_{\{z,z'\}},$ each of these evolutions being independent from the others. In the following, we denote by ${\mathcal{D}}$ the set of càdlàg functions from ${\mathbb{R}}_{+}$ to $\mathcal{P}({\mathbb{Z}^d})$: it is the set of trajectories for Markov processes with state space $\mathcal{P}({\mathbb{Z}^d})$. To define the contact process in the environment $\lambda\in\Lambda$, we use Harris’ construction [@MR0488377]. It allows to make a coupling between contact processes starting from distinct initial configurations by building them from a single collection of Poisson measures on ${\mathbb{R}}_+$. ### Graphical construction {#graphical-construction .unnumbered} We endow ${\mathbb{R}}_+$ with the Borel $\sigma$-algebra $\mathcal B({\mathbb{R}}_+)$, and we denote by $M$ the set of locally finite counting measures $m=\sum_{i=0}^{+\infty} \delta_{t_i}$. We endow this set with the $\sigma$-algebra $\mathcal M$ generated by the maps $m\mapsto m(B)$, where $B$ describes the set of Borel sets in ${\mathbb{R}}_+$. We then define the measurable space $(\Omega, \mathcal F)$ by setting $$\Omega=M^{{\mathbb{E}^d}}\times M^{{\mathbb{Z}^d}} \text{ and } \mathcal F=\mathcal{M}^{\otimes {\mathbb{E}^d}} \otimes \mathcal{M}^{\otimes {\mathbb{Z}^d}}.$$ On this space, we consider the family $({\mathbb{P}}_{\lambda})_{\lambda\in\Lambda}$ of probability measures defined as follows: for every $\lambda=(\lambda_e)_{e \in {\mathbb{E}^d}} \in \Lambda$, $${\mathbb{P}}_{\lambda}=\left(\bigotimes_{e \in {\mathbb{E}^d}} \mathcal{P}_{\lambda_{e}}\right) \otimes \mathcal{P}_1^{\otimes{\mathbb{Z}^d}},$$ where, for every $\lambda\in{\mathbb{R}}_+$, $\mathcal{P}_{\lambda}$ is the law of a Poisson point process on ${\mathbb{R}}_+$ with intensity $\lambda$. If $\lambda \in {\mathbb{R}}_+$, we write ${\mathbb{P}}_\lambda$ (rather than ${\mathbb{P}}_{(\lambda)_{e \in {\mathbb{E}^d}}}$) for the law in deterministic environment with constant infection rate $\lambda$. For every $t\ge 0$, we denote by $\mathcal{F}_t$ the $\sigma$-algebra generated by the maps $\omega\mapsto\omega_e(B)$ and $\omega\mapsto\omega_z(B)$, where $e$ ranges over all edges in ${\mathbb{E}^d}$, $z$ ranges over all points in ${\mathbb{Z}^d}$, and $B$ ranges over all Borel sets in $[0,t]$. We build the contact process in environment $\lambda\in\Lambda$ from this family of Poisson process, as detailed in Harris [@MR0488377] for the classical contact process and in [@GM-contact] for the random environment case. Note especially that the process is attractive $$(A \subset B) \Rightarrow (\forall t \ge 0\quad \xi_t^A \subset \xi_t^B),$$ and Fellerian; then it enjoys the strong Markov property. ### Time translations {#time-translations .unnumbered} For $t \ge 0$, we define the translation operator $\theta_t$ on a locally finite counting measure $m=\sum_{i=1}^{+\infty} \delta_{t_i}$ on ${\mathbb{R}}_+$ by setting $$\theta_t m=\sum_{i=1}^{+\infty} {1\hspace{-1.3mm}1}_{\{t_i\ge t\}}\delta_{t_i-t}.$$ The translation $\theta_t$ induces an operator on $\Omega$, still denoted by $\theta_t$: for every $\omega \in \Omega$, we set $$\theta_t \omega=((\theta_t \omega_e)_{e \in {\mathbb{E}^d}}, (\theta_t \omega_z)_{z \in {\mathbb{Z}^d}}).$$ ### Spatial translations {#spatial-translations .unnumbered} The group ${\mathbb{Z}^d}$ can act on the process and on the environment. The action on the process changes the observer’s point of view: for $x \in {\mathbb{Z}^d}$, we define the translation operator $T_x$ by $$\forall \omega \in \Omega\quad T_x \omega=(( \omega_{x+e})_{e \in {\mathbb{E}^d}}, ( \omega_{x+z})_{z \in {\mathbb{Z}^d}}),$$ where $x+e$ the edge $e$ translated by vector $x$. Besides, we can consider the translated environment ${{x}.{\lambda}}$ defined by $({{x}.{\lambda}})_e=\lambda_{x+e}$. These actions are dual in the sense that for every $\lambda \in \Lambda$, for every $x \in {\mathbb{Z}^d}$, $$\begin{aligned} \label{translationspatiale} \forall A\in\mathcal{F}\quad{\mathbb{P}}_{\lambda}(T_x \omega \in A) & = & {\mathbb{P}}_{{{x}.{\lambda}}}(\omega \in A).\end{aligned}$$ Consequently, the law of $\xi^x$ under ${\mathbb{P}}_\lambda$ coincides with the law of $\xi^0$ under ${\mathbb{P}}_{x.\lambda}$. ### Essential hitting times and associated translations {#essential-hitting-times-and-associated-translations .unnumbered} For a set $A \subset {\mathbb{Z}^d}$, we define the lifetime $\tau^A$ of the process starting from $A$ by $$\tau^A=\inf\{t\ge0: \; \xi_t^A=\varnothing\}.$$ For $A \subset {\mathbb{Z}^d}$ and $x \in {\mathbb{Z}^d}$, we also define the first infection time $t^A(x)$ of the site $x$ from the set $A$ by $$t^A(x)=\inf\{t\ge 0: \; x \in \xi_t^A\}.$$ If $y\in{\mathbb{Z}^d}$, we write $t^y(x)$ instead of $t^{\{y\}}(x)$. Similarly, we simply write $t(x)$ for $t^0(x)$. In our previous paper [@GM-contact], we introduced a new quantity $\sigma(x)$: it is a time when the site $x$ is infected from the origin $0$ and also has an infinite lifetime. This essential hitting time is defined from a family of stopping times as follows: we set $u_0(x)=v_0(x)=0$ and we define recursively two increasing sequences of stopping times $(u_n(x))_{n \ge 0}$ and $(v_n(x))_{n \ge 0}$ with $u_0(x)=v_0(x)\le u_1(x)\le v_1(x)\le u_2(x)\dots$ as follows: - Assume that $v_k(x)$ is defined. We set $u_{k+1}(x) =\inf\{t\ge v_k(x): \; x \in \xi^0_t \}$.\ If $v_k(x)<+\infty$, then $u_{k+1}(x)$ is the first time after $v_k(x)$ where site $x$ is once again infected; otherwise, $u_{k+1}(x)=+\infty$. - Assume that $u_k(x)$ is defined, with $k \ge 1$. We set $v_k(x)=u_k(x)+\tau^x\circ \theta_{u_k(x)}$.\ If $u_k(x)<+\infty$, the time $\tau^x\circ \theta_{u_k(x)}$ is the lifetime of the contact process starting from $x$ at time $u_k(x)$; otherwise, $v_k(x)=+\infty$. We then set $$\label{definitiondeK} K(x)=\min\{n\ge 0: \; v_{n}(x)=+\infty \text{ or } u_{n+1}(x)=+\infty\}.$$ This quantity represents the number of steps before the success of this process: either we stop because we have just found an infinite $v_n(x)$, which corresponds to a time $u_n(x)$ when $x$ is occupied and has infinite progeny, or we stop because we have just found an infinite $u_{n+1}(x)$, which says that after $v_n(x)$, site $x$ is nevermore infected. We proved that $K(x)$ is almost surely finite, which allows to define the essential hitting time $\sigma(x)$ by setting $\sigma(x)=u_{K(x)}$. It is of course larger than the hitting time $t(x)$ and can been seen as a regeneration time. Note however that $\sigma(x)$ is not necessary the first time when $x$ is occupied and has infinite progeny: for instance, such an event can occur between $u_1(x)$ and $v_1(x)$, being ignored by the recursive construction. At the same time, we define the operator $\tilde \theta_x$ on $\Omega$ by: $$\tilde \theta_x = \begin{cases} T_{x} \circ \theta_{\sigma(x)} & \text{if $\sigma(x)<+\infty$,} \\ T_x &\text{otherwise,} \end{cases}$$ or, more explicitly, $$(\tilde \theta_x)(\omega) = \begin{cases} T_{x} (\theta_{\sigma(x)(\omega)} \omega) & \text{if $\sigma(x)(\omega)<+\infty$,} \\ T_x (\omega) &\text{otherwise.} \end{cases}$$ We will mainly deal with the essential hitting time $\sigma(x)$ that enjoys, unlike $t(x)$, some good invariance properties in the survival-conditioned environment. Moreover, the difference between $\sigma(x)$ and $t(x)$ was controlled in [@GM-contact]; this will allow us to transpose to $t(x)$ the results obtained for $\sigma(x)$. ### Contact process in the survival-conditioned environment {#contact-process-in-the-survival-conditioned-environment .unnumbered} For $\lambda \in \Lambda$, we define the probability measure ${{\overline{\mathbb{P}}}}_\lambda$ on $(\Omega, \mathcal F)$ by $$\forall E\in\mathcal{F}\quad {{\overline{\mathbb{P}}}}_\lambda(E)={\mathbb{P}}_\lambda(E\|\tau^0=+\infty).$$ It is thus the law of the family of Poisson point processes, conditioned to the survival of the contact process starting from $0$. Let then $\nu$ be a probability measure on the set of environments $\Lambda$. On the same space $(\Omega, \mathcal F)$, we define the corresponding annealed probabilities ${\overline{\mathbb{P}}}$ and ${\mathbb{P}}$ by setting $$\forall E\in\mathcal{F}\quad {{\overline{\mathbb{P}}}}(E)=\int_\Lambda {{\overline{\mathbb{P}}}}_\lambda(E)\ d\nu(\lambda) \quad{ and } \quad {{\mathbb{P}}}(E)=\int_\Lambda {{\mathbb{P}}}_\lambda(E)\ d\nu(\lambda).$$ Previous results ---------------- We recall here the results established in [@GM-contact] for the contact process in random environment. \[magic\] Let $x,y \in {\mathbb{Z}^d}\backslash \{0\}$, $\lambda\in\Lambda$, $A$ in the $\sigma$-algebra generated by $\sigma(x)$, and $B\in \mathcal F$. Then $$\forall \lambda \in \Lambda \quad {\overline{\mathbb{P}}}_\lambda(A \cap (\tilde{\theta}_x)^{-1}(B))={\overline{\mathbb{P}}}_\lambda(A) {\overline{\mathbb{P}}}_{{{x}.{\lambda}}}(B).$$ \[invariancePbarre\] As consequences we have: - The probability measure ${\overline{\mathbb{P}}}$ is invariant under the translation $\tilde \theta_x$. - Under ${\overline{\mathbb{P}}}_\lambda$, $\sigma(y)\circ\tilde{\theta}_x$ and $\sigma(x)$ are independent. Moreover, the law of $\sigma(y)\circ\tilde{\theta}_x$ under ${\overline{\mathbb{P}}}_\lambda$ is the same as the law of $\sigma(y)$ under ${\overline{\mathbb{P}}}_{{{x}.{\lambda}}}$. - The random variables $(\sigma(x) \circ (\tilde \theta_{x})^j)_{j \ge 0}$ are independent under ${\overline{\mathbb{P}}}_\lambda$. \[propmoments\] There exist $A,B,C>0$ and, for every $p\ge 1$, a constant $C_p>0$ such that for every $x\in{\mathbb{Z}^d}$ and every $\lambda\in\Lambda$, $$\begin{aligned} \label{moms} {\overline{\mathbb{E}}}_{\lambda} [\sigma(x)^p ]& \le& C_p (1+\\|x\\|)^{p},\\ \label{asigma} \forall t\ge 0 \quad ( \\|x\\|\le t) & \Longrightarrow & \left({\overline{\mathbb{P}}}_{\lambda}(\sigma(x)> Ct) \le A\exp(-Bt^{1/2})\right).\end{aligned}$$ \[systemeergodique\] For every $x\in\operatorname{Erg}(\nu)$, the measure-preserving dynamical system $(\Omega,\mathcal{F},{\overline{\mathbb{P}}},\tilde{\theta}_x)$ is ergodic. We then proved that ${\overline{\mathbb{P}}}$ almost surely, for every $x \in {\mathbb{Z}^d}$, $\frac{\sigma(nx)}n$ converges to a deterministic real number $\mu(x)$. The function $x\mapsto \mu(x)$ can be extended to a norm on ${\mathbb{R}^d}$, that characterizes the asymptotic shape. Let $A_{\mu}$ be the unit ball for $\mu$. We define $$\begin{aligned} H_t & = & \{x\in{\mathbb{Z}^d}: \; t(x)\le t\},\\ G_t & = & \{x\in{\mathbb{Z}^d}: \; \sigma(x)\le t\},\\ K'_t & = & \{x\in{\mathbb{Z}^d}: \;\forall s\ge t \quad \xi^0_s(x)=\xi^{{\mathbb{Z}^d}}_s(x)\},\end{aligned}$$ and we denote by $\tilde{H}_t,\tilde{G}_t,\tilde{K}'_t$ their “fattened” versions: $$\tilde{H}_t=H_t+[0,1]^d, \; \tilde{G}_t=G_t+[0,1]^d \text{ and } \tilde{K}'_t=K'_t+[0,1]^d.$$ We can now state the asymptotic shape result: \[thFA\] For every ${\varepsilon}>0$, ${\overline{\mathbb{P}}}-a.s.$, for every $t$ large enough, $$\label{leqdeforme} (1-{\varepsilon})A_{\mu}\subset \frac{\tilde K'_t\cap \tilde G_t}t\subset \frac{\tilde G_t}t\subset\frac{\tilde H_t}t\subset (1+{\varepsilon})A_{\mu}.$$ In order to prove the asymptotic shape theorem, we established exponential controls uniform in $\lambda \in \Lambda$. We set $$B_r^x=\{y \in {\mathbb{Z}^d}: \; \\|y-x\\|_{\infty} \le r\},$$ and we write $B_r$ instead of $B_r^0$. \[propuniforme\] There exist $A,B,M,c,\rho>0$ such that for every $\lambda\in\Lambda$, for every $y \in {\mathbb{Z}^d}$, for every $ t\ge0$ $$\begin{aligned} {\mathbb{P}}_\lambda(\tau^0=+\infty) & \ge & \rho, \label{uniftau} \\ {\mathbb{P}}_\lambda(H^0_t \not\subset B_{Mt} ) & \le & A\exp(-Bt), \label{richard} \\ {\mathbb{P}}_\lambda ( t<\tau^0<+\infty) &\le& A\exp(-Bt), \label{grosamasfinis} \\ {\mathbb{P}}_{\lambda}\left( t^0(y)\ge \frac{\\|y\\|}c+t,\; \tau^0=+\infty \right) & \le & A\exp(-Bt), \label{retouche}\\ {\mathbb{P}}_{\lambda}(0\not\in K'_t, \; \tau^0=+\infty) &\le &A\exp(-B t). \label{petitsouscouple} \end{aligned}$$ \[momtprime\] There exist $A,B,C>0$ such that for every $x\in{\mathbb{Z}^d}$ and every $\lambda\in\Lambda$, $$\label{momtprimeeq} \forall t\ge 0 \quad ( \\|x\\|\le t) \Longrightarrow \left({\overline{\mathbb{P}}}_{\lambda}(t'(x)> Ct) \le A\exp(-Bt^{1/2})\right).$$ For every $\lambda \in \Lambda$, for every $ x \in {\mathbb{Z}^d}$, $$\begin{aligned} {\overline{\mathbb{P}}}_{\lambda}(t'(x)>\sigma(x)+s) & = & {\overline{\mathbb{P}}}_{\lambda}(x \not\in K'_{\sigma(x)+s}\cap G_{\sigma(x)+s})\nonumber \\ & = & {\overline{\mathbb{P}}}_{\lambda}(x \not\in K'_{\sigma(x)+s})\nonumber \\ & \le & {\overline{\mathbb{P}}}_{\lambda}( x \not\in x+(K'_s) \circ \tilde{\theta}_x)= {\overline{\mathbb{P}}}_{x.\lambda}(0 \not\in K'_s\nonumber)\\ & \le & A \exp(-Bs),\label{demai}\end{aligned}$$ with (\[uniftau\]) and (\[petitsouscouple\]). With (\[asigma\]), this estimate gives the announced result. An abstract restart procedure ----------------------------- We formalize here the restart procedure for Markov chains. Let $E$ be the state space where our Markov chains $(X^x_n)_{n\ge 0}$ evolve, $x \in E$ being the starting point of the chain. We suppose that we have on our disposal a set $\tilde{\Omega}$, an update function $f:E\times \tilde{\Omega}\to E$, and a probability measure $\nu$ on $\tilde{\Omega}$ such that on the probability space $(\Omega, \mathcal{F}, {\mathbb{P}})=(\tilde{\Omega}^{{\mathbb N}^},{\mathcal{B}(\tilde{\Omega}^{{\mathbb N}^})},\nu^{\otimes{\mathbb N}^})$, endowed with the natural filtering $(\mathcal{F}_n)_{n\ge 0}$ given by $\mathcal{F}_n=\sigma(\omega\mapsto \omega_k: \;k\le n)$, the chains $(X^x_n)_{n\ge 0}$ starting from the different states enjoy the following representation: $$\begin{aligned} \begin{cases} X^x_0(\omega)=x \\ X^x_{n+1}(\omega)=f(X^x_n(\omega),\omega_{n+1}). \end{cases}\end{aligned}$$ As usual, we define $\theta:\Omega\to\Omega$ which maps $\omega=(\omega_n)_{n\ge 1}$ to $\theta\omega=(\omega_{n+1})_{n\ge 1}$. We assume that for each $x\in E$, we have defined a $(\mathcal{F}_n)_{n\ge 0}$-adapted stopping time $T^x$, a $\mathcal{F}_{T^x}$-measurable function $G^x$ and a $\mathcal{F}$-measurable function $F^x$. Now, we are interested in the following quantities: $$\begin{aligned} T_0^x=0 \text{ and } T^x_{k+1} & = & \begin{cases} +\infty & \text{if }T^x_{k}=+\infty\\ T_k^x+T^{x_k}(\theta_{T_k^{x}}) & \text{with $x_k=X^x_{\theta_{T_k^x}}$ otherwise;} \end{cases} \\ K^x & = & \inf\{k\ge 0:\;T_{k+1}^x=+\infty\}; \\ M^x & = & \sum_{k=0}^{K^x-1} G^{x_k}(\theta_{T_k^x})+F^{X^{x_K}}(\theta_{T^x_{K}}).\end{aligned}$$ We wish to control the exponential moments of the $M^x$’s with the help of exponential bounds for $G^x$ and $F^x$. In numerous applications to directed percolation or to the contact process, $T^x$ is the extinction time of the process (or of some embedded process) starting from the smallest point (in lexicographic order) in the configuration $x$. \[restartabstrait\] We suppose that there exist real numbers $A>0$, $c<1$, $p>0$, $\beta>0$ such that the real-valued functions $(G^x)_{x\in E},(F^x)_{x\in E}$ defined above satisfy $$\forall x\in E\quad \left\lbrace \begin{array}{l} \mathbf{G}(x)={\mathbb{E}}[\exp(\beta G^x){1\hspace{-1.3mm}1}_{\{T^x<+\infty\}}]\le c;\\ \mathbf{F}(x)={\mathbb{E}}[{1\hspace{-1.3mm}1}_{\{T^x=+\infty\}} \exp(\beta F^x)]\le A;\\ \mathbf{T}(x)={\mathbb{P}}(T^x=+\infty)\ge p. \end{array} \right.$$ Then, for each $x \in E$, $K^x$ is ${\mathbb{P}}$-almost surely finite and $${\mathbb{E}}[ \exp(\beta M^x)]\le \frac{A}{1-c} <+\infty.$$ Oriented percolation -------------------- We work, for $d \ge1$, on the following graph: - The set of sites is ${{\mathbb{V}}^{d+1}}=\{(z,n)\in {\mathbb{Z}^d}\times {\mathbb N}\}$. - We put an oriented edge from $(z_1,n_1)$ to $(z_2,n_2)$ if and only if $n_2=n_1+1$ and $\\|z_2-z_1\\|_1\le1$; the set of these edges is denoted by ${\overrightarrow{\mathbb{E}}^{d+1}_{\text{alt}}}$. Define ${\overrightarrow{\mathbb{E}}^d}$ in the following way: in ${\overrightarrow{\mathbb{E}}^d}$, there is an oriented edge between two points $z_1$ and $z_2$ in ${\mathbb{Z}^d}$ if and only if $\\|z_1-z_2\\|_1\le 1$. The oriented edge in ${\overrightarrow{\mathbb{E}}^{d+1}_{\text{alt}}}$ from $(z_1,n_1)$ to $(z_2,n_2)$ can be identified with the couple $((z_1,z_2),n_2)\in{\overrightarrow{\mathbb{E}}^d}\times{\mathbb N}^$. Thus, we identify ${\overrightarrow{\mathbb{E}}^{d+1}_{\text{alt}}}$ and ${\overrightarrow{\mathbb{E}}^d}\times{\mathbb N}^$. We consider $\Omega=\{0,1\}^{{\overrightarrow{\mathbb{E}}^{d+1}_{\text{alt}}}}$ endowed with its Borel $\sigma$-algebra: the edges $e$ such that $\omega_e=1$ are said to be open, the other ones are closed. For $v, w$ in ${\mathbb{Z}^d}\times{\mathbb N}$, we denote by $v \to w$ the existence of an oriented path from $v$ to $w$ composed of open edges. We denote by ${\overrightarrow{p_c}^{\text{alt}}}(d+1)$ the critical parameter for the Bernoulli oriented percolation on this graph ([i.e.* ]{}all edges are independently open with probability $p$). We set, for $n \in {\mathbb N}$ and $(x,0)\in {{\mathbb{V}}^{d+1}}$, $$\begin{aligned} \bar{\xi}^x_n & = & \{y \in {\mathbb{Z}}: \; (x,0)\to(y,n)\}, \\ \bar{\tau}^x & = & \max\{n \in {\mathbb N}:\; \bar{\xi}^x_n \neq \varnothing\}.\end{aligned}$$ We recall results from [@GM-dop] for a class $\mathcal{C}_d(M,q)$ of dependent oriented percolation models on this graph. The parameter $M$ controls the range of the dependence while the parameter $q$ controls the probability for an edge to be open. Let $d\ge1$ be fixed. Let $M$ be a positive integer and $q\in (0,1)$. Let $(\Omega,\mathcal{F},{\mathbb{P}})$ be a probability space endowed with a filtration $(\mathcal{G}_n)_{n\ge 0}$. We assume that, on this probability space, a random field $(W^n_e)_{e\in{\overrightarrow{\mathbb{E}}^d},n\ge 1}$ taking its values in $\{0,1\}$ is defined. This field gives the states – open or closed – of the edges in ${\overrightarrow{\mathbb{E}}^{d+1}_{\text{alt}}}$. We say that the law of the field $(W^n_e)_{e\in{\overrightarrow{\mathbb{E}}^d},n \ge 1}$ is in $\mathcal{C}_d(M,q)$ if it satisfies the two following conditions. - $\forall n\ge 1,\forall e \in {\overrightarrow{\mathbb{E}}^d}\quad W^n_e\in\mathcal{G}_n$; - $\forall n \ge 0,\forall e \in {\overrightarrow{\mathbb{E}}^d}\quad {\mathbb{P}}[W^{n+1}_e=1\|\mathcal{G}_n\vee \sigma(W^{n+1}_f, \; d(e,f)\ge M)]\ge q$, where $\sigma(W^{n+1}_f, \; d(e,f)\ge M)$ is the $\sigma$-field generated by the random variables $W^{n+1}_f$, with $d(e,f)\ge M$. Note that if $0\le q\le q'\le 1$, we have $\mathcal{C}_d(M,q')\subset \mathcal{C}_d(M,q)$. We can control the probability of survival and also the lifetime for these dependent oriented percolations. \[petitmomentexpo\] Let ${\varepsilon}>0$ and $M>1$. There exist $\beta>0$ and $q<1$ such that for each $\chi\in\mathcal{C}_d(M,q)$, $$\forall x \in {\mathbb{Z}^d}\quad {\mathbb{E}}_{\chi}[{1\hspace{-1.3mm}1}_{\{\bar{\tau}^x<+\infty\}}\exp(\beta\bar{\tau}^x)]\le {\varepsilon}\quad \text{ and } \quad \chi(\bar{\tau}^x=+\infty) \ge 1-{\varepsilon}.$$ A point $(y,k) \in {\mathbb{Z}^d}\times {\mathbb N}$ such that $(x,0)\to(y,k)\to\infty$ is called an immortal descendant of $x$. We will need estimates on the density of immortal descendants of $x$ above some given point $y$ in oriented dependent percolation. So we define $$\begin{aligned} \bar{G}(x,y)&=& \{k\in{\mathbb N}\quad (x,0)\to(y,k)\to\infty\}, \\ \bar{\gamma}(\theta,x,y) & = & \inf\{n\in{\mathbb N}: \quad \forall k\ge n \quad {\vert \{0,\dots,k\}\cap \bar{G}(x,y) \vert}\ge\theta k\}.\end{aligned}$$ \[lineairegamma\] Let $M>1$. There exist $q_0<1$ and positive constants $A,B,\theta,\alpha$ such that for each $\chi\in\mathcal{C}_d(M,q_0)$, we have $$\forall x,y\in{\mathbb{Z}^d}\quad\forall n\ge 0\quad \chi(+\infty>\gamma(\theta,x,y)> \alpha \\|x-y\\|_1+n)\le Ae^{-Bn}.$$ Quenched upper large deviations =============================== The aim is now to prove the quenched upper large deviations of Theorem \[theoGDUQ\]. In order to exploit the subadditivity, we show that $\sigma(x)$ admits exponential moments uniformly in $\lambda \in \Lambda$: \[theomomexpsigma\] There exist positive constants $\gamma_1,\beta_1$ such that $$\label{momexpsigma} \forall x \in {\mathbb{Z}^d}\quad \forall \lambda \in \Lambda \quad {\overline{\mathbb{E}}}_{\lambda}(e^{\gamma_1 \sigma(x)}) \le e^{\beta_1\\|x\\|_{1}}.$$ As an immediate consequence, we get \[sauveur\] There exist positive constants $A,B,c,$ such that for each $\lambda\in\Lambda$, each $x \in {\mathbb{Z}^d}$ and every $ t\ge0$ $$\begin{aligned} {\overline{\mathbb{P}}}_{\lambda}\left( t'(x)\ge \frac{\\|x\\|}c+t \right) & \le & A\exp(-Bt).\end{aligned}$$ $${\overline{\mathbb{P}}}_{\lambda}\left( t'(x)\ge \frac{\\|x\\|}c+t\right)\le {\overline{\mathbb{P}}}_{\lambda}\left( \sigma(x)\ge \frac{\\|x\\|}c+t/2\right)+{\overline{\mathbb{P}}}_{\lambda}(t'(x)-\sigma(x)\ge t/2).$$ The second term is controlled by Inequality and Theorem \[theomomexpsigma\] gives the desired result with $c=\frac{\gamma_1}{\beta_1}$. The rest of this section is organized as follows. We first prove how the subadditive properties and the existence of exponential moments for $\sigma$ given by Theorem \[theomomexpsigma\] imply the large deviations inequalities of Theorem \[theoGDUQ\]. Next we show how Theorem \[lemme-pointssourcescontact\] gives Theorem \[theomomexpsigma\]. Finally, the last (and most important) part will be devoted to the proof of Theorem \[lemme-pointssourcescontact\]. Proof of Theorem \[theoGDUQ\] from Theorem \[theomomexpsigma\] -------------------------------------------------------------- Let ${\varepsilon}>0$. Let $\beta_1$ and $\gamma_1$ be the constants given by (\[momexpsigma\]), and let $$C>2 \beta_1/\gamma_1. \label{jechoisisC}$$ Theorem \[thFA\] gives the almost sure convergence of $\sigma(x)/\mu(x)$ to $1$ when $\\|x\\|$ tends to $+\infty$, and Proposition \[propmoments\] ensures that the family $(\sigma(x)/\mu(x))_{x \in {\mathbb{Z}^d}}$ is bounded in $L^2({\overline{\mathbb{P}}})$, therefore uniformly integrable: then the convergence also holds in $L^1({\overline{\mathbb{P}}})$. Let then $M_0$ be such that $$\label{je choisisM0} ( \mu(x) \ge M_0) \quad \Rightarrow \quad \left( \frac{{\overline{\mathbb{E}}}(\sigma(x))}{\mu(x)}\right) \le 1+{\varepsilon}/8.$$ We assumed that $\{ay: \;a\in{\mathbb{R}}_+,y\in\operatorname{Erg}(\nu)\}$ is dense in ${\mathbb{R}^d}$. Its range by $x\mapsto \frac{x}{\mu(x)}$ is therefore dense in $\{x\in{\mathbb{R}^d}: \;\mu(x)=1\}$, thus the set $\{\frac{y}{\mu(y)}: \;y\in\operatorname{Erg}(\nu), \,\mu(y)\ge M_0\}$ is also dense in $\{x\in{\mathbb{R}^d}:\;\mu(x)=1\}$. By a compactness argument, one can find a finite subset $F$ in $\{\frac{y}{\mu(y)}:\;y\in\operatorname{Erg}(\nu), \,\mu(y)\ge M_0\}$ such that $$\forall \hat{x}\in {\mathbb{R}^d}\text{ such that } \mu(\hat{x})=1 \quad\exists y\in F, \; \left\\|\frac{y}{\mu(y)}-\hat{x}\right\\|_1\le {\varepsilon}/C.$$ We let $M=\max\{\mu(y):\;y\in F\}.$ For $y\in F$, note $\tilde{\sigma}(y)=\sigma(y)-(1+\frac{{\varepsilon}}4)\mu(y)$. Since, with (\[momexpsigma\]), $\tilde{\sigma}(y)$ admits exponential moments, the asymptotics ${\overline{\mathbb{E}}}[e^{t\tilde{\sigma}(y)}]=1+t{\overline{\mathbb{E}}}[\tilde{\sigma}(y)]+o(t)$ holds in the neighborood of $0$. Since ${\overline{\mathbb{E}}}[\tilde{\sigma}(y)]<0$, we have ${\overline{\mathbb{E}}}[e^{t\tilde{\sigma}(y)}]<1$ when $t$ is small enough. Since $F$ is finite, we can find some constants $\alpha>0$ and $c_\alpha<1$ such that $$\begin{aligned} \forall y\in F && {\overline{\mathbb{E}}}[ \exp(\alpha (\sigma(y)-(1+{\varepsilon}/4)))]\le c_{\alpha}.\end{aligned}$$ Let $x\in{\mathbb{Z}^d}$. We associate to $x$ a point $y\in F$ and an integer $n$ such that $$\label{jechoisisxn} \left\\| \frac{x}{\mu(x)}-\frac{y}{\mu(y)}\right\\|_1\le \frac{{\varepsilon}}C \text{ and } \left\|n-\frac{\mu(x)}{\mu(y)}\right\|\le 1.$$ By the definition of $t(x)$, for each $\lambda \in \Lambda$, we have $$\begin{aligned} && {\overline{\mathbb{P}}}_{\lambda} \left( t(x) \ge (1+{\varepsilon})\mu(x) \right) \nonumber \\ & \le & {\overline{\mathbb{P}}}_{\lambda} \left( \sum_{i=0}^{n-1} \sigma(y)\circ\tilde{\theta}_y^i + \sigma(x-ny)\circ\tilde{\theta}_y^n \ge (1+{\varepsilon})\mu(x) \right) \nonumber \\ & \le & {\overline{\mathbb{P}}}_{\lambda} \left( \sum_{i=0}^{n-1} \sigma(y)\circ\tilde{\theta}_y^i \ge \left(1+\frac{{\varepsilon}}2 \right)\mu(x)\right) + {\overline{\mathbb{P}}}_{\lambda} \left(\sigma(x-ny)\circ\tilde{\theta}_y^n \ge \frac{{\varepsilon}}2 \mu(x) \right). \label{deuxtermes}\end{aligned}$$ Let first consider the first term in (\[deuxtermes\]). With Proposition \[invariancePbarre\] and estimate (\[momexpsigma\]), it follows that $$\begin{aligned} {\overline{\mathbb{P}}}_{\lambda} \left(\sigma(x-ny)\circ\tilde{\theta}_y^n \ge \frac{{\varepsilon}}2 \mu(x) \right) & = & {\overline{\mathbb{P}}}_{ny. \lambda} \left(\sigma(x-ny) \ge \frac{{\varepsilon}}2 \mu(x) \right) \\ & \le & \exp\left( -\frac{\gamma_1 {\varepsilon}\mu(x)}2 \right) {\overline{\mathbb{E}}}_{ny.\lambda}( \exp(\gamma_1 \sigma(x-ny)))\\ & \le & \exp\left( -\frac{\gamma_1 {\varepsilon}\mu(x)}2 \right) \exp(\beta_1 \\|x-ny\\|_1).\end{aligned}$$ Our choices (\[jechoisisxn\]) for $y$ and $n$ and the definition of $M$ ensure that $$\\|x-ny\\|_1\le \left\\| x -\frac{\mu(x)}{\mu(y)} y \right\\|_1+\left\| \frac{\mu(x)}{\mu(y)} - n \right\| \\|y\\|_1 \le \frac{{\varepsilon}\mu(x)}C+M.$$ Our choice (\[jechoisisC\]) for $C$ gives then the existence of two positive constants $A_1$ and $B_1$ such that for each $\lambda \in \Lambda$ and each $x \in {\mathbb{Z}^d}$, $${\overline{\mathbb{P}}}_{\lambda}\left(\sigma(x-ny)\ge \frac{{\varepsilon}}{2}\mu(x)\right)\le A_1\exp(-B_1\\|x\\|).$$ Let us move to the first term of (\[deuxtermes\]). Our choices (\[jechoisisxn\]) for $y$ and $n$ ensure that $$\left\| \frac{\mu(x)}{n \mu(y)}-1\right\| \le \frac1n \le \left(\frac{\mu(x)}{M}-1\right)^{-1}.$$ Then, we can find $T$ sufficiently large to have, for $\mu(x) \ge T$, that $$\frac{\mu(x)}{n \mu(y)} \ge \frac{1+{\varepsilon}/4}{1+{\varepsilon}/2}.$$ Suppose now that $\mu(x) \ge T$. Proposition \[invariancePbarre\] ensures that the variables $\sigma(y)\circ\tilde{\theta}_y^i$ are independent under ${\overline{\mathbb{P}}}_\lambda$ and moreover that the law of $\sigma(y)\circ\tilde{\theta}_y^i$ under ${\overline{\mathbb{P}}}_\lambda$ coincides with the law of $\sigma(y)$ under ${\overline{\mathbb{P}}}_{iy.\lambda}$ : thus $$\begin{aligned} &&{\overline{\mathbb{P}}}_{\lambda} \left( \sum_{i=0}^{n-1} \sigma(y)\circ\tilde{\theta}_y^i \ge \left(1+\frac{{\varepsilon}}2 \right)\mu(x)\right)\\& \le & {\overline{\mathbb{P}}}_{\lambda} \left( \sum_{i=0}^{n-1} \sigma(y)\circ\tilde{\theta}_y^i\ge (1+\frac{{\varepsilon}}4) n\mu(y)\right) \\ &\le & {\overline{\mathbb{P}}}_{\lambda} \left( \prod_{i=0}^{n-1} \exp \left( \alpha [\sigma(y)\circ\tilde{\theta}_y^i - (1+\frac{{\varepsilon}}4)\mu(y)]\right) \ge 1 \right) \\ & \le & \prod_{i=0}^{n-1} {\overline{\mathbb{E}}}_{iy.\lambda} \left[ \exp \left( \alpha(\sigma(y)-(1+\frac{{\varepsilon}}4)\mu(y)) \right) \right].\end{aligned}$$ Applying the Ergodic Theorem to the system $(\Lambda,\mathcal{B}(\Lambda),\nu, y.)$ and to the function $\lambda \mapsto \log {\overline{\mathbb{E}}}_{\lambda} \left( \exp[\alpha(\sigma(y)-(1+{\varepsilon}/4)\mu(y))] \right)$, we get that for $\nu$-almost every $\lambda$ and for each $y\in F$, $$\begin{aligned} && {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ {\overline{\lim}}\\ {\scriptstyle n\to+\infty} \end{array} \renewcommand{\arraystretch}{1}} \frac1{n}\log {\overline{\mathbb{P}}}_{\lambda} \left( \frac{1}{n\mu(y)} \sum_{i=0}^{n-1} \sigma(y)\circ\tilde{\theta}_y^i \ge 1+{\varepsilon}/4 \right) \\ & \le & \int_{\Lambda} \log {\overline{\mathbb{E}}}_{\lambda} \left(\exp[\alpha(\sigma(y)-(1+{\varepsilon}/4)\mu(y))] \right) d\nu(\lambda)\\ & \le & \log \int_{\Lambda} {\overline{\mathbb{E}}}_{\lambda} \left( \exp[\alpha(\sigma(y)-(1+{\varepsilon}/4)\mu(y))]\right) d\nu(\lambda) \le \log c_{\alpha}<0.\end{aligned}$$ Using the norm equivalence theorem and noting that the choices (\[jechoisisxn\]) for $n$ and $y$ ensure that $$\frac{n}{\mu(x)} \le \frac{1}{M} +\frac{1}{T},$$ we deduce that $${\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ {\overline{\lim}}\\ {\scriptstyle \\|x\\|\to +\infty} \end{array} \renewcommand{\arraystretch}{1}}\frac{\log {\overline{\mathbb{P}}}_{\lambda}(t(x)\ge\mu(x)(1+{\varepsilon}))}{\\|x\\|}\le -C_{{\varepsilon}},$$ with $C_{{\varepsilon}}=\max(-\log c_{\alpha},B_1)$. Inequality (\[venus\]) of Theorem \[theoGDUQ\] follows (with another $C_{{\varepsilon}}$, if necessary). Let us move to the proof of inequality (\[tcouple\]) of Theorem \[theoGDUQ\]. Let $T= \sum_{i=0}^{n-1} \sigma(y)\circ\tilde{\theta}_y^i + \sigma(x-ny)\circ\tilde{\theta}_y^n$. Using Corollary \[invariancePbarre\] repeatedly, the same reasoning as in the proof of Lemma \[momtprime\] gives ${\overline{\mathbb{P}}}_{\lambda}(t'(x)>T+{\varepsilon}\mu(x))\le {\overline{\mathbb{P}}}_{x.\lambda}(0\not\in K'_{{\varepsilon}\mu(x)})\le A\exp(-B\mu(x))$. Thus, since ${\overline{\mathbb{P}}}_{\lambda}(t'(x)>(1+2{\varepsilon})\mu(x))\le {\overline{\mathbb{P}}}_{\lambda}(T>(1+{\varepsilon})\mu(x))+{\overline{\mathbb{P}}}_{\lambda}(t'(x)>T+{\varepsilon}\mu(x))$ and $T$ has already been controlled, inequality follows. Let us prove inequality of Theorem \[theoGDUQ\]. Since $t\mapsto K'_t\cap H_t$ is non-decreasing, it is sufficient to prove that there exist constants $A,B>0$ such thay $$\forall n\in{\mathbb N}\quad {\overline{\mathbb{P}}}((1-{\varepsilon})nA_{\mu}\not\subset K'_n\cap H_n)\le A\exp(-Bn).$$ The proof of the last inequality is classic. For points that have a small norm, we use inequality and Corollary \[sauveur\]; for the other ones, we use inequalities and . Proof of Theorem \[theomomexpsigma\] from Theorem \[lemme-pointssourcescontact\] -------------------------------------------------------------------------------- Theorem \[lemme-pointssourcescontact\] ensures that with a probability exceeding $1-A\exp(-Bt)$, the Lebesgue measure of the times $s\le C\\|x\\|+t$ when $(0,0) \to (x,s) \to \infty$ is at least $\theta t$. If $\sigma(x)\ge C\\|x\\|_\infty+t$, it means that all these times are ignored by the recursive construction of $\sigma(x)$: those times necessarily belong to ${\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle K(x)-1}\\ \cup\\ {\scriptstyle i=1} \end{array} \renewcommand{\arraystretch}{1}}[u_k(x),v_k(x)] $. Thus, we choose $\theta,C$ as in Theorem \[lemme-pointssourcescontact\] and get $$\begin{aligned} && {\overline{\mathbb{P}}}_{\lambda}(\sigma(x)\ge C\\|x\\|+t) \\ & \le &{\overline{\mathbb{P}}}_{\lambda} \left( \{s\le C\\|x\\|+t:\; (0,0)\to (x,s)\to\infty\}\subset{\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle K(x)-1}\\ \cup\\ {\scriptstyle i=1} \end{array} \renewcommand{\arraystretch}{1}}[u_k(x),v_k(x)]\right)\\ & \le & {\overline{\mathbb{P}}}_{\lambda}(\operatorname{Leb}(\{s\le C\\|x\\|+t:\; (0,0)\to (x,s)\to\infty\})\le\theta t)\\ && +{\overline{\mathbb{P}}}_{\lambda} \left({\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle K(x)-1}\\ \sum\\ {\scriptstyle i=1} \end{array} \renewcommand{\arraystretch}{1}}(v_k(x)-u_k(x))>\theta t \right).\end{aligned}$$ Lemma \[lemme-pointssourcescontact\] allows to control the first term. To control the second one with a Markov inequality, it is sufficient to prove the existence of exponential moments for $\displaystyle {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle K(x)-1}\\ \sum\\ {\scriptstyle i=1} \end{array} \renewcommand{\arraystretch}{1}}(v_k(x)-u_k(x))$. To do so, we apply the abstract restart Lemma \[restartabstrait\]. We define, for each subset $B$ in ${\mathbb{Z}^d}$, $F^B=0$ and $$\begin{aligned} T^B & = & \inf\{t >\tau^x: \; x \in \xi_t^B\}, \\ G^B & = & \tau^x.\end{aligned}$$ Estimate (\[uniftau\]) ensures that for each $\lambda \in \Lambda$, $${\mathbb{P}}_\lambda(T^B=+\infty)\ge {\mathbb{P}}_\lambda( \tau^x=+\infty)\ge \rho>0,$$ and estimate (\[retouche\]) ensures the existence of $\alpha>0$ and $c<1$ – that do not depend on $B$ – such that for each $\lambda \in \Lambda$, $${\mathbb{E}}_\lambda[\exp(\alpha G^B){1\hspace{-1.3mm}1}_{\{T^B<+\infty\}}]\le{\mathbb{E}}_\lambda[\exp(\alpha \tau^x) {1\hspace{-1.3mm}1}_{\{\tau^x<+\infty\}}] = {\mathbb{E}}_{x.\lambda}[\exp(\alpha \tau^0) {1\hspace{-1.3mm}1}_{\{\tau^0<+\infty\}}]\le c.$$ Then, with the notation of Lemma \[restartabstrait\], we have $$\begin{aligned} && {\mathbb{E}}_\lambda \left[ \exp \left( \alpha {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle K(x)-1}\\ \sum\\ {\scriptstyle i=1} \end{array} \renewcommand{\arraystretch}{1}}(v_k(x)-u_k(x)) \right) \right] = {\mathbb{E}}_\lambda \left[ \exp \left( \alpha {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle K(x)-1}\\ \sum\\ {\scriptstyle i=0} \end{array} \renewcommand{\arraystretch}{1}} \tau^x \circ T_k \right) \right] \le \frac{1}{1-c}.\end{aligned}$$ To conclude, we note, using (\[uniftau\]), that ${\overline{\mathbb{E}}}_\lambda(.) \le {\mathbb{E}}_\lambda(. )/\rho$. Proof of Theorem \[lemme-pointssourcescontact\] ----------------------------------------------- We will include in the contact process a block event percolation: sites will correspond to large blocks in ${\mathbb{Z}}^d \times [0,\infty)$, and the opening of the bonds will depend of the occuring of good events that we define now. ### Good events Let $C_1>0$ and $M_1>0$ be fixed.\ Let $I \in {\mathbb N}^$, $L \in {\mathbb N}^$ and $\delta>0$ such that $I \le L$ and $\delta<C_1L$. For $\bar{n_0} \in {\mathbb{Z}^d}$, $x_0,x_1\in [-L,L[^d$ and $u \in {\mathbb{Z}^d}$ such that $\\|u\\|_1\le1$, we define the following event: $$\begin{aligned} A(\bar{n_0},u,x_0,x_1) & = & A_{I,L,\delta}^{C_1,M_1}(\bar{n_0},u,x_0,x_1) \\ & = & \left\{ \begin{array}{c} \exists t \in [0,C_1L-\delta] \quad 2L\bar{n_0}+x_1 \in \xi_t^{2L\bar{n_0}+x_0+[-I,I]^d} \\ \omega_{2L\bar{n_0}+x_1}([t,t+\delta])=0\\ \exists s \in 2L(\bar{n_0}+u) +[-L,L]^d \quad s+[-I,I]^d \subset \xi^{2L\bar{n_0}+x_1}_{C_1L-t}\circ \theta_t \\ \bigcup_{t \in [0,C_1L]} \xi_t^{2L\bar{n_0}+[-L-I,I+L]^d} \subset 2L\bar{n_0}+[-M_1L,M_1L]^d \end{array} \right\}.\end{aligned}$$ We let then $T=C_1L$. When the event $A(\bar{n_0},u,x_0,x_1)$ occurs, we denote by $s(\bar{n_0},u,x_0,x_1)$ a point $s$ satisfying the last condition that defines the event. Else, we let $s(\bar{n_0},u,x_0,x_1)=\infty$. If this event occurs, then: - Starting from an area of size $I$ centered at a starting point $2L\bar{n_0}+x_0$ in the box with spatial coordinate $\bar{n_0}$, the process at time $T$ colonizes an area of size $I$ centered around the exit point $2L(\bar{n_0}+u)+s(\bar{n_0},u,x_0,x_1)$ in the box with spatial coordinate $\bar{n_0}+u$. - Moreover, the point $2L\bar{n_0}+x_1$ is occupied between time $0$ and time $T$ in a time interval with duration at least $\delta$. - The realization of this event only depends on what happens in the space-time box $(2L\bar{n_0}+[-M_1L,M_1L])\times[0,T]$. Let us give a summary of the different parameters: -------------------------- ------------------------------------------------------------------------- $L$ spatial scale of the macroscopic boxes $I$ size of the entrance area and of the exit area $(I\le L$) $T$ temporal size of the macroscopic boxes ($T=C_1L$) $\delta$ minimum duration for the infection of $x_1$ $\bar{n_0}$ macroscopical spatial coordinate (coordinate of the big box) $u$ direction of move ($\\|u\\|_1\le 1$) $x_0$ relative position of the entrance area in the box ($x_0\in [-L,L[^d$) $x_1$ relative position of the target point ($x_1\in [-L,L[^d$) $s(\bar{n_0},u,x_0,x_1)$ relative position of the exit area in the box with coordinate $(\bar{n_0}+u)$ ($s(\bar{n_0},u,x_0,x_1) \in [-L,L[^d$) -------------------------- ------------------------------------------------------------------------- \[bonev1\] We can find constants $C_1>0$ and $M_1>0$ such that we have the following property. For each $\varepsilon>0$, we can choose, in that specific order, two integers $I \le L$ large enough and $\delta>0$ small enough such that for every $\lambda\in \Lambda$, $\bar{n_0} \in {\mathbb{Z}^d}$, and each $u \in {\mathbb{Z}^d}$ with $\\|u\\|_1\le 1$, $$\forall x_0,x_1\in [-L,L[^d\quad {\mathbb{P}}_\lambda(A(\bar{n_0},u,x_0,x_1))\ge 1-\varepsilon.$$ Moreover, as soon as $\\|\bar{n_0}-\bar{n_0'}\\|_\infty \ge 2M_1+1$, for every $u,u',x_0,x_0',x_1$, $$\text{ the events } A(\bar{n_0},u,x_0,x_1) \text{ and } A(\bar{n_0}',u',x_0',x_1) \text{ are independent.}$$ Let us first note that $${\mathbb{P}}_\lambda(A(\bar{n_0},u,x_0,x_1))={\mathbb{P}}_{2L\bar{n_0}.\lambda}(A(0,u,x_0,x_1)),$$ which permits to assume that $\bar{n_0}=0$. Let ${\varepsilon}>0$ be fixed. We first choose $I$ large enough to have $$\label{choixI} \forall x \in {\mathbb{Z}^d}\quad {\mathbb{P}}_{\lambda_{\min}}(\tau^{x+[-I,I]^d}=+\infty) \ge 1-{\varepsilon}/4.$$ We let ${\varepsilon}'={\varepsilon}/(2I+1)^d$.\ By the FKG Inequality, ${\mathbb{P}}_{\lambda_{\min}}(\forall y\in [-I,I]^d, \tau^y=+\infty)>0$. Translation invariance gives then $${\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \lim\\ {\scriptstyle L\to +\infty} \end{array} \renewcommand{\arraystretch}{1}} {\mathbb{P}}_{\lambda_{\min}}(\exists n\in [0,L]: \; \forall y\in ne_1+[-I,I]^d, \tau^y=+\infty)=1.$$ Let then $L_1$ be such that $${\mathbb{P}}_{\lambda_{\min}}(\exists n\in [0,L]; \forall y\in ne_1+[-I,I]^d, \tau^y=+\infty)>1-\frac{{\varepsilon}'}{12}{\mathbb{P}}_{\lambda_{\min}}(\tau^0=+\infty).$$ By a time-reversal argument, we have for each $t>0$, $$\begin{aligned} & & {\mathbb{P}}_{\lambda_{\min}}(\exists n\in [0,L]:\; ne_1+[-I,I]^d\subset \xi^{{\mathbb{Z}^d}}_t)\\ & = & {\mathbb{P}}_{\lambda_{\min}}(\exists n\in [0,L]:\; \forall y\in ne_1+[-I,I]^d, \tau^y\ge t)> 1-\frac{{\varepsilon}'}{12}{\mathbb{P}}_{\lambda_{\min}}(\tau^0=+\infty).\end{aligned}$$ We have for each $t\ge 0$ and each $\lambda\in\Lambda$: $$\begin{aligned} & &{\mathbb{P}}_{x_1.\lambda}(\tau^{0}=+\infty,\; \forall n\in [0,L], \, 2Lu-x_1+ne_1+[-I,I]\not\subset\xi^{0}_t)\\ & \le & {\mathbb{P}}_{x_1.\lambda}(\forall n\in [0,L],\, 2Lu-x_1+ne_1+[-I,I]\not\subset\xi^{{\mathbb{Z}^d}}_t)\\ & & \quad \quad + {\mathbb{P}}_{x_1.\lambda}(\tau^{0}=+\infty,\,[-(I+4L),(I+4L)]^d\not\subset K'_t)\\ & \le & {\mathbb{P}}_{\lambda_{\min}}(\forall n\in [0,L], \, ne_1+[-I,I]\not\subset\xi^{{\mathbb{Z}^d}}_t)\\ & & \quad \quad + {\mathbb{P}}_{x_1.\lambda}(\tau^0=+\infty,\,[-(4L+I),(4L+I)]^d\not\subset K'_t).\end{aligned}$$ Let $C>0$ be large enough to satisfy properties and . Then, with , we can find $L_2\ge L_1$ such that for $L\ge L_2$ and $t\ge 5CL$, we have $${\overline{\mathbb{P}}}_{x_1.\lambda}(\exists n\in [0,L]; 2Lu-x_1+ne_1+[-I,I]\subset\xi^{0}_t)\ge 1-{\varepsilon}'/6.$$ Let $\delta>0$ such that $1-e^{-\delta}\le {\mathbb{P}}_{\lambda_{\min}}(\tau^0=+\infty){\varepsilon}'/6$ and $\delta<5CL$: if we let $$F_t=\big\{\omega_{0}([0,\delta])=0;\; \exists n\in [0,L], \, 2Lu-x_1+ne_1+[-I,I]\subset\xi^{0}_t\big\},$$ we also have, for each $\lambda\in\Lambda$ and each $t\ge 5CL$, that ${\overline{\mathbb{P}}}_{x_1.\lambda}(F_t)\ge 1-{\varepsilon}'/3$. Then, with Proposition \[magic\], one deduces that if $y\in x+[-I,I]^d$, then $${\overline{\mathbb{P}}}_{y.\lambda}(\sigma(x_1-y)\le 4CL, \; \tilde{\theta}_{x_1-y}^{-1} (F_{9CL-\sigma(x_1-y)})) \ge {\overline{\mathbb{P}}}_{y.\lambda}(\sigma(x_1-y)\le 4CL)(1-{\varepsilon}'/3).$$ Considering estimate (\[asigma\]), we can choose $L_3\ge L_2$ such that for $L\ge L_3$, we have $${\overline{\mathbb{P}}}_{y.\lambda}(\sigma(x_1-y)\le 4CL, \;\tilde{\theta}_{x_1-y}^{-1} (F_{9CL-\sigma(x_1-y)}))\ge 1-{\varepsilon}'/2.$$ Let $C_1=9C$. With and the definition of ${\varepsilon}'$, we get $${\mathbb{P}}_{\lambda}\left( \begin{array}{c} \exists t \in [0,C_1L-\delta]: \quad x_1 \in \xi_t^{x_0+[-I,I]^d} \\ \omega_{x_1}([t,t+\delta])=0\\ \exists s \in 2Lu +[-L,L]^d \quad s+[-I,I]^d \subset \xi^{x_1}_{C_1L-t}\circ \theta_t \end{array} \right)\ge 1-3{\varepsilon}/4.$$ Finally, one takes for $M$ the constant given by equation (\[richard\]) and lets $M_1=MC_1+2$. With (\[richard\]), we can find $L \ge L_3$ sufficiently large to have for each $\lambda \in \Lambda$: $$\label{choixM} {\mathbb{P}}_{\lambda_{\text{max}}}\left( \bigcup_{0 \le t \le C_1L} \xi^{[-L-I,L+I]^d}_t \subset [-M_1L,M_1L]^d \right)\ge 1-{\varepsilon}/4;$$ this fixes the integer $L$. The local dependence of the events comes from the third condition in their definition. This concludes the proof of the lemma. ### Dependent macroscopic percolation We fix $C_1,M_1$ given by Lemma \[bonev1\]. We choose $I \in {\mathbb N}^$, $L \in {\mathbb N}^$ and $\delta>0$ such that $I \le L$ and $\delta<C_1L$ and we let $T=C_1L$. Let $x$ in ${\mathbb{Z}^d}$ be fixed. We write $x=2L[x]+\{x\}$, with $\{x\}\in [-L,L[^d$ and $[x]\in{\mathbb{Z}^d}$. We will first, from the events defined in the preceding subsection, build a field $(^xW^n_{(\bar{k},u)})_{n \ge 0, \bar{k} \in {\mathbb{Z}^d}, \\|u\\|_1\le1}$. The idea is to construct a macroscopic oriented percolation on the bonds of ${\overrightarrow{\mathbb{E}}^d}\times{\mathbb N}^$, looking for the realizations, floor by floor, of translates of good events of type $A(.)$. We start from an area centered at $0$ in the box with coordinate $\bar{0}$; for each $u$ such that $\\|u\\|_1\le 1$, say that the bond between $(\bar{0},0)$ and $(u, 1)$ is open if $A(\bar{0},u,0,\{x\})$ holds; in that case we obtain an infected square centered at the exit point $s(\bar{0},u,0,\{x\})$; all bonds in this floor that are issued from another point than $\bar{0}$ are open, with fictive exit points equal to $\infty$. Then we move to the upper floor: for a box $(\bar{y},1)$, look if it contains exit points of bonds that were open at the preceding step. If it is the case, we choose one of these, denoted by $d^x_1(\bar{y})$, open the bond between $(\bar{y},1)$ and $(\bar{y}+u, 2)$ if $A(\bar{y},u,d^x_1(\bar{y}),\{x\})\circ \theta_{T}$ happens and close it otherwise; in the other case we open all bonds issued from that box, and so on for every floor. Precisely, we let $d^x_0(\bar{0})=0$ and also $d^x_0(\bar{y})=+\infty$ for every $\bar{y} \in{\mathbb{Z}^d}$ that differs from $0$. Then, for each $\bar{y}\in{\mathbb{Z}^d}$, each $u \in {\mathbb{Z}^d}$ such that $\\|u\\|_1\le1$ and for each $n\ge 0$, we recursively define: - If $d^x_n(\bar{y})=+\infty$, $^xW^n_{(\bar{y},u)}=1$. - Otherwise, $$\begin{aligned} ^xW^n_{(\bar{y},u)} & = & {1\hspace{-1.3mm}1}_{A(\bar{y},u,d^x_n(\bar{y}),\{x\})} \circ \theta_{nT}, \\ d^x_{n+1}(\bar{y}) & = & \min\{ s(\bar{y}+u,-u,d^x_n(\bar{y}+u),\{x\})\circ \theta_{nT}: \; \\|u\\|_1\le 1, \; d^x_n(\bar{y}+u)\ne+\infty \}.\end{aligned}$$ Recall that the definition of the function $s$ has been given with the one of a good event in the preceding subsection. Then, $d^x_{n+1}(\bar{y})$ represents the relative position of the entrance area for the $^xW^{n+1}_{(\bar{y},u)}$’s, with $\\|u\\|_1\le1$. We may have several candidates, that are the relative positions of the exit areas of the $^xW^n_{(\bar{y}+u,-u)}$’s; the $\min$ only plays the role of a choice function. We thus obtain an oriented percolation process. Among open bonds, only those corresponding to the realization of good events are relevant for the underlying contact process. Let us note however that the percolation cluster starting at $\bar{0}$ only contains bonds that correspond to the propagation of the contact process. \[domistoc\] Again, we work with $C_1,M_1$ given by Lemma \[bonev1\]. For each $q<1$, we can choose parameters $I,L, \delta$ such that for each $\lambda \in \Lambda$, and each $x \in {\mathbb{Z}^d}$, $$\text{the law of }(^xW_e^n)_{(e,n) \in{\overrightarrow{\mathbb{E}}^d}\times{\mathbb N}^}\text{under }{\mathbb{P}}_{\lambda}\text{ is in }\mathcal{C}(2M_1+1,q).$$ For each $n \in {\mathbb N}$, let $\mathcal G_n=\mathcal F_{nT}$, with $T=C_1L$. Let us note that, for each $x,\overline{k} \in {\mathbb{Z}^d}$ and $n \ge 1$, the quantity $d^x_n(\overline{k})$ is $\mathcal{G}_n$-mesurable, and so does ${}^xW^n_{(\overline{k},u)}$. Lemma \[bonev1\] ensures that the events $A(\bar{k},u,x_0,\{x\})$ and $A(\bar{l},v,x_0',\{x\})$ are independent as soon as $\\|\bar{k}-\bar{l}\\|_1 \ge 2M_1+1$; so we deduce that, conditionally to $\mathcal G_n$, the random variables ${}^xW^{n+1}_{(\overline{k},u)}$ and ${}^xW^{n+1}_{(\overline{l},v)}$ are independent as soon as $\\|\overline{k}-\overline{l}\\|_1 \ge 2M_1+1$. Let now $x,\overline{k} \in {\mathbb{Z}^d}$, $n \ge 0$ and $u \in {\mathbb{Z}^d}$ such that $\\|u\\|_1\le 1$: $$\begin{aligned} && {\mathbb{E}}_{\lambda}[{}^xW^{n+1}_{(\overline{k},u)}\|\mathcal{G}_n\vee \sigma({}^xW^{n+1}_ {(\overline{l},v)}, \; \\|v\\|_1\le 1, \; \\|\overline{l}-\overline{k}\\|_1 \ge 2M_1+1)] \\ & = & {\mathbb{E}}_{\lambda}[{}^xW^{n+1}_{(\overline{k},u)}\|\mathcal{G}_n] \\ & = & {1\hspace{-1.3mm}1}_{\{d^x_n(\overline{k})=+\infty\}}+ {1\hspace{-1.3mm}1}_{\{d^x_n(\overline{k})<+\infty\}}{\mathbb{P}}_{\lambda}[{}^xW^{n+1}_{(\overline{k},u)}=1\|d^x_n(\overline{k})<+\infty] \\ & = & {1\hspace{-1.3mm}1}_{\{d^x_n(\overline{k})=+\infty\}}+ {1\hspace{-1.3mm}1}_{\{d^x_n(\overline{k})<+\infty\}}{\mathbb{P}}_{\lambda}[A(\overline{k},u,d^x_n(\overline{k}), \{x\})].\end{aligned}$$ With Lemma \[bonev1\], we can choose integers $I <L$ and $\delta>0$ in such a way that $${\mathbb{E}}_{\lambda}[{}^xW^{n+1}_{(\overline{k},u)}\|\mathcal{G}_n\vee \sigma({}^xW^{n+1}_ {(\overline{l},v)}, \; \\|v\\|_1\le 1, \; \\|\overline{l}-\overline{k}\\|_1 \ge 2M_1+1)] \ge q.$$ This concludes the proof of the lemma. For this percolation process, we denote by $\overline{\tau}^{\bar{k}}$ and $\overline{\gamma}(\theta,\bar{k},\bar{l})$ the lifetime starting from $\bar{k}$ and the essential hitting times of $\bar{l}$ starting from $\bar{k}$ in the dependent oriented percolation induced by the Bernoulli random field $(^xW_e^n)_{(e,n) \in{\overrightarrow{\mathbb{E}}^d}\times{\mathbb N}^}$. \[controledep\] We can choose the parameters $I,L,\delta$ such that the following holds: - $\forall \lambda\in\Lambda\quad{\mathbb{P}}_{\lambda}(\overline{\tau}^0=+\infty)\ge \frac12$. - $\forall \lambda\in\Lambda\quad {\varphi}(\lambda)={\mathbb{E}}_{\lambda}[e^{\alpha \overline{\tau}^0}{1\hspace{-1.3mm}1}_{\{\overline{\tau}^0<+\infty\}}] \le 1/2$ - there exist strictly positive constants $\alpha_0>0,\overline{C}$ such that for every $x,y\in{\mathbb{Z}^d}$ $$\forall \alpha\in [0,\alpha_0]\quad\forall \lambda\in\Lambda\quad \ell(\lambda,\alpha,x,y)={\mathbb{E}}_{\lambda} [{1\hspace{-1.3mm}1}_{\{\overline{\tau}^{x}=+\infty\}} e^{\alpha \overline{\gamma}(\theta,x,y))}]\le 2 e^{\overline{C}\alpha \\|x-y\\|}.$$ By Lemma \[petitmomentexpo\], we know that there exist $q<1$ and $\alpha>0$ such that we have $${\mathbb{E}}[e^{\alpha \overline{\tau}^{\bar{0}}}{1\hspace{-1.3mm}1}_{\{\overline{\tau}^{\bar{0}}<+\infty\}}]\le 1/2$$ for each field in $\mathcal{C}(2M_1+1,q)$. By Lemma \[domistoc\], we can choose $I,L,\delta$ such that $(^xW_e^n)_{(e,n) \in{\overrightarrow{\mathbb{E}}^d}\times{\mathbb N}^} \in \mathcal{C}(2M_1+1,q),$ which gives the two first points. Then, from Lemma \[lineairegamma\], we get constants $A,B,C$ such that for every $x,y\in{\mathbb{Z}^d}$, every $n\ge 0$ and each $\lambda\in\Lambda$, we have $${\mathbb{P}}_{\lambda}(+\infty>\overline{\gamma}(\theta,x,y)> C\\|x-y\\|_1+n)\le Ae^{-Bn}.$$ We can then find $B'>0$ independent from $x$ and $\lambda$ such that the Exponential law with parameter $B'$ stochastically dominates $(\overline{\gamma}(\theta,x,y)-C\\|x-y\\|_1){1\hspace{-1.3mm}1}_{\{\overline{\gamma}(\theta,x,y)<+\infty\}}$. Let then $\alpha\le B'/2$: we have $$\begin{aligned} \ell(\lambda,\alpha,x,y) & = &e^{\alpha C\\|x-y\\|_1}{\mathbb{E}}_{\lambda}[{1\hspace{-1.3mm}1}_{\{\overline{\tau}^x=+\infty\}}e^{\alpha ((\overline{\gamma}(\theta,x,y)-C\\|y-x\\|_1))}]\\ & \le & e^{\alpha C\\|x-y\\|_1} \frac{B'}{B'-\alpha}\\ & \le & 2 e^{\alpha C\\|x-y\\|_1}.\end{aligned}$$ ### Proof of Theorem \[lemme-pointssourcescontact\] We first choose $I,L, \delta$ in order to satisfy the inequalities of Lemma \[controledep\], and we let $T=C_1L$. We use a restart argument. The idea is as follows: fix $\lambda \in \Lambda$ and $x \in {\mathbb{Z}^d}$; if the lifetime $\tau^0$ of the contact process in random environment is infinite, then one can find by the restart procedure an instant $T_K$ such that - $\xi^0_{T_K}$ contains an area $z+[-2L,2L]^d$, which allows to activate a block oriented percolation, as defined in the previous subsection, from some $\bar{z}_0\in {\mathbb{Z}^d}$ such that $2\bar{z}_0L+[-L,L]^d \subset z+[-2L,2L]^d$, - the block oriented percolation issued from $\bar{z}_0$ infinitely survives. Then, with Lemma \[lineairegamma\], we give a lower bound for the proportion of time when $\bar{x}_0=[x]$ is occupied by descendents having themselves infinite progeny. By the definition of good events, this will allow to bound from below the measure of $\{t\ge 0; (0,0)\to (x,t)\to \infty\}$ in the contact process. Indeed, recall that the definition of the event $A(\bar{x}_0,u,x_0,\{x\})$ targets $\{x\}$ and ensures that each time the site $\bar{x}_0=[x]$ is occupied in the macroscopic oriented percolation, then the contact process occupies the site $2L\bar{x}_0+\{x\}=x$ during $\delta$ units of time. We define the following stopping times: for each non-empty subset $A\subset{\mathbb{Z}^d}$, $$\begin{aligned} U^A & = & \left\{ \begin{array}{l} T \text{ if } \forall z \in {\mathbb{Z}^d}\; z+[-2L,2L]^d \not\subset \xi^A_T, \\ T(1+\bar{\tau}^0 \circ T_{2\bar{x}^AL} \circ \theta_{T}) \text{ otherwise}\\ \quad \quad \quad \text{with } \bar{x}^A=\inf\{\bar{m} \in {\mathbb{Z}^d}: \; 2\bar{m}L+[-L,L]^d\subset \xi^A_T\} \end{array} \right. \\ \text{and }U^\varnothing & = & +\infty.\end{aligned}$$ In other words, starting from a set $A$, we ask if the contact process contains an area in the form $2\bar{m}L+[-L,L]^d$ at time $T$, : if the answer is no, we stop, otherwise we consider the lifetime of the macroscopic percolation issued from the macroscopic site corresponding to that area. Particularly, if $A \neq \varnothing$ and $U^A=+\infty$, then there exists, at time $T$, in the contact process issued from $A$, an area $2\bar{x}^AL+[-L,L]^d$ which is fully occupied, and such that the macroscopic oriented percolation issued from thae macroscopic site $\bar{x}^A$ percolates. We then search in that infinite cluster not too large a time when the proportion of individuals living at $\bar{x}_0=[x]$ and having infinite progeny becomes sufficiently large: if $A \neq \varnothing$ and $U^A=+\infty$, we note $$R^A=R^A(x)= \left\{ \begin{array}{ll} T(1+\overline{\gamma}(\theta,\bar{x}^A,\bar{x}_0)) & \text{ if }A \neq \varnothing \text{ and }U^A=+\infty; \\ 0 & \text{ otherwise }. \end{array} \right.$$ Thus, when $U^A=+\infty$, the variable $R^A$ represents the first time (in the scale of the contact process, not that of the macroscopic oriented percolation) when the proportion of individuals living at $\bar{x}_0=[x]$ and having infinite progeny becomes sufficiently large. There exist constants $\alpha>0$, $q>0$, $c<1$, $A',h>0$ such that for each $\lambda \in \Lambda$, each $A\subset {\mathbb{Z}^d}$, and each $x \in {\mathbb{Z}^d}$, $$\begin{aligned} {\mathbb{P}}_{\lambda}(U^A=+\infty) & \ge & q; \label{restart-q} \\ {\mathbb{E}}_{\lambda}[\exp(\alpha U^A){1\hspace{-1.3mm}1}_{\{U^A<+\infty\}}] & \le & c; \label{restart-c} \\ {\mathbb{E}}_{\lambda}[\exp(\alpha R^A(x)){1\hspace{-1.3mm}1}_{\{U^A=+\infty\}}] & \le & A'e^{\alpha h(\\|\bar{x}_0\\|_\infty+ \\|A\\|_\infty)}. \label{restart-A}\end{aligned}$$ We easily get (\[restart-q\]) from a stochastic comparison: for each $\lambda \in \Lambda$ and each non-empty $A$, $${\mathbb{P}}_{\lambda}(U^A=+\infty) \ge {\mathbb{P}}_{\lambda_{\text{min}}}([-2L,2L]^d \subset\xi^0_T){\mathbb{P}}(\bar{\tau}^0=+\infty)=q>0.$$ Now, if $\alpha>0$ , $A \subset {\mathbb{Z}^d}$ is non-empty and $\lambda \in \Lambda$, we have with Lemma \[controledep\], $$\begin{aligned} && {\mathbb{E}}_{\lambda}[\exp(\alpha U^A){1\hspace{-1.3mm}1}_{\{U^A<+\infty\}}] \\ & = & e^{\alpha T}\left( 1-{\mathbb{P}}_{\lambda}(\exists z \in {\mathbb{Z}^d}, \; z+[-2L,2L]^d \subset \xi^A_T)\left(1- {\mathbb{E}}[e^{\alpha T \bar{\tau}^0} {1\hspace{-1.3mm}1}_{\{\bar{\tau}^0<+\infty\}}]\right)\right) \\ & \le & e^{\alpha T}\left( 1-\frac12{\mathbb{P}}_{\lambda}(\exists z \in {\mathbb{Z}^d}, \; z+[-2L,2L]^d \subset \xi^A_T) \right) \\ & \le & e^{\alpha T}\left( 1-\frac12{\mathbb{P}}_{\lambda_{\text{min}}}(\exists z \in {\mathbb{Z}^d}, \; z+[-2L,2L]^d \subset \xi^A_T) \right)=c <1\end{aligned}$$ provided that $\alpha>0$ is small enough; this proves (\[restart-c\]). By the strong Markov property and Lemma \[controledep\], if we choose $\alpha>0$ small enough, then for each $\lambda \in \Lambda$, $$\begin{aligned} && {\mathbb{E}}_{\lambda} [ \exp(\alpha R^A) {1\hspace{-1.3mm}1}_{\{U^A=+\infty\}} \| \mathcal{F}_{T} ] \nonumber \\ & = & {1\hspace{-1.3mm}1}_{\{\exists z \in {\mathbb{Z}^d}, \; z+[-2L,2L]^d \subset \xi^A_T\}} e^{\alpha T} {\mathbb{E}}[ \exp(\alpha T \overline{\gamma}(\theta,\bar{x}^A,\bar{x}_0)){1\hspace{-1.3mm}1}_{\{\bar{\tau}^{\bar{x}^A}=+\infty\}}]\nonumber\\ & \le & 2 e^{\alpha T} \exp(\overline{C} \alpha T \\|\bar{x}^A-\bar{x}_0\\|_\infty) \nonumber \\ &\le & 2 e^{\alpha T(1+\overline{C}\\|\bar{x}_0\\|_\infty)}\exp(\overline{C} \alpha T \\|\xi^A_T\\|_\infty). \label{laun}\end{aligned}$$ We use the comparison with Richardson’s model to bound the mean of the last term: let us choose the positive constants $M,\beta$ such that $$\forall s,t\ge 0\quad {\mathbb{P}}_{\lambda_{\max}}(\\|\xi^0_s\\|_{\infty}\ge Ms+t)\le e^{-\beta t}.$$ Then, for each non-empty finite set $A$, each $t>0$, and each $\lambda \in \Lambda$, $$\begin{aligned} {\mathbb{P}}_{\lambda}(\\|\xi^A_T\\|_{\infty}\ge 2\\|A\\|_\infty + MT+t) & \le & {\mathbb{P}}_{\lambda_{\max}}(\max_{a \in A} \\|\xi^a_T -a \\|_{\infty}\ge \\|A\\|_\infty+MT+t) \\ & \le & {\vert A \vert} {\mathbb{P}}_{\lambda_{\max}}(\\|\xi^0_T\\|_{\infty}\ge MT+\\|A\\|_\infty+t)\\ & \le & \\|A\\|_\infty^d e^{-\beta (\\|A\\|_\infty+t)}\le \alpha' \exp(-\beta t). \end{aligned}$$ Then, for $\alpha$ small enough, $$\begin{aligned} {\mathbb{E}}_\lambda[\exp(\overline{C}\alpha T \\|\xi^A_T\\|_\infty) ] & \le & e^{\overline{C}\alpha T(2\\|A\\|_\infty + MT)} \left(1+\frac{\overline{C}\alpha T\alpha'}{\beta-\overline{C}\alpha T} \right) \nonumber \\ & \le & 2e^{\overline{C}\alpha T(2\\|A\\|_\infty + MT)}. \label{ladeux}\end{aligned}$$ Inequality (\[restart-A\]) immediately follows from (\[laun\]) and (\[ladeux\]). Let $$\begin{aligned} T_0=0 \text{ and } T_{k+1} & = & \begin{cases} +\infty & \text{if }T_k=+\infty\\ T_k+U^{\xi_0^{T_k}} \circ \theta_{T_k} & \text{otherwise;} \end{cases} \\ K & = & \inf\{k\ge 0:\;T_{k+1}=+\infty\}. $$ The restart lemma, applied with $T^.=G^.=U^.$ and $F^.=0$, ensures that $$\begin{aligned} {\mathbb{E}}_\lambda[\exp(\alpha T_K)] & \le & \frac{A'}{1-c}.\end{aligned}$$ Applying now the restart lemma with $G^.=0$ and $F^.=R^.$, we get that $$\begin{aligned} {\mathbb{E}}_\lambda[\exp(\alpha (R^{\xi_0^{T_K}} \circ\theta_{T_{K}}-(h\\|\bar{x}_0\\|_\infty+\\|\xi_0^{T_K}\\|_\infty )))] & \le & \frac{A'}{1-c} .\end{aligned}$$ Particularly, it holds that for each $s>0$ and $t>0$, $$\begin{aligned} {\mathbb{P}}_\lambda(T_K>s) & \le & \frac{A'}{1-c} \exp(-\alpha s); \label{queueTK}\\ {\mathbb{P}}_\lambda \left( \begin{array}{c} R^{\xi_0^{T_K} \circ\theta_{T_{K}}}\ge t/2, \\ T_K \le s, \; H^0_s \subset B^0_{Ms} \end{array} \right) & \le & \frac{A'}{1-c} \exp(\alpha (h (\\|\bar{x}_0\\|_\infty+Ms )- t/2)) \label{queueR}.\end{aligned}$$ On the event $\{\tau^0=+\infty\}$, one can be sure that the contact process is non-empty at each step of the restart procedure : the restart Lemma ensures that at time $T_K+T$, one can find some area from which the directed block percolation percolates, and, by construction, that for every $t \ge T_K+R^{\xi_0^{T_K} \circ\theta_{T_{K}}}$, $$\operatorname{Leb}(\{s \in [T_K+T,t]: \; (0,0) \to(x,s)\to \infty \})\ge \delta \theta \text{Int}(\frac{t-(T_K+T)}{T})\ge \frac{\delta \theta}{2T} t$$ as soon as $T_K\le t/2-1$. Let $C=\frac{2h}L$. Let now be $x \in {\mathbb{Z}^d}$, and $t \ge C\\|x\\|_\infty$. $$\begin{aligned} & & {\mathbb{P}}_\lambda \left(\tau^0=+\infty, \operatorname{Leb}(\{s \in [0,t]: \; (0,0) \to(x,s)\to \infty \})<\frac{\delta \theta}{2T} t \right) \\ & \le & {\mathbb{P}}_\lambda(T_K> t/2-1)+{\mathbb{P}}_\lambda(T_K\le t/2-1, \; t<T_K+R^{\xi_0^{T_K}} \circ\theta_{T_{K}})\\ & \le & {\mathbb{P}}_\lambda(T_K> t/2-1)+{\mathbb{P}}_\lambda(R^{\xi_0^{T_K}} \circ\theta_{T_{K}}>t/2).\end{aligned}$$ We control the first term with (\[queueTK\]). For the second one, we take $s=\frac{t}{8hM}$: $$\begin{aligned} &&{\mathbb{P}}_\lambda(R^{\xi_0^{T_K}} \circ\theta_{T_{K}}>t/2) \\ & \le & {\mathbb{P}}_\lambda(R^{\xi_0^{T_K} \circ\theta_{T_{K}}}> t/2, \; T_K \le s, \; H^0_s \subset B^0_{Ms}) + {\mathbb{P}}_\lambda(T_K > s)+{\mathbb{P}}_\lambda(H^0_s \not\subset B^0_{Ms}).\end{aligned}$$ We control the last two terms with (\[queueTK\]) and (\[richard\]); for the first one, we use (\[queueR\]): since $\\|\bar{x}_0\\|_\infty \le \frac1{2L} \\|x\\|_\infty+1$, $$\begin{aligned} {\mathbb{P}}_\lambda \left( \begin{array}{c} R^{\xi_0^{T_K} \circ\theta_{T_{K}}}> t/2, \\ T_K \le s, \; H^0_s \subset B^0_{Ms} \end{array} \right) & \le & \frac{A'}{1-c} \exp(\alpha (h (\\|\bar{x}_0\\|_\infty+Ms )- t/2)) \\ & \le & \frac{A'e^{\alpha h}}{1-c} \exp\left( \alpha \left( \left( \frac{h}{2L}\\|x\\|_\infty-\frac{t}4\right) -\frac{t}{8} \right) \right)\\ & \le & \frac{A'e^{\alpha h}}{1-c}\exp(-\alpha t/8), \end{aligned}$$ which concludes the proof. Lower large deviations ====================== Duality ------- We have seen that the hitting times $\sigma(nx)$ enjoy superconvolutive properties. In a deterministic frame, Hammersley [@MR0370721] has proved that the superconvolutive property allows to express the large deviation functional in terms of the moments generating function, as in Chernoff’s Theorem. We will see that this property also holds in an ergodic random environment. The following proof is inspired by Kingman [@MR0438477]. Since $\{t(x)\le t,\; \tau^x\circ \theta_{t(x)}=+\infty\}\subset\{\sigma(x)\le t\}\subset\{t(x)\le t\},$ the Markov property ensures that $${\overline{\mathbb{P}}}_{\lambda}(t(x)\le t){\mathbb{P}}_{\lambda}(\tau^x=+\infty)\le{\overline{\mathbb{P}}}_{\lambda}(\sigma(x)\le t)\le{\overline{\mathbb{P}}}_{\lambda}(t(x)\le t).$$ Thus, letting $R=-\log {\mathbb{P}}_{\lambda_{\min}}(\tau^0=+\infty)$, we have $$\label{equationunun} -\log({\overline{\mathbb{P}}}_{\lambda}(t(x)\le t))\le -\log({\overline{\mathbb{P}}}_{\lambda}(\sigma(x)\le t))\le -\log({\overline{\mathbb{P}}}_{\lambda}(t(x)\le t))+R.$$ Similarly, $${\mathbb{E}}_{\lambda}[e^{-t(x)}] \ge {\mathbb{E}}_{\lambda}[e^{-\theta\sigma(x)}] \ge {\mathbb{E}}_{\lambda}[{1\hspace{-1.3mm}1}_{\{\tau^x\circ\theta_{t(x)}=+\infty\}} e^{-\theta t(x)}]={\mathbb{E}}_{\lambda}[ e^{-\theta t(x)}]{\mathbb{P}}_{\lambda}(\tau^x=+\infty),$$ which leads to $$\label{equationquatrequatre} -\log {\overline{\mathbb{E}}}_{\lambda}[e^{-t(x)}] \le -\log {\overline{\mathbb{E}}}_{\lambda}[e^{-\theta\sigma(x)}] \le -\log {\overline{\mathbb{E}}}_{\lambda}[ e^{-\theta t(x)}]+R.$$ Then, having a large deviation principle in mind, working with $\sigma$ or $t$ does not matter. We will work here with $\sigma$, which gives simpler relations. We know that $$\\ \label{sousaddd} t((n+p)x)\le \sigma(nx)+\sigma(px)\circ\tilde{\theta}_{nx},$$ that $\sigma(nx)$ and $\sigma(px)\circ\tilde{\theta}_{nx}$ are independent under ${\overline{\mathbb{P}}}_{\lambda}$, and that the law of $\sigma(px)\circ\tilde{\theta}_{nx}$ under ${\overline{\mathbb{P}}}_{\lambda}$ is the law of $\sigma(px)$ under ${\overline{\mathbb{P}}}_{nx.\lambda}$ (see Proposition \[magic\]). Then $$\label{equationdeuxdeux} -\log {\overline{\mathbb{P}}}_{\lambda}(t((n+p)x)\le nu+pv)\le -\log {\overline{\mathbb{P}}}_{\lambda}(\sigma(nx)\le nu)-\log {\overline{\mathbb{P}}}_{nx.\lambda}(\sigma(px)\le pv).$$ Let $g_n^{x}(\lambda,u)=-\log {\overline{\mathbb{P}}}_{\lambda}(\sigma(nx)\le nu)+R\text{ and }G_n^{x}(u)=\int_{\Lambda} g^{x}_{n}(\lambda,u)\ d\nu(\lambda)$. Inequalities (\[equationunun\]) and (\[equationdeuxdeux\]) ensure that $$\label{equationtroistrois} g^{x}_{n+p}(\lambda,u)\le g^{x}_n(\lambda,u)+g^{x}_p( T_{x}^n\lambda,u).$$ Since $0\le g^{x}_1(\lambda,u) \le -\log {\overline{\mathbb{P}}}_{\lambda_{\min}}(\sigma(x)\le u)+R<+\infty$, Kingman’s subadditive ergodic theorem ensures that $\frac{g^{x}_{n}(u,\lambda)}{n}$ converges to $$\Psi_x(u)=\inf_{n\ge 1}\frac1{n} G^x_n(u)=\lim_{n\to+\infty} \frac1{n}G^x_n(u).$$ for $\nu$-almost every $\lambda$. Note that (\[equationtroistrois\]) ensures that for every $n,p\in{\mathbb N}$ and every $u,v>0$, $$\Psi_x\left(\frac{nu+pv}{n+p}\right)\le \frac1{n+p}G^x_{n+p}\left(\frac{nu+pv}{n+p}\right)\le \frac{n}{n+p} \frac{G^x_n(u)}{n}+\frac{p}{n+p} \frac{G^x_p(v)}{p}.$$ Let $\alpha \in ]0,1[$. Since $\Psi_x$ is non-increasing, considering some sequence $n_k,p_k$ such $\frac{n_k}{n_k+p_k}$ tends to $\alpha$ from above, we get $$\Psi_x(\alpha u+(1-\alpha)v)\le \alpha\Psi_x(u)+(1-\alpha)\Psi_x(v).$$ So $\Psi$ is convex. Similarly, let $h_n^{x}(\lambda,\theta)=-\log {\overline{\mathbb{E}}}_{\lambda}[e^{-\theta \sigma(nx)}]+R\text{ and }H^x_n({\theta})=\int h^{x}_{n}(\lambda,\theta)\ d\nu(\lambda)$. As previously, with (\[equationquatrequatre\]) and the subadditive relation (\[sousaddd\]), we have $$\begin{aligned} {\overline{\mathbb{E}}}_{\lambda}[e^{-\theta\sigma((n+p)x)}]&\ge & e^{-R}{\overline{\mathbb{E}}}_{\lambda}[ e^{-\theta t((n+p)x)}]\\ & \ge & e^{-R}{\overline{\mathbb{E}}}_{\lambda}[ e^{-\theta (\sigma(nx)+\sigma(px)\circ\tilde{\theta}_{nx})}] = e^{-R}{\overline{\mathbb{E}}}_{\lambda}[ e^{-\theta \sigma(nx)}]{\overline{\mathbb{E}}}_{\lambda}[e^{-\theta\sigma(px)}],\end{aligned}$$ and then the inequality $$h^{x}_{n+p}(\lambda,{\theta})\le h^{x}_n(\lambda,{\theta})+h^{x}_p (T_{x}^n\lambda,{\theta}).$$ Since $0\le h^{x}_1(\lambda,{\theta})\le -\log {\overline{\mathbb{E}}}_{\lambda_{\min}}[e^{-\theta\sigma(x)}]<+\infty$, Kingman’s subadditive ergodic theorem ensures that for $\nu$-almost every $\lambda$, $\frac{h^{x,{\theta}}_{n}(\lambda,{\theta})}{n}$ converges to $$K_x({\theta})=\inf_{n\ge 1}\frac1{n} H^x_n({\theta})=\lim_{n\to+\infty} \frac1{n}H^x_n({\theta}).$$ Let now $\theta\ge 0$ and $u>0$. By the Markov inequality, we observe that $$\begin{aligned} {\overline{\mathbb{P}}}_{\lambda}(\sigma(nx)\le nu)\le e^{\theta nu}{\overline{\mathbb{E}}}_{\lambda}[e^{-\theta\sigma(nx)}], & {\emph{i.e. }}& -g_n^{x}(.,u)\le\theta nu-h_n^{x,\theta}(.,u), \nonumber \\ & {\emph{i.e. }}& G_n^x(u)\ge -\theta nu +H_n(\theta), \nonumber \\ & {\emph{i.e. }}& \Psi_x(u)\ge-\theta u+K_x(\theta).\end{aligned}$$ Thus, we easily get $$\begin{aligned} \label{lune} \forall u>0\quad \Psi_x(u)& \ge& \sup_{\theta\ge 0}(K_x(\theta)-\theta u), \\ \label{lunea} \forall \theta>0 \quad K_x(\theta) & \le & \inf_{u>0} (\Psi_x(u)+\theta u).\end{aligned}$$ It remains to prove both reversed inequalities. Let us first prove $$\label{lautre} \forall \theta>0 \quad K_x(\theta)\ge \inf_{u>0} \{\Psi_x(u)+\theta u\}.$$ Let $\theta>0$. Define $M={\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \inf\\ {\scriptstyle u>0} \end{array} \renewcommand{\arraystretch}{1}} \{\Psi_x(u)+\theta u\}$ and note that for each $u$ and each integer $n$ $$G_n^x(u)+n\theta u\ge n\Psi_x(u)+n\theta u\ge nM.$$ Fix ${\varepsilon}>0$. Define $E_{n,{\varepsilon}}=\{\lambda: g_n^{x,u}(\lambda)\ge G^x_n(u)-n{\varepsilon}\}$. We have $$\begin{aligned} H^x_n(\theta) & \ge & \int_{E_{n,{\varepsilon}}} h^x_n(\theta)\ d\nu(\lambda) = \int_{E_{n,{\varepsilon}}} (R-\log {\overline{\mathbb{E}}}_{\lambda}[e^{-\theta \sigma(nx)})]\ d\nu(\lambda)\\ & = & \int_{E_{n,{\varepsilon}}} -\log \left[\int_0^{+\infty}n\theta e^{-\theta nu}e^{-R}{\overline{\mathbb{P}}}_{\lambda}(\sigma(nx)<nu)\ du\right]\ d\nu(\lambda). $$ For every $\lambda\in E_{n,{\varepsilon}}$ and $b>0$, one has $$\begin{aligned} \int_0^{+\infty} n\theta e^{-\theta nu}e^{-R}{\overline{\mathbb{P}}}_{\lambda}(\sigma(nx)<nu)\ du & \le & e^{-\theta n b}+ \int_0^{b} n\theta e^{-\theta nu}e^{-R}{\overline{\mathbb{P}}}_{\lambda}(\sigma(nx)<nu)\ du\\ & \le & e^{-\theta n b}+\int_0^{b} n\theta e^{-\theta nu}e^{-g_n^{x,u}(\lambda)}\ du\\ & \le & e^{-\theta n b}+\int_0^{b} n\theta e^{-\theta nu}e^{-G^x_n(u)+n{\varepsilon}}\ du\\ & \le & e^{-\theta n b}+n\theta be^{-n(M-{\varepsilon})}\\ & \le & (nM+1)e^{-n(M-{\varepsilon})}\quad\text{with }b=M/\theta.\end{aligned}$$ Finally, $$\frac{H^x_n(\theta)}{n} \ge \nu(E_{n,{\varepsilon}})\left(-\frac{\log(1+nM)}n+M-{\varepsilon}\right).$$ Since $\nu(E_{n,{\varepsilon}})$ tends to $1$ when $n$ goes to infinity, one deduces that $$K_x(\theta)=\lim\frac1{n}H^x_n(\theta)\ge M-{\varepsilon}.$$ Letting ${\varepsilon}$ tend to $0$, we get (\[lautre\]). Let us finally prove $$\label{lautreb} \forall u>0\quad \Psi_x(u) \le \sup_{\theta\ge 0}(K_x(\theta)-\theta u).$$ Let $u>0$. It is sufficient to prove that there exists $\theta_u\ge 0$, with $\Psi_x(u)\le -\theta_u u+K_x(\theta_u)$. Since $\Psi_x$ is convex and non-increasing, there exists a slope $-\theta_u\le 0$ such that $\Psi_x(v)\ge\Psi_x(u)-\theta_u(v-u)$. Then $$K_x(\theta_u)=\inf_{v>0} \{\Psi_x(v)+\theta_u v\} \ge \inf_{v>0} \{\Psi_x(u)-\theta_u(v-u)+\theta_u v\} \ge \Psi_x(u)+\theta_u u,$$ which completes the proof of (\[lautreb\]) and of the reciprocity formulas. The function $-K_x(-\theta)$ corresponds to $\Psi_x$ in the Fenchel-Legendre duality: therefore, it is convex. Particularly, the functions $\Psi_x$ and $K_x$ are continuous on $]0,+\infty[$. By the definition of $\Psi_x$ and $K_x$, there exists $\Lambda'\subset\Lambda$ with $\nu(\Lambda')=1$ and such that for each $u\in{\mathbb{Q}}\cap (0,+\infty)$ and each $\theta\in{\mathbb{Q}}\cap [0,+\infty)$, we have $$\begin{aligned} && \lim_{n\to +\infty} -\frac1{n}\log {\overline{\mathbb{P}}}_{\lambda}(\sigma(nx)\le nu) = \Psi_x(u), \\ \text{and } && \lim_{n\to +\infty} -\frac1{n}\log {\overline{\mathbb{E}}}_{\lambda}e^{-\theta\sigma(nx)} = K_x(\theta).\end{aligned}$$ Since the functions $\theta\mapsto h^{x,\theta}_n$ and $u\mapsto h^{x,u}_n$ are monotonic and their limits $\Psi_x$ and $K_x$ are continuous, it is easy to check that the convergences also hold for every $\lambda\in\Lambda'$, $u>0$ and $\theta\ge 0$. Lower large deviations ---------------------- We prove here Theorem \[dessouscchouette\]. Remember that ${{\mathbb{P}}}(.)=\int_\Lambda {{\mathbb{P}}}_\lambda(.)\ d\nu(\lambda)$. The main step is actually to prove the following: \[GDdessous\] Assume that $\nu=\nu_0^{\otimes {\mathbb{E}^d}}$ and that the support of $\nu_0$ is included in $[\lambda_{\text{min}}, \lambda_{\text{max}}]$. For every ${\varepsilon}>0$, there exist $A,B>0$ such that $$\forall t \ge 0 \quad {\mathbb{P}}(\xi^0_t \not \subset (1+{\varepsilon})t A_\mu)\le A\exp(-Bt).$$ Using the norm equivalence on ${\mathbb{R}^d}$, we introduce constants $C^-_\mu,C^+_\mu>0$ such that $$\label{normes} \forall z \in {\mathbb{R}^d}\quad C^-_\mu\\|z\\|_\infty \le \mu(z) \le C^+_\mu \\|z\\|_{\infty}.$$ Let $\alpha,L,N,{\varepsilon}>0$. We define the following event, relatively to the space-time box $B_N=B_N(0,0)=[-N,N]^d\times[0,2N]$: $$\begin{aligned} A^{\alpha,L,N,{\varepsilon}}= \left\{\forall (x_0,t_0) \in B_N\quad \xi^{x_0}_{\alpha L N-t_0}\circ \theta_{t_0} \subset x_0+(1+{\varepsilon})(\alpha L N-t_0)A_{\mu}\right\}\cap\\ \left\{\forall (x_0,t_0) \in B_N {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \cup\\ {\scriptstyle 0\le s\le\alpha L N-t_0} \end{array} \renewcommand{\arraystretch}{1}} \xi^{x_0}_{s}\circ \theta_{t_0} \subset ]-LN,LN[^d\right\}.\end{aligned}$$ The first part of the event ensures that the descendants, at time $\alpha L N$, of any point $(x_0,t_0)$ in the box $B_N$ are included in $x_0+(1+{\varepsilon})(\alpha L N)A_{\mu}$: it is a sharp control, requiring the asymptotic shape theorem. The second part ensures that the descendants, at all times in $[0,\alpha L N] $, of the whole box $B_N$ are included in $]-LN,LN[^d$: the bound is rough, only based on the (at most) linear growth of the process. We say that the box $B_N$ is good if $A^{\alpha,L,N,{\varepsilon}}$ occurs. We also define, for $k \in {\mathbb{Z}^d}$ and $n \in {\mathbb N}$, the event $A^{\alpha,L,N,{\varepsilon}}(k,n)=A^{\alpha,L,N,{\varepsilon}} \circ T_{2kN} \circ \theta_{2nN}$ and we say that the box $B_N(k,n)$ is good if the event $A^{\alpha,L,N,{\varepsilon}}(k,n)$ occurs. The proof of the lower large deviation inequalities is close to the one by Grimmett and Kesten [@grimmett-kesten] for first passage-percolation. If a point $(x,t)$ is infected too early, it means that its path of infection has “too fast” portions when compared with the speed given by the asymptotic shape theorem. For this path, we build a sequence of boxes associated with path portions, and the existence of a “too fast portion” forces the corresponding box to be bad. But we are going to see that we can choose the parameters to ensure that - the probability under ${\mathbb{P}}$ for a box to be good is as close to $1$ as we want, - the events “$B_N(k,0)$ is good” are only locally dependent. We then conclude the proof by a comparison with independent percolation with the help of the Liggett–Schonmann–Stacey Lemma [@LSS] and a control of the number of possible sequences of boxes. \[catendvers12\] We have - The events $(\{B_N(k,0) \text{ is good}\})_{k\in {\mathbb{Z}^d}}$ are identically distributed under ${\mathbb{P}}$. - There exists $\alpha>0$ such that for every ${\varepsilon}\in (0,1)$, there exists an integer $L$ (that can be taken as large as we want) such that $$\lim_{N \to +\infty} {\mathbb{P}}(A^{\alpha,L,N,{\varepsilon}})=1.$$ - If moreover $\nu=\nu_0^{\otimes {\mathbb{E}^d}}$, then the events $(\{B_N(k,0) \text{ is good}\})_{k\in {\mathbb{Z}^d}}$ are $(L+1)$-dependent under ${\mathbb{P}}$. The first and last points are clear. Let us prove the second point. The idea is to find a point $(0,-k)$, with $k$ large enough, such that - the descendants of $(0,-k)$ are infinitely many and behave correctly (without excessive speed) - the coupled region of $(0,-k)$ contains a set of points that is necessarily crossed by any infection path starting from the box $B_N$. Indeed, this will allow to find, for all the descendants of $B_N$, a unique common ancestor, and thus to control the growth of all the descendants of $B_N$ by simply controlling the descendants of this ancestor. A control on a number of points of the order of the volume of $B_N$ will thus be replaced by a control on a single point. See Figure \[uneautrefigure\]. Let ${\varepsilon}>0$ be fixed. We first control the positions of the descendants of the box $B_N$ at time $4N$. Let $A,B,M$ be the constants given by Proposition \[propuniforme\]. We recall that $\omega_x$, for $x\in {\mathbb{Z}^d}$, and $\omega_e$, for $e \in {\mathbb{E}^d}$ are the Poisson point processes giving respectively the death times for $x$ and the potential infection times through edge $e$. We define, for every integer $N$: $$\begin{aligned} \tilde{A}_1^N & = & \{H^0_{4N}\not \subset [-(4M+1)N,(4M+1)N]^d\}, \\ A_1^N & = & \left\{ \sum_{x \in [-N,N]^d}\int {1\hspace{-1.3mm}1}_{\{ \tilde{A}_1^N \circ T_x \circ \theta_t \} }\ d\left(\delta_0+\sum_{e \ni x}\omega_e\right)(t) =0 \right\}.\end{aligned}$$ Note in particular that $$\label{inclusion1} A_1^N \subset \left\{\forall (x_0,t_0) \in B_N\quad \xi^{x_0}_{4N-t_0}\circ \theta_{t_0} \subset [-(4M+1)N,(4M+1)N]^d \right\} .$$ We have with , $$\begin{aligned} && {\mathbb{E}}\left(\sum_{x \in [-N,N]^d}\sum_{e \ni x}\int_0^{2N} {1\hspace{-1.3mm}1}_{\{ \tilde{A}_1^N \circ T_x \circ \theta_t \} } \ d(\delta_0+\omega_e)(t) \right) \\ & \le & (2N+1)^d2d(1+2N\lambda_{\max}) {\mathbb{P}}_{\lambda_{\max}}(\tilde{A}_1^N) \\ & \le & (2N+1)^d2d(1+2N\lambda_{\max})A\exp(-4BN), \end{aligned}$$ and thus, with the Markov inequality, $$\label{probab1} \lim_{N\to +\infty} {\mathbb{P}}(A_1^N) =1.$$ With (\[inclusion1\]), we deduce that with a large probability, if $N$ is large enough, the descendants of $B_N$ at time $4N$ are included in $[-(4M+1)N,(4M+1)N]^d$. Now, we look for points with a good growth (we will look for the common ancestor of $B_N$ among these candidates): $$\begin{aligned} \tilde{A}_2^t & = & \{\tau^0=+\infty, \; \forall s\ge t\quad K'_s\supset (1-{\varepsilon})s A_{\mu}\text{ and } \xi^0_s\subset (1+{\varepsilon}/2)sA_{\mu}\}, \\ {A}_2^{t,N} & = & {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle N-1}\\ \cup\\ {\scriptstyle k=0} \end{array} \renewcommand{\arraystretch}{1}} \tilde{A}_2^{t}\circ \theta_{-k}.\end{aligned}$$ The first event says that the point $(0,0)$ lives forever and has a good growth after time $t$ (at most linear growth, and at least linear growth for its coupled zone), while the second event says that there exists a point $(0,-k)$ with a good growth and such that $k \in [0..N-1]$. Theorem 3 in Garet-Marchand [@GM-contact] ensures that ${\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \lim\\ {\scriptstyle t\to +\infty} \end{array} \renewcommand{\arraystretch}{1}}{\overline{\mathbb{P}}}(\tilde{A}_2^t)=1$. But $$\begin{aligned} {\mathbb{P}}(\tilde{A}_2^t) & = & \int {\mathbb{P}}_{\lambda}(\tilde{A}_2^t) d\nu(\lambda) = \int {\overline{\mathbb{P}}}_{\lambda}(\tilde{A}_2^t){\mathbb{P}}_{\lambda}(\tau^0=+\infty) d\nu(\lambda)\\ & \ge & \int {\overline{\mathbb{P}}}_{\lambda}(\tilde{A}_2^t){\mathbb{P}}_{\lambda_{\min}}(\tau^0=+\infty) d\nu(\lambda) \ge {\mathbb{P}}_{\lambda_{\min}}(\tau^0=+\infty){\overline{\mathbb{P}}}(\tilde{A}_2^t).\end{aligned}$$ So there exists $t_2$ such that ${\mathbb{P}}(\tilde{A}_2^{t_2})>0$. As the time translation $\theta_{-1}$ is ergodic under ${\mathbb{P}}$, we get $$\label{probab2} \lim_{N \to +\infty} {\mathbb{P}}\left( {A}_2^{t_2,N}\right)=\lim_{n\to +\infty}{\mathbb{P}}\left( {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle n-1}\\ \cup\\ {\scriptstyle k=0} \end{array} \renewcommand{\arraystretch}{1}} \tilde{A}_2^{t_2}\circ \theta_{-k}\right)=1.$$ In other words, with a large probability, if $N$ is large enough there exists $k \in [0..N-1]$ such that the point $(0,-k)$ has a good growth. ![Coupling from the past[]{data-label="uneautrefigure"}](couplage "fig:") (-230,10)[$-k$]{}(-223,35)[$0$]{}(-226,75)[$2N$]{}(-230,115)[$4N$]{}(-170,120)[$y$]{}(-150,50)[$x_0$]{}(-223,50)[$t_0$]{} (-155,3)[$0$]{}(-130,193)[$x$]{}(-226,193)[$t$]{} Take $L_1=L_1({\varepsilon})>0$ such that $$\label{inclusion}\forall N\ge 1\quad (L_1+1)N(1-{\varepsilon})A_{\mu}\supset [-(4M+1)N,(4M+1)N]^d.$$ Thus, if we find an integer $k\ge \max(t_2,L_1 N)$ such that $A_{t_2}\circ \theta_{-k}$ occurs, then the descendants of the box $B_N$ at time $4N$ are in the coupled region of $(0,-k)$. Denote by $\overleftarrow{\tau}^y$ the life time of $(y,0)$ for the contact process when we reverse time. As the contact process is self-dual, $\overleftarrow{\tau}^y$ as the same law as $\tau^y$. Set $$A_3^N=\left\{\forall y\in [-(4M+1)N,(4M+1)N]^d\quad \overleftarrow{\tau}^y\circ{\theta}_{4N}=+\infty \text{ or } \overleftarrow{\tau}^y\circ{\theta}_{4N}<2N\right\}.$$ The control (\[grosamasfinis\]) of large lifetimes ensures that $$\label{probab3} \lim_{N \to +\infty} {\mathbb{P}}(A_3^N)=1.$$ Assume now that $N\ge t_2/L_1$. Thus $L_1N\ge t_2$. Let us see that on $\displaystyle A_1^{N}\cap (A_2^{t_2,N}\circ \theta_{-L_1N}) \cap A_3^N$, we have $$\label{attrape} \forall t\ge 4N \quad {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \cup\\ {\scriptstyle (x_0,t_0)\in B_N} \end{array} \renewcommand{\arraystretch}{1}} \xi_{t-t_0}^{x_0}\circ\theta_{t_0} \subset ((L_1+1)N+t)(1+{\varepsilon}/2)A_{\mu}.$$ Indeed, let $t\ge 4N$ and $x\in {\mathbb{Z}^d}$ be such that $(x,t)$ is a descendant of $(x_0,t_0)\in B_N$. Let $(y,4N)$ be an ancestor of $(x,t)$ and a descendant of $(x_0,t_0)$. On the event $A_1^{N}$, the point $y$ is in $[-4MN,4MN]^d$. But, on $A_3^N$, the definition of $y$ ensures that $\overleftarrow{\tau}^y\circ{\theta}_{4N}=+\infty$: so $(y,4N)$ has a living ancestor a time $-k$, for each $k$ such that $L_1 N \le k \le (L_1+1) N-1$. But, on $A_2^{t_2,N}\circ \theta_{-L_1N}$, inclusion (\[inclusion\]) ensures that $(y,4N)$ is in the coupled region of $(0,-k)$ for one of these $k$, and so $(y,4N)$ is a descendant of this $(0,-k)$. Finally, $(x,t)$ is also a descendant of $(0,-k)$, and, always on $A_2^{t_2,N}\circ \theta_{-L_1N}$, $$\mu(x)\le (k+t)(1+{\varepsilon}/2)\le ((L_1+1)N-1+t)(1+{\varepsilon}/2),$$ which proves (\[attrape\]). We then choose $\alpha \in (0,1)$ and an integer $L$ such that $$\begin{aligned} \alpha & < & \frac{2C_\mu^-}{3} \le \frac{C_\mu^-}{1+{\varepsilon}/2}, \label{alpha} \\ L & \ge & \max\left\{ \frac4{\alpha}, \; \frac{L_1+1}{ C_\mu^--\alpha(1+{\varepsilon}/2)}, \; 4M+1, \; \frac{2}{\alpha {\varepsilon}}((L_1+1)(1+{\varepsilon}/2)+C_\mu^++2 \right\}. \label{L}\end{aligned}$$ If $N \ge t_2/L_1$, as $\alpha L N \ge 4N$, we can use (\[attrape\]) with $t\in[4N,\alpha LN]$; thus our choices for $\alpha,L$ and (\[inclusion1\]) ensure that on the event $\displaystyle A_1^{N}\cap (A_2^{t_2,N}\circ \theta_{-L_1N}) \cap A_3^N$, for every $ (x_0,t_0) \in B_N$ $$\begin{aligned} {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \cup\\ {\scriptstyle 4N\le s\le\alpha L N-t_0} \end{array} \renewcommand{\arraystretch}{1}} \xi^{x_0}_{s}\circ \theta_{t_0} \subset ((L_1+1+\alpha L)N)(1+{\varepsilon}/2)A_{\mu} & \subset& [-LN,LN]^d, \\ {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \cup\\ {\scriptstyle 0\le s\le\alpha 4N} \end{array} \renewcommand{\arraystretch}{1}} \xi^{x_0}_{s}\circ \theta_{t_0} \subset [-(4M+1)N,(4M+1)N]^d& \subset& [-LN,LN]^d, \\ \xi^{x_0}_{\alpha L N-t_0}\circ \theta_{t_0} \subset (L_1+1+\alpha L )N(1+{\varepsilon}/2)A_{\mu}& \subset & x_0+(1+{\varepsilon})(\alpha L N-t_0)A_{\mu}.\end{aligned}$$ Finally, if $N \ge t_2/L_1$, $$A_1^{N}\cap (A_2^{t_2,N}\circ \theta_{-L_1N}) \cap A_3^N \subset A^{\alpha,L,N,{\varepsilon}},$$ and we conclude with (\[probab1\]), (\[probab2\]) and (\[probab3\]). We first prove the existence of $C>0$ such that, with a large probability, the point $(0,0)$ can not give birth to more than $Ct$ generations before time $t$: \[histoirelineaire\] There $A,B,C>0$ such that for every $\lambda\in [0,\lambda_{\max}]^{{\mathbb{E}^d}}$, for every $t,\ell\ge 0$: $${\mathbb{P}}_\lambda \left( \begin{array}{c} \exists (x,s)\in {\mathbb{Z}^d}\times [0,t] \text{ and an infection path from }(0,0) \\ \text{ to } (x,s)\text{ with more than }Ct+\ell\text{ horizontal edges }\end{array} \right) \le A \exp(-B\ell).$$ Let $\alpha>0$ be fixed. For every path $\gamma$ in ${\mathbb{Z}^d}$ starting from $0$ and eventually self-intersecting, we set $$X_{\gamma}={1\hspace{-1.3mm}1}_{\{\gamma\text{ is the projection on ${\mathbb{Z}^d}$ of an infection path starting from }(0,0)\}}e^{-\alpha t(\gamma)},$$ where $t(\gamma)$ is the time when the extremity is infected after visiting successively the previous points. More formally, if the sequence of points in $\gamma$ is $(0=x_0,\dots,x_n)$ and if we set $T_0=0$, and for $k\in\{0,\dots,n-1\}$, $$T_{k+1}=\inf\left\{t>T_k; \omega_{\{x_k,x_{k+1}\}}([T_k,t])=1\text{ and }\omega_{x_k}([T_k,t])=0\right\},$$ we have $t(\gamma)=T_n$. The random variable $t(\gamma)$ is a stopping time (it is infinite if $\gamma$ is not the projection of an infection path). Let $\gamma$ be a path in ${\mathbb{Z}^d}$ starting from $0$ and let $f$ be an edge at the extremity of $\gamma$. If we denote by $\gamma.f$ the concatenation of $\gamma$ with $f$, the strong Markov property at time $t(\gamma)$ ensures that $${\mathbb{E}}_{\lambda} [X_{\gamma.f}\|\mathcal{F}_{t(\gamma)}]\le X_{\gamma} \frac{\lambda_{\max}}{\alpha+\lambda_{\max}}, \text{ and so } {\mathbb{E}}[X_{\gamma}]\le \left( \frac{\lambda_{\max}}{\alpha+\lambda_{\max}} \right)^{\|\gamma\|}.$$ Now, $$\begin{aligned} && {\mathbb{P}}_\lambda \left( \begin{array}{c} \exists (x,s)\in {\mathbb{Z}^d}\times [0,t] \text{ is an infection path from }(0,0) \\ \text{ to } (x,s)\text{ with more than }Ct+\ell\text{ horizontal edges}\end{array} \right) \\ & = & {\mathbb{P}}_\lambda \left({\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \cup\\ {\scriptstyle \gamma:\|\gamma\|\ge Ct+\ell} \end{array} \renewcommand{\arraystretch}{1}} \{X_{\gamma}\ge e^{-\alpha t}\} \right) \\ & \le & e^{\alpha t} {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \sum\\ {\scriptstyle \gamma:\|\gamma\|\ge Ct+\ell} \end{array} \renewcommand{\arraystretch}{1}} \left(\frac{\lambda_{\max}}{\alpha+\lambda_{\max}}\right)^{\|\gamma\|} \le e^{\alpha t} {\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ \sum\\ {\scriptstyle n\ge Ct+\ell} \end{array} \renewcommand{\arraystretch}{1}} \left(\frac{2d\lambda_{\max}}{\alpha+\lambda_{\max}}\right)^{n} .\end{aligned}$$ To conclude, we take $\alpha=2d\lambda_{\max}$, and then $C$ such that $(\frac{2d}{2d+1})^{C}=e^{-\alpha }$. Let ${\varepsilon}>0$ and $t>0$ be fixed. Obviously $$\begin{aligned} \label{majgross} {\mathbb{P}}(\xi^0_t \not \subset (1+{\varepsilon})tA_{\mu})&\le &{\mathbb{P}}(\xi^0_t \not \subset (1+{\varepsilon})tA_{\mu},\xi^0_t \subset [-Mt,Mt]^d)\\&&+{\mathbb{P}}(\xi^0_t \not \subset [-Mt,Mt]^d)\nonumber.\end{aligned}$$ The second term is controlled by equation Assume that $\xi^0_t \not \subset (1+{\varepsilon})tA_{\mu}$: let $x \in \xi^0_t$ be such that $\mu(x) \ge (1+{\varepsilon})t$, $\\|x\\|_{\infty}\le Mt$ and let $\gamma$ be an infection path from $(0,0)$ to $(x,t)$. With Lemma \[histoirelineaire\], we choose $C>1,A_2,B_2>0$ such that for every $t \ge 0$, $$\label{pastropdaretes} {\mathbb{P}}\left( \begin{array}{c} \text{there exists an infection path from } (0,0) \text{ to } {\mathbb{Z}^d}\times\{t\} \\ \text{with more than } Ct \text{ horizontal edges} \end{array} \right) \le A_2 \exp(-B_2t).$$ With the last estimate, we can assume that $\gamma$ has less than $Ct$ horizontal edges. We take $0<\alpha<1$ and $L=L(\alpha,{\varepsilon})$ large enough to apply Lemma \[catendvers12\] and such that $$\label{choiceL} \frac{4C_\mu^+C}{\alpha L-1}\le \frac{{\varepsilon}}3, \quad \alpha L\ge 2\quad \text{ and } L\ge 3. $$ We fix an integer $N$ and we cut the space-time ${\mathbb{Z}^d}\times {\mathbb{R}}_+$ into space-time boxes: $$\forall k \in {\mathbb{Z}^d}\quad \forall n \in {\mathbb N}\quad B_N(k,n)=(2Nk+[-N,N]^d)\times(2Nn+[0,2N]).$$ We associate to the path $\gamma$ a finite sequence $\Gamma=(k_i,n_i,a_i,t_i)_{0 \le i \le {\ell}}$, where the $(k_i,n_i)\in{\mathbb{Z}}^d\times {\mathbb N}$ are the coordinates of space-time boxes and the $(a_i,t_i)$ are points in ${\mathbb{Z}^d}\times {\mathbb{R}}_+$ in the following manner: - $k_0=0$, $n_0=0$, $a_0=0$ and $t_0=0$: $B_N(k_0,n_0)$ is the box containing the starting point $(a_0,t_0)=(0,0)$ of the path $\gamma$. - Assume we have chosen $(k_i,n_i,a_i,t_i)$, where $(a_i,t_i)$ is a point in $\gamma$ and $(k_i,n_i)$ are the coordinates of the space-time box containing $(a_i,t_i)$. To the box $B_N(k_i,n_i)$, we add the larger box $(2Nk_i+[-LN,LN]^d)\times (2Nn_i+[0,\alpha LN])$, we take for $(a_{i+1},t_{i+1})$ the first point – if it exists – along $\gamma$ after $(a_i,t_i)$ to be outside this large box, and we take for $(k_{i+1},n_{i+1})$ the coordinates of the space-time box that containing $(a_{i+1},t_{i+1})$. Otherwise, we stop the process. The idea is to extract from the path a sequence of large portions, [i.e. ]{}the portions of $\gamma$ between $(a_i,t_i)$ and $(a_{i+1},t_{i+1})$. We have the following estimates: $$\begin{aligned} &&\forall i\in[0..{\ell}-1] \quad \\|a_{i+1}-a_i\\|_\infty\le (L+1)N \text{ and } \\|a_l-x\\|_\infty \le (L+1)N, \label{absolument} \\ &&\forall i\in[0..{\ell}-1] \quad 0 \le t_{i+1}-t_i\le \alpha LN \text{ and } 0 \le t-t_l \le \alpha LN, \label{cequetuveux}\\ && 1\le {\ell} \le \frac{Ct}{(L-1)N}+\frac{t}{(\alpha L-1)N}+2 \le\frac{2Ct}{(\alpha L-1)N}+2 . \label{majoratl}\end{aligned}$$ The two first estimates just say that – spatially for and in time for – the point $(a_{i+1},t_{i+1})$ remains in the large box centered around $B_N(k_{i},n_{i})$, which contains $(a_i,t_i)$. Now consider the third estimate. We note that a path can get out of a large box either with its time coordinate – and the number of such exits is smaller than $\frac{t}{(\alpha L-1) N}+1$ –, or by the space coordinate – , and the number of such exits is smaller than $\frac{Ct}{(L-1)N}+1$. The last inequality comes from $C>1$ and $\alpha<1$. To ensure that the space coordinates of the boxes associated to the path are all distinct, we extract a subsequence $\overline{\Gamma}=(k_{\varphi(i)})_{0 \le i \le \overline{{\ell}}}$ with the loop-removal process described by Grimmett–Kesten [@grimmett-kesten]: - $\varphi(0)=\max\{j\ge 0: \; \forall i \in [0..j] \; k_i=0\}$; - Assume we chose $\varphi(i)$, then we take, if it is possible, $$\begin{aligned} j_0(i) & = & \inf\{j >\varphi(i): k_j \neq k_{\varphi(i)}\}, \\ \varphi(i+1) & = & \max\{j\ge j_0(i): \; k_j=k_{j_0(i)}\}.\end{aligned}$$ and we stop the extraction process otherwise. Then, as in [@grimmett-kesten] $$\begin{aligned} &&\\|a_{\varphi(\overline{{\ell}})}-x\\|_\infty \le (L+1)N, \\ && 0 \le t-t_{\varphi(\overline{{\ell}})} \le \alpha LN, \\ && \forall i \in [0..\overline{{\ell}}-1] \quad \\|a_{\varphi(i)+1}-a_{\varphi(i+1)}\\|_{\infty} \le 2N,\\ && \forall i \in [0..\overline{{\ell}}-1] \quad \|t_{\varphi(i)+1}-t_{\varphi(i+1)}\| \le 2N.\end{aligned}$$ Moreover, the upper bound (\[majoratl\]) for ${\ell}$ ensures that $$\label{majoratlbar} 1\le \overline{{\ell}}\le {\ell} \le \frac{2Ct}{(\alpha L-1)N}+2.$$ On the other hand, as $\displaystyle \mu(x)-\mu(a_{\varphi(\overline{{\ell}})}-x) \le \mu(a_{\varphi(\overline{l})})$, we have with (\[majoratlbar\]): $$\begin{aligned} (1+{\varepsilon})t -C^+_{\mu}(L+1)N & \le &\mu \left( \sum_{i=0}^{\overline{{\ell}}-1}a_{\varphi(i+1)}- a_{\varphi(i)}\right) \\ & \le &\sum_{i=0}^{\overline{{\ell}}-1} \mu(a_{\varphi(i+1)}-a_{\varphi(i)+1}) +\sum_{i=0}^{\overline{{\ell}}-1} \mu(a_{\varphi(i)+1}-a_{\varphi(i)}) \\ & \le &2N C_\mu^+\overline{{\ell}} +\sum_{i=0}^{\overline{{\ell}}-1} \mu(a_{\varphi(i)+1}-a_{\varphi(i)})\\ & \le &2NC_\mu^+\left(\frac{2Ct}{(\alpha L-1)N}+2\right) +\sum_{i=0}^{\overline{{\ell}}-1} \mu(a_{\varphi(i)+1}-a_{\varphi(i)}).\end{aligned}$$ This ensures, with the choice (\[choiceL\]) we made for $\alpha,L$, that $$\label{senex} \sum_{i=0}^{\overline{{\ell}}-1} \mu(a_{\varphi(i)+1}-a_{\varphi(i)}) \ge (1+2{\varepsilon}/3)t -2C^+_{\mu}(L+1)N.$$ In other words, even after the extraction process, the sum of the lengths of the crossings remains of order $(1+2{\varepsilon}/3)t$. Let $k \in {\mathbb{Z}^d}$ and $n \in {\mathbb N}$. We say now that $B_N(k,n)$ is good if $$\text{the event }A^{\alpha,L,N,{\varepsilon}/3} \circ T_{2kN} \circ \theta_{2nN} \text{ occurs},$$ and bad otherwise. If $B_N(k_{\varphi(i)},n_{\varphi(i)})$ is good, then the path exits the corresponding large box by the time coordinate, and thus $\mu(a_{\varphi(i)+1}-a_{\varphi(i)})\le (1+{\varepsilon}/3)(t_{\varphi(i)+1}-t_{\varphi(i)})$; this ensures that $$\begin{aligned} \mu \left(\sum_{i: \; B_N(k_{\varphi(i)},n_{\varphi(i)}) \text{ good}}(a_{{\varphi(i)}+1}-a_{\varphi(i)}) \right) & \le & \sum_{i: \; B_N(k_{\varphi(i)},n_{\varphi(i)})\text{ good}} \mu(a_{{\varphi(i)}+1}-a_{\varphi(i)}) \\ & \le & (1+\frac{{\varepsilon}}3)\sum_{i:\; B_N(k_{\varphi(i)},n_{\varphi(i)}) \text{ good}}(t_{{\varphi(i)}+1}-t_{\varphi(i)}) \\ & \le & (1+\frac{{\varepsilon}}3) t.\end{aligned}$$ With , it implies that $$\sum_{i: \; B_N(k_{\varphi(i)},n_{\varphi(i)}) \text{ bad}}\mu(a_{{\varphi(i)}+1}-a_{\varphi(i)}) \ge \frac{{\varepsilon}}3t-2C^+_{\mu}(L+1)N,$$ and then, with $${\vert \{i: \; B_N(k_{\varphi(i)},n_{\varphi(i)}) \text{ bad}\} \vert} \ge \frac{{\varepsilon}t}{3C^+_\mu (L+1)N}-2.$$ In other words, if $t>0$, if $x$ is such that $\mu(x) \ge (1+{\varepsilon})t$, if there exists an infection path $\gamma$ from $(0,0)$ to $(x,t)$ with less than $Ct$ horizontal edges, the associated sequence $\overline{\Gamma}$ has a number of bad boxes proportional to $t$. Note that Lemma \[catendvers12\] says that for any deterministic family $n=(n_k)_{k \in {\mathbb{Z}^d}} \in {\mathbb N}^{{\mathbb{Z}^d}}$, the field $(\eta^n_k)_{k \in {\mathbb{Z}^d}}$, defined by $\eta^n_{k}={1\hspace{-1.3mm}1}_{\{ B_N(k,n_{k}) \text{ good}\}}$ is locally dependent and that $$\lim_{N \to +\infty} {\mathbb{P}}(B_N(0,0) \text{ good})=1.$$ By the extraction process, the spatial coordinates of the boxes in $\overline{\Gamma}$ are all distinct. With the comparison theorem by Liggett–Schonmann–Stacey [@LSS], we can, for any $p_1<1$, take $N$ large enough to ensure that for any family $n=(n_k)_{k \in {\mathbb{Z}^d}} \in {\mathbb N}^{{\mathbb{Z}^d}}$, the law of the field $(\eta^n_k)_{k \in {\mathbb{Z}^d}}$ under ${\mathbb{P}}$ stochastically dominates a product on ${\mathbb{Z}^d}$ of Bernoulli laws with parameter $p_1$. Thus, if $x$ is such that $\mu(x) \ge (1+{\varepsilon})t$, then $$\begin{aligned} & &{\mathbb{P}}\left(\begin{array}{c} \text{there exists an infection path $\gamma$ from $(0,0)$ to $(x,t)$}\\ \text{with less than $Ct$ horizontal edges and such that $\overline{\Gamma}=\overline{\Gamma}(\gamma)$ has}\\ \text{at least $\frac{{\varepsilon}t}{3C^+_\mu (L+1)N}-2$ bad boxes} \end{array} \right) \\ & \le & \sum_{{\ell}=1}^{\frac{2Ct}{(\alpha L-1)N}+2} \sum_{{\vert \overline{\Gamma} \vert}={\ell}}2^{\ell}(1-p_1)^{\frac{{\varepsilon}t}{3C^+_\mu (L+1)N}-1} \\ &=& (1-p_1)^{\frac{{\varepsilon}t}{3C^+_\mu (L+1)N}-1}\sum_{{\ell}=1}^{\frac{2Ct}{(\alpha L-1)N}+2}2^{\ell}\textrm{Card}{\{\overline{\Gamma};{\vert \overline{\Gamma} \vert}=\ell\}}\end{aligned}$$ A classical counting argument gives the existence of a constant $K=K(d,\alpha,L)$ independent of $N$ such that $$\forall \ell\ge 1\quad \textrm{Card}{\{\overline{\Gamma};{\vert \overline{\Gamma} \vert}=\ell\}}\le K^{\ell}.$$ We get then an upper bound for our probability of the form $$A \frac{t}N \left( (1-p_1)^{\frac{{\varepsilon}}{3C_\mu^+(L+1)}} (2K)^{\frac{2C}{\alpha L-1}}\right)^{t/N},$$ which leads to a bound of the form $A_3\exp(-B_3 t)$ as soon as $p_1$ is close enough to $1$. Summing over all $x\in [-Mt,Mt]^d$, we have again an exponential bound. With this last upper bound, (\[majgross\]) and (\[pastropdaretes\]), we end the proof of Theorem \[GDdessous\]. We first prove there exist $A,B>0$ such that $$\label{GDTgrand} \forall T > 0\quad {\mathbb{P}}(\exists t\ge T\quad \xi^0_t \not \subset (1+{\varepsilon})tA_\mu)\le A\exp(-BT).$$ Indeed, $$\begin{aligned} & & {\mathbb{P}}(\exists t\ge T\quad \xi^0_t \not \subset (1+{\varepsilon})tA_\mu))\\ & \le & {\mathbb{P}}(\exists n\in{\mathbb N}\quad \xi^0_{T+n} \not \subset (1+{\varepsilon}/2)(T+n)A_\mu)\\ & & + {\mathbb{P}}(\exists n\in{\mathbb N}\quad \exists t\in [0,1]\quad \xi^0_{T+n} \subset (1+{\varepsilon}/2)(T+n)A_\mu, \xi^0_{T+n+t} \not \subset ((1+{\varepsilon})(T+n)A_\mu) \\ & \le & \sum_{n\ge 0} {\mathbb{P}}(\xi^0_{T+n} \not \subset (1+{\varepsilon}/2)(T+n)A_\mu)\\ & & +\sum_{n\ge 0}{\mathbb{P}}(\exists t\in [0,1]\quad \xi^0_{T+n} \subset (1+{\varepsilon}/2)(T+n)A_\mu, \xi^0_{T+n+t} \not \subset ((1+{\varepsilon})(T+n)A_\mu).\end{aligned}$$ The first sum can be controlled with Theorem \[GDdessous\]. For the second sum, the Markov property gives for any $\lambda\in\Lambda$, $$\begin{aligned} & &{\mathbb{P}}_{\lambda}(\exists t\in [0,1]\quad \xi^0_{T+n} \subset (1+{\varepsilon}/2)(T+n)A_\mu, \; \xi^0_{T+n+t} \not \subset ((1+{\varepsilon})(T+n)A_\mu)\\ & \le &\sum_{x\in (1+{\varepsilon}/2)(T+n)A_\mu}{\mathbb{P}}_{x.\lambda}(\exists t\in [0,1]\quad \xi^0_{t} \not \subset ({\varepsilon}/2)(T+n)A_\mu)\\ & \le &{\vert (1+{\varepsilon}/2)(T+n)A_\mu \vert}{\mathbb{P}}(H_1^0 \not \subset ({\varepsilon}/2)(T+n)A_\mu)\le A \exp(-B (T+n)),\end{aligned}$$ where the last upper bound comes from a comparison with the Richardson model. We conclude the proof of (\[GDTgrand\]) by integrating with respect to $\lambda$. Let us prove now the existence of $A,B>0$ such that $$\label{GDHpetit} \forall r>0\quad {\mathbb{P}}(H^0_r \not\subset(1+{\varepsilon})r A_{\mu})\le A\exp(-Br).$$ With , we can find $A_1,B_1>0$ and $c<1$ such that ${\mathbb{P}}(H^0_{cr} \not\subset r A_{\mu})\le A_1\exp(-B_1r).$ Now, $$\begin{aligned} {\mathbb{P}}(H^0_r \not\subset(1+{\varepsilon})r A_\mu) & \le & {\mathbb{P}}(H^0_{cr} \not\subset r A_\mu)+{\mathbb{P}}(\exists t \in( cr,r) \quad \xi^0_t \not\subset (1+{\varepsilon})r A_\mu) \\ & \le & A_1\exp(-B_1r) +{\mathbb{P}}(\exists t \ge cr \quad \xi^0_t \not\subset (1+{\varepsilon})t A_\mu),\end{aligned}$$ and we conclude the proof of with (\[GDTgrand\]). To obtain , we just need to note that $t\mapsto H_t$ is non-decreasing. Finally, for $x \in {\mathbb{Z}^d}\backslash\{0\}$, $${\mathbb{P}}(t(x) \le (1-{\varepsilon})\mu(x)) \le {\mathbb{P}}(H^0_{(1-{\varepsilon})\mu(x)} \not\subset \mu(x)A_\mu).$$ Applying (\[GDHpetit\]), we end the proof of , and thus of Theorem \[dessouscchouette\]. About the order of the deviations {#bonnevitesse} ================================= By Theorems \[theoGDUQ\] and \[dessouscchouette\], we have for $\nu$-almost every $\lambda$ and each ${\varepsilon}>0$: $${\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ {\overline{\lim}}\\ {\scriptstyle x\to +\infty} \end{array} \renewcommand{\arraystretch}{1}}\frac1{\\|x\\|}\log {\overline{\mathbb{P}}}_{\lambda}\left(\frac{t(x)}{\mu(x)}\not\in [1-{\varepsilon},1+{\varepsilon}]\right)<0.$$ To see that the exponential decrease in $\\|x\\|$ is optimal, we need too see that ${\renewcommand{\arraystretch}{0.6} \begin{array}{c} {\scriptstyle }\\ {\underline{\lim}}\\ {\scriptstyle x\to +\infty} \end{array} \renewcommand{\arraystretch}{1}}\frac1{\\|x\\|}\log {\overline{\mathbb{P}}}_{\lambda}\left(\frac{t(x)}{\mu(x)}\not\in [1-{\varepsilon},1+{\varepsilon}]\right)>-\infty.$ In fact, we will prove here that for every $(s,t)$ with $0<s<t$, there exists a constant $\gamma>0$ such that for each $\lambda\in\Lambda$ and each $x\in{\mathbb{Z}^d}$, $$\begin{aligned} {\mathbb{P}}_\lambda(t(x)\in [s,t]\\|x\\|_1) & \ge & \exp(-\gamma\\|x\\|_1).$$ Let $s,t$ with $0 <s <t$. For each $u \in {\mathbb{Z}^d}$ such that $\\|u\\|_1=1$, we define $T_u=\inf\{t \ge 0: \; \xi^0_t=\{u\},\quad \forall s\in [0,t)\quad \xi^0_s=\{0\}\}$. We are going to prove that $$\exists \gamma>0 \quad \forall \lambda \in \Lambda \quad \forall u \in {\mathbb{Z}^d}, \; \\|u\\|_1=1 \quad {\mathbb{P}}_\lambda(T_u \in [s,t]) \ge e^{-\gamma}.$$ In order to ensure that $T_u\in [s,t]$, it it sufficient to satisfy - The lifetime of the particle at $(0,0)$ is strictly between $(s+t)/2$ and $t$, which happens with probability $e^{-(s+t)/2}-e^{-t}$ under ${\mathbb{P}}_\lambda$ ; - The first opening of the bond between $0$ and $u$ happens strictly between $s$ and $(s+t)/2$, which happens with probability $$\exp(-\lambda_{\{0,u\}} s)-\exp(-\lambda_{\{0,u\}} (s+t)/2) \ge \exp(-\lambda_{\max} s)(1-\exp(-\lambda_{\min}(t-s)/2))$$ under ${\mathbb{P}}_\lambda$; - There is no opening between time $0$ and time $t$, on the set $J$ constituted by the $4d-2$ bonds that are neighour of $0$ or $u$ and differ from $\{0,u\}$, which happens under ${\mathbb{P}}_\lambda$ with probability $$\prod_{j\in J} \exp(-\lambda_j t) \ge \exp(-(4d-2)\lambda_{\max} t);$$ - There is no death at site $u$ between $0$ and $t$, which happens under ${\mathbb{P}}_\lambda$ with probability $e^{-t}$. Then, using the independence of the Poisson processes, we get $$\begin{aligned} & &{\mathbb{P}}_{\lambda}(T_u \in [s,t])\\ & \ge & (e^{-(s+t)/2)}-e^{-t})e^{-t} e^{-(4d-2)\lambda_{\max} t} e^{-\lambda_{\max} s}(1-e^{-\lambda_{\min}(t-s)/2})=e^{-\gamma}.\end{aligned}$$ Moreover, $T_u$ is obviously a stopping time. Then, applying the strong Markov property $\\|x\\|_1$ times, we get, $${\mathbb{P}}_{\lambda}(t(x)\in [s,t]\\|x\\|_1)\ge \exp(-\gamma\\|x\\|_1).$$ This gives the good speed for both upper and lower large deviations. Note that the order of the large deviations is the same for upper and lower deviations, as happens for the chemical distance in Bernoulli percolation (see Garet–Marchand [@GM-large]). Conversely, it is known that these orders may differ for first-passage percolation (see Kesten [@kesten] and Chow–Zhang [@chow-zhang]). [10]{} R. Cerf and Marie Th[é]{}ret. 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Large deviations for the chemical distance in supercritical [B]{}ernoulli percolation. , 35(3):833–866, 2007. Olivier Garet and R[é]{}gine Marchand. Annealed upper large deviations for the asymptotic shape of the contact process in $\mathbb{Z}^d$. , 2012. Olivier Garet and R[é]{}gine Marchand. Asymptotic shape for the contact process in random environment. , to appear. Olivier [Garet]{} and Régine. [Marchand]{}. . , febr 2012. Geoffrey Grimmett. Large deviations in subadditive processes and first-passage percolation. In [Particle systems, random media and large deviations ([B]{}runswick, [M]{}aine, 1984)]{}, volume 41 of [Contemp. Math.]{}, pages 175–194. Amer. Math. Soc., Providence, RI, 1985. Geoffrey Grimmett and Harry Kesten. First-passage percolation, network flows and electrical resistances. , 66(3):335–366, 1984. J. M. Hammersley. Postulates for subadditive processes. , 2:652–680, 1974. J. M. Hammersley and D. J. A. Welsh. First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In [Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif.]{}, pages 61–110. Springer-Verlag, New York, 1965. T. E. Harris. Additive set-valued [M]{}arkov processes and graphical methods. , 6(3):355–378, 1978. Harry Kesten. Aspects of first passage percolation. In [École d’été de probabilités de Saint-Flour, XIV—1984]{}, volume 1180 of [Lecture Notes in Math.]{}, pages 125–264. Springer, Berlin, 1986. J. F. C. Kingman. Subadditive ergodic theory. , 1:883–909, 1973. With discussion by D. L. Burkholder, Daryl Daley, H. Kesten, P. Ney, Frank Spitzer and J. M. Hammersley, and a reply by the author. J. F. C. Kingman. Subadditive processes. In [École d’Été de Probabilités de Saint-Flour, V–1975]{}, pages 167–223. Lecture Notes in Math., Vol. 539. Springer, Berlin, 1976. T.M. Liggett, R.H. Schonmann, and A.M. Stacey. Domination by product measures. , 25:71–95, 1997. R. Rossignol and Marie Th[é]{}ret. Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation. , 46(4):1093–1131, 2010. R. Rossignol and Marie Th[é]{}ret. Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation. , to appear:(electronic), 2011. Rapha[ë]{}l Rossignol and Marie Th[é]{}ret. Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation. , 120(6):873–900, 2010. Timo Sepp[ä]{}l[ä]{}inen and J. E. Yukich. Large deviation principles for [E]{}uclidean functionals and other nearly additive processes. , 120(3):309–345, 2001. Marie Th[é]{}ret. Upper large deviations for the maximal flow in first-passage percolation. , 117(9):1208–1233, 2007. Marie Th[é]{}ret. On the small maximal flows in first passage percolation. , 17(1):207–219, 2008.
--- abstract: 'Precise extractions of $\alpha_s$ from ${\tau\to {\rm (hadrons)}+\nu_\tau}$ and from ${e^+e^-\to {\rm (hadrons)}}$ below the charm threshold rely on finite energy sum rules (FESR) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-$\beta_0$ limit and in full QCD. In large-$\beta_0$, we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the $C$-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-$\beta_0$. Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations.' author: - 'D. Boito' - 'F. Oliani' bibliography: - 'References.bib' date: - - --- Introduction ============ Extractions of the strong coupling, $\alpha_s$, at lower energies can be very precise due to increased sensitivity to the higher-order corrections, as long as the non-perturbative contributions are under good control. The prominent example of this type of $\alpha_s$ determination is the extraction from inclusive hadronic decays of the $\tau$ lepton, which have been used since the 90s as a reliable source of information about QCD dynamics [@Braaten:1988hc; @Braaten:1991qm]. Although the decay rate receives a non-negligible contribution from non-perturbative effects, it is largely dominated by perturbative QCD, which renders feasible a competitive extraction of the strong coupling [@Salam:2017qdl; @Boito:2014sta; @Davier:2013sfa; @Pich:2016bdg]. Recently, a similar $\alpha_s$ determination was introduced [@Boito:2018yvl] making use of a compilation of data for $e^+e^-\to {\rm (hadrons)}$ below the charm threshold [@Keshavarzi:2018mgv]. An attractive feature of this new analysis is that the systematics is under very good control, although the error due to the data is still somewhat large.[^1] Both analyses rely on finite energy sum rules (FESR) where, on the experimental side, one has weighted integrals of the experimentally accessible hadronic spectral functions. Exploiting the analyticity properties of the quark-current correlators one is able to express the theoretical counterpart of the sum rules as an integral in a closed contour on the complex plane of the variable $s$ — which represents the invariant mass of the final-state hadrons — thereby circumventing the breakdown of perturbative QCD at low energies. In this framework, the perturbative contribution is obtained from the complex integration of the Adler function in the chiral limit, which nowadays is exactly known up to $\alpha_s^4$ [@Baikov:2008jh; @Herzog:2017dtz]. When performing this integration, one must adopt a procedure to set the renormalisation scale. The two most widely used ones are Fixed Order Perturbation Theory (FOPT) [@Beneke:2008ad], in which the scale is kept fixed, and Contour Improved Perturbation Theory (CIPT) [@Pivovarov:1991rh; @LeDiberder:1992zhd], where the scale varies along the contour resumming the running of the coupling. The procedures lead to different series and to values of $\alpha_s$ that are different. This difference remains one of the dominant uncertainties in the $\alpha_s$ extraction from $\tau$ decays [@Boito:2014sta; @Pich:2016bdg]. In the case of $e^+e^-\to {\rm (hadrons)}$, the difference is significantly smaller, but still non-negligible [@Boito:2018yvl]. In the discussion of perturbative expansions in QCD one must take into account a basic but important fact: the perturbative series are divergent and, at best, they are asymptotic expansions — as discovered by Dyson in the context of QED in 1952 [@Dyson:1952tj]. The series is better understood in terms of its Borel transform, which suppresses the factorial growth of the perturbative coefficients and allows for an understanding of the higher-order behaviour in terms of singularities along the real axis in the Borel plane. These singularities are the renormalons of perturbation theory [@Beneke:1998ui]. An optimal use of an asymptotic series of this type can be achieved (most often) by truncating it at the smallest term [@Boyd1999]. In this procedure, the error one makes is parametrically of the form $e^{-p/\alpha}$ where $p>0$ is a constant and $\alpha$ the expansion parameter. In QCD, the expansion parameter, $\alpha_s(Q^2)$, runs logarithmically which implies that the truncation error is $\sim \left(\Lambda^2_{\rm QCD}/Q^2 \right)^p$, where $Q^2$ is the Euclidian momentum. These power corrections are a necessary feature of perturbative QCD and are, of course, related to the higher-dimension terms in the Operator Product Expansion (OPE). In the Borel plane, their manifestation is the appearance of renormalon singularities along the real axis at specific locations related to their dimensionality. In realistic $\alpha_s$ analyses the non-perturbative contributions must be taken into account. These include the OPE condensates as well as duality violations (DVs) which are due to resonances and are not encoded in the OPE expansion [@Shifman:2000jv; @Cata:2005zj; @Peris:2016jah; @Boito:2017cnp] . In order to extract from the data $\alpha_s$ and the non-perturbative parameters in a self-consistent way, i.e. without relying on external information, one resorts to the use of several (pseudo) observables. Those are built using the fact that any analytic weight function gives rise to a valid FESR, with an experimental side that can be computed from the empirical spectral functions and a theoretical counterpart that can be obtained from the integral along the complex contour. The main guiding principle behind the judicious choice of weight functions that enter a given analysis has been, for a long time, the suppression of non-perturbative contributions. The different analyses of hadronic tau decay data can be divided into two categories. In one of the analysis strategies, one strongly suppresses the poorly known higher-order OPE condensates [@Boito:2014sta; @Boito:2018yvl]. In this case, duality violations are larger and one must include them; this is done relying on a parametrization that can be connected with fundamental properties of QCD [@Boito:2017cnp]. In the other, only moments that suppress duality violations are used [@Davier:2013sfa; @Pich:2016bdg]. The price to pay in this case is the contamination of the results by the neglected higher-order OPE condensates [@Boito:2016oam]. Apart from issues related to non-perturbative contaminations, since the work of Ref. [@Beneke:2012vb], it is known that the different weight functions lead to distinct perturbative series that are not equally well behaved. Some of those used in the literature [@Davier:2013sfa; @Pich:2016bdg] have a poor perturbative convergence and are therefore not the ideal choice in precise $\alpha_s$ analyses. The main purpose of this work is to understand the perturbative behaviour of the different integrated spectral function moments at intermediate and high orders by studying the renormalon singularities appearing in their Borel transform. The perturbative behaviour of the different moments is intricate, in fact, each of the moments is a different asymptotic expansion with different renormalon contributions and conclusions about their perturbative behaviour have to be drawn almost case by case. As is customary, we will use the large-$\beta_0$ limit of QCD as a guide. In this limit, all renormalon singularities are double poles, with the exception of the leading IR singularity, which is simple. A number of facts can be established. First, the Borel transform of polynomial moments of the Adler function is always less singular than the Borel transform of the Adler function itself. An infinite number of renormalon poles become simple poles. Second, the renormalon poles corresponding to the OPE condensate(s) to which the moment is maximally sensitive are not reduced (or cancelled). What we mean by “maximally sensitive” will become clearer in the remainder, but these two facts are enough to draw interesting conclusions about the behaviour of the different moments and help explaining the instabilities (or “run-away behaviour”) identified in Ref. [@Beneke:2012vb]. We will show that the leading IR renormalon is largely responsible for the unstable behaviour of moments that are highly sensitive to the gluon condensate. We also show that an absence of the leading IR pole and partial cancellations of the renormalon singularities are behind the good perturbative behaviour of some of the moments. Turning to QCD, the situation is more complicated, mainly because the renormalon poles become branch points. The Borel transform has superimposed branch cuts. We will show that most of the difficulties in QCD can be circumvented by a convenient scheme transformation, to the so-called $C$ scheme [@Boito:2016pwf], together with the use of a modified Borel transform introduced in Ref. [@Brown:1992pk].[^2] In this framework, the Borel transform of the moments can be calculated exactly in terms of the Borel transform of the Adler function — which is one of the main results of this paper, Eq. (\[eq:modBoreldelta\]). The parallel with the large-$\beta_0$ limit is apparent and the results are formally identical. In fact, we show that the leading IR singularity is also a simple pole in this case. The enhancement and suppression of renormalon singularities identified in large-$\beta_0$ is, therefore, also present in QCD which explains the similarity between the perturbative behaviour of moments in the two cases. We then study the behaviour of a few emblematic moments in QCD, using a recent reconstruction of the higher-order terms based on Padé approximants [@Boito:2018rwt]. Finally, we show how to optimize the truncation of the moments with good perturbative behaviour in the spirit of an asymptotic series exploiting scheme transformations. The procedure we employ has been suggested for the $\tau$ hadronic width in Ref. [@Boito:2016pwf] but had never been investigated systematically for different integrated moments. This work is organised as follows. In Sec. \[theory\], we present the theoretical framework. In Sec. \[renormalons\], we discuss the renormalon content of polynomial moments and their phenomenological consequences, both in large-$\beta_0$ and in QCD. In Sec. \[optimization\], we discuss the optimized truncation of the moments with good perturbative behaviour through scheme transformations. In Sec. \[conclusions\], we present our conclusions. Finally, in App. \[betafunction\] we present our conventions for the QCD $\beta$ function; App. \[borelint\] contains further details about the Borel integral of the moments discussed in this work. Theoretical framework {#theory} ===================== In the low-$Q^2$ $\alpha_s$ determinations from hadronic $\tau$ decays and from ${e^+e^-\to {\rm (hadrons)}}$ one uses FESRs constructed from integrated moments of the experimental hadronic spectral functions. In the case of ${e^+e^-\to {\rm (hadrons)}}$, one has access to the electromagnetic vacuum polarization spectral function, which mixes isospin 0 and 1. Below the charm threshold one can safely work in the chiral limit, apart from the inclusion of perturbative corrections arising from the strange-quark mass. In hadronic $\tau$ decays, the decay width of the $\tau$ lepton into hadrons normalized to the decay width of $\tau\to \nu_\tau e^- \bar\nu_e$ can be separated experimentally into three distinct components: the vector and axial vector, $R_{\tau, V/A}$, arising from the $(\bar ud)$-quark current, and the contributions with net strangeness, intermediated by the $(\bar us)$-quark current. In the extractions of the strong coupling $\alpha_s$, the focus is on the non-strange contributions since they have a smaller contamination from non-perturbative effects and the quark masses in this case can safely be neglected. Since the FESRs we discuss here, and in particular the choice of moments, were primarily introduced in the context of $\tau$ decays, we will present them in this context. The translation to ${e^+e^-\to {\rm (hadrons)}}$ is straightforward and the perturbative contribution, in particular, is essentially identical [@Boito:2018yvl]. We define a generalized observable $R_{\tau, V/A}^{(w_i)}(s_0)$ that can be written as a weighted integral over the experimentally accessible spectral functions as R\_[, V/A]{}\^[(w\_i)]{}(s\_0) = 12S\_[EW]{} \|V\_[ud]{}\|\^2 \_0\^1 dx w\_i(x) ,\[eq:expmoment\] where $w_i(x)$ is any analytic weight function and $x=s/s_0$. The correlators $\Pi^{(1)}_{V/A}$ and $\Pi^{(0)}_{V/A}$ are the transverse and longitudinal parts of \_[V/A]{}\^(p) i dx e\^[i p x]{} \| T { J\_ [V/A]{}\^(x) J\_[V/A]{}\^(0)\^ } \| ,\[eq:Corr\] formed from the quark currents $J_{V/(A)}^\mu = (\bar u \gamma^\mu(\gamma_5) d)(x)$; we define $\Pi^{(1+0)}_{V/A}=\Pi^{(1)}_{V/A}+\Pi^{(0)}_{V/A}$. Setting $s_0=m_\tau^2$ in Eq. (\[eq:expmoment\]) and with the particular choice of weight function w\_(x) = (1-x)\^2(1+2x) dictated by kinematics we have $R_{\tau,{V/A}}\equiv R_{\tau,V/A}^{(w_\tau)}(m_\tau^2)$, the hadronic decay width normalized to the decay width of $\tau^- \to \nu_\tau e^- \bar{\nu}_e $. In precise extractions of $\alpha_s$ from $\tau$ decays it has become customary to exploit other analytic weight functions, conveniently chosen in order to suppress or enhance the different contributions to the decay rate. The generalized observable $R_{\tau,V/A}^{(w_i)}(s_0)$ can be decomposed as R\_[,V/A]{}\^[(w\_i)]{}(s\_0) = S\_[EW]{} \|V\_[ud]{}\|\^2 ,\[eq:Rtaudecomposition\] where $N_c$ is the number of colours, $S_{\rm EW}$ is an electroweak correction, and $V_{ud}$ is the quark-mixing matrix element. The perturbative terms are represented by $\delta_{w_i}^{\rm tree}$ and $\delta_{w_i}^{(0)}(s_0)$, where the former corresponds to the partonic result, while the latter encode the $\alpha_s$ corrections computed in the chiral limit. The OPE corrections of dimension $D$ are collected in the terms $\delta_{w_i,V/A}^{(D)}(s_0) $ and, finally, duality violation corrections are given by $\delta^{\rm DV}_{w_i,V/A}(s_0)$. The leading contribution to $R_{\tau,V/A}^{(w_i)}(s_0)$ stems from perturbative QCD. It is obtained from the perturbative expansion of the correlators of Eq. (\[eq:Corr\]), which is the dimension zero term in the OPE expansion that can be written, for $\Pi^{(1+0)}(s)$, as \_[OPE]{}\^[(1+0)]{} = \_[D=0,2,4...]{}\^,\[PiOPE\] where the sum is done over all the contributions from gauge invariant operators of dimension $D$. The case $D=0$ is the perturbative part and $D=2$ are small mass corrections. The first non-perturbative contribution starts at $D=4$ and is dominated by the gluon condensate. The $s$ dependence in the Wilson coefficients $C_D(s)$ arise from the logarithms in their perturbative description and is higher-order in $\alpha_s$. In the case of the gluon condensate the leading logarithm is known but, to an excellent approximation, the coefficient $C_4(s)$ can be treated as a constant [@Boito:2011qt]. Little is known for the logarithms in the higher-dimension condensates, but it is customary, based on the experience with $D=4$, to neglect their $s$ dependence as well and treat all $C_D$ as effective coefficients with no $s$ dependence. The theoretical treatment of the observables $R_{\tau, V/A}^{(w_i)}$ is done in the framework of FESRs, relating the experimental results to counter-clockwise contour integrals along the circle $\|s\|=s_0$ in the complex plane of the variable $s$. To eliminate the conventions related to renormalisation it is convenient to work with the Adler function D(s) = -s \^[(1+0)]{}(s). In terms of the Adler function, the perturbative correction of Eq. (\[eq:Rtaudecomposition\]) can be written as [@Beneke:2008ad] \^[(0)]{}\_[w\_i]{} = \_[\|x\|=1]{} W\_i(x) D\_[pert]{}(s\_0 x),\[eq:delta0\] where $W_i(x)=2\int_x^1 dz \ w_i(z)$ is the weight function. The reduced Adler function, $\widehat D$, which intervenes in Eq. (\[eq:delta0\]), is defined in order to separate the partonic contribution 1+ D(Q\^2) = D(Q\^2), where $Q^2 \equiv -s $. Accordingly, the perturbative expansion of the function $\widehat D$ starts at order $\alpha_s$ and can be written as D\_[pert]{}(s) = \_[n = 1]{}\^[a\^[n]{}\_]{} \_[k = 1]{}\^[n+1 ]{}[k c\_[n,k]{}\^[k-1]{}]{},\[eq:AdlerExp\] where $a_\mu=\alpha_s(\mu)/\pi$. The only independent coefficients in this expansion are the $c_{n,1}$; all the others can be written with the used of Renormalisation Group (RG) equations in terms of the $c_{n,1}$ and $\beta$-function coefficients. At present, the coefficients of the expansion are known up to $c_{4,1}$ (five loops) [@Baikov:2008jh; @Herzog:2017dtz]. Resumming the logarithms with the choice $\mu^2=-s$ the result is (for $n_f=3$) D\_[pert]{}(Q\^[2]{}) = \_[n=1]{}\^c\_[n,1]{} a\_Q\^[n]{} = a\_Q + 1.640 a\_Q\^2 +6.371 a\_Q\^3+ 49.08 a\_Q\^4+,\[eq:DCIPT\] from which the known independent coefficients can be read off. (Henceforth we will often omit the subscript “pert” in perturbative quantities.) The perturbative series of Eq. (\[eq:AdlerExp\]) is divergent. It is assumed that it must be an asymptotic series [@Beneke:1998ui] to the true (unknown) value of the function being expanded. The divergence stems from the factorial growth of the $c_{n,1}$ coefficients at large order and it is, therefore, convenient to work with the Borel-Laplace transform of the series B\[D\](t) \_[n=0]{}\^[r\_[n]{} ]{}, \[BorelDef\] which has a finite radius of convergence and where $r_n = c_{n+1,1}/\pi^{n+1}$. The original expansion is then, by construction, the asymptotic series to the inverse Borel transform (the usual Laplace transform) given by D () \_[0]{}\^[dt \^[-t/]{}B\[R\](t)]{} \[BorelInt\]. On the assumption that the integral exists, the last equation defines unambiguously the Borel sum of the asymptotic series. However, the divergence of the original series is related to singularities in the $t$ variable known as renormalons. They appear at both positive and negative integer values of the variable $u=\frac{\beta_1 t}{2\pi}$ (with the exception of $u=1$). In particular, the IR renormalons, that lie on the positive real axis, obstruct the integration in the Borel sum. A prescription to circumvent these poles becomes necessary, which entails an ambiguity in the Borel sum of the series. This remaining ambiguity is expected on general grounds to be cancelled by corresponding ambiguities in the power corrections of the OPE. At large orders, the UV pole at $u=-1$, being the closest to the origin, dominates the behaviour of the series. The coefficients of the series are, therefore, expected to diverge with sign alternation at sufficiently high orders. The calculation of the perturbative contribution to FESR observables requires that one performs the integral of Eq. (\[eq:delta0\]). A prescription for the renormalisation scale $\mu$ — which enters through the logarithms of Eq. (\[eq:AdlerExp\]) — must be adopted in the process. In the procedure known as Contour-Improved Perturbation Theory (CIPT) [@Pivovarov:1991rh; @LeDiberder:1992zhd] a running scale $\mu^2=Q^2$ is adopted and the running of $\alpha_s$ is resummed along the contour with the QCD beta function. With this procedure the perturbative contribution is cast as \^[(0)]{}\_[[CI]{},w\_i]{} = \_[n=1]{}\^ c\_[n,1]{}J\^[(n)]{}\_[[CI]{},w\_i]{}(s\_0), J\^[(n)]{}\_[[CI]{},w\_i]{}(s\_0)= \_[\|x\|=1]{} W\_i(x)a\^[n]{}(-s\_0 x). A strict fixed order prescription, known as Fixed Order Perturbation Theory (FOPT) corresponds to the choice of a fixed scale $\mu=s_0$. The coupling can then be taken outside the integrals which are now performed over the logarithms of Eq. (\[eq:AdlerExp\]) as \^[(0)]{}\_[[FO]{},w\_i]{} = \_[n=1]{}\^[a\_[s\_0]{}\^[n]{}]{} \_[k=1]{}\^[n]{}[k c\_[n,k]{}]{} J\_[[FO]{},w\_i]{}\^[(k-1)]{}, J\_[[FO]{},w\_i]{}\^[(n)]{} \_[\|x\|=1]{}[W\_i(x)\^[n]{}(-x)]{}.\[FOPTdef\] The FOPT series can be written as an expansion in the coupling as \_[[FO]{},w\_i]{}\^[(0)]{} = \_[n=1]{}\^d\_n\^[(w\_i)]{} a\_Q\^n, where the coefficients now depend on the choice of weight function. The chosen prescription for the renormalisation group improvement of the series affects, in practice, the precise extraction of the strong coupling from hadronic $\tau$ decays. It remains, as of today, one of the main sources of theoretical uncertainty in these $\alpha_s$ determinations [@Pich:2016bdg; @Davier:2013sfa; @Boito:2014sta; @Boito:2018yvl]. The two prescriptions define two different asymptotic series with rather different behaviours. Inevitably, the analysis of the reliability of the two procedures requires knowledge about higher orders of the series. In particular, some of the arguments often put forward in favour of CIPT — in an attempt to leave aside the issue with the higher orders — mention a “radius of convergence" [@Braaten:1991qm; @Pich:2013lsa], a notion that contradicts the fact that the series are both asymptotic. Here we employ the estimate for the higher-order coefficients of the series obtained from a careful and systematic use of Padé approximants [@Boito:2018rwt]. The results of Ref. [@Boito:2018rwt] are model independent and corroborate to a large extent the results obtained in the context of renormalon models, in which the series is modelled by a small number of dominant renormalon singularities employing the available knowledge about their nature [@Beneke:2008ad; @Beneke:2012vb; @Cvetic:2018qxs], as well as those obtained from conformal mappings that make use of the location of renormalon sigularities [@Caprini:2019kwp]. We will also exploit scheme variations as a method to improve convergence of the perturbative series and discuss their usefulness in realistic extractions of $\alpha_s$ from hadronic $\tau$ decays. Renormalons in spectral function moments {#renormalons} ======================================== Several moments of the spectral functions have been used in low-$Q^2$ $\alpha_s$ determinations from hadronic $\tau$ data and ${e^+e^-\to {\rm (hadrons)}}$ [@Boito:2011qt; @Davier:2013sfa; @Boito:2014sta; @Boito:2016oam; @Pich:2016bdg]. Since the FESR requires the weight function to be analytic it is customary to employ polynomials, which we denote in terms of their expansions in monomials as w\_i(x) = \_[k=0]{}b\^[(w\_i)]{}\_kx\^k. Of particular importance are the weight functions that are “pinched", i.e. weight functions that are zero at $x=1$, and that have $b_0^{(w_i)}=1$ such as the kinematic moment w\_= (1-x)\^2(1+2x) = 1-3x\^2+2x\^3\[wtau\]. Another important class of weight functions identified in [@Beneke:2012vb] are those that contain the linear term in $x$. We will discuss these two classes of moments in detail below. The series for $\delta^{(0)}_{w_i}$ inherits the divergence of the Adler function expansion and accordingly is also amenable to a treatment in terms of its Borel transform. However, the renormalon content of the Borel transformed $\delta^{(0)}_{w_i}$ is different from the Adler function counterpart, as we discuss in the remainder of the section. Results in large-beta0 ---------------------- We start investigating the renormalons in $\delta^{(0)}_{w_i}$ in the large-$\beta_0$ limit of QCD [@Beneke:1998ui]. These results are obtained by first considering a large number of fermion flavours, $N_f$, but keeping $N_f\alpha_s$ constant. The $q\bar q$ bubble corrections to the gluon propagator are order one in this power counting and must be summed to all orders. This dressed gluon propagator is used to obtain all the leading $N_f$ corrections, at every $\alpha_s$ order, to a given observable. In the end, $N_f$ is replaced by the leading $\beta$ function coefficient, effectively incorporating a set of non-abelian contributions [@Beneke:1994qe]. Accordingly, the $\alpha_s$ evolution is performed at one-loop. In this limit, the Borel transformed Adler function is known to all orders in perturbation theory and it can be written in a compact form as [@Beneke:1998ui; @Beneke:1992ch; @Broadhurst:1992si] B\[D\]= \_[k=2]{}\^ , \[eq:BorelAdlerLb0\] where $C$ is a parameter which depends on the renormalisation scheme. For $C=0$ we have $\overline{\text{MS}}$. This result displays explicitly the renormalon poles. They are all double poles with the exception of the leading IR pole at $u=2$, which is simple. The IR poles are particularly important in the subsequent discussion, and in particular their connection to OPE condensates. Each of the IR poles that appear at a given position $u=p$ in the Borel transform of the Adler function can be mapped to the existence of contributions of dimension $D=2p$ in the OPE [@Beneke:1998ui]. This explains, for example, the absence of a pole at $u=1$ since there is no gauge-invariant $D=2$ condensate in the OPE. This non-trivial connection between perturbative and non-perturbative physics will also be manifest in the Borel transform of $\delta_{w_i}^{(0)}$. Using this result, the Borel transform of $\delta_{w_i}^{(0)}$ can be obtained from Eq. (\[eq:delta0\]) employing the Borel integral representation of the Adler function, Eq. (\[BorelInt\]). One can then write \^[(0)]{}\_[w\_i]{} =\_0\^[2]{}d W\_i(e\^[i]{}) \_0\^dt e\^[-t/\_s(-s\_0e\^[i]{})]{}B\[\](t),\[eq:delta0intrepre\] where we performed the change of variables $x=e^{i \phi}$. In the large-$\beta_0$ limit, the $\beta$ function is truncated at its first term[^3] =().\[eq:alphasrunning1loop\] The exponential in Eq. (\[eq:delta0intrepre\]) can be written as e\^[-t/\_s(-s\_0x)]{}=e\^[-t/\_s(s\_0)]{}e\^[-iu(-)]{}. Inverting the order of integration and using Eq. (\[BorelInt\]) one can read off the Borel transform of $\delta^{(0)}_{w_i}$ B\[\^[(0)]{}\_[w\_i]{}\] = B\[D\](u). \[BTdLb1\] The prefactor of Eq. (\[BTdLb1\]) can be obtained analytically for polynomial weight functions. For the monomial $w_i=x^n$ one finds[^4] B\[\^[(0)]{}\_[x\^n]{}\] = B\[D\](u). \[BTdLb2\] One immediately sees that the $\sin(\pi u)$ reduces an infinite number of UV and IR double poles in $B[\widehat D](u)$ to simple poles. In this sense, one can say that $B[\delta^{(0)}_{w_i}]$ is significantly less singular than the Adler function counterpart, a fact that has been exploited in Ref. [@Boito:2018rwt]. The prefactor of Eq. (\[BTdLb2\]) is also highly non-trivial. It cancels the zero at $u=1+n$ in $\sin(\pi u)$, which means that the pole at $u=1+n$ of $B[\widehat D](u)$ remains double (or single, in the case of $u=2$). This is clearly not a coincidence and is related to the non-perturbative contributions to $R_{\tau,V/A}$. To expose this connection, consider the contribution of $D\geq 4$ in the OPE expansion, Eq. (\[PiOPE\]), to $R_{\tau,V/A}$ which can be cast as \_[w\_i,V/A]{}\^[(D)]{}= dx C\_D(xs\_0). For a monomial $w_i(x)=x^n$ — and to the extent that the $s$ dependence of the coefficients $C_D$ can be neglected, as discussed previously — this reduces to \_[x[\^n]{},V/A]{}\^[(D)]{}= C\_D dx .\[OPEdeltaD\] For positive integer values of $n$, the integral in the last equation is only non-vanishing for $-n+D/2=1$. Therefore, as is well known, under these assumptions, for $w_i=x^n$ the only contribution comes from the condensates with $D=2(n+1)$ which, in turn, is related to the pole in the Borel transform of the Adler function at $u=n+1$. It becomes apparent that the prefactor of Eq. (\[BTdLb2\]) is not accidental: the pole in $B[\widehat D](u)$ that corresponds to the condensate that contributes maximally to moments of $w=x^n$ is [not]{} cancelled by the prefactor of Eq. (\[BTdLb2\]). For monomials $x^n$ with $n\geq0$ three cases can be distinguished: - If $n=0$ all poles become simple poles, since there is no contribution from OPE condensates, apart from the $\alpha_s$-suppressed terms, under the assumptions of Eq. (\[OPEdeltaD\]). In particular, the pole at $u=2$ which was simple is exactly cancelled and the function is regular at $u=2$. - If $n=1$, the dominant contribution from the OPE is the one from $D=4$. The pole at $u=2$ related to this OPE contribution is not canceled and all other IR and UV poles become simple poles. This is a distinct situation because it is the only case where $B[\delta^{(0)}_{x^n}]$ is singular at $u=2$, in all other cases the leading IR singularity is located at $u=3$. - Finally, if $n\geq 2$, all IR poles for $u>2$ become simple poles, with the exception of the pole at $u=n+1$, which remains double and is now the only double pole in $B[\delta^{(0)}_{x^n}]$ — all others are reduced to simple poles by the zeros of $\sin(\pi u)$. In this case, the pole at $u=2$ that corresponds to the contributions due to the gluon condensate is again exactly cancelled by $\sin(\pi u)$ and the function is analytic at $u=2$. $w(z)$ $u=-2$ $u=-1$ $u=1$ $u=2$ $u=3$ $u=4$ $u=5$ -------- ---------------------- ---------------------- ------- --------- ----------------- ------------------ ------------------ -- -- $1$ $8.411\times10^{-4}$ $2.672\times10^{-2}$ $0$ $0$ $-21.00$ $-18.53$ $-29.43$ $z$ $6.309\times10^{-4}$ $1.781\times10^{-2}$ $0$ $17.85$ $-41.99$ $-27.79$ $-39.24$ $z^2$ $5.047\times10^{-4}$ $1.336\times10^{-2}$ $0$ $0$ $\boxed{6.999}$ $-55.58$ $-58.86$ $z^3$ $4.206\times10^{-4}$ $1.069\times10^{-2}$ $0$ $0$ $41.99$ $\boxed{-32.42}$ $-117.7$ $z^4$ $3.605\times10^{-4}$ $8.907\times10^{-3}$ $0$ $0$ $21.00$ $55.58$ $\boxed{-104.0}$ $z^5$ $3.154\times10^{-4}$ $7.634\times10^{-3}$ $0$ $0$ $14.00$ $27.79$ $117.7$ : Residues for the dominant poles in the Borel transform of $\delta^{(0)}_{w_i}$ for the first six monomials. Boxed numbers refer to residues of double poles, all other poles are simple poles. \[tab:w\_monomial\] In Tab. \[tab:w\_monomial\], we show the residues of the dominant UV and IR poles for the first six monomials, which are the building blocks for most of the moments used in the literature. Residues of the double poles are shown as boxed numbers. The results of this table can be used to understand a few features of specific cases. For example, in the Borel transform of the kinematic moment, Eq. , a partial cancellation of the leading UV renormalon is manifest: its residue is reduced by a factor of 3.3. The perturbative series associated with this moment is expected to display a more tamed behaviour, with the asymptotic nature setting in later. We are now in a position to reassess some of the findings of Ref. [@Beneke:2012vb] in the light of these results. One of the main observations of Ref. [@Beneke:2012vb] is that the perturbative series for moments of weight functions that contain the monomial $x$ tend to be badly behaved, in the sense that the series never stabilise around the true value of the function, it displays what was called a “run-away behaviour". This can be directly linked to the fact that the Borel transform of these moments are the only ones that have the singularity at $u=2$. The contribution of this renormalon to the coefficients of the series is fixed sign and it is large at higher orders. In order to establish the correspondence between the leading renormalons and the behaviour of the perturbative series, we will make use of an even simpler model. Since the series is dominated by the leading renormalons, we can construct an approximation to the result in large-$\beta_0$ using only the leading UV pole and the first two IR poles, which corresponds to truncating the sum in Eq. at its first term. We know from the works of Ref. [@Beneke:2008ad; @Beneke:2012vb; @Boito:2018rwt] that such a minimalistic model should be largely sufficient to capture the main features of the full result in large-$\beta_0$. In Fig. \[fig:Adler\_Lb\] we confirm this expectation by plotting the results for the Adler function in large-$\beta_0$ and in its truncated version, normalized to the value of the Borel integral in each case, which removes an overall normalization effect that is immaterial here (throughout this paper we use $\alpha_s(m_\tau^2)=0.316(10)$ [@Patrignani:2016xqp]). In Fig. \[fig:Adler\_Lb\], one sees that the results are essentially identical for our purposes. However, the simplicity of this model prevents the study of moments that are maximally sensitive to condensates with dimension $D \geq 8$, because the corresponding renormalon poles are not included. [![Adler function order by order in $\alpha_s$ in large-$\beta_0$ and in the truncated version that includes only the leading UV and the first two IR renormalon poles. Both results are normalised to the respective Borel integrals. The horizontal band gives the ambiguity arising from the IR poles, in the prescription of Ref. [@Beneke:2008ad]. Here and elsewhere we use $\alpha_s(m_\tau^2)=0.316(10)$ [@Patrignani:2016xqp].[]{data-label="fig:Adler_Lb"}](Figs/LB0-complete-Adler "fig:"){width=".7\columnwidth"}]{} We start by considering the moment of $w(x)=x$. The perturbative expansion of $\delta^{(0)}_x$ in the truncated model for FOPT and CIPT are displayed in Fig. \[fig:TruncatedLbweqx\]. The FOPT series shows the “run-away behaviour” identified in Ref. [@Beneke:2012vb]. It overshoots the true value, at first, and later crosses it and runs into the asymptotic regime with almost no stable region. The CIPT series is better behaved but also overshoots the true value and then runs into the asymptotic behaviour, with sign alternating coefficients, much earlier than FOPT. To understand this pattern we can use the Borel transform of $\delta^{(0)}_{x}$ which is rather simple in the truncated model, $$B[\delta^{(0)}_x]_T (u) = \frac{4 \text{e}^{5u/3}\sin(\pi u)}{27\pi^2}\Big{[} \frac{8}{u} + \frac{8(5u-4)}{3(2-u)^2} - \frac{(13u+19)}{3(1+u)^2} - \frac{(17 u - 57)}{(3-u)^2} \Big{]}.$$ The result exhibits the UV pole at $u=-1$, as well as the IR poles at $u=2$ and $u=3$. All poles are simple due to the zeros of the $\sin(\pi u)$ in the prefactor. We also note that the Borel transform has a regular part, which stems from the first term within square brackets (the would-be pole at zero is also canceled by the prefactor). In Fig. \[fig:TruncatedLbPolesweqx\] we show the breakdown of the different contributions to the perturbative series in FOPT. The series is dominated by the regular contribution which initially overshoots the true value. At higher orders, the first IR and UV poles dictate the tendency and the series never stabilizes around the true value. The IR contribution is negative and is responsible for the run-away behaviour, with a superimposed sign alternation from the UV pole. We now turn to the pinched moments without the term in $x$. In Fig. \[fig:TruncatedLbweq1mx2\], we show the results for $w=1-x^2$. The Borel transform of $\delta^{(0)}_{1-x^2}$ is regular at $u=2$ and the leading UV pole is partially cancelled, as we can infer from the results of Tab. \[tab:w\_monomial\]. This translates into a smoother series. Now the FOPT series nicely approaches the true value and remains stable around it for several orders until eventually entering the asymptotic regime, when the leading UV pole takes over. The result for CIPT, on the other hand, is less accurate (red dashed line in Fig. \[fig:TruncatedLbweq1mx2\]). It approaches the Borel sum of the series only when the asymptotic behaviour has already set in. Finally, we comment on the results for $w(x)=1$. This moment lies somewhere in between the two extreme cases we discussed above. It also benefits from being regular at $u=2$ but the partial cancellation of singularities that happens in pinched moments is not present. In this case, FOPT is able to approach the result although at the expense of overshooting it for the first four orders or so. We omit the plots in this case for the sake of brevity (the result in the context of Borel models can be found in Ref. [@Beneke:2012vb]). One should finally remark that, in general, the perturbative series for $\delta^{(0)}_{w_i}$ are better behaved than the Adler function series, shown in Fig. \[fig:Adler\_Lb\]. This fact is a consequence of the Borel transform of $\delta^{(0)}_{w_i}$ being significantly less singular than the Adler function counterpart. The sign alternation in the Adler function starts already at $\mathcal{O}(\alpha_s^5)$ and the perturbative series never stabilises around the Borel sum. This does not prevent, however, the pinched moments without the linear term from having a very good perturbative behaviour, as exemplified in Fig. \[fig:TruncatedLbweq1mx2\]. ### Partial conclusions We are in a position to draw a few conclusions from the study of spectral function moments in large-$\beta_0$ and its truncated form: - The Borel transform of $\delta^{(0)}_{w_i}$ for polynomial weight functions is less singular than the Borel transform of the Adler function. The prefactor that appears in $B[\delta^{(0)}_{w_i}]$ cancels an infinite number of poles. Precisely for this reason, the perturbative expansions of $\delta^{(0)}_{w_i}$ are in general smoother than the Adler function counter part. - The pattern of the remaining poles in the Borel transform of $\delta^{(0)}_{w_i}$ can be understood in terms of the contributions from the OPE condensates. The Borel transform has a pole at $u=2$ if and only if the weight function contains a term proportional to $x$. The behaviour of the perturbative series associated with these moments is qualitatively different and the true value of series is not well approached neither by FOPT nor by CIPT, as already discussed in Ref. [@Beneke:2012vb]. - The Borel transform of the moments from the monomials $w(x)=x^{n}$, with $n>1$, has only one double pole at $u=n+1$, related to the OPE condensate with $D=2(n+1)$ to which the moment is maximally sensitive, in the sense of Eq. (\[OPEdeltaD\]). - Moments that are pinched and do not contain the term proportional to $x$ are particularly stable. Their Borel transform is regular at $u=2$ and there is a partial cancellation of the leading UV renormalon, which translates into a smoother series. The series, in these cases, is well described by FOPT while CIPT struggles to approach the Borel sum and runs into the asymptotic behaviour already at $\mathcal{O}(\alpha_s^{4})$ or $\mathcal{O}(\alpha_s^5)$. In the remainder we will discuss the case of QCD. With the use of a convenient scheme transformation and a redefinition of the Borel transform one is able to show that results in QCD are very similar to the ones obtained in large-$\beta_0$. Results in QCD {#resultsQCD} -------------- Two main ingredients enter the discussion of the previous section. First, we have full knowledge about the renormalon structure of the Adler function. In particular we know which poles exist and if they are double or simple poles, exactly. Second, in the derivation of Eq. (\[BTdLb2\]), because we work in the large-$\beta_0$ limit, we made use of the one-loop running of $\alpha_s$. In the case of QCD, on the other hand, we have to be content with a partial knowledge about the renormalon structure of the Adler function. The positions of the singularities are unchanged, but now they are no longer poles and become branch cuts. The running of the coupling is also much more involved when terms beyond one loop are included in the $\beta$ function. In order to be able to obtain an analytical expression for the Borel transform of $\delta^{(0)}_{w_i}$, it is useful to leave the ${{\overline{\rm MS}}}$ scheme and work in another class of schemes which have a particularly simple $\beta$ function. Without loss of generality, we will employ the $C$ scheme introduced in Ref. [@Boito:2016pwf] in the derivation we perform below. The implementation of the $C$ scheme is based on the fact that, when going from an input scheme, say the ${{\overline{\rm MS}}}$, to another scheme that we denote with hatted quantities, the QCD scale parameter $\Lambda$ changes as [@Celmaster:1979km] = \_ e\^[c\_1/\_1]{}, where the coefficient $c_1$ is the first non-trivial coefficient in the perturbative expansion of the coupling $\hat a\equiv \hat \alpha_s/\pi$ in terms of $a\equiv \alpha_s^{{{\overline{\rm MS}}}}/\pi$: a = a +c\_1a\^2+c\_2a\^3+. With the expression of the scale-invariant QCD $\Lambda$ parameter one can then relate the two schemes with a continuous parameter $C$, that measures the shift in $\Lambda$, by + a\_Q = + C + a\_Q - \_1 \_0\^[a\_Q]{} , where $C=\frac{-2c_1}{\beta_1}$, we defined $$\frac{1}{\tilde\beta(a)} \,\equiv\, \frac{1}{\beta(a)} - \frac{1}{\beta_1 a^2} + \frac{\beta_2}{\beta_1^2 a},$$ and we have made explicit the renormalisation scale dependence in $a_Q$. A relation that is important in the remainder is the analogue of Eq. (\[eq:alphasrunning1loop\]) in the $C$ scheme which reads =() + -[a]{}\_Q.\[eq:oneoverahat\] In this scheme, the $\beta$-function is known exactly and reads -Q ([a]{}\_Q) = .\[betainCscheme\] This fact enormously simplifies the task of obtaining a closed form for the Borel transform of $\delta^{(0)}_{w_i}$ in QCD. Finally, we remark that the dependence on the scheme parameter $C$ is, in fact, governed by the same function -2 = . The coupling becomes smaller for larger values of $C$ and the theory ceases to be perturbative for $C\approx -1.5$ (using the ${{\overline{\rm MS}}}$ scheme as input) [@Boito:2016pwf]. This means that the coupling in the $C$-scheme depends on a particular combination of the scale and scheme parameters $\alpha_s\equiv \alpha_s(Q^2e^C)$. Scale and scheme variations become, therefore, completely equivalent. The explicit expressions for the perturbative coefficients relating the ${{\overline{\rm MS}}}$ and the $C$ schemes, together with further details, can be found in the original publications [@Boito:2016pwf; @Boito:2016feb]. Finally, we remark that there is no value of $C$ that corresponds strictly to the ${{\overline{\rm MS}}}$, but for $C\approx 0$ the results are very similar (at one loop, $C=0$ corresponds to the ${{\overline{\rm MS}}}$ exactly). For schemes in which the $\beta$-function takes the form of Eq. (\[betainCscheme\]) it is convenient to work with a modified Borel transform defined as [@Brown:1992pk] $$\mathcal{B}[\widehat D](t) = \sum\limits_{n=1}^\infty \frac{\Gamma(1+{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}t)}{\Gamma(n+1+{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}t)} n \, \frac{\hat c_{n,1}}{\pi^n} \ t^n, \label{eq:modified-borel-transform-adler}$$ where ${{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}=\beta_2/(\beta_1\pi)$ and $\hat c_{n,1}$ are the Adler function coefficients in the $C$ scheme. With this definition, the Borel sum of the series now reads $$\widehat D(\hat \alpha_s (Q^2)) = \int\limits_0^\infty \frac{dt}{t} e^{-t/\hat \alpha_s(Q^2)} \frac{[t/\hat \alpha_s (Q^2)]^{{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}t}}{\Gamma(1+{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}t)} \mathcal{B}[\widehat D](t). \label{eq:modified-borel-int-adler}$$ The asymptotic expansion to the latter result is obtained using Eq. (\[eq:modified-borel-transform-adler\]) in (\[eq:modified-borel-int-adler\]) and gives, as expected [@Brown:1992pk], D = \_[n=1]{}\^c\_[n,1]{} [a]{}\^n\_Q. The modified Borel transform has renormalon singularites at the same location as the usual Borel transform, but their exponent is shifted. As demonstrated in Ref. [@Brown:1992pk], if the usual Borel transform has a singularity of the form $$B[\widehat D](u)\sim \frac{1}{(p-u)^\alpha},$$ the modified Borel transforms behaves for $u\sim p$ as $$\mathcal{B}[\widehat D](u) \sim \frac{1}{(p-u)^{\alpha-\frac{2\pi p}{\beta_1}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}}},\label{eq:shift}$$ with the exponent of the singularity shifted by $\frac{2\pi p}{\beta_1}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}=+2p(\beta_2/\beta_1^2)$ and, as before, $u=\frac{\beta_1 t}{2\pi}$. Let us now calculate the modified Borel transform of $\delta^{(0)}_{w_i}$ in the C scheme. The calculation is very similar to what was done in large-$\beta_0$. Using Eq. (\[eq:modified-borel-int-adler\]) into Eq. (\[eq:delta0\]) we obtain $$\delta^{(0)}_{w_i}=\frac{1}{2 \pi }\int_{0}^{2\pi}d\phi \ W_i(e^{i\phi}) \ \int\limits_0^\infty \frac{dt}{t} \ e^{-t/\hat \alpha_s(-s_0 e^{i\phi})} \frac{[t/\hat \alpha_s(-s_0 e^{i\phi})]^{{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}t}}{\Gamma(1+{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}t)} \label{eq:delta0modborel} \mathcal{B}[\widehat D](t).$$ With the use of Eq. (\[eq:oneoverahat\]) one finds \^[-t/\_s(-s\_0 e\^[i]{})]{} = e\^[-t/\_s(s\_0)]{}e\^[-iu(-)]{}()\^[[[0.75mu’26-9.75mu]{}]{}t]{}, and inverting the order of the integration in Eq. (\[eq:delta0modborel\]) one obtains, for the monomial weight function $w(x)=x^n$, the following result $$\mathcal{B}[\delta^{(0)}_{x^n}](u) = \frac{2}{1+n-u}\frac{\sin(\pi u)}{\pi u}\mathcal{B}[\widehat D](u). \label{eq:modBoreldelta}$$ This shows that the relation of Eq. (\[BTdLb2\]) is, in fact, much more general, since any scheme can be brought to the $C$ scheme without loss of generality. The prefactor is the same in QCD and in large-$\beta_0$, and so is the enhancement of the renormalon associated with the contribution with dimension $D=2(n+1)$ in the OPE.[^5] The main difference is that in QCD the singularities of $\mathcal{B}[\widehat D](u)$ are, in general, branch points and are no longer poles. The exponent of the singularities is related to the anomalous dimension of the associated operator contributing to the OPE. To make further progress, let us look at the explicit structure of the IR singularities. In the notation of [@Beneke:2008ad], the singularities of the usual Borel transform are written as $$B[\widehat D_p^{\rm IR}] \equiv \frac{d_p^{\rm IR}}{(p-u)^{1+\tilde \gamma }}\left[1+\tilde b_1^{(p)}(p-u)+\tilde b_2^{(p)}(p-u)^2+\cdots \right],$$ where the constants $\tilde \gamma$ and $\tilde b_i^{(p)}$ depend on the anomalous dimension of the associated operator in the OPE as well as on $\beta$-function coefficients. The explicit expression for the exponent $\tilde \gamma$ is $$\tilde \gamma = 2p \frac{\beta_2}{\beta_1^2} - \frac{\gamma_{O_d}^{(1)}}{\beta_1}\label{eq:gammatilde},$$ where the anomalous dimension associated with the operator $O_d$ is defined as $$-\mu \frac{d}{d\mu} O_d(\mu) = \left(\gamma_{O_d}^{(1)}a_\mu +\gamma_{O_d}^{(2)}a_\mu^2+\cdots \right)O_d(\mu).$$ For the modified Borel transform we have then $$\mathcal{B}[\widehat D_p^{\rm IR}] \sim \frac{1}{(p-u)^{1+\tilde \gamma - \frac{2\pi p}{\beta_1}{{\mkern0.75mu\mathchar '26\mkern -9.75mu\lambda}}}} = \frac{1}{(p-u)^{1 - \gamma_{O_d}^{(1)}/\beta_1}},$$ where the first factor in the r.h.s. Eq. (\[eq:gammatilde\]) is exactly cancelled by the shift in the singularity of Eq. (\[eq:shift\]). For the discussion of the Borel transformed $\delta^{(0)}_{w_i}$ it is crucial to inspect the leading IR renormalon. This renormalon is related to the gluon condensate which can be expressed in terms of the scale invariant combination $\langle aG^2\rangle$ [@Pich:1999hc]. In this case, the leading IR singularity of the Adler function in the $C$-scheme, and using the modified Borel transform, reduces simply to $$\mathcal{B}[\widehat D_2^{\rm IR}] \sim \frac{1}{(2-u)},\label{eq:IRCscheme}$$ which is a simple pole, exactly as in the large-$\beta_0$ limit. This is remarkable because, with Eq. (\[eq:modBoreldelta\]), one can directly translate many of our conclusions from large-$\beta_0$ to QCD, in particular, $\mathcal{B}[\delta^{(0)}_{w_i}]$ has a pole at $u=2$ if and only if the weight function contains a term proportional to $x$. The conclusion that $\mathcal{B}[\delta^{(0)}_{w_i}]$ is less singular also remains valid, and it is again true that the singularity associated with the contributions in the OPE to which the moment is maximally sensitive are not altered by the prefactor of Eq. (\[eq:modBoreldelta\]). In view of the above discussion, and the results of Eqs. (\[eq:modBoreldelta\]) and (\[eq:IRCscheme\]), we learn that the same mechanisms of suppression, enhancement, and cancellation of renormalon singularities identified in large-$\beta_0$ are also at work in QCD. Similarities between the results in the two cases were identified in Ref. [@Beneke:2012vb] although the explicit connection with the renormalon singularities of $\mathcal{B}[\delta^{(0)}_{w_i}]$ was not investigated in that work. Although in QCD we have only partial information about the renormalon singularities, and in particular about their numerator, we are in a position to speculate that the reason behind the good or bad perturbative behaviour of the different moments is rooted in the same interplay between the renormalons of $\mathcal{B}[\delta^{(0)}_{w_i}]$. Here, we study the QCD perturbative series for different moments using the model-independent reconstruction of the higher-order coefficients of Ref. [@Boito:2018rwt], where the mathematical method of Padé approximants was used to describe the series. In Fig. \[fig:QCDwPAs\], we show results for four emblematic moments, in FOPT and CIPT, with $s_0=m_\tau^2$ and in the ${{\overline{\rm MS}}}$; the shaded bands represent an uncertainty that stems from the Padé approximant method, as discussed in [@Boito:2018rwt]. In Figs. \[fig:QCDweqtau\] and \[fig:QCDweq1mx2\], the results for two moments that display good perturbative behaviour are shown: $w_\tau$ and $w(x)=1-x^2$, respectively. The horizontal yellow bands represent an estimate for the Borel integral of the moments within the Padé-approximant description. Again, the FOPT series approaches the true value, as predicted by the Padé approximants, and is rather stable around it until at least the 8-th order. (We relegate to App. \[borelint\] a more detailed discussion about the Borel integrals together with a comparison with results from the model of Refs. [@Beneke:2008ad; @Beneke:2012vb], which are similar to ours.) In Fig. \[fig:QCDweqx\], we show the results for the monomial $w(x)=x$, which exacerbates the run-away behaviour that stems from the leading IR singularity, as in the large-$\beta_0$ case of Fig. \[fig:TruncatedLbweqx\]. Here CIPT is relatively good agreement with the true results, but this is not the case for other moments containing the linear term, such as $w(x)=1-x$, shown in Fig. \[fig:QCDweq1mx\]. This moment inherits the run-away behaviour of the monomial and both FOPT and CIPT are rather unstable, never stabilizing around the true result. Finally, one can corroborate our conclusion that the bad perturbative behaviour of moments containing the linear term is related to the leading IR singularity by considering the “alternative model" of Ref. [@Beneke:2012vb]. In this case, a model for the QCD Adler function is constructed without the leading IR singularity. In the $C$-scheme and using the modified Borel transform, this means that, for this model, the Borel transform of $\delta^{(0)}_{w_i}$ is regular at $u=2$, since in the prefactor of Eq. (\[eq:modBoreldelta\]) the pole is cancelled by the zero in $\sin(\pi u)$. As shown in [@Beneke:2012vb], the run-away behaviour is not present in this case, which shows, once more, that it stems from the leading IR singularity in $B[\delta^{(0)}_{w_i}]$. In conclusion, with the use of the $C$-scheme and the modified Borel transform, the relation between the Borel transform Adler function and the Borel transform of $\delta^{(0)}_{w_i}$ are formally the same in QCD and in the large-$\beta_0$ limit. The singularities related to the contributions in the OPE are equally enhanced or suppressed and, in general, $\delta^{(0)}_{w_i}$ is significantly less singular than the Adler function. In the case of the leading IR renormalon the parallel is strict since within this framework it is a simple pole both in QCD and in large-$\beta_0$. The phenomenological consequences are then the same: moments with a linear term in $x$ display an unstable perturbative behaviour. Finally, the moments with good perturbative behaviour in large-$\beta_0$ are also well behaved in QCD, at least if FOPT is used. The results from the model-independent Padé approximant reconstruction of the series are qualitatively similar to the “Borel model" of Refs. [@Beneke:2008ad; @Beneke:2012vb] which attests the robustness of our conclusions. Optimal truncation with scheme variations {#optimization} ========================================= We close this work with a discussion of the optimal truncation of the (asymptotic) series associated with the moments that display good perturbative behaviour. In Ref. [@Boito:2016pwf; @Boito:2016yom] it has been suggested that, in the spirit of an asymptotic series, the optimal truncation for the perturbative expansion of the Adler function and of integrated moments is achieved by choosing the scheme (or scale) in which the last known coefficient of the series vanishes. In this case, by construction, the smallest term of the series, which is zero, is precisely the last known term, which makes it the ideal point for the optimal truncation of the asymptotic series.[^6] Through this procedure, one expects to make maximum use of the available information from perturbative QCD. Here we show that this type of optimization works very well in FOPT for all the moments with good perturbative behaviour, within the reconstruction of the series provided by the Padé approximants of Ref. [@Boito:2018rwt]. We will work in the $C$-scheme and perform variations of the continuous scheme parameter $C$. However, as discussed in Sec. \[resultsQCD\], in this scheme, scale and scheme transformations are essentially equivalent and the same results can be achieved by renormalisation scale variations. Let us illustrate the procedure with the help of a concrete case. The FOPT expansion for $w(x)=1-x^2$ and $s_0=m_\tau^2$ in the $C$-scheme is given by $$\begin{aligned} \delta^{(0)}_{1-x^2} &= 1.333 \ \hat a_Q + (6.186+3C) \ \hat a_Q^2 + (27.77 + 33.17C + 6.750C^2) \ \hat a_Q^3 \nn \\ & + (119.4 + 246.4C + 124.0C^2 + 15.19C^3) \ \hat a_Q^4 \nn \\ & + (90.73 + 1512C + 1329C^2 + 398.9C^3 + 34.17C^4 + 1.333\,c_{5,1}) \ \hat a_Q^5 + \cdots ,\end{aligned}$$ where we show the four exactly known contributions (up to and including $\alpha_s^4$) plus the first unknown contribution, proportional to $c_{5,1}$. At order $\alpha_s^5$, the terms without $c_{5,1}$ depend only on $\beta$-function coefficients and lower $c_{n,1}$ and are known exactly. It is customary to include the fifth term in realistic $\alpha_s$ analysis through an estimate of $c_{5,1}$ which here is taken to be $c_{5,1}=277\pm 51$ [@Boito:2018rwt] — but we will see that the results do not depend strongly on the value of $c_{5,1}$. The optimized truncation is obtained then by finding the value(s) of $C$ for which the coefficient of $\hat a_Q^5$ vanishes. In the case of $1-x^2$, for FOPT with our central value for $c_{5,1}$, one finds two such values: $C_1= -1.463$ and $C_2=-0.4763$. The former leads to a rather unstable series, that we discard, since the coupling is already entering the non-perturbative regime \[$\hat \alpha_s(C_1,m_\tau^2)=0.531$\] while the latter is still in the perturbative regime \[$\hat \alpha_s(C_2,m_\tau^2)=0.355$\] and gives rise to the optimized result. In Fig. \[fig:QCDweq1mx2\_SV\] we compare the optimized series for $\delta^{(0)}_{1-x^2}$ (green dot-dashed line) with the usual ${{\overline{\rm MS}}}$ result (solid blue line) using the higher-order coefficients and the Borel sum from the description of Ref. [@Boito:2018rwt]. One sees that the optimized FOPT series approaches the true value faster than the ${{\overline{\rm MS}}}$ result, already at $\mathcal{O}(\alpha_s^3)$, and remains rather stable around it. This optimization is related to the larger value of $\hat \alpha_s$ which leads to a series that “converges" faster than the ${{\overline{\rm MS}}}$ one.[^7] With the optimized series, an estimate of the true result is obtained with the truncation at $\mathcal{O}(\alpha_s^5)$ which gives \^[(0)]{}\_[1-x\^2]{} (a\_,C=-0.4763) = 0.23580.0017,\[eq:delta1mx2truncated\] where the error is due to the variation of $c_{5,1}$ within one sigma. It is clear from Fig \[fig:QCDweq1mx2\_SV\] that this leads to an excellent agreement with the true result — as predicted from the results of [@Boito:2018rwt] — which reads $0.2364\pm 0.0020$. One should also remark that the procedure is rather independent of the value of $c_{5,1} $ that is used. An uncertainty due to the value of $\alpha_s$, for example, would be about one order of magnitude larger than the uncertainty shown in Eq. (\[eq:delta1mx2truncated\]). An attempt to apply the same procedure to the CIPT series does not lead to any significant improvement with respect to the (already bad) result obtained in the ${{\overline{\rm MS}}}$, as shown in the red and purple lines in Fig. \[fig:QCDwPAs\_SV\]. The optimization can also be applied to the kinematic moment, $w_\tau$. The result is again very good and the acceleration of the series is even more obvious, as displayed in Fig. \[fig:QCDweqtau\_SV\]. For illustration, we also show, in grey, the series in a scheme with larger value of $C$, namely $C=0.8$ for which $\hat \alpha_s(C=0.8,m_\tau^2)=0.2554$. One sees that in a scheme with a very small value of the coupling the convergence is smooth but very slow for practical purposes, where only the first few terms are available. Similar results can be obtained for the other moments that have a good perturbative behaviour. As an example, in Fig. \[fig:QCDweq1m4x3p3x4\_SV\] we show the result of the optimization of one of the pinched moments introduced in Ref. [@Pich:2016bdg]. It is also interesting to analyse a borderline case, namely that of $w(x)=1$. This is not a moment with a bad perturbative behaviour (it does not have a linear term in $x$), but it is also not among the most stable peturbative series, since it does not benefit from the partial cancellation of renormalons. The result in this case is shown in Fig. \[fig:QCDweq1\_SV\]. Here the ${{\overline{\rm MS}}}$ series overshoots the true value up to $\mathcal{O}(\alpha_s^4)$, as shown in Fig. \[fig:QCDweq1\_SV\]. In this case, the value of $C$ that optimizes the truncation turns out positive and the optimal scheme has a smaller value of $\alpha_s$ than in ${{\overline{\rm MS}}}$. The optimization is achieved by avoiding the overshooting of the true result that is prominent in the ${{\overline{\rm MS}}}$ series. The final result is more stable than that in the ${{\overline{\rm MS}}}$ and one could expect a smaller error from the truncation of the series, but the acceleration is not very significant. Finally, moments with bad perturbative behaviour do not improve in any significant way when we apply the optimization described here. A more stable perturbative expansion for these moments can be achieved with the method of conformal mappings, making use of the information about the location of the renormalon singularities [@Caprini:2011ya; @Abbas:2013usa; @Caprini:2017ikn; @Caprini:2018agy]. Even with this technique, in some cases, the series approaches the true value only at high orders. Conclusions =========== In this work, we have discussed in detail the perturbative behaviour of integrated spectral function moments and the connection with the renormalon singularities of their Borel transformed series, denoted $B[\delta^{(0)}_{w_i}]$. The understanding of the perturbative expansion of such moments is important in guiding the choice of moments employed in realistic $\alpha_s$ determinations from low-$Q^2$ FESRs. Moments with tamed perturbative expansions are more reliable and lead to smaller uncertainties from the truncation of perturbation theory. In large-$\beta_0$, one can easily establish the relation between the renormalons of the Adler function and those of the integrated moments in the ${{\overline{\rm MS}}}$ scheme. An infinite number of renormalon poles of the Adler function is cancelled and $B[\delta^{(0)}_{w_i}]$ is significantly less singular. In particular, for polynomial moments, the leading IR pole is exactly cancelled unless the weight function contains a term proportional to $x$. The weight functions with this term are therefore the only ones that are singular at $u=2$ and they display an unstable perturbative behaviour that stems from the contribution of this IR pole to the perturbative series. For the pinched moments that had been identified as having a good perturbative behaviour in Ref. [@Beneke:2012vb], we found additional cancellations of renormalon singularities, which are related to a better behaviour at higher orders and postpone the asymptotic regime of the series. Using the $C$ scheme and a modified Borel transform we have been able to show, in Eq. (\[eq:modBoreldelta\]), that the relation between Borel transformed moments and the Borel transformed Adler function is the same in QCD and in large-$\beta_0$. In Eq. (\[eq:IRCscheme\]), we have also shown that the leading IR singularity in this framework is again a simple pole. These are the main results of this paper since they allow us to conclude that the same mechanisms of enhancement, suppression, and partial cancellation of renormalon singularities responsible for the behaviour of the perturbative moments in large-$\beta_0$ are operative in QCD as well. The similar behaviour of the integrated spectral function moments in the two cases is therefore no surprise and again the pinched moments without the linear term are the best ones (as pointed out in Ref. [@Beneke:2012vb]). The instabilities related to the leading IR pole are also present in QCD. Finally, we have shown that it is possible to use renormalisation scheme (or scale) variations to accelerate the convergence of the moments that display good perturbative behaviour. This had been suggested in Ref. [@Boito:2016pwf] for the $R_\tau$ ratio but it had never been investigated systematically before. In conclusion, we have been able to understand the instabilities and stabilities of the perturbative expansions of integrated spectral function moments in terms of their renormalons. Apart from the implications for the choice of moments in precise $\alpha_s$ analysis, our results can be used in the context of Borel models for the Adler function, since we have shown that scheme transformations and the modified Borel transformed can be used in order to simplify the structure of the leading IR singularity, related to the gluon condensate, which becomes a simple pole. In fact, the results in large-$\beta_0$ and QCD are therefore much more similar than previously thought. Our findings also suggest that alternative expansions that suppress some of the renormalons may lead to much more stable results, and we plan to investigate this issue further in the near future. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Irinel Caprini, Matthias Jamin, and Santi Peris for comments on a previous version of the manuscript. Discussions and email exchanges about the modified Borel transform with Santi Peris and Matthias Jamin are gratefully acknowledged. The work of DB is supported by the São Paulo Research Foundation (FAPESP) grant No. 2015/20689-9 and by CNPq grant No. 309847/2018-4. The work of FO is supported by CNPq grant No. 141722/2018-5. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior — Brasil (CAPES) — Finance Code 001. Conventions for the QCD beta-function {#betafunction} ===================================== We define the QCD $\beta$ function as $$\beta(a_\mu) \equiv -\mu \frac{d a_\mu}{d\mu} = \beta_1 a_\mu^2 + \beta_2 a_\mu^3 + \beta_3 a_\mu^4 + \beta_4 a_\mu^5 + \beta_5 a_\mu^6 +\cdots,$$ where the first five coefficients are known analytically [@Baikov:2016tgj; @Luthe:2017ttg]. It is important to highlight that $\beta_1$ and $\beta_2$ are scheme independent and, in our conventions, they are given by $$\beta_1 = \frac{11}{2} - \frac{1}{3}N_f, \qquad \beta_2 = \frac{51}{4}-\frac{19}{12}N_f,$$ with $N_f$ being the number of flavours. In the particular case of $N_f=3$, relevant here, we have $$\beta_1 = \frac{9}{2}, \qquad \beta_2=8.$$ Details on the Borel integrals from Padé approximants {#borelint} ===================================================== In this appendix we discuss in further detail how the Borel integrals, or “true values", of the perturbative series are obtained. We also compare our results with those of Ref. [@Beneke:2012vb]. Our results are based on the reconstruction of the higher-order coefficients performed in Ref. [@Boito:2018rwt], using Padé approximants. Several methods have been studied in [@Boito:2018rwt], using different variants of rational approximants, and constructing the approximants to the Borel transformed Adler function, to the Borel transformed $\delta^{(0)}_{w_\tau}$, as well as to the FOPT expansion of $\delta^{(0)}_{w_\tau}$. Due to the less singular structure of $B[\delta^{(0)}_{w_\tau}]$, Padé approximants built to this Borel transform converge faster and were the basis for the main result of Ref. [@Boito:2018rwt], which we use here. The coefficients of the Adler function are then extracted, indirectly, from the series of $\delta^{(0)}_{w_\tau}$. With these coefficients, given in Tab. 6 of Ref. [@Boito:2018rwt], one can obtain the expansion of any moment, in FOPT or CIPT, rather accurately up to order $\mathcal{O}(\alpha_s^{10})$. The main advantage of the use of Padé approximants is that the method is almost completely model independent. In this framework, however, no unique representation of the Borel transformed Adler function is obtained, which makes the task of calculating the Borel integrals for each moment less straightforward. In order to estimate the Borel integrals we have constructed new Padé approximants, following the same methods of Ref. [@Boito:2018rwt], to each of the Borel transformed $\delta^{(0)}_{w_i}$. In all cases where the moments have good perturbative behaviour, the approximants converge very fast, only three coefficients suffice to obtain a rather stable result. This means that the prediction from these Padé approximants are based only on the exactly known QCD results. From the Borel transform described by the Padés one can then easily calculate the Borel integral. Of course, more than one Padé can be built from the same input and we have constructed many different approximants, belonging to different sequences and also using Dlog Padés [@Boito:2018rwt], in order to estimate the horizontal error band shown in Figs. \[fig:QCDwPAs\] and \[fig:QCDwPAs\_SV\]. The moments containing the linear term $x$, however, lead to less stable results. In order to obtain a stable description of the Borel transform it is necessary to use more coefficients in the construction of the Padé approximants — which make these results less model independent since they require input from the higher-order coefficients predicted in [@Boito:2018rwt]. The final uncertainties in the horizontal (yellow) bands take into account the dispersion of the results from the use of different Padé approximants as well as the original uncertainty in the prediction of the coefficients of Ref. [@Boito:2018rwt]. In most cases, the former dominates. Finally, it is interesting to compare our Borel integrals with the ones from the description of Ref. [@Beneke:2012vb; @Beneke:2008ad]. In these works, the Borel transformed Adler function is modelled with its first three dominant renormalons, the leading UV and the first two IR singularities. The residues of the singularities are fixed such as to reproduce the known QCD results. The main advantage of this procedure is that one obtains a unique description of the Borel Adler function, from which all the results are derived. The disadvantage is a possible residual model dependence which could lead to unaccounted systematics. The results from the Padés are, however, in very good agreement with the “reference model" of [@Beneke:2012vb], although the uncertainties in the latter case, stemming only from the imaginary ambiguities in the Borel integral, are usually smaller. In Fig. \[fig:QCDwPAsRM\], we compare the two approaches to the Borel integral, for two exemplary moments, and show that they lead to very similar results. The Borel integral from the reference model of [@Beneke:2008ad] is shown as a green band, with a horizontal offset with respect to the results from Padé approximants, in yellow. In both cases, the FOPT series is preferred. [^1]: This new type of $\alpha_s$ extraction has been recently included in the 2019 update of the PDG world average, under the “low-$Q^2$ category" [@Tanabashi:2018oca]. [^2]: The use of modified Borel transforms in combination with the $C$ scheme in similar contexts has been suggested by M. Jamin and S. Peris [@communication]. [^3]: For our conventions regarding the QCD $\beta$ function we refer to App. \[betafunction\] [^4]: This result has been used in Refs. [@Boito:2018rwt; @Caprini:2019kwp]. [^5]: An approximate relation between the Borel transformed Adler function and the Borel transform of $\delta^{(0)}_{w_\tau}$ can be found in [@Caprini:2019kwp]. The result of Eq. (\[eq:modBoreldelta\]) is fully general. [^6]: It is not guaranteed that the truncation at the smallest term, which is known as superasymptotic approximation, is always the optimal truncation, but experience shows that it is very often the case [@Boyd1999]. [^7]: This can be seen as a manifestation of Carrier’s rule: “Divergent series converge faster than convergent series because they don’t have to converge" [@Boyd1999].
--- abstract: 'Integrated single-photon detectors open new possibilities for monitoring inside quantum photonic circuits. We present a concept for the in-line measurement of spatially-encoded multi-photon quantum states, while keeping the transmitted ones undisturbed. We theoretically establish that by recording photon correlations from optimally positioned detectors on top of coupled waveguides with detuned propagation constants, one can perform robust reconstruction of the density matrix describing the amplitude, phase, coherence and quantum entanglement. We report proof-of-principle experiments using classical light, which emulates single-photon regime. Our method opens a pathway towards practical and fast in-line quantum measurements for diverse applications in quantum photonics.' author: - Kai Wang - 'Sergey V. Suchkov' - 'James G. Titchener' - Alexander Szameit - 'Andrey A. Sukhorukov' title: 'In-line detection and reconstruction of multi-photon quantum states' --- Quantum properties of multiple entangled photons underpin a broad range of applications [@Zeilinger:2017-72501:PS] encompassing enhanced sensing, imaging, secure communications, and information processing. Accordingly, approaches for measurements of multi-photon states are actively developing, from conventional quantum tomography with multiple bulk optical elements [@James:2001-52312:PRA] to integrated photonic circuits [@Shadbolt:2012-45:NPHOT]. Latest advances in nanofabrication enable the integration of multiple single photon detectors based on superconducting nanowires [@Natarajan:2012-63001:SCST; @Kahl:2017-557:OPT]. Beyond the miniaturization, integrated detectors open new possibilities for photon monitoring inside photonic circuits [@Pernice:2012-1325:NCOM]. However there remains an open question of how to perform in-line measurements of the quantum features of multi-photon states encoded in their density matrices, while ideally keeping the transmitted states undisturbed apart from weak overall loss. Such capability would be highly desirable similar to the classical analogues [@Mueller:2016-42:OPT], yet it presents a challenging problem since traditional approaches for quantum measurements are not suitable. In particular, direct measurement methods [@Lundeen:2012-70402:PRL; @Thekkadath:2016-120401:PRL] are difficult to employ due to complex measurement operators. On the other hand, the conventional state tomography [@James:2001-52312:PRA], a most widely used reconstruction method, requires reconfigurable elements to apply modified projective measurements in different time windows. Recently, the reconstruction was achieved in static optical circuits [@Titchener:2016-4079:OL; @Oren:2017-993:OPT; @Titchener:2018-19:NPJQI] or metasurfaces [@Wang:1804.03494:ARXIV], however it comes at a cost of spreading out quantum states to a larger number of outputs, which is incompatible with in-line detection principle. In this work, we present a new conceptual approach for practical in-line measurement of multi-photon states using integrated detectors which is suitable for multi-port systems. We illustrate it for two waveguides in Fig. \[fig1\](a). The coupled waveguides (coupling coefficient $C$) have different propagation constants (detuned by $\beta$) and the length $L$ is exactly a revival period. An even number $M$ of weakly coupled single-photon click detectors are placed at $M/2$ cross-sections, starting from $z_1$ with equal distances $2L/M$ between each other. As we demonstrate in the following, the measurement of $N$-fold nonlocal correlations by averaging the coincidence events enables a full reconstruction of the density matrix $\rho$ for $N$-photon states. ![Conceptual sketch of in-line detection and reconstruction of $N$-photon state ${\rho}$ with two spatial modes. (a) Two waveguides with coupling constant $C$ and detuning $\beta$ in propagation constants. An even number $M$ of single-photon click detectors are positioned at $M/2$ equidistant cross-sections, illustrated for $M=6$ with D1–D6 labels. (b) Examples of simulated single-photon counts ($N=1$) and $N$-fold correlations (for $N=2$) for $z_1=0$ and $\beta=C / \sqrt{2}$, which enable full reconstruction of the input density matrix $\rho$.[]{data-label="fig1"}](Fig1-v2){width="\linewidth"} ![Robust reconstruction with waveguide detuning. (a,b) The analysis states on a Bloch sphere for (a) identical waveguides ($\beta=0$) and (b) introduced detuning ($\beta=C/\sqrt{2}$). Curves show the evolution along $z$, and arrows correspond to six detectors as indicated in Fig. \[fig1\](a). Red and blue colors denote waveguide a and b, respectively. (c,d) Inverse condition number $\kappa^{-1}$ versus (c) normalized detuning $\beta/C$ and (d) number of detectors $M$ for the optimal detuning $\beta=C/\sqrt{2}$. []{data-label="fig2"}](Fig2-v2){width="\linewidth"} The quantum state evolution along the waveguides, under the assumption of weakly coupled detectors, is governed by the Hamiltonian $$\hat{H}= \beta\hat{a}_1^\dagger\hat{a}_1 -\beta\hat{a}_2^\dagger\hat{a}_2 +C\hat{a}_1^\dagger\hat{a}_2 + C\hat{a}_2^\dagger\hat{a}_1,$$ where $\hat{a}_{q}^\dagger$ ($\hat{a}_{q}$) are the photon creation (annihilation) operators in waveguide number $q$, $\beta$ describes the propagation constant detuning, and $C$ is the coupling constant. We use Heisenberg representation [@Bromberg:2009-253904:PRL; @Peruzzo:2010-1500:SCI] and map time $t$ to the propagation distance $-z$. Hence the operator evolution reads $$\hat{a}_q(z) = \sum_{q'=1,2} \mathbf{T}^{\ast}_{q,q'}(z) \hat{a}_{q'}(0), \quad \hat{a}^\dagger_q(z) = \sum_{q'=1,2} \mathbf{T}_{q,q'}(z) \hat{a}_{q'}^\dagger(0).$$ Here the linear transfer matrix elements of the waveguide coupler are $\mathbf{T}_{q,q}(z) = \cos(\eta z) + i (-1)^{q} {\beta}{\eta^{-1}}\sin(\eta z)$ and $\mathbf{T}_{q,3-q} = -i{C}{\eta^{-1}}\sin(\eta z)$, where $\eta = (C^2 + \beta^2)^{1/2}$. We note that the input state is restored at the revival length $L = 2 \pi / \eta$, since $\hat{a}^\dagger_q(L) = \hat{a}^\dagger_q(0)$. While the in-line detectors introduce small loss, due to their symmetric positions the revival effect will remain, and the output density matrix would only exhibit an overall loss $\mu \rho$, where $\mu$ can be close to unity. We consider the measurement of states with a fixed photon number $N$, which is a practically important regime [@James:2001-52312:PRA; @Titchener:2016-4079:OL; @Oren:2017-993:OPT; @Titchener:2018-19:NPJQI]. We base the analysis on the most common type of click detectors that cannot resolve the number of photons arriving simultaneously at the same detector, and cannot distinguish which photon is which. Then, the measurement of a single photon by detector number $m$ can be described by an operator $\hat{A}_m = \hat{a}_{q_m}^\dagger(z_m) \hat{a}_{q_m}(z_m)$, where $q_m$ is the waveguide number and $z_m$ is the coordinate along the waveguide at the detector position. Accordingly, the photon correlations corresponding to the simultaneous detection of $N$ photons by a combination of $N$ detectors with numbers $m_1, m_2, \ldots, m_N$ are found as: $$\label{eq:correl} \Gamma(m_1, m_2, \ldots, m_N) = {\rm Tr}( \rho \hat{A}_{m_1} \hat{A}_{m_2} \cdots \hat{A}_{m_N} ) .$$ For a single photon ($N=1$), the measurement corresponds to a direct accumulation of counts at each detector without correlations, and we show an example in Fig. \[fig1\](b, left) for a pure input state $\ket{\psi_i}=[1,1]^\mathrm{T}/\sqrt{2}$ with the density matrix $\rho = \ket{\psi_i}\bra{\psi_i}$. We present an example of coincidence counts for a two-photon N00N state ($N=2$) in Fig. \[fig1\](b, right), corresponding to the events of clicks at different combinations of two detectors. We outline the principle of input $N$-photon density matrix $\rho$ reconstruction from the correlation measurements. The vectors $\ket{\psi_q}=[\mathbf{T}_{q,1}, \mathbf{T}_{q,2}]^\mathrm{T}$ are the analysis states, which define the measurements at the different detector positions. We visualize their evolution along $z$ on a Bloch sphere, where the projector $\ket{\psi_q}\bra{\psi_q}$ is decomposed into the Pauli matrices $\hat{\sigma}_i\ (i=x,y,z)$ to generate the coordinates $S_i=\mathrm{Tr}(\hat{\sigma}_i \ket{\psi_q}\bra{\psi_q})$. First we present the trajectories for the case of no detuning ($\beta=0$) in Fig. \[fig2\](a), where we see that the analysis states simply trace out a circle along the Bloch sphere. Considering for example $M=6$ detectors as sketched in Fig. \[fig1\](a), all the analysis states corresponding to detectors shown by arrows are in one plane. Such a configuration cannot probe states beyond that plane, therefore one cannot perform state reconstruction with a static, non-detuned directional coupler. In contrast, for detuned waveguides ($\beta \ne 0$) the circular trajectories in the first and second waveguides become non-degenerate, see the red and blue curves in Fig. \[fig2\](b). We indicate the analysis states with arrows for $M=6$, and note that they are spread out to different directions in the sphere and can thus be utilized to probe all information about the states and enable the density matrix reconstruction. The input state reconstruction is performed as follows. First, we introduce an index $p = 1, 2, \ldots, P$ with $P=M!/[N!(M-N)!]$ to enumerate all possible $N$ combinations of $M$ detectors, $(m_1, m_2, \ldots, m_N)_p$. Second, we select $S=(N+3)!/(N! 3!)$ real-valued parameters $r_1, r_2, \ldots, r_S$, which represent independent real and imaginary parts of the density matrix elements following the procedure described in the Supplementary of Ref. [@Titchener:2018-19:NPJQI]. Here, we consider the indistinguishable detection scheme that does not tell which photon is which. Then, we reformulate Eq. (\[eq:correl\]) in a matrix form: $$\label{eq:matrix} \Gamma_p \equiv \Gamma(m_1, m_2, \ldots, m_N)_p = \sum_{s=1}^S {\bf B}_{p,s} \, r_{s} ,$$ where the elements of matrix ${\bf B}$ are expressed through the transfer matrices $\mathbf{T}$ at the detector positions. The density matrix parameters $\{r_s\}$ can be reconstructed from the correlation measurements by performing pseudo-inversion of Eq. (\[eq:correl\]) provided $P \ge S$, i.e. when the number of detectors is $$\label{eq:Mmin} M \ge N+3 .$$ We now analyze the robustness of reconstruction with respect to possible experimental inaccuracies in the correlation measurements, such as shot noise. This can be quantified by the condition number $\kappa$ of the matrix ${\bf B}$ [@Foreman:2015-263901:PRL; @Titchener:2018-19:NPJQI]. Then, the most accurate reconstruction corresponds to the smallest condition number. We numerically calculate the condition numbers for different combinations of $M$ and $N$, considering a symmetric arrangement of detectors as sketched in Fig. \[fig1\](a). We find that accurate reconstruction can be achieved for the number of click detectors $M > N+3$. However, the condition number is infinite and the reconstruction cannot be performed for $M = N+3$. This happens because the $M$ analysis states are not fully independent, but actually constitute $M/2$ pairs of orthogonal states, as illustrated in Fig. \[fig2\](b). There is no contradiction with Eq. (\[eq:Mmin\]), since it establishes only a necessary condition for reconstruction. We determine that a detuning of the waveguide propagation constants ($\beta$) is essential to perform the reconstruction. We illustrate in Fig. \[fig2\](c) that the variation of $\beta$ strongly changes the reconstruction condition for a different number of photons $N=1,2,3$, considering the minimum possible even number of detectors. With no detuning, for $\beta=0$, $\kappa^{-1}$ goes to zero, meaning that the reconstruction is ill-conditioned and cannot be performed in practice. The optimal measurement frames occur at $\beta/C \simeq 1/\sqrt{2}$. In Fig. \[fig2\](d) we fix $\beta/C=1/\sqrt{2}$ and plot $\kappa^{-1}$ versus the number of detectors $M$. We find that $\kappa = 1 / \sqrt{3}$ for $N=1$ and any $M \ge 6$, in agreement with classical measurement theory [@Foreman:2015-263901:PRL]. For multi-photon states, the inverse condition number slightly increases for a larger number of detectors. This happens because we consider click detectors which do not resolve multiple photons. The reconstruction is possible even in absence of measurement counts corresponding to several photons arriving to the same detector, yet as for larger number of detectors such events become less probable, the reconstruction accuracy might be slightly improved. However this improvement could be offset by a higher number of dark counts, so the optimal detector number will depend on their characteristics. We demonstrate the scheme with [proof-of-principle experiments]{} for classical light. Mathematically, this is equivalent to the single-photon problem [@Perets:2008-170506:PRL; @Peruzzo:2010-1500:SCI], since monochromatic laser light can be described by a density matrix of the same dimensionality as for a single photon, $2 \times 2$ corresponding to two waveguides. We employ laser-written waveguides in fused silica to perform a reconstruction with a source of coherent laser light. In the fabrication process, we use different laser energies to write the coupled waveguides, and thereby achieve a pre-determined offset ($\pm\beta$) of the propagation constants for the fundamental modes in the two waveguides [@Szameit:2010-163001:JPB]. We launch 633 nm laser light into one of the waveguides, and perform in-line measurements of the evolution of intensity by observing the fluorescence (around 650 nm wavelength) emitted from the color centers of the glass material under a microscope [@Szameit:2007-241113:APL; @Szameit:2010-163001:JPB]. We present a characteristic fluorescence image in Fig. \[fig3\](a, top), which features a clear beating pattern due to light coupling between the waveguides along the propagation direction $z$. The extracted normalized power in each waveguide is shown in Fig. \[fig3\](a, bottom). We observe a good agreement with the theory (dashed line) for the coupled waveguide modes with $C=0.0885\ \mathrm{mm^{-1}}$ and propagation constants detuning $\beta=0.0240\ \mathrm{mm^{-1}}$. This confirms that the fluorescence indeed acts as in-line detection, which is conceptually equivalent to putting many weakly coupled detectors homogeneously along both waveguides. ![Experimental in-line measurement and reconstruction of $N=1$ states emulated with classical laser light in detuned directional couplers. (a) A fluorescence image (top), showing light intensity along the coupler. Bottom – the corresponding normalized power evolution in each waveguide (solid curves) compared to theory (dashed curves). Shading indicates a section $(z,z+L)$, where in-line measurements are used to reconstruct the full density matrix at the section input, $\rho(z)$. (b) The reconstructed real and imaginary parts of the density matrix elements for different combinations of the waveguide numbers (WG No.) at $z=10\ \mathrm{mm}$, the beginning of the shaded section in (a). (c) Experimentally reconstructed $\rho(z)$ from a set of different positions $z$ (crosses) compared with theoretical predictions (green curve) on a Bloch sphere, demonstrating an average fidelity of $99.65\%$. []{data-label="fig3"}](Fig3-v4){width="\linewidth"} We then test our reconstruction method using the fluorescence image of the waveguide coupler. In the 80-mm long section of the fabricated coupler shown in Fig.\[fig3\](a), light periodically couples several times between the waveguides over this distance. We truncate a section along $z$ with one revival period $L$ to mimic the in-line quantum detection. Then, the state $\rho(z)$ can be predicted by the coupled wave equation using the characterized parameters as described above. We consider different $z$ as starting positions, which allows us to effectively change the input state, and verify it’s reconstruction accuracy from the fluorescence image in the truncated section $(z,z+L)$. We employ a maximum-likelihood method to perform a pseudo-inversion of Eq. (\[eq:matrix\]) and thereby find an input density matrix that best fits the evolution of the fluorescence power in the two waveguides, see an example in Fig. \[fig3\](b). We analyze 34 different input states, truncated from successive $z$ with 1 mm increments. The reconstructed states are plotted on a Bloch sphere with crosses in Fig. \[fig3\](c), and compared to theoretical modelling results shown with a green curve. The coordinates are obtained via decomposing the density matrices to the Pauli matrices, i.e. $S_i=\mathrm{Tr}(\hat{\sigma_i}\rho)$ with $i$ spanning $x,y,z$. We observe an excellent consistency between direct theoretical modelling and reconstruction from the experimental fluorescence images of the full density matrix, where we reach a very high average fidelity of $99.65\%$. ![Minimization of detector numbers to the limit $M=N+3$: (a) Conceptual sketch of introducing a shift $\Delta z$ for all detectors in one waveguide and skip the last detector if $M$ is an odd number. (b) Normalized $\Delta z$ (to $2L/M$) that achieves the best reconstruction condition with $\beta=C/\sqrt{2}$ for $N=1,2,3$. (c) The corresponding inversed condition number for the three cases in (b).[]{data-label="fig4"}](Fig4nv2){width="\linewidth"} Finally, we explore the possibility to perform reconstruction with the minimal number of detectors, $M=N+3$ according to Eq. (\[eq:Mmin\]). We perform extensive numerical modelling by optimizing $\beta$ and the individual positions of all detectors. Although asymmetric positions may lead to a slight modification of the output state, this could be minimized by reducing the detector coupling. We find that well-conditioned reconstruction with $M=N+3$ can be achieved for $N=1,2,3$ with the detung $\beta/C=1/\sqrt{2}$ and by shifting all detectors in one waveguide by $\Delta z$, as sketched in Fig. \[fig4\](a) for $N=2$ and $M=5$. We present in Fig. \[fig4\](b) the optimal shifts $\Delta z$, normalized to the separation $2L/M$ between neighboring detectors in a waveguide. The corresponding inverse condition numbers are shown in Fig. \[fig4\](c). For $N=1$ the best case appears at $\Delta z M/2L=0.5$, which means the four detectors are arranged in a zig-zag manner, giving rise to an optimal inverse condition number $\kappa^{-1}=1/\sqrt{3}$, the same as for a larger number of detectors as shown in Fig. \[fig2\](d). For multi-photon states with $N=2,3$, the optimal values of $\Delta z$ correspond to asymmetric detector positions, while the inverse condition numbers are lower (worse) compared to larger detector numbers \[c.f. Fig. \[fig2\](d)\]. We note that for higher photon numbers $N \ge 4$, reconstruction with $M=N+3$ appears impossible for any detector positions. We anticipate that this limitation can be overcome by modulating the waveguide coupling and detuning along the propagation direction, which would require further investigation beyond the scope of this work. In conclusion, we proposed a practical and efficient approach for in-line detection and measurement of single and multi-photon quantum states in coupled waveguides with integrated photon detectors, suitable for various applications in quantum photonics. We showed proof-of-principle results with laser-written waveguides for classical light emulating single-photon regime. We presented the theory and experiments for a two-port system, and this can be scaled to multiple coupled waveguides. Moreover, our approach has a potential for translation from spatial to frequency and time-domain measurements. We gratefully thank financial support from Australian Research Council (DP160100619); the Australia-Germany Joint Research Cooperation Scheme; Erasmus Mundus (NANOPHI 2013 5659/002-001); the Alexander von Humboldt-Stiftung; German Research Foundation (SZ 276/9-1, SZ 276/12-1, BL 574/13-1, SZ 276/15-1, SZ 276/20-1). 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--- abstract: 'We use Andrews’ $q$-analogues of Watson’s and Whipple’s $_3F_2$ summation theorems to deduce two formulas for products of specific basic hypergeometric functions. These constitute $q$-analogues of corresponding product formulas for ordinary hypergeometric functions given by Bailey. The first formula was obtained earlier by Jain and Srivastava by a different method.' address: 'Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria' author: - 'Michael J. Schlosser' title: '$q$-Analogues of two product formulas of hypergeometric functions by Bailey' --- Introduction {#secintro} ============ We refer to Slater’s text [@Sl] for an introduction to hypergeometric series, and to Gasper and Rahman’s text [@GR] for an introduction to basic hypergeometric series, whose notations we follow. Throughout, we assume $\|q\|<1$ and $\|z\|<1$. In [@A], George Andrews proved the following two theorems: \[qwatson\] $${}_4\phi_3\!\left[\begin{matrix}a,b,c^{\frac 12},-c^{\frac 12}\\ (abq)^{\frac 12},-(abq)^{\frac 12},c \end{matrix}\,;q,q\right]=a^{\frac n2} \frac{(aq,bq,cq/a,cq/b;q^2)_\infty} {(q,abq,cq,cq/ab;q^2)_\infty},$$ where $b=q^{-n}$ and $n$ is a nonnegative integer. \[qwhipple\] $$\label{qwhippleeq} {}_4\phi_3\!\left[\begin{matrix}a,q/a,c^{\frac 12},-c^{\frac 12}\\ -q,e,cq/e\end{matrix}\,;q,q\right]=q^{\binom{n+1}2} \frac{(ea,eq/a,caq/e,cq^2/ae;q^2)_\infty} {(e,cq/e;q)_\infty},$$ where $a=q^{-n}$ and $n$ is a nonnegative integer. By a standard polynomial argument also holds when $a$ is a complex variable but $c=q^{-2n}$ with $n$ being a nonnegative integer. (This is the case we will make use of.) Theorems \[qwatson\] and \[qwhipple\] are $q$-analogues of Watson’s and of Whipple’s $_3F_2$ summation theorems, listed as Equations (III.23) and (III.24) in [@Sl p. 245], respectively. Two product formulas for basic hypergeometric functions ======================================================= We now have the following two product formulas which are derived using Theorems \[qwatson\] and \[qwhipple\]. The first one in Theorem 3 was already given earlier by Jain and Srivastava [@JS Equation (4.9)] (as Slobodan Damjanović has kindly pointed out to the author, after seeing an earlier version of this note), who established the result by specializing a general reduction formula for double basic hypergeometric series. The second formula in Theorem 4 appears to be new. \[qwatsonprod\] $${}_2\phi_1\!\left[\begin{matrix}a,-a\\ a^2\end{matrix}\,;q,z\right] {}_2\phi_1\!\left[\begin{matrix}b,-b\\ b^2\end{matrix}\,;q,-z\right] ={}_4\phi_3\!\left[\begin{matrix}ab,-ab,abq,-abq\\ a^2q,b^2q,a^2b^2\end{matrix};q^2,z^2\right].$$ \[qwhippleprod\] \[qwhippleprodid\] $$\begin{aligned} {}_2\phi_1\!\left[\begin{matrix}a,q/a\\ -q\end{matrix}\,;q,z\right] {}_2\phi_1\!\left[\begin{matrix}b,q/b\\ -q\end{matrix}\,;q,-z\right] =\sum_{j=0}^\infty\frac{(q^{2-j}/ab,aq^{1-j}/b;q^2)_j} {(q^2;q^2)_j}q^{\binom j2}(bz)^j&\\ ={}_4\phi_3\!\left[\begin{matrix}ab,q^2/ab,aq/b,bq/a\\ -q^2,q,-q\end{matrix}\,;q^2,z^2\right] \qquad\qquad\qquad\qquad\qquad\quad&\notag\\[.1em] {}-\frac{(a-b)(1-q/ab)}{1-q^2}\,z\, {}_4\phi_3\!\left[\begin{matrix}abq,q^3/ab,aq^2/b,bq^2/a\\ -q^2,q^3,-q^3\end{matrix}\,;q^2,z^2\right]&.\end{aligned}$$ To prove Theorem \[qwatsonprod\], compare coefficients of $z^n$. The resulting identity is equivalent to Theorem \[qwatson\]. The proof of Theorem \[qwhippleprod\] is similar. Comparison of coefficients of $z^n$ gives an identity which is equivalent to Theorem \[qwhipple\] (where in the latter theorem the restriction $a=q^{-n}$ is replaced by $c=q^{-2n}$, as mentioned). The second identity in Equation follows from splitting the sum over $j$ into two parts depending on the parity of $j$. (This is motivated by the particular numerator factors in the $j$-th summand.) The technical details – elementary manipulation of $q$-shifted factorials – are routine and thus omitted. Theorem \[qwatsonprod\] is a $q$-analogue of Bailey’s formula in [@B p. 246, Equation (2.11)]: $$\label{qwatsonprodido} {}_1F_1\!\left[\begin{matrix}a\\ 2a\end{matrix}\,;z\right] {}_1F_1\!\left[\begin{matrix}b\\ 2b\end{matrix}\,;-z\right] ={}_2F_3\!\left[\begin{matrix}\frac 12(a+b),\frac 12(a+b+1)\\ a+\frac 12,b+\frac 12,a+b\end{matrix};\frac 14z\right].$$ To obtain from Theorem \[qwatsonprod\], replace $(a,b,z)$ by $(q^a,q^b,(1-q)z/2)$, and let $q\to 1$. Similarly, Theorem \[qwhippleprod\] is a $q$-analogue of Bailey’s formula in [@B p. 245, Equation (2.08)]: $$\begin{aligned} \label{qwhippleprodido} {}_2F_0\!&\left[\begin{matrix}a,1-a\\ -\end{matrix}\,;z\right] {}_2F_0\!\left[\begin{matrix}b,1-b\\ -\end{matrix}\,;-z\right]\notag\\[.2em] &{}={}_4F_1\!\left[\begin{matrix}\frac 12(1+a-b),\frac 12(1-a+b), \frac 12(a+b),\frac 12(2-a-b)\\[.1em] \frac 12\end{matrix}\,;4z^2\right]\notag\\[.1em] &\qquad{}-(a-b)(a+b-1)\,z\notag\\ &\qquad\times{}_4F_1\!\left[\begin{matrix}\frac 12(2+a-b),\frac 12(2-a+b), \frac 12(1+a+b),\frac 12(3-a-b)\\[.1em] \frac 32\end{matrix}\,;4z^2\right].\end{aligned}$$ To obtain from Theorem \[qwhippleprod\], replace $(a,b,z)$ by $(q^a,q^b,2z/(1-q))$ and let $q\to 1$. Related results in the literature ================================= A different product formula for basic hypergeometric functions was established by Srivastava [@S1 Eq. (21)] (see also [@S2 Eq. (3.13)]): $${}_2\phi_1\!\left[\begin{matrix}a,b\\ -ab\end{matrix}\,;q,z\right] {}_2\phi_1\!\left[\begin{matrix}a,b\\ -ab\end{matrix}\,;q,-z\right] ={}_4\phi_3\!\left[\begin{matrix}a^2,b^2,ab,abq\\ a^2b^2,-ab,-abq\end{matrix};q^2,z^2\right].$$ This formula is a $q$-extension of Bailey’s formula in [@B p. 245, Equation (2.08)] (or, equivalently, of an identity recorded by Ramanujan [@R Ch. 13, Entry 24]). Finally, we mention that in 1941 F.H. Jackson [@J] had derived the identity $${}_2\phi_1\!\left[\begin{matrix}a^2,b^2\\ a^2b^2q\end{matrix}\,;q^2,z\right] {}_2\phi_1\!\left[\begin{matrix}a^2,b^2\\ a^2b^2q\end{matrix}\,;q^2,qz\right] ={}_4\phi_3\!\left[\begin{matrix}a^2,b^2,ab,-ab\\ a^2b^2,abq^{\frac 12},-abq^{\frac 12}\end{matrix};q,z\right],$$ which is a $q$-analogue of Clausen’s formula of 1828, $$\left({}_2F_1\!\left[\begin{matrix}a,b\\ a+b+\frac 12\end{matrix}\,;z\right]\right)^2 ={}_3F_2\!\left[\begin{matrix}2a,2b,a+b\\ 2a+2b,a+b+\frac 12\end{matrix};z\right].$$ Another $q$-analogue of Clausen’s formula was delivered by Gasper in [@G]. While it has the advantage that it expresses a square of a basic hypergeometric series as a basic hypergeometric series, it only holds provided the series terminate: $$\label{qClausen} \left({}_4\phi_3\!\left[\begin{matrix}a,b,aby,ab/y\\ abq^{\frac 12},-abq^{\frac 12},-ab\end{matrix}\,;q,q\right]\right)^2 ={}_5\phi_4\!\left[\begin{matrix}a^2,b^2,ab,aby,ab/y\\ a^2b^2,abq^{\frac 12},-abq^{\frac 12},-ab\end{matrix};q,q\right].$$ See [@GR Sec. 8.8] for a nonterminating extension of and related identities. Acknowledgement {#acknowledgement .unnumbered} =============== I would like to thank George Gasper for his interest and for informing me of the papers [@S1; @S2] by Srivastava. I am especially indebted to Slobodan Damjanović for pointing out that Theorem 3 was already given by Jain and Srivastava [@JS Equation (4.9)]. [99]{} G.E. Andrews, “On $q$-analogues of the Watson and Whipple summations”, SIAM J. Math. Anal. 7 (3) (1976), 332–336. W.N. Bailey, “Products of generalized hypergeometric series”, Proc. London Math. Soc. S2-28 (1) (1928), 242–254. T. Clausen, “Ueber die Fälle, wenn die Reihe von der Form $\ldots$ ein Quadrat von der Form $\ldots$ hat”, J. reine angew. Math. 3 (1828), 89–91. G. Gasper, “$q$-Extensions of Clausen’s formula and of the inequalities used by de Branges in his proof of the Bieberbach, Robertson and Milin conjectures”, SIAM J. Math. Anal. 20 (1989), 1019–1034. G. Gasper and M. Rahman, [Basic hypergeometric series]{}, second edition, Encyclopedia of Mathematics and Its Applications 96, Cambridge University Press, Cambridge, 2004. F.H. Jackson, “Certain $q$-identities”, Quart. J. Math. (Oxford) 12 (1941), 167–172. V.K. Jain and H.M. Srivastava, “$q$-Series identities and reducibility of basic double hypergeometric functions”, Canad. J. Math. 38 (1986), 215–231. S. Ramanujan, [Notebooks of Srinivasa Ramanujan, Vols. I and II]{}, Tata Institute of Fundamental Research, Bombay, 1957. L.J. Slater, [Generalized hypergeometric functions]{}, Cambridge University Press, Cambridge, 1966. H.M. Srivastava, “Some formulas of Srinivasa Ramanujan involving products of hypergeometric functions”, Indian J. Math. 29 (1987), 91–100. H.M. Srivastava, “Srinivasa Ramanujan and generalized basic hypergeometric functions”, Serdica Bulgariacae mathematicae publicationes 19 (1993), 191–197.
--- abstract: 'We study the photoreaction in the delta energy region using the QMD approach. The proton and pion cross-sections are calculated and compared with experimental data. Through this work we examine the multistep contributions in the cross-sections and the [$\pi$-$\Delta$]{} dynamics.' address: \| $^{1}$Advanced Science Research Center, Japan Atomic Energy Research Institute,\ Tokai-mura, Naka-gun, Ibaraki-ken 319-11, Japan\ $^{2}$Research Organization for Information Science and Technology\ Tokai-mura, Naka-gun, Ibaraki-ken 319-11, Japan\ $^{3}$Nuclear Data Center, Japan Atomic Energy Research Institute,\ Tokai-mura, Naka-gun, Ibaraki-ken 319-11, Japan\ author: - \| Tomoyuki Maruyama$^{(1)}$, Koji Niita$^{(1,2)}$,\ Satoshi Chiba$^{(1,3)}$, Toshiki Maruyama$^{(1)}$ and Akira Iwamoto$^{(1)}$ title: \| Analysis of Photoreaction in the Delta Energy Region\ by the Quantum Molecular Dynamics Approach --- = 10000 Introduction ============ $\Delta$ properties in medium has been investigated both in experiments and in theories. J. Chiba et al. [@chibaj] suggested experimentally that the [$\Delta$]{} mass becomes lower in nuclear medium than that in the free space. Horikawa et al. [@horikawa] gave that the depth of the [$\Delta$]{}-potential is about 30 MeV in normal nuclei. On the other hand the recent total photoabsorption experiments [@totab] showed the broadening of the [$\Delta$]{} peak but no shift of the peak-position. These works indicated that [$\Delta$]{} properties are modified in nuclear medium, but this medium correction has not been understood definitely. Experiments with photon in the [$\Delta$]{} energy region are expected to has some advantages, compared with other experiments, to get information of [$\Delta$]{} properties in medium. First the photon can directly produce [$\Delta$]{} in the bulk region since the electromagnetic interaction is weak, while the proton- and pion-induced reactions produce it mainly in the surface region. Second the produced [$\Delta$]{} does not have so large momentum because a value of momentum transfer is same as that of energy transfer for photoreaction. In the proton-induced reaction the momentum transfer is much larger than the energy transfer, so that the produced [$\Delta$]{} must have a large momentum, and in-medium properties do not strongly affect observables. The inclusive experimental data of photoabsorption is not sufficient to identify the elementary absorption process uniquely. We need to observe a outgoing nucleon and/or a pion to investigate in-medium [$\Delta$]{} properties. In fact TAGX collaboration at INS [@TAGX; @Emura] showed in experiments of photoreaction with the $^{3,4}$He target that one must observe several outgoing particles coincidentally so as to identify the pure [$\Delta$]{} channel. These particles, however, interact with other nucleons before they escape the nucleons and have lost information of the photoabsorption process at the beginning. Thus we have to analyze the multistep collisional processes after the photoabsorption. Recently we developed a framework of QMD [@Aich] plus statistical decay model (SDM) [@niita95], and applied systematically this QMD $+$ SDM to nucleon- (N-) induced reactions. It was shown [@niita95] that this framework could reproduce quite well the measured double-differential cross sections of (N,xN’) type reactions from 100 MeV to 3 GeV incident energies in a systematic way. In the subsequent papers [@chadwick95; @chiba96], we gave detailed analysis of the pre-equilibrium (p,xp’) and (p,xn) reactions in terms of the QMD in the energy region of 100 to 200 MeV. In these analysis, a single set of parameters was used, and no readjustment was attempted. The reaction process after the initial photoabsorption is almost the same as the preequibrilium process of nucleon-induced reactions. It should then be natural to apply the QMD $+$ SDM approach to the analysis of the electron scattering and photoreaction. Of course we have the other methods to calculate the multistep contributions such as the PICA code [@PICA], a Monte-Carlo (MC) model [@carrasco] and BUU [@hombach; @effenberger; @BUU4]. The PICA does not have a [$\Delta$]{}-degree of freedom explicitly, and it is not very useful to study the $\pi$-$\Delta$ dynamics. The MC calculation is performed only in the momentum-space, and cannot describe the refraction from the mean-field. The BUU approach has succeeded to describe the particle production in the intermediate energy heavy-ion collisions. However this approach can describe only the one-body dynamics, and then it cannot distinguish single nucleons from clusters and cannot calculate coincident observables. Anyway none of these models can treat the nucleon-induced reactions, heavy-ion collisions and photoreactions (electron scattering) in the uniform way. The ability of the uniform description is one of the strongest advantages of the QMD approach. We then carry out an analysis of photoreactions with the same formula and the same set of parameters as the previous works of the nucleon-induced reactions[@niita95; @chadwick95; @chiba96]. This enables us to check and improve further the elementary collisional processes included in the QMD approach from another point of view. In this paper, we focus only on the photoreaction in the energy region above the [$\Delta$]{} threshold. In the next section, a brief explanation of the QMD plus SDM approach is given. The comparison of the calculation with the experimental data and discussions on the reaction mechanisms are given in section \[results\]. Summary of this work is given in section \[summary\]. Brief explanation of The Quantum Molecular Dynamics {#brief} =================================================== Equation of motion ------------------ We start from representing each nucleon (denoted by a subscript [i]{}) by a Gaussian wave packet in both the coordinate and momentum spaces. The total wave function is assumed to be a direct product of these wave functions. Thus the one-body distribution function is obtained by the Wigner transformation of the wave function, $$f({\bf r}, {\bf p}) = \sum_i f_i({\bf r}, {\bf p}) = \sum_i 8 \cdot \exp \left[ -\frac{({\bf r}-{\bf R}_i)^2}{2L} -\frac{2L(\bf{p}-\bf{P}_i)^2}{ \hbar ^2} \right] \label{eq:phase-space}$$ where ${\it L}$ is a parameter representing the spatial spread of a wave packet, ${\bf R}_{\it i}$ and ${\bf P}_i$ corresponding to the centers of a wave packet in the coordinate and momentum spaces, respectively. The equation of motion of ${\bf R}_i$ and ${\bf P}_i$ is given, on the basis of the time-dependent variational principle, by the Newtonian equation: $$\dot{\bf R}_i=\frac{\partial H}{\partial {\bf P}_i}, ~~~~~ \dot{\bf P}_i=-\frac{\partial H}{\partial {\bf R}_i}, \label{eq:eos}$$ and the stochastic N-N collision term [@niita95]. We have adopted the Hamiltonian $\it{H}$ consisting of the relativistic kinetic and mass energies and the Skyrme-type effective N-N interaction[@skyrme] plus Coulomb and symmetry energy terms: $$\begin{aligned} H~=~& &\sum_{i}\sqrt{m^2_i~+~\bf{P}^2_i} \nonumber \\ & &{} +\frac{1}{2} \frac{A}{\rho_0} \sum_{i}<\rho_i> +\frac{1}{1+\tau} \frac{B}{\rho_0^\tau} \sum_{i}<\rho_i>^\tau \nonumber \\ & &{} +\frac{1}{2}\sum_{i,j(\neq i)}\frac{e_i e_j} {\mid \bf{R}_i-\bf{R}_j \mid} {\rm erf} \left( \mid \bf{R}_i-\bf{R}_j \mid /\sqrt{4 {\it L}} \right) \nonumber \\ & &{} +\frac{C_s}{2 \rho_0} \sum_{i,j(\neq i)} c_i c_j \rho_{ij}, \label{eq:hamiltonian}\end{aligned}$$ where “erf” denotes the error function, the $e_i$ is the charge of the $i$-th particle, and the $c_i$ is 1 for proton, -1 for neutron and 0 for other particles. With the definition $$\begin{aligned} \rho_i({\bf r})&\equiv &\int \frac{d{\bf p}}{(2 \pi \hbar)^3}f_i({\bf r},{\bf p})\nonumber \\ {}&=& (2 \pi L)^{-3/2} \exp \left[-({\bf r}-{\bf R}_i)^2/2L \right] , \label{eq:rhoi}\end{aligned}$$ the other symbols in eq.(\[eq:hamiltonian\]) are given as: $$\begin{aligned} <\rho_i>&\equiv & \sum_{j (\ne i)} \rho_{ij} \equiv \sum_{j (\ne i)} \int d {\bf r} \rho_i({\bf r}) \cdot \rho_j({\bf r})\nonumber \\ {}&=& \sum_{j (\ne i)}(4 \pi L)^{-3/2} \exp \left[-({\bf R}_i-{\bf R}_j)^2/4L \right]. \label{eq:hoho}\end{aligned}$$ The symmetry energy coefficient $C_s$ is taken to be 25 MeV. The four remaining parameters, the saturation density $\rho_0$, Skyrme parameters $A$, $B$ and $\tau$ are chosen to be 0.168 fm$^{-3}$, $-124$ MeV, 70.5 MeV and 4/3, respectively. These values give the binding energy/nucleon of 16 MeV at the saturation density $\rho_0$ and the compressibility of 237.7 MeV (soft EOS) for nuclear matter limit. The only arbitrary parameter in QMD, i.e., the width parameter [ L]{}, is fixed to be 2 fm$^2$ to give stable ground state of target nuclei in a wide mass range. These values and the details of the other description are just the same as those according to our previous paper[@niita95]. As for the isobar resonances such as $\Delta$ and $N^$, we use the same interactions as nucleons though the symmetry force dose not work for them. At each collision process we satisfy the energy conservation by varying slightly an absolute value of relative momentum between colliding two particles. The initial state of the photoabsorption ---------------------------------------- The QMD calculation is started at the moment when the photon is absorbed by the target nucleus. As an initial state of the simulation, we have to assume the photoabsorption channels. For 200 - 400 MeV/c incident photon momenta, the one-pion production process is dominant in the photoabsorption. We introduce the following three channels for the photoabsorption, i.e., $$\begin{aligned} \gamma + N \rightarrow & \Delta, ~~~ & ~~(C1) \nonumber \\ \gamma + N \rightarrow & N^, ~~~ & ~~(C2) \nonumber \\ \gamma + N \rightarrow & N + \pi. ~ & ~~(C3)\end{aligned}$$ The nuclear resonances $\Delta$ and $N^{}(1440)$ in the above equations can decay into nucleon and pion according to the decay width and the isospin selection. In addition, the $N^$ can also decay into $\Delta$ and $\pi$ as in our prescription [@niita95], which means that the (C2) channel includes implicitly two-pion production process. The channel of the pure ${\pi}N$-pair production (C3) is adopted only for the charged pion and the isospin symmetry is assumed for the cross section, i.e., $\sigma(\gamma + p) = \sigma(\gamma + n)$. The cross sections of the above three channels are determined to reproduce the experimental one-pion production data of $\gamma + p \rightarrow N + \pi$ [@eldata; @eldata2]. The results are given in Fig. \[gnpi\]. The upper part (a) of Fig. \[gnpi\] denotes the cross-section of $\gamma + p \rightarrow p + \pi^0$ and the lower part (b) denotes that of $\gamma + p \rightarrow n + \pi^{+}$. The long dashed, dashed and thin solid lines indicate contributions of the $\Delta$ resonance (C1), $N^{}(1440)$ (C2) and the pure ${\pi}N$-pair production (Born term) (C3), respectively. For the future discussion, we define the alternative channel (C4) that the all photon is absorbed through the pure ${\pi}N$-pair with the same amount of the cross-section as the sum of $C1 + C2 + C3$ channels. In each event, a nucleon which absorbs the photon is selected randomly. The photoabsorption channel is also randomly chosen according to the rate of each cross section. We assume that the angular distribution of the pion emission is isotropic in the center of mass system of the pion and the nucleon. Decomposition into step-wise contribution in multistep reactions ---------------------------------------------------------------- For the later discussion of the multistep reaction, we define here the step number $s$ which indicates the number of collisions responsible for emission of a particle in the QMD calculation as following. First we assign the step number zero to each nucleon in the target nucleus. After a nucleon absorbs the incident photon and becomes a resonance, we set the step number of the resonance to be one. For the case of the pure ${\pi}N$-pair creation, the step numbers of both $\pi$ and $N$ are one. The rule of the change of the step number for each nucleon is that, if two nucleons [i]{} and [j]{} having step numbers $s_i$ and $s_j$ make a collision, the step numbers of both particles are modified to be $s_i + s_j + 1$. If a pion $i$ and nucleon $j$ becomes a resonance, the step number of the resonance is also $s_i + s_j + 1$. When a resonance $i$ decays into a pion and a nucleon, on the other hand, both the step numbers of the pion and the nucleon are set to be $s_i$; namely this process does not change the step number. We prohibit successive collisions with the same partner and the collisions between two nucleons with 0-step number. We attach n-step contribution to the ($\gamma, N$) or ($\gamma, \pi$) reactions when a nucleon or a pion emitted from the target nucleus has a step number $n$. Calculation of the Cross-Section {#cross-section} -------------------------------- In the simulation, the double-differential cross-section of the emitted particle is calculated as $$\frac{d^{2}\sigma}{d E \ d \Omega}~=~\frac{1}{N_{event}} \sum_{i} \sigma^{T}_{\gamma}(i) M(E,~\Omega,~i) \label{eq:cross-section}$$ where $i$ indicates the event number, $N_{event}$ is the total number of the events, $\sigma^{T}_{\gamma}(i)$ shows the total photoabsorption cross-section in the $i$-th event, and $M(E,~\Omega,~i)$ denotes the multiplicity of the particle under interest emitted in the unit energy-angular interval around $E$ and $\Omega$ for the $i$-th event. Typically, 400000 events were generated to get a reasonable statistics in the step-wise double-differential cross-section. In the calculation, the parameters have been fixed to the same values as in Ref. [@niita95] without any adjustment. Results and Discussion {#results} ====================== In Fig. \[gC375\], we show our results of proton (left-hand-side) and $\pi^0$ (right-hand-side) double differential cross sections at $\theta = 30^{\circ}, 60^{\circ}$ and $90^{\circ}$ from $\gamma (375 MeV/c) + {\rm C}$ reaction. The experimental data at $\theta = 30^{\circ}$ for proton, which are denoted by the full circles with error bars, are taken from Ref. [@Kanazawa]. In these figures, we decompose the total cross sections into the step-wise contributions. The total results of the QMD+SDM simulation are shown by the thick lines. In the same figures, we draw the individual contributions from the 1-step process (long-dashed lines), 2-step (dashed lines), 3-step (thin lines) and SDM process (dotted lines), respectively. The contribution from SDM process is shown only for proton. First we look at the contribution from the 1-step process. The peak of the 1-step process, which is usually called quasi-free (QF) peak, is clearly separated from the multistep contribution in the forward angle ($\theta = 30^{\circ}$), while it overlaps to other contributions at larger angles, particularly for proton. For the more detailed analysis of the step-wise contributions, we plot again the the results of the proton cross-section at $\theta = 30^{\circ}$ from $\gamma (375 MeV/c) + {\rm C}$ reactions in Fig. \[gC375ch\]. The upper column (Fig. \[gC375ch\]a) is just the same as that in Fig.\[gC375\]. We can see in this figure that the position of the calculated QF peak almost coincides with the experimental peak position, though the absolute value of this peak overestimates the data. This peak comes mainly from the 1-step process, namely from the elementary process of $\gamma + N \rightarrow \pi + N$. There are two other peaks in our results. The peak in the lower momentum region shows the contribution of SDM, the evaporation from the residual excited nuclei. The higher momentum peak, on the other hand, mainly comes from 2-step process. There is a dip region between this peak and the QF peak, which are not observed experimentally. In order to understand the meaning of the higher momentum peak, we also decompose contributions to the proton cross-section of the events with zero pion and with one pion at the final state in Fig. \[gC375ch\]b. This figure clearly shows that the cross-section around the higher momentum peak comes only from the zero pion events . >From this analysis we can easily know that the cross-section around the higher peak comes from the following 2-step process: $$\begin{aligned} \gamma + N \rightarrow & \Delta & , \nonumber \\ & \Delta & \; + \; N \rightarrow N + N .\end{aligned}$$ This 2-step process is effectively equivalent to the two-nucleons ($2N$) photoabsorption process. In Fig. \[gC375ch\], we give also the result (chain-dotted line) of QMD calculation including only the pure ${\pi}N$ pair as the initial channel (C4 channel mentioned before). It is seen that this result does not show the higher peak of $2N$ photoabsorption any more. Hombach et al. have commented in Ref. [@hombach] that these two channels do not make large difference in $\pi$-productions. At least proton spectra, however, there is important diffrence between the two initial channels. In Fig. \[g390p\], we show the results of proton energy-spectrum from the photoreaction at the photon-momentum $q = 390 MeV/c$ with the target $^{12}$C (Fig. \[g390p\] (a) ) and $^{48}$Ti (Fig. \[g390p\] (b) ). We draw the full results of the QMD+SDM simulation and the individual contributions from the 1-, 2- and 3- step processes as in the previous figures. The experimental data are taken from Ref. [@Arends91]. In order to compare each contribution in detail, we plot the results with the logarithmic scale. In this reaction the 1-step and 2-step processes make almost same contributions to the cross section at low proton energy, and the QF peak cannot be seen clearly. Our QMD+SDM results underestimate experimental data around the QF peak and also $2N$-photoabsorption energy regions. The same behavior was seen in the results of PICA for this reaction [@Arends91]. In the above two comparisons with the experimental data, it is seen that around the QF peak energy region our calculations overestimate the experimental data at $\theta = 30^{\circ}$ in Fig. \[gC375ch\], but underestimate at $\theta = 52^{\circ}$ in Fig. \[g390p\]. It has been already known from the analysis of the proton-induced reaction [@niita95; @chadwick95; @chiba96] that the QF contribution strongly depends on the detailed of the elementary process. In this energy region, $q \approx 380 MeV/c$, the angular distribution of emitted pion has a sideward peak in the CM system of the ${\pi}N$-pair [@eldata]. We have not fit the angular distribution of the pion photoproduction cross-section to this experimental data in the elementary process, and this might be the reason for the disagreement near the QF peak. For the higher momentum region, on the other hand, our results seem to agree with the data and imply that the multistep process, mainly the 2-step process, can explain the data for the photon energy above the pion threshold. Next we explore the origin of the dip region between the above two momentum region. The experimental analysis [@Emura] with $^{3}$He target indicated that the three-nucleons ($3N$) photoabsorption process also contributes to the final results. The $3N$-photoabsorption process must contribute the proton spectrum around this dip momentum region. We then check the $3N$-photoabsorption process for this experiment. Of course the semi-classical approach cannot well describe the structure of so small nuclei such as $^{3}$He. However a cross-section of each step contribution is almost determined by the geometrical position of nucleon. The interaction range of the two-body collisions is about 1 - 2 fm while the root-mean square radius of $^{3}$He is about 1.8fm. Hence there is no serious trouble to make a rough estimation with the QMD approach. In Fig. \[gHe280\] we give a cross-section of two-protons emission from the photoabsorption by $^3$He as a function of the undetected neutron momentum. Experimental data is taken from Ref. [@Emura]. In order to separate contributions of the two-protons $(2N(pp))$-photoabsorption process experimentally [@Emura], they chose the events without pions at the final state and by a experimental trigger that $p \ge 300$MeV/c and $\theta = 15^{\circ} - 165^{\circ}$ for the two emitted protons. This experimental condition selects the process in which at least two protons are concerned with the photoabsorption. The calculation also follows this condition. In this figure the absolute value is arbitrarily because the normalization of the experimental data has not been determined. We again decompose the total yield to 0-step (dotted line) and higher step contributions (dashed line) by the step-number of the undetected neutron. In this case, the 0-step process (dotted line) means that the photon is absorbed by the two protons, and that the neutron is a spectator. In the other case of the multistep contributions above 1-step, three nucleons are all related with the photoabsorption process. Though we only treat the sequential binary process, this multistep process is effectively regarded as the $3N$-photoabsorption process, while the 0-step process is the $2N(pp)$-photoabsorption process. In Ref. [@Emura], they also decompose the total yield to the two contributions, one from the spectator neutron for $2N(pp)$ photoabsorption and the other from the emitted neutron for the $3N$-photoabsorption processes. We also plot their decomposition of the $2N(pp)$ process (thin dashed line) and the $3N$ process (long dashed line) in the same figure. The 0-step contribution appears in the low momentum region as a narrow peak. The peak position of the 0-step process ($2N(pp)$-photoabsorption) is lower than that from the experimental analysis. This peak is sensitive to the momentum distribution of the neutron in the $^3$He, since the neutron is a spectator. The semi-classical approach cannot describe correctly the momentum distribution of nucleons particularly the high momentum component in small nuclei such as $^3$He. This is the reason of this discrepancy. As for the $3N$-photoabsorption process, we can see that the second peak is much lower than the first peak in our calculation, which disagrees with the experimental result. Namely our simulation does not make so large $3N$-photoabsorption process. The $3N$-photoabsorption process should contribute the cross-section between the QF and the $2N$-photoabsorption processes. The shortage of the $3N$-photoabsorption is the origin of the dip between the QF contribution and the $2N(pp)$ contribution, seen in Fig. \[gC375\]. Since both the $2N$- and $3N$-photoabsorption processes make no pion events at the final state, the $\pi$-absorption must play an important role in these absorption processes. Then we can get some information on a reason of the dip by studying the pion photoproduction. In Fig. \[tot-pi0\] we show the results of the integrated cross-sections for neutral pion photoproduction in the $^{12}$C, $^{27}$Al and $^{63}$Cu targets, and compare them with the experimental data [@Arends86]. The results of full QMD are denoted by the solid lines, while the long-dashed lines indicate the result without $\pi$-absorption. The calculated total cross-sections overestimate the data at the peak energy of the $\Delta$ resonance and decrease faster at lower and higher energies. This behavior is also seen in other calculations [@carrasco; @hombach; @Arends86]. The overestimate of the calculation becomes larger with increasing mass number $A$. These results show that the $\pi$-absorption is too small in our approach. In order to examine the effects of the $\pi$-absorption, we make another test calculations by using a fixed pion mean-free path for the $\pi$-absorption process. The pion mean-free path $\lambda$ is estimated to be 5 fm [@BUU4]. >From the mean-free path we estimate the $\pi$-absorption and the photoproduction cross-section in the following way. We factorize the absorption rate $\exp(- <l> /{\lambda})$ to the results without the $\pi$-absorption, where $<l>$ is a mean distance of pion propagating in nuclear medium. Assuming hard sphere with a radius $R$ for nucleus, we can estimate $<l>$ as $$<l> = \frac{ \int_{\Omega} d{\bf r}_1 \int_{S} d{\bf r}_2 \|{\bf r}_1 - {\bf r}_2\|} { \int_{\Omega} d{\bf r}_1 \int_{S} d{\bf r}_2 } = \frac{6}{5} R,$$ where $\Omega$ and $S$ indicate the volume and surface integrals in the hard sphere with radius $R$, respectively. Here the pion is considered to be produced randomly at the position inside the target nucleus, and the $\Delta$ is not directly taken into account in this estimation. The results of this test calculation is shown by the dashed lines. They nicely reproduce the experimental result around the [$\Delta$]{} peak, though it is slightly lower at lower and higher energies. We then need another $\pi$-absorption process. It may be a pure $2N$ $\pi$-absorption process which is not taken into account in our approach. This pure $2N$ $\pi$-absorption process is given by the detailed valance of the s-wave pion production. The production rate of this $s$-wave process is small, but the absorption rate of the $2N$ process is not so small [@eng94]. This fact suggests us one of the answers of the too small $3N$-photoabsorption process in Fig. 5. If the pion is absorbed by two nucleons, as a result, three nucleons share the photon energy. This is an additional $3N$-photoabsorption process. However it is not so easy to introduce the pure $2N$ $\pi$-absorption in the actual simulation. We have to treat three body collisions, which is not impossible but difficult in the actual simulation, Engel et al. [@eng94] suggested an easy method of the effective treatment of the $2N$ $\pi$-absorption process by factorizing a density-dependent factor to the cross-section of $N + \Delta \rightarrow N + N$; later Hombach et al. [@hombach] also used the same method in the photoreaction. Here we test this approximate method. In order to estimate this effect in a rough way, we increase this cross section by four times; we call this test simulation “QMD/T1”. We then calculate the total $\pi^0$ production cross-sections at the photon momentum $q = 375$MeV/c as a function of the target mass. The results are shown in Fig. \[tot-A\]. The solid, dashed and broken lines indicate the results of QMD, QMD/T1, and no absorption, respectively. The full square show experimental data taken from [@Arends86]. The QMD/T1 slightly improves the result but still overestimates experimental data. Furthermore we calculate again the proton and pion momentum-spectra for $\gamma (375 MeV/c) + {\rm C}$ reactions in Fig. \[gC375\] with QMD/T1. The results are given in Fig. \[gCd\] by dashed lines. The QMD/T1 slightly reduces the $\pi^0$ cross-section overall, and enhances the proton cross-section around $2N$-absorption energy region. However the dip between the QF peak and the $2N$-absorption peak exist in the QMD/T1 calculations. The enlarged ${\sigma_{{\Delta}N \rightarrow NN}}$ mainly enhances the [$\Delta$]{} absorption process at the second step which contributes the $2N$-photoabsorption part, but it does not contribute higher multisteps such as the $3N$-photoabsorption part. When a pion propagates in a nucleus, it stays at a pure $\pi$ state much longer than a $\Delta$ state because the $\Delta$ life-time is very short. Then the enlarged ${\sigma_{{\Delta}N \rightarrow NN}}$ does not largely change the $\pi$-absorption. One may have an idea that we can increase the $\pi$-absorption by enlarging the cross-section of ${\pi}N \rightarrow {\Delta}$ as well as that of ${{\Delta}N \rightarrow NN}$. This approximate method must increase the pion-absorption, but it cannot solve the problem that the enlarged ${\sigma_{{\Delta}N \rightarrow NN}}$ still overestimates the proton cross-section in the $2N$-photoabsorption energy region. Hence the enlargement of ${\sigma_{{\Delta}N \rightarrow NN}}$ and ${\sigma_{{\pi}N \rightarrow {\Delta}}}$ together cannot explain the experimental results of the proton emission and the pion production at the same time. Thus we need an additional absorption process without a $\Delta$ state, i.e. the pure $2N$-absorption of pion. Finally we examine effects of the [$\Delta$]{}-potential to observables because the photoreactions in the [$\Delta$]{} energy region are studied for the purpose of the determination of [$\Delta$]{} properties in nuclear medium. Along this line we calculate positive pion spectra with and without the [$\Delta$]{}-potential in the $\gamma (213 {\rm MeV/c}) + ^{40}$Ca reaction. In Fig. \[gCadel\] we show our results at $\theta_\pi = 81^\circ, 109^\circ {\rm and} 141^\circ$; the solid and dashed lines indicate the results with the [$\Delta$]{}-potential same as nucleon one and those with no [$\Delta$]{}-potential, respectively. Experimental data are taken from Ref.[@fissum]. >From this figure, we see that our results of the pion cross sections are not so largely different from experimental data though the peak position is slightly shifted to higher energy at $\theta_\pi = 81^\circ$ and $109^\circ$ when the $\Delta$-potential is switched off. In the case of no [$\Delta$]{}-potential the QF peaks are further shifted to higher energy, but this difference is not significant; the [$\Delta$]{}-potential does not strongly affect the pion spectrum. This result suggests that it is difficult to know in-medium [$\Delta$]{} properties from only simply observing such as a pion spectrum. We should investigate more complicated coincident observables. Summary ======= In this paper we have calculated the emitted proton and produced pion cross-sections in the photoreaction within the framework of the QMD approach, and compared them with experimental data. We focus on the examination of applying the QMD approach to the photoreaction, and use rather rough initial photoabsorption in the present analysis. Through this work, however, we can analyze the multistep contribution and check the ${\pi}$-${\Delta}$ dynamics. As for the proton spectrum, the multistep contributions are not negligibly small even in the forward direction where the contribution from the QF process is dominate. As increasing the emission angle, the multistep contribution becomes larger, and overlaps with the QF contribution. It is found that the high momentum parts in the proton spectrum comes mainly from by the 2-step process without pion, which is effectively identical to the $2N$-photoabsorption process. Thus it is very important to take account of the multistep contribution when comparing a theoretical result with experimental data. We have examined how the [$\Delta$]{}-potential affects the pion spectrum. We cannot find any significant difference between two kinds of results of the inclusive pion spectra with and without the $\Delta$-potential. The cross-section around QF peak of proton are overestimated at $\theta_{p} = 30^{\circ}$ and underestimated at $\theta_{p} = 52^{\circ}$. The discrepancy of the cross-section around the QF-peak may be improved by using the realistic angular distribution of ${\pi}$-production at the initial channels. In order to include it, we have to introduce the anisotropic decay of [$\Delta$]{}. It is, however, not easy because the spin degree is not involved in our semi-classical approach. We then need to consider a technical extension of our approach such as the p-wave decay of [$\Delta$]{} given by Engel et al. [@eng94]. Furthermore we have the other problems: too large pion-photoproduction and a dip between the QF-peak and the $2N$-absorption peak, which is not observed in the experimental data. These two problems are supposed to be caused by the same reason, i.e. lack of the pure $2N$ $\pi$-absorption process which dose not involve the intermediate [$\Delta$]{}. It is shown that the artificial enhancement of the $N{\Delta} \rightarrow NN$ cross-section (QMD/T1 calculation) does not give sufficient absorption of pion. >From both results of the proton emission and the pion production, thus, we can conclude that one should treat directly the pure $2N$ $\pi$-absorption process, i.e. three body collisional process, in the dynamical simulation. 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--- abstract: \| In a RAC drawing of a graph, vertices are represented by points in the plane, adjacent vertices are connected by line segments, and crossings must form right angles. Graphs that admit such drawings are RAC graphs. RAC graphs are beyond-planar graphs and have been studied extensively. In particular, it is known that a RAC graph with $n$ vertices has at most $4n-10$ edges. We introduce a superclass of RAC graphs, which we call arc-RAC graphs. In an [arc-RAC]{}drawing, edges are drawn as circular arcs whereas crossings must still form right angles. We provide a Turán-type result showing that an [arc-RAC]{}graph with $n$ vertices has at most $14n-12$ edges and that there are $n$-vertex [arc-RAC]{}graphs with $4.5n - O(\sqrt{n})$ edges. author: - Steven Chaplick - Henry Förster - Myroslav Kryven - Alexander Wolff bibliography: - 'abbrv.bib' - 'refs.bib' title: \| Drawing Graphs\ with Circular Arcs and Right Angle Crossings --- Introduction ============ A drawing of a graph in the plane is a mapping of its vertices to distinct points and each edge $uv$ to a curve whose endpoints are $u$ and $v$. Planar graphs, which admit crossing-free drawings, have been studied extensively. They have many nice properties and several algorithms for drawing them are known, see, e.g., [@juenger-book; @kaufmann-book]. However, in practice we must also draw non-planar graphs and crossings make it difficult to understand a drawing. For this reason, graph classes with restrictions on crossings are studied, e.g., graphs that can be drawn with at most $k$ crossings per edge (known as $k$-planar graphs) or where the angle formed by each crossing is “large”. These classes are categorized as beyond-planar graphs and have experienced increasing interest in recent years [@DBLP:journals/csur/DidimoLM19]. As introduced by Didimo et al. [@del-dgrac-TCS11], a prominent beyond-planar graph class that concerns the crossing angles is the class of $k$-bend right-angle-crossing graphs, or [RAC$_{k}$]{} graphs for short, that admit a drawing where all crossings form $90^\circ$ angles and each edge is a poly-line with at most $k$ bends. Using right-angle crossings and few bends is motivated by several cognitive studies suggesting a positive correlation between large crossing angles or small curve complexity and the readability of a graph drawing [@Huang2007; @Huang2014; @Huang2008]. Didimo et al. [@del-dgrac-TCS11] studied the edge density of [RAC$_{k}$]{} graphs. They showed that [RAC$_{0}$]{} graphs with $n$ vertices have at most $4n-10$ edges (which is tight), that [RAC$_{1}$]{} graphs have at most $O(n^{\frac{4}{3}})$ edges, that [RAC$_{2}$]{} graphs have at most $O(n^{\frac{7}{4}})$ edges and that all graphs are [RAC$_{3}$]{}. Dujmović et al. [@dgmw-nlacg-2010] gave an alternative simple proof of the $4n-10$ bound for [RAC$_{0}$]{} graphs using charging arguments similar to those of Ackerman and Tardos [@ackerman-tardos] and Ackerman [@ackerman]. Arikushi et al. [@afkmt-garacd-CGTA12] improved the upper bounds to $6.5n$ for [RAC$_{1}$]{} graphs and to $74.2n$ for [RAC$_{2}$]{} graphs. The bound of $6.5n-13$ for [RAC$_{1}$]{} graphs was also obtained by charging arguments. They also provided a [RAC$_{1}$]{} graph with $4.5n - O(\sqrt{n})$ edges. The best known lower and upper bound for the edge density of [RAC$_{1}$]{} graphs of $5n-10$ and $5.5n-11$, respectively, are due to Angelini et al. [@abfk-rdgobe-2018]. We extend the class of [RAC$_{0}$]{} graphs by allowing edges to be drawn as circular arcs but still requiring $90^\circ$ crossings. An angle at which two circles intersect is the angle between the two tangents to each of the circles at an intersection point. Two circles intersecting at a right angle are called orthogonal. For any circle $\gamma$, let ${C({\gamma})}$ be its center and let ${r({\gamma})}$ be its radius. ![Circles $\alpha$ and $\beta$ are orthogonal if and only if $\triangle X{C({\alpha})}{C({\beta})}$ is orthogonal. []{data-label="fig:digon"}](lenses) \[obs:basic\] Let $\alpha$ and $\beta$ be two circles. Then $\alpha$ and $\beta$ are orthogonal if and only if ${r({\alpha})}^2 + {r({\beta})}^2 = \|{C({\alpha})}{C({\beta})}\|^2$; see Figure \[fig:digon\]. Similarly, two circular arcs $\alpha$ and $\beta$ are orthogonal if they intersect properly (that is, ignoring intersections at endpoints) and the underlying circles (that contain the arcs) are orthogonal. For the remainder of this paper, all arcs will be circular arcs. We consider any straight-line segment an arc with infinite radius. We call a drawing of a graph where the edges are drawn as arcs such that any pair of intersecting arcs is orthogonal an [arc-RAC]{} drawing; see Figure \[fig:example\]. A graph that admits an [arc-RAC]{}drawings is called an [arc-RAC]{}graph. ![An [arc-RAC]{}drawing of a graph. This graph is not [RAC$_{0}$]{} [@bbhnr-nicpg-DAM17].[]{data-label="fig:example"}](nic-arcrac-notrac) The idea of drawing graphs with arcs dates back to at least the work of the artist Mark Lombardi who drew social networks, featuring players from the political and financial sector [@lh-mkgn-IC-2003]. Indeed, user studies [@phnk-otuflgd-gd-2013; @xrph-ausoceigv-DRAI2-12] state that users prefer edges drawn with curves of small curvature; not necessarily for performance but for aesthetics. Drawing graphs with arcs can help to improve certain quality measures of a drawing such as angular resolution [@cdgk-dpgwca-DCG01; @aacjr-twca-gd-2012] or visual complexity [@s-dgfa-JGAA15; @krw-dgfa-JGAA19]. An immediate restriction on the edge density of [arc-RAC]{}graphs is imposed by the following known result. \[lem:no4arcs\] In an [arc-RAC]{}drawing, there cannot be four pairwise orthogonal arcs. It follows from Lemma \[lem:no4arcs\] that [arc-RAC]{}graphs are [$4$-quasi-planar]{}, that is, an [arc-RAC]{}drawing cannot have four edges that pairwise cross. This implies that an [arc-RAC]{}graph with $n$ vertices can have at most $72(n - 2)$ edges [@ackerman]. Our main contribution is that we reduce this bound to $14n-12$ using charging arguments similar to those of Ackerman [@ackerman] and Dujmović et al. [@dgmw-nlacg-2010]; see Section \[sec:upper-bound\]. For us, the main challenge was to apply these charging arguments to a modification of an [arc-RAC]{}drawing and to exploit, at the same time, geometric properties of the original [arc-RAC]{}drawing to derive the bound. We also provide a lower bound of $4.5n - O(\sqrt{n})$ on the edge density of [arc-RAC]{}graphs based on the construction of Arikushi et al. [@afkmt-garacd-CGTA12]; see Section \[sec:lower-bound\]. We conclude with some open problems in Section \[sec:open-problems\]. Throughout the paper our notation won’t distinguish between the entities (vertices and edges) of an abstract graph and the geometric objects (points and curves) representing them in a drawing. Edge Density Upper Bound {#sec:upper-bound} ======================== Let $G$ be a [$4$-quasi-planar]{}graph, and let $D$ be a [$4$-quasi-planar]{}drawing of $G$. In his proof of the upper bound on the edge density of [$4$-quasi-planar]{}graphs, Ackerman [@ackerman] first modified the given drawing as to remove faces of small degree. We use a similar modification that we now describe. Consider two edges $e_1$ and $e_2$ in $D$ that intersect multiple times. A region in $D$ bounded by pieces of $e_1$ and $e_2$ that connect two consecutive intersection points is called a lens. If one of the intersection points is a vertex of $G$, then we call such a lens a 1-lens, otherwise a 0-lens. (From now on, by intersection point we always mean a proper intersection point.) A lens that does not contain a vertex of $G$ is empty. ![A simplification step resolves a smallest empty 0-lens; if two edges $e_1$ and $e_2$ change the order in which they share points with the edge $e$, they form an empty 0-lens intersecting $e$ before the step, and thus, in the original [$4$-quasi-planar]{}drawing.[]{data-label="fig:simplification-step"}](simplification-2) Every drawing with 0-lenses has a smallest empty 0-lens, that is, an empty 0-lens that does not contain any other empty 0-lenses in its interior. We can swap [@prt-rptg-2006; @ackerman] the two curves that bound a smallest empty 0-lens; Figure \[fig:simplification-step\]. We do it repeatedly until there are no empty 0-lenses left. We call the new empty-0-lens-free drawing $D'$ of $G$ a simplified drawing and the empty-0-lens-removal process simplification. As mentioned above, Ackerman [@ackerman] used a similar modification to prepare a [$4$-quasi-planar]{}drawing for his charging arguments; note, that unlike Ackerman, we do not resolve 1-lenses. We look at simplification in more detail, in particular, we consider how it changes the order in which edges share points. Note that during simplification, no new pairs of crossing edges are introduced, because each iteration only resolves a smallest empty lens. Therefore, the drawing remains [$4$-quasi-planar]{}. Because the drawing remains [$4$-quasi-planar]{}and because we always resolve a smallest empty 0-lens we have the following observation. \[obs:change-order\] If two edges $e_1$ and $e_2$ change the order in which they share points with another edge $e$ after a simplification step, then they form an empty 0-lens intersecting $e$ in the original [$4$-quasi-planar]{}drawing; see Figure \[fig:simplification-step\]. We now focus on the special type of [$4$-quasi-planar]{}drawings we are interested in. Suppose that $G$ is an [arc-RAC]{}graph, $D$ is an [arc-RAC]{}drawing of $G$, and $D'$ is a simplified drawing of $D$. If two edges $e_1$ and $e_2$ share an intersection point in $D'$, then they do not form a 0-empty lens in $D$, as otherwise simplification would remove both intersection points of these two edges. If $e_1$ and $e_2$ share a vertex of $G$ then they also do not form a 0-empty lens in $D$, as otherwise they would share three points in $D$ (the two intersection points of the lens and the common vertex of $G$). Thus, we have the following observation. \[obs:share-point-no-lens\] If two edges $e_1$ and $e_2$ share a point in the simplified drawing, then they do not form an empty 0-lens in the original [arc-RAC]{}drawing. In the following, we first state the main theorem of this section and provide the structure of its proof (deferring one small lemma and the main technical lemma until later). Then, we carefully prove the remaining technical details in Lemmas \[lem:1-lenses\] to \[lem:0-pentagon-at-most-4-triangles-main\] to establish the result. \[thm:density\_upper\_bound\] An [arc-RAC]{}graph with $n$ vertices can have at most $14n-12$ edges. Let $G=(V, E)$ be the given [arc-RAC]{}graph with an [arc-RAC]{}drawing $D$, let $D'$ be a simplified drawing of $D$, and let $G' = (V', E')$ be the planarization of $D'$. Our charging argument consists of three steps. First, each face $f$ of $G'$ is assigned an initial charge ${ch({f})} = \|f\| + v(f) - 4$, where $\|f\|$ is the degree of $f$ in the planarization and $v(f)$ is the number of vertices of $G$ on the boundary of $f$. Applying Euler’s formula several times, Ackerman and Tardos [@ackerman-tardos] showed that $\sum_{f\in G'} {ch({f})} = 4n-8$, where $n$ is the number of vertices of $G$. In addition, we set the charge ${ch({v})}$ of a vertex $v$ of $G$ to $16/3$. Hence the total charge of the system is $4n-8 + 16n/3=28n/3-8$. In the next two steps (described below), similarly to Dujmović et al. [@dgmw-nlacg-2010], we redistribute the charges among faces of $G'$ and vertices of $G$ so that, for every face $f$, the final charge ${ch_{\mathrm{fin}}({f})}$ is at least $v(f)/3$ and the final charge of each vertex is non-negative. Observing that $$28n/3-8 \ge \sum_{f\in G'} {ch_{\mathrm{fin}}({f})} \ge \sum_{f\in G'} v(f)/3 = \sum_{v\in G} \frac{\deg{(v)}}{3} = 2\|E\|/3$$ yields that the number of edges of $G$ is at most $14n-12$ as claimed. After the first charging step above it is easy to see that ${ch({f})} \ge v(f)/3$ holds if $\|f\| \ge 4$. We call a face $f$ of $G'$ a $k$-triangle, $k$-quadrilateral, or $k$-pentagon if $f$ has the corresponding shape and $v(f) = k$. Similarly, we call a face of degree two a digon. Note that any digon is a 1-digon, because it must be incident to a vertex of $G$. In the second charging step, we need to find ${4}/{3}$ units of charge for each digon, one unit of charge for each 0-triangle, and ${1}/{3}$ unit of charge for each 1-triangle. To charge a digon $d$ incident to a vertex $v$ of $G$ we decrease ${ch({v})}$ by $4/3$ and increase ${ch({d})}$ by $4/3$; see Figure \[fig:charging-vertex\]. We say that $v$ contributes charge to $d$. To charge triangles, we proceed similarly to Ackerman [@ackerman] and Dujmović et al. [@dgmw-nlacg-2010 Theorem 7]. Consider a $1$-triangle $t_1$. Let $v$ be the unique vertex incident to $t_1$, and let $s_1\in E'$ be the edge of $t_1$ opposite of $v$; see Figure \[fig:flank\]. Note that the endpoints of $s_1$ are intersection points in $D'$. Let $f_1$ be the face on the other side of $s_1$. If $f_1$ is a $0$-quadrilateral, then we consider its edge $s_2 \in E'$ opposite to $s_1$ and the face $f_2$ on the other side of $s_2$. We continue iteratively until we meet a face $f_k$ that is not a $0$-quadrilateral. If $f_k$ is a triangle, then all the faces $t_1,f_1,f_2, \dots, f_k$ belong to the same empty 1-lens $l$ (recall that $D'$ does not contain empty 0-lenses) incident to the vertex $v$ of $t_1$. In this case, we decrease ${ch({v})}$ by $1/3$ and increase ${ch({t_1})}$ by $1/3$; see Figure \[fig:charging-vertex\]. Otherwise, $f_k$ is not a triangle and $\|f_k\| + v(f_k) - 4 \ge 1$ (see Figure \[fig:flank\]). In this case, we decrease ${ch({f_k})}$ by $1/3$ and increase ${ch({t_1})}$ by $1/3$. We say that the face $f_k$ contributes charge to the triangle $t_1$ over its side $s_k$. For a 0-triangle $t_0$, we repeat the above charging over each side. If the last face on our path is a triangle $t'$, then $t_0$ and $t'$ are contained in an empty 1-lens and $t'$ is a 1-triangle incident to a vertex $v$ of $G$. In this case, we decrease ${ch({v})}$ by $1/3$ and increase ${ch({t_0})}$ by $1/3$; see Figure \[fig:vertex-charging-0-triangle\]. Thus, at the end of the second step, each digon or triangle $f$ has charge of at least $v(f)/3$. Note that the charge for $f$ comes either from a higher-degree face or from a vertex $v$ incident to an empty 1-lens containing $f$. In the third step, we do not modify the charging any more, but we need to ensure that (i) ${ch({f})} \ge v(f)/3$ still holds for each face $f$ of $G'$ with $\|f\| \ge 4$ and (ii) ${ch({v})} \ge 0$ for each $v$ of $G$. We first show statement (i). Ackerman [@ackerman] noted that a face $f$ can contribute charges over each of its edges at most once. Therefore, if $\|f\| + v(f) \ge 6$, then $f$ still has a charge of at least $v(f)/3$. It remains to verify that 1-quadrilaterals and 0-pentagons, which initially had only one unit of charge, have a charge of at least $1/3$ or zero, respectively, at the end of the second step. A 1-quadrilateral $q$ can contribute charge to at most two triangles since the endpoints of any edge of $G'$ over which a face contributes charge must be intersection points in $D'$; see Figure \[fig:charging-1-quadrilateral\] and recall that $q$ now plays the role of $f_k$ in Figure \[fig:flank\]. A 0-pentagon cannot contribute charge to more than three triangles; see Lemma \[lem:0-pentagon-at-most-4-triangles-main\]. Now we show statement (ii). Recall that a vertex $v$ can contribute charge to a digon incident to $v$ or to at most two triangles contained in an empty 1-lens incident to $v$. Observe that two empty 1-lenses with either triangles or a digon taking charge from $v$ cannot overlap; see Figure \[fig:charging-vertex\]. We show in Lemma \[lem:1-lenses\] that $v$ cannot be incident to more than four such empty 1-lenses. In the worst case, $v$ contributes ${4}/{3}$ units of charge to each of the at most four incident digons representing these empty 1-lenses. Thus, $v$ has non-negative charge at the end of the second step. [0.24]{} ![Transferring charge from vertices and high-degree faces to small-degree faces[]{data-label="fig:charging"}](charging-vertex "fig:") [0.18]{} ![Transferring charge from vertices and high-degree faces to small-degree faces[]{data-label="fig:charging"}](flank "fig:") [0.19]{} ![Transferring charge from vertices and high-degree faces to small-degree faces[]{data-label="fig:charging"}](flank "fig:") [0.22]{} ![Transferring charge from vertices and high-degree faces to small-degree faces[]{data-label="fig:charging"}](charging-1-quadrilateral "fig:") \[lem:1-lenses\] In any simplified [arc-RAC]{}drawing, each vertex is incident to at most four non-overlapping empty 1-lenses. Let $v$ be a vertex incident to some 1-lenses. Let $l$ be one of these 1-lenses. Then $l$ forms an angle of $90^\circ$ between the two edges incident to $v$ that form $l$; see Figure \[fig:1-lenses\]. This is due to the fact that the other “endpoint” of $l$ is a proper intersection point where the two edges must meet at $90^\circ$. Thus $v$ is incident to at most four empty 1-lenses. ![The operator ${\Pi({\cdot\,}; {\,\cdot})}$ is not meant to describe all intersection points along an edge; here ${\Pi({e}; {e_1, e_2, e_3, e_4,e_3, e_5, e_6})}$, ${\Pi({e}; {e_1, e_3, e_4, e_5})}$, and ${\Pi({e}; {e_2, e_4, e_3, e_5})}$ all hold at the same time.[]{data-label="fig:order"}](1-lenses) ![The operator ${\Pi({\cdot\,}; {\,\cdot})}$ is not meant to describe all intersection points along an edge; here ${\Pi({e}; {e_1, e_2, e_3, e_4,e_3, e_5, e_6})}$, ${\Pi({e}; {e_1, e_3, e_4, e_5})}$, and ${\Pi({e}; {e_2, e_4, e_3, e_5})}$ all hold at the same time.[]{data-label="fig:order"}](order) We now set the stage for proving Lemma \[lem:0-pentagon-at-most-4-triangles-main\], which shows that a 0-pentagon in a simplified drawing does not contribute charge to more than three triangles. The proof goes by a contradiction. First we consider the edges of the graph forming a 0-pentagon contributing charge to at least four triangles in the simplified drawing. Then we describe the order in which these edges share points in the simplified drawing and show that this order is preserved in the original [arc-RAC]{}drawing. Finally, we use some geometric arguments to show that such an [arc-RAC]{}drawing of the edges does not exist; see Lemma \[lem:0-pentagon-at-most-4-triangles\]. Let $D$ be an [arc-RAC]{}drawing of some [arc-RAC]{}graph $G = (V, E)$, let $D'$ be its simplified drawing, and let $p$ be a 0-pentagon that contributes charge to at least four triangles. Let $s_0, s_1, \dots, s_4$ be the sides of $p$ in clockwise order and denote the edges of $G$ that contain these sides as $e_0, e_1, \dots, e_4$ so that edge $e_0$ contains side $s_0$ etc. Since $p$ contributes charge over at least four sides, these sides are consecutive around $p$. Without loss of generality, we assume that $s_4$ is the side over which $p$ does not necessarily contribute charge. For $i \in \{ 0, 1, 2, 3\}$, let $t_i$ be the triangle that gets charge from $p$ over the side $s_i$. The triangle $t_i$ is bounded by the edges $e_{i-1}$ and $e_{i+1}$. (Indices are taken modulo $5$.) Note that all faces bounded by $e_{i-1}$ and $e_{i+1}$ that are between $t_i$ and $p$ must be 0-quadrilaterals. If $t_i$ is a 1-triangle, then $e_{i-1}$ and $e_{i+1}$ share a vertex of the graph at a vertex of the triangle, otherwise $t_i$ is a 0-triangle and $e_{i-1}$ and $e_{i+1}$ intersect at a vertex of the triangle. Let $A'_{i-1, i+1}$ denote this common point of $e_{i-1}$ and $e_{i+1}$, and let $E_p=\{e_0, \dots, e_4\}$; see Figure \[fig:0-pentagon-D’\]. [0.48]{} ![A $0$-pentagon cannot contribute charge to more than three triangles.[]{data-label="fig:0-pentagon"}](our-contradiction "fig:") [0.44]{} ![A $0$-pentagon cannot contribute charge to more than three triangles.[]{data-label="fig:0-pentagon"}](cases "fig:") We now describe the order in which the edges in $E_p$ share points in $D'$. To this end, we orient the edges in $E_p$ so that this orientation conforms with the orientation of a clockwise walk around the boundary of $p$ in $D'$. In addition, we write ${\Pi({e_k}; {e_{i_1}, e_{i_2}, \dots, e_{i_l}})}$ if the edge $e_k$ shares points (either intersection points or vertices of the graph) with the edges $e_{i_1}, e_{i_2}, \dots, e_{i_l}$ in this order with respect to the orientation of $e_k$; see Figure \[fig:order\]. (Note that we can have ${\Pi({e_k}; {e_i, e_j, e_i})}$ as edges may intersect twice. We will not consider more than two edges sharing the same endpoint.) Due to the order in which we numbered the edges in $E_p$, it holds in $D'$ that ${\Pi({e_0}; {e_{4}, e_{1}, e_{2}})}$, ${\Pi({e_3}; {e_{1}, e_{2}, e_{4}})}$, and, for $i \in \{1,2,4\}$, ${\Pi({e_i}; {e_{i-2}, e_{i-1}, e_{i+1}, e_{i+2}})}$; see Figure \[fig:0-pentagon-D’\]. Now we show that this order is preserved in $D$. Obviously every pair of edges $(e_{i-1}, e_{i+1})$ that shares a vertex of $G$ in $D'$ also shares a vertex of $G$ in $D$. Furthermore, every pair of intersecting edges $(e_i, e_{i+1})$ or $(e_{i-1}, e_{i+1})$ also intersects in $D$ because simplification does not introduce new pairs of crossing edges. \[lem:edges-0-3\] In the drawing $D$, the edges $e_{0}$ and $e_{3}$ do not cross. Assume that the edges $e_{0}$ and $e_{3}$ cross in $D$ and notice that each of the pairs of edges $(e_{0}, e_{1})$, $(e_{1}, e_{2})$, and $(e_{2}, e_{3})$ forms a crossing in $D'$ (see Figure \[fig:0-pentagon-D’\]), and hence in $D$, too. For any arc $e$, let ${\bar{e}}$ denote the circle containing $e$. Recall that a family of Apollonian circles [@excursions; @cfkw-oaooc-GD19] consists of two sets of circles such that each circle in one set is orthogonal to each circle in the other set. Thus, the pairs of circles $({\bar{e}}_{1}, {\bar{e}}_{3})$ and $({\bar{e}}_{0}, {\bar{e}}_{2})$ belong to such a family; the pair $({\bar{e}}_{1}, {\bar{e}}_{3})$ belongs to one set of the family and $({\bar{e}}_{0}, {\bar{e}}_{2})$ belongs to the other set. If not all of the circles in the family share the same point, which is the case for the circles ${\bar{e}}_{0}$, ${\bar{e}}_{1}$, ${\bar{e}}_{2}$, and ${\bar{e}}_{3}$, then one such set consists of disjoint circles. So either the pair $({\bar{e}}_{0}, {\bar{e}}_{2})$ or the pair $({\bar{e}}_{1}, {\bar{e}}_{3})$ must consist of disjoint circles. This is a contradiction because each of the two pairs shares a point in $D'$ (see Figure \[fig:0-pentagon-D’\]), and thus, in $D$. \[lem:edges-order\] In the drawing $D$, ${\Pi({e_0}; {e_{4},e_{1},e_{2}})}$, ${\Pi({e_3}; {e_{1},e_{2},e_{4}})}$, and, for $i \in \{1,2,4\}$, ${\Pi({e_i}; {e_{i-2},e_{i-1},e_{i+1},e_{i+2}})}$ hold. Recall that in the drawing $D'$, ${\Pi({e_0}; {e_{4},e_{1},e_{2}})}$, ${\Pi({e_3}; {e_{1},e_{2},e_{4}})}$, and, for $i \in \{1,2,4\}$, ${\Pi({e_i}; {e_{i-2},e_{i-1},e_{i+1},e_{i+2}})}$ hold; see Figure \[fig:0-pentagon-D’\]. Choose different indices $i,j,k \in\{0,1,2,3,4\}$ so that the edges $e_i$ and $e_j$ share points with $e_k$ in this order in $D'$, that is, ${\Pi({e_k}; {e_{i}, e_{j}})}$ in $D'$. We will show that the edges $e_i$ and $e_j$ share points with $e_k$ in the same order in $D$, that is, ${\Pi({e_k}; {e_{i}, e_{j}})}$ in $D$. Thus, the order in which the edges $E_p$ share points in $D'$ is preserved in $D$. If $(i,j) \in \{(0,3),(3,0)\}$, then, according to Lemma \[lem:edges-0-3\], the edges $e_i$ and $e_j$ do not cross in $D$, so they do not form an empty 0-lens in $D$, and thus, by Observation \[obs:change-order\], $e_i$ and $e_j$ share points with $e_k$ in the same order in $D$ as in $D'$, that is, ${\Pi({e_k}; {e_{i}, e_{j}})}$ in $D$. Otherwise the edges $e_i$ and $e_j$ share a point in $D'$; see Figure \[fig:0-pentagon-D’\]. Therefore, according to Observation \[obs:share-point-no-lens\], $e_i$ and $e_j$ do not form an empty 0-lens in $D$, and thus, according to Observation \[obs:change-order\], $e_i$ and $e_j$ share points with $e_k$ in the same order in $D$ as in $D'$, that is, ${\Pi({e_k}; {e_{i}, e_{j}})}$ in $D$. Thus, we have shown that the order in which the edges in $E_p$ share points in $D'$ is preserved in $D$; see Figure \[fig:0-pentagon-D\]. We show now that such an [arc-RAC]{}drawing does not exist; see Lemma \[lem:0-pentagon-at-most-4-triangles\]. This is the main ingredient to prove Lemma \[lem:0-pentagon-at-most-4-triangles-main\], which says that a 0-pentagon in a simplified arc-RAC drawing contributes charge to at most three triangles. For simplicity of presentation and without loss of generality, we assume that the points $A'_{i-1, i+1}$ are vertices of $G$, which we denote by $v_{i-1, i+1}$. To prove Lemma \[lem:0-pentagon-at-most-4-triangles\], we need the following simple observation. \[obs:tangent-center\] For two orthogonal circles the tangent to one circle at one of the intersection points goes through the center of the other circle; see Figure \[fig:digon\]. In particular, a line is orthogonal to a circle if the line goes through the center of the circle. \[lem:0-pentagon-at-most-4-triangles\] The edges in $E_p$ do not admit an [arc-RAC]{}drawing where it holds that ${\Pi({e_0}; {e_{4}, e_{1}, e_{2}})}$, ${\Pi({e_3}; {e_{1}, e_{2}, e_{4}})}$, and, for $i \in \{1,2,4\}$, ${\Pi({e_i}; {e_{i-2}, e_{i-1}, e_{i+1}, e_{i+2}})}$. Assume that the edges in $E_p$ admit an [arc-RAC]{}drawing where they share points in the order indicated above. For $i \in \{0,\dots,4\}$, let $P_{i, i+1}$ be the intersection point of $e_{i}$ and $e_{i+1}$; see Figure \[fig:0-pentagon-D\]. Note that on $e_i$, the point $P_{i-1, i}$ is before the point $P_{i, i+1}$ (due to ${\Pi({e_i}; {e_{i-1}, e_{i+1}})}$). Recall that an inversion [@excursions] with respect to a circle $\alpha$, the inversion circle, is a mapping that takes any point $P \neq {C({\alpha})}$ to a point $P'$ on the ray ${C({\alpha})}P$ so that $\|{C({\alpha})}P'\|\cdot\|{C({\alpha})}P\| = {r({\alpha})}^2$. Inversion maps each circle not passing through ${C({\alpha})}$ to another circle and each circle passing through ${C({\alpha})}$ to a line. The center of the inversion circle is mapped to the “point at infinity”. It is known that inversion preserves angles and the topology of the original drawing. Therefore, the image of an inversion of an [arc-RAC]{}drawing is also an [arc-RAC]{}drawing and the order in which the edges in $E_p$ share points is preserved. We invert the drawing of the edges in $E_p$ with respect to a small inversion circle centered at $v_{24}$. Let ${{e}^\circ}_{i}$ be the image of $e_{i}$, ${{v}^\circ}_{i-1,i+1}$ be the image of $v_{i-1,i+1}$ (${{v}^\circ}_{24}$ is the point at infinity), and ${{P}^\circ}_{i,i+1}$ be the image of $P_{i,i+1}$. We consider two cases regarding whether the edges $e_{2}$ and $e_{4}$ belong to two different circles or not. Case I: $e_{2}$ and $e_{4}$ belong to two different circles. One of the intersection points of their circles is $v_{24}$, and we let $X$ denote the other intersection point. Here we have that ${{e}^\circ}_{2}$ and ${{e}^\circ}_{4}$ are two straight line rays whose lines intersect at the point ${{X}^\circ}$ which is the image of $X$; see Figure \[fig:0-pentagon-inversion-different\]. [0.48]{} ![Illustration for the proof of Lemma \[lem:0-pentagon-at-most-4-triangles\] when $e_{2}$ and $e_{4}$ belong to two different circles. Image of the inversion with respect to the red circle in Figure \[fig:0-pentagon-D\].[]{data-label="fig:0-pentagon-inversion-different"}](inversion-1 "fig:") [0.48]{} ![Illustration for the proof of Lemma \[lem:0-pentagon-at-most-4-triangles\] when $e_{2}$ and $e_{4}$ belong to two different circles. Image of the inversion with respect to the red circle in Figure \[fig:0-pentagon-D\].[]{data-label="fig:0-pentagon-inversion-different"}](inversion-2 "fig:") We now assume for a contradiction that the arc ${{e}^\circ}_{1}$ forms a concave side of the triangle $\Delta_1={{P}^\circ}_{12}{{v}^\circ}_{41}{{X}^\circ}$; see Figure \[fig:0-new-proof\] where the triangle is filled gray. (Symmetrically, we can show that the arc ${{e}^\circ}_{0}$ cannot form a concave side of the triangle $\Delta_0={{P}^\circ}_{40}{{v}^\circ}_{02}{{X}^\circ}$.) By Observation \[obs:tangent-center\], ${C({{{e}^\circ}_{1}})}$ must lie on the ray ${{e}^\circ}_2$. Since we assume that the arc ${{e}^\circ}_{1}$ forms a concave side of the triangle $\Delta_1$, ${C({{{e}^\circ}_{1}})}$ and ${{v}^\circ}_{02}$ are separated by ${{P}^\circ}_{12}$ on ${{e}^\circ}_2$. Consider the tangent $l_0$ to ${{e}^\circ}_{0}$ at ${{P}^\circ}_{01}$. Again in light of Observation \[obs:tangent-center\], $l_0$ has to go through ${C({{{e}^\circ}_{1}})}$ because ${{e}^\circ}_{0}$ and ${{e}^\circ}_{1}$ are orthogonal. On the one hand, ${{v}^\circ}_{02}$ is to the same side of $l_0$ as ${{P}^\circ}_{12}$; see Figure \[fig:0-new-proof\]. On the other hand, $l_0$ separates ${{P}^\circ}_{12}$ and ${{v}^\circ}_{41}$ due to ${\Pi({e_1}; {e_4,e_0,e_2})}$. Moreover, $l_0$ does not separate ${{v}^\circ}_{41}$ and ${{P}^\circ}_{40}$ since it intersects the line of ${{e}^\circ}_4$ when leaving the gray triangle $\Delta_1$. So the two points ${{v}^\circ}_{02}$ and ${{P}^\circ}_{40}$ of the same arc ${{e}^\circ}_{0}$ are separated by $l_0$, which is a tangent of this arc; contradiction. Thus, the arc ${{e}^\circ}_{1}$ forms a convex side of the triangle $\Delta_1$, and ${{e}^\circ}_{0}$ forms a convex side of $\Delta_0$; see Figure \[fig:0-new-proof-old-part\]. Now, due to Observation \[obs:tangent-center\], ${C({{{e}^\circ}_{0}})}$ is between ${{v}^\circ}_{41}$ and ${{P}^\circ}_{40}$, and ${C({{{e}^\circ}_{1}})}$ is between ${{v}^\circ}_{02}$ and ${{P}^\circ}_{12}$, because that is where the tangents $l_1$ of ${{e}^\circ}_1$ and $l_0$ of ${{e}^\circ}_0$ in ${{P}^\circ}_{01}$ intersect the lines of ${{e}^\circ}_{4}$ and ${{e}^\circ}_{2}$, respectively. Taking into account that ${C({{{e}^\circ}_{3}})} = {{X}^\circ}$, because ${{e}^\circ}_{3}$ is orthogonal to both ${{e}^\circ}_{2}$ and ${{e}^\circ}_{4}$, we obtain that the points ${C({{{e}^\circ}_{3}})}$, ${C({{{e}^\circ}_{1}})}$, ${{P}^\circ}_{12}$, ${{P}^\circ}_{23}$ appear on the line of ${{e}^\circ}_{2}$ in this order. Thus, the circle of ${{e}^\circ}_{1}$ is contained within the circle of ${{e}^\circ}_{3}$. This is a contradiction because ${{e}^\circ}_{3}$ and ${{e}^\circ}_{1}$ must share a point; namely ${{v}^\circ}_{13}$. Case II: $e_{2}$ and $e_{4}$ belong to the same circle. Here ${{e}^\circ}_{2}$ and ${{e}^\circ}_{4}$ are two disjoint straight-line rays on the same line $l$ (meeting at infinity at ${{v}^\circ}_{24}$); see Figure \[fig:0-pentagon-inversion-same\]. We direct $l$ as ${{e}^\circ}_4$ and ${{e}^\circ}_2$ (from right to left in Figure \[fig:0-pentagon-inversion-same\]). Because ${{e}^\circ}_{0}$, ${{e}^\circ}_{1}$, and ${{e}^\circ}_{3}$ are orthogonal to $l$, their centers have to be on $l$. Due to our initial assumption, we have ${\Pi({e_4}; {e_2,e_3,e_0,e_1})}$ and ${\Pi({e_2}; {e_0,e_1,e_3,e_4})}$. Hence, along $l$, we have ${{P}^\circ}_{34}$, ${{P}^\circ}_{40}$, ${{v}^\circ}_{41}$, (on ${{e}^\circ}_4$) and then ${{v}^\circ}_{02}$, ${{P}^\circ}_{12}$, ${{P}^\circ}_{23}$ (on ${{e}^\circ}_2$). Therefore, the circle of ${{e}^\circ}_1$ is contained in that of ${{e}^\circ}_3$. Hence, ${{e}^\circ}_{1}$ does not share a point with ${{e}^\circ}_{3}$; a contradiction. ![Illustration to the proof of Lemma \[lem:0-pentagon-at-most-4-triangles\] when $e_{2}$ and $e_{4}$ belong to the same circle. Image of the inversion with respect to the red circle in Figure \[fig:0-pentagon-D\].[]{data-label="fig:0-pentagon-inversion-same"}](our-contradiction-2) \[lem:0-pentagon-at-most-4-triangles-main\] A 0-pentagon in a simplified [arc-RAC]{}drawing contributes charge to at most three triangles. As discussed above, if a 0-pentagon formed by edges $e_0, e_1, \dots, e_4$ contributes charge to more than three triangles in a simplified drawing (see Figure \[fig:0-pentagon-D’\]), then this implies the existence of an [arc-RAC]{}drawing where it holds that ${\Pi({e_0}; {e_{4}, e_{1}, e_{2}})}$, ${\Pi({e_3}; {e_{1}, e_{2}, e_{4}})}$ and, for $i \in \{1, 2, 4\}$, ${\Pi({e_i}; {e_{i-2}, e_{i-1}, e_{i+1}, e_{i+2}})}$; see Figure \[fig:0-pentagon-D\]. This, however, contradicts Lemma \[lem:0-pentagon-at-most-4-triangles\]. With the proofs of Lemmas \[lem:1-lenses\] and \[lem:0-pentagon-at-most-4-triangles-main\] now in place, the proof of Theorem \[thm:density\_upper\_bound\] is complete. Edge Density Lower Bound {#sec:lower-bound} ======================== We give a lower bound on the number of edges by using the construction of Arikushi et al. [@arikushi] that they used to give the lower bound on edge density for [RAC$_{1}$]{} graphs. Let $G$ be an embedded graph where the vertices of $G$ are points of the hexagonal lattice clipped in a rectangle; see Figure \[fig:tiling\]. The edges of $G$ are the edges of the lattice and, inside each hexagon that is bounded by cycle $(P_0,\ldots,P_5)$, six additional edges $(P_i, P_{i+2})$. We refer to a part of the drawing made up of a single hexagon and its diagonals as a tile. In Theorem \[thm:density\_lower\_bound\] we show that each hexagon can be drawn as a regular hexagon and its diagonals can be drawn as two sets of arcs $A = \{\alpha_0, \alpha_1, \alpha_2\}$ and $B= \{\beta_0, \beta_1, \beta_2\}$, so that the arcs in $A$ are pairwise orthogonal and for each arc in $B$ intersecting another arc in $A$ the two arcs are orthogonal; we use this construction to establish Theorem \[thm:density\_lower\_bound\]. In particular, the arcs in $A$ form the triangle $(P_0,P_2,P_4)$ while the arcs in $B$ form the triangle $(P_1,P_3,P_5)$. We first define the radii and centers of the arcs in a tile and show that they form only orthogonal crossings. We use the geometric center of the tile as the origin of our coordinate system in the following analysis. We now discuss the arcs in $A$; then we turn to the arcs in $B$. For each $j \in \{0, 1, 2\}$, the arc $\alpha_j$ has radius $r_A = 1$ and center ${C({\alpha_j})} = (x_A\cos(\pi/6 + j \frac{2\pi}{3}), x_A\sin(\pi/6 + j \frac{2\pi}{3}))$ where $x_A=\sqrt{2/3}$; see Figure \[fig:confA\]. \[lem:3circles\] The arcs in $A$ are pairwise orthogonal. Consider the equilateral triangle $\triangle {C({\alpha_0})}{C({\alpha_1})}{C({\alpha_2})}$ on the centers of the three arcs. Because the origin is in the center of the triangle it is easy to see that the length of the side of the triangle is $2 x_A\cos{\pi/6} = \sqrt{2}$, and so the distance between the centers of any two arcs is $\sqrt{2}$. Because, in addition, the radius of each arc is 1, by Observation \[obs:basic\], every two arcs are orthogonal. As in Figure \[fig:confB\], for each $j \in \{0,1,2\}$, the arc $\beta_j$ has radius $r_B = \sqrt{\frac{70 + 40\sqrt{3}}{6}}$ and center ${C({\beta_j})} = (x_B\cos(\frac{\pi}{2} + \frac{(j+1)2\pi}{3}), x_B\sin(\frac{\pi}{2} + \frac{(j+1)2\pi}{3}))$, where $x_B = \frac{\sqrt{2}}{2\sqrt{3}} + \sqrt{\frac{73 + 40\sqrt{3}}{6}}$. \[lem:3big\_circles\] If an arc in $B$ intersects an arc in $A$, then the two arcs are orthogonal. Let $i, j \in \{0,1,2\}$. If $j = i$, then $\\|{C({\alpha_i})} -{C({\beta_j})}\\| = \sqrt{\frac{112+64\sqrt{3}}{6}} > 1 + \sqrt{\frac{70 + 40\sqrt{3}}{6}} = r_A + r_B$, so the arcs $\alpha_i$ and $\beta_j$ do not intersect, otherwise $\\|{C({\alpha_i})} -{C({\beta_j})}\\|^2 = \frac{76 + 40\sqrt{3}}{6} = 1 + \frac{70 + 40\sqrt{3}}{6} = r_A^2 + r_B^2$, so by Observation \[obs:basic\] the arcs $\alpha_i$ and $\beta_j$ are orthogonal. [0.48]{} ![Tiling used for the lower bound construction](density2-old "fig:") [0.48]{} ![Tiling used for the lower bound construction](density2-old "fig:") [0.48]{} ![Construction for the lower bound on the edge density of arc-RAC graphs[]{data-label="fig:density_lower_bound"}](density2-old "fig:") [0.48]{} ![Construction for the lower bound on the edge density of arc-RAC graphs[]{data-label="fig:density_lower_bound"}](density2-old "fig:") \[thm:density\_lower\_bound\] There exist arc-RAC graphs with $4.5n - O(\sqrt{n})$ edges. We start by constructing a tile and then show that its drawing is indeed a valid [arc-RAC]{}drawing. After that it is easy to construct the drawing of the embedded graph $G$ with the claimed edge-density. Consider two intersecting circles $\alpha$ and $\beta$ so that one of their intersection points is closer to the origin than the other, we denote by $X^-_{\alpha\beta}$ the intersection point which is closer to the origin and by $X^+_{\alpha\beta}$ the intersection point which is further from the origin. Let the vertices of the hexagon in a tile be $P_0 = X^+_{\alpha_0\alpha_1}$, $P_1 = X^-_{\beta_2\beta_0}$, $P_2 = X^+_{\alpha_1\alpha_2}$, $P_3 = X^-_{\beta_0\beta_1}$, $P_4 = X^+_{\alpha_2\alpha_0}$, and $P_5 = X^-_{\beta_1\beta_2}$. Due to the symmetric definitions of the arcs the angle between two consecutive vertices of the hexagon is $\frac{\pi}{3}$. Moreover, by a simple computation, we see that for each $j\in \{0, 1, 2\}$ and with $b = \frac{\sqrt{2}}{2}(\frac{1}{\sqrt{3}}+1)$ we have: - $X^+_{\alpha_j\alpha_{j+1}} = (b\cos(\frac{\pi}{2} + j \frac{2\pi}{3}), b\sin(\frac{\pi}{2} + j \frac{2\pi}{3}))$ and - $X^-_{\beta_j\beta_{j+1}} = (b\cos(\frac{\pi}{6} + (j+2) \frac{2\pi}{3}), b\sin(\frac{\pi}{6} + (j+2) \frac{2\pi}{3}))$. Thus, all the vertices of the hexagon are equidistant from its center, so the hexagon is regular. Now let the arcs of $A$ and $B$ be exactly the arcs with radii and centers specified above, such that they are completely contained in the regular hexagon. According to Lemmas \[lem:3circles\] and \[lem:3big\_circles\] all the crossings of arcs belonging to the same tile are orthogonal. Moreover, the arcs do not intersect the interior of the edges of the hexagon. To see this take, for example, the arc $\alpha_2$. Its center ${C({\alpha_2})}$ is below the segment connecting $P_2$ and $P_4$. This segment makes an orthogonal angle in the hexagon with the side of the hexagon $P_2P_1$, therefore, the tangent of $\alpha_2$ at $P_2$ intersects the interior of the hexagon. Thus, $\alpha_2$ cannot intersect an edge of the hexagon. Similarly we can show that the arcs in $B$ do not intersect the interior of the edges of the hexagon. Therefore, the drawing of a tile, and hence of $G$, is valid. As mentioned by Arikushi et al. [@arikushi] almost all vertices of the lattice with the exception of at most $O(\sqrt{n})$ vertices at the lattice’s boundary have degree 9. Hence the number of edges is $4.5n - O(\sqrt{n})$. As an immediate corollary of Theorem \[thm:density\_lower\_bound\] we obtain the following result. The class of [arc-RAC]{}graphs is a proper superclass of [RAC$_{0}$]{} graphs. Open Problems and Conjectures {#sec:open-problems} ============================= An obvious open problem is to tighten the bounds on the edge density of [arc-RAC]{}graphs in Theorems \[thm:density\_upper\_bound\] and \[thm:density\_lower\_bound\]. Another immediate question is the relation to [RAC$_{1}$]{} graphs, which also extend the class of [RAC$_{0}$]{} graphs. This is especially intriguing as the best known lower bound for the edge density of [RAC$_{1}$]{} graphs is indeed larger than our lower bound for [arc-RAC]{}graphs whereas potentially there might be [arc-RAC]{}graphs that are denser than the densest [RAC$_{1}$]{} graphs. The relation between [RAC$_{k}$]{} graphs and 1-planar graphs is well understood [@afkmt-garacd-CGTA12; @bbhnr-nicpg-DAM17; @bdlmm-oracdo1pg-TCS17; @bdeklm-radicpg-TCS16; @clwz-cdo1pgwracafb-CG19; @el-racga1p-DAM13]. What about the relation between [arc-RAC]{}graphs and 1-planar graphs? In particular, is there a 1-planar graph which is not [arc-RAC]{}? We are also interested in the area required by [arc-RAC]{}drawings. Are there [arc-RAC]{}graphs that need exponential area to admit an [arc-RAC]{}drawing? (A way to measure this off the grid is to consider the ratio between the longest and the shortest edge in a drawing.) Finally, the complexity of recognizing [arc-RAC]{}graphs is open, but likely NP-hard.
--- abstract: \| Let $X$ be a finite set such that $\|X\|=n$. Let ${\mathcal{T}_{n}}$ and ${\mathcal{S}_{n}}$ denote the transformation monoid and the symmetric group on $n$ points, respectively. Given $a\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$, we say that a group $G{\leqslant}{\mathcal{S}_{n}}$ is $a$-normalizing if $$\langle a,G\rangle \setminus G=\langle g^{{-1}}ag\mid g\in G\rangle,$$ where $\langle a, G\rangle$ and $\langle g^{{-1}}ag\mid g\in G\rangle$ denote the subsemigroups of ${\mathcal{T}_{n}}$ generated by the sets $\{a\}\cup G$ and $\{g^{-1}ag \mid g\in G\}$, respectively. If $G$ is $a$-normalizing for all $a\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$, then we say that $G$ is normalizing. The goal of this paper is to classify the normalizing groups and hence answer a question of Levi, McAlister, and McFadden. The paper ends with a number of problems for experts in groups, semigroups and matrix theory. address: - \| Universidade Aberta and Centro de Álgebra\ Universidade de Lisboa\ Av. Gama Pinto, 2, 1649-003 Lisboa\ Portugal - \| Department of Mathematics\ School of Mathematical Sciences at Queen Mary\ University of London - 'Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland\' - 'Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, Scotland' author: - João Araújo - 'Peter J. Cameron' - James Mitchell - Max Neunhöffer title: The Classification of Normalizing Groups --- [Date:]{} 10 August 2011\ [Key words and phrases:]{} Transformation semigroups, permutation groups, primitive groups, GAP\ [2010 Mathematics Subject Classification:]{} 20B30, 20B35, 20B15, 20B40, 20M20, 20M17.\ [Corresponding author: João Araújo]{} Introduction and Preliminaries ============================== For notation and basic results on group theory we refer the reader to [@cam; @dixon]; for semigroup theory we refer the reader to [@Ho95]. Let ${\mathcal{T}_{n}}$ and ${\mathcal{S}_{n}}$ denote the monoid consisting of mappings from $[n]:=\{1,\ldots ,n\}$ to $[n]$ and the symmetric group on $[n]$ points, respectively. The monoid ${\mathcal{T}_{n}}$ is usually called the full transformation semigroup. In [@lm], Levi and McFadden proved the following result. Let $a\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$. Then 1. $\langle g^{-1}ag\mid g\in {\mathcal{S}_{n}}\rangle$ is idempotent generated; 2. $\langle g^{-1}ag\mid g\in {\mathcal{S}_{n}}\rangle$ is regular. Using a beautiful argument, McAlister [@mcalister] proved that the semigroups $\langle g^{-1}ag\mid g\in {\mathcal{S}_{n}}\rangle$ and $\langle a, {\mathcal{S}_{n}}\rangle\setminus {\mathcal{S}_{n}}$ (for $a\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$) have exactly the same set of idempotents; therefore, as $\langle g^{-1}ag\mid g\in {\mathcal{S}_{n}}\rangle$ is idempotent generated, it follows that $$\langle g^{-1}ag\mid g\in {\mathcal{S}_{n}}\rangle=\langle a, {\mathcal{S}_{n}}\rangle\setminus {\mathcal{S}_{n}}.$$ Later, Levi [@levi96] proved that $\langle g^{-1}ag\mid g\in {\mathcal{S}_{n}}\rangle= \langle g^{-1}ag\mid g\in {\mathcal{A}_{n}}\rangle$ (for $a\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$), and hence the three results above remain true when we replace ${\mathcal{S}_{n}}$ by ${\mathcal{A}_{n}}$. The following list of problems naturally arises from these considerations. 1. Classify the groups $G{\leqslant}{\mathcal{S}_{n}}$ such that for all $a\in {\mathcal{T}_{n}}\setminus{\mathcal{S}_{n}}$ we have that the semigroup $\langle g^{-1}ag\mid g\in G\rangle$ is idempotent generated. 2. Classify the groups $G{\leqslant}{\mathcal{S}_{n}}$ such that for all $a\in {\mathcal{T}_{n}}\setminus{\mathcal{S}_{n}}$ we have that the semigroup $\langle g^{-1}ag\mid g\in G\rangle$ is regular. 3. Classify the groups $G{\leqslant}{\mathcal{S}_{n}}$ such that for all $a\in {\mathcal{T}_{n}}\setminus{\mathcal{S}_{n}}$ we have $$\langle a,G\rangle\setminus G = \langle g^{-1}ag\mid g\in G\rangle.$$ The two first questions were solved in [@ArMiSc] as follows: If $n{\geqslant}1$ and $G$ is a subgroup of ${\mathcal{S}_{n}}$, then the following are equivalent: 1. The semigroup ${\ensuremath{\langle\: g^{{-1}}ag \mid g\in G \:\rangle}}$ is idempotent generated for all $a\in {\mathcal{T}_{n}}\setminus{\mathcal{S}_{n}}$. 2. One of the following is valid for $G$ and $n$: 1. $n=5$ and $G$ is ${\mbox{\rm AGL}}(1,5)$; 2. $n=6$ and $G$ is ${\mbox{\rm PSL}}(2,5)$ or ${\mbox{\rm PGL}}(2,5)$; 3. $G$ is ${\mathcal{A}_{n}}$ or ${\mathcal{S}_{n}}$. \[th2\] If $n{\geqslant}1$ and $G$ is a subgroup of ${\mathcal{S}_{n}}$, then the following are equivalent: 1. The semigroup ${\ensuremath{\langle\: g^{{-1}}ag \mid g\in G \:\rangle}}$ is regular for all $a\in {\mathcal{T}_{n}}\setminus{\mathcal{S}_{n}}$. 2. One of the following is valid for $G$ and $n$: 1. $n=5$ and $G$ is $ C_5,\ {\ensuremath{D_{5}}},$ or ${\mbox{\rm AGL}}(1,5)$; 2. $n=6$ and $G$ is $ {\mbox{\rm PSL}}(2,5)$ or ${\mbox{\rm PGL}}(2,5)$; 3. $n=7$ and $G$ is ${\mbox{\rm AGL}}(1,7)$; 4. $n=8$ and $G$ is ${\mbox{\rm PGL}}(2,7)$; 5. $n=9$ and $G$ is ${\mbox{\rm PSL}}(2,8)$ or ${\mbox{\rm P}\Gamma {\rm L}}(2,8)$; 6. $G$ is ${\mathcal{A}_{n}}$ or ${\mathcal{S}_{n}}$. These results leave us with the third problem. Given $a\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$, we say that a group $G{\leqslant}{\mathcal{S}_{n}}$ is $a$-normalizing if $$\langle a,G\rangle \setminus G=\langle g^{{-1}}ag\mid g\in G\rangle.$$ If $G$ is $a$-normalizing for all $a\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$, then we say that $G$ is normalizing. Recall that the rank of a transformation $f$ is just the number of points in its image; we denote this by ${\operatorname{rank}}(f)$. For a given $k$ such that $1{\leqslant}k< n$, we say that $G$ is $k$-normalizing if $G$ is $a$-normalizing for all rank $k$ maps $a\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$. Levi, McAlister and McFadden [@lmm p.464] ask for a classification of all pairs $(a,G)$ such that $G$ is $a$-normalizing, and in [@ArMiSc] is proposed the more tractable problem of classifying the normalizing groups. The aim of this paper is to provide such a classification. \[main\] If $n{\geqslant}1$ and $G$ is a subgroup of ${\mathcal{S}_{n}}$, then the following are equivalent: 1. The group $G$ is normalizing, that is, for all $a\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$ we have $$\langle a,G\rangle \setminus G = \langle g^{{-1}}ag\mid g\in G\rangle;$$ 2. One of the following is valid for $G$ and $n$: 1. $n=5$ and $G$ is $ {\mbox{\rm AGL}}(1,5)$; 2. $n=6$ and $G$ is $ {\mbox{\rm PSL}}(2,5)$ or ${\mbox{\rm PGL}}(2,5)$; 3. $n=9$ and $G$ is ${\mbox{\rm PSL}}(2,8)$ or ${\mbox{\rm P}\Gamma {\rm L}}(2,8)$; 4. $G$ is $\{1\}$, ${\mathcal{A}_{n}}$ or ${\mathcal{S}_{n}}$. Main result =========== The goal of this section is to prove Theorem \[main\] for all groups of degree at least $10$. This proof is carried out in a sequence of lemmas. The groups of degree less than $10$ will be handled in the next section. The results of this section hold for all $n$ unless otherwise stated. If $G$ is trivial, then $G$ is obviously normalizing, so we always assume that $G$ is non-trivial. We start by stating an easy lemma whose proof is self-evident, and that will be used without further mention. A subset $X$ of $[n]$ is said to be a section of a partition $\mathcal{P}$ of $[n]$ if $X$ contains precisely one element in every class of $\mathcal{P}$. The kernel of $a\in {\mathcal{T}_{n}}$ is the equivalence relation ${{\rm ker}}(a)=\{(x,y)\in [n]:(x)a=(y)a\}$. \[tiny\] Let $G$ be a subgroup of ${\mathcal{S}_{n}}$ and let $a\in{\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$. Then, if for some $g,h\in G$ we have ${\operatorname{rank}}(h^{-1}ahg^{-1}ag\ldots)={\operatorname{rank}}(a)$, then exists $h_{1}:=hg^{{-1}}\in G$ such that $h_{1}$ maps the image of $a$ to a section of the kernel of $a$. The following lemma is probably well-known: it is an easy generalization of a result of Birch et al. [@birch]. \[enormous\] Let $G$ be a transitive permutation group on $X$, where $\|X\|=n$. Let $A$ and $B$ be subsets of $X$ with $\|A\|=a$ and $\|B\|=b$. Then the average value of $\|Ag\cap B\|$, for $g\in G$, is $ab/n$. In particular, if $\|Ag\cap B\|=c$ for all $g\in G$, then $c=ab/n$. Count triples $(x,y,g)$ with $x\in A$, $y\in B$, and $xg=y$. There are $a$ choices for $x$ and $b$ choices for $y$, and then $\|G\|/n$ choices for $g$. Choosing $g$ first, there are $\|Ag\cap B\|$ choices for $(x,y)$ for each $g$. The result follows. Let $G \le S_n$ be normalizing and non-trivial. Then - $G$ is transitive; - $G$ is primitive. Regarding (i), let $A$ be an orbit of $G$ which is not a single point, and suppose that $\|A\| < n$. Let $a$ be an (idempotent) map which acts as the identity on $A$ and maps the points outside $A$ to points of $A$ in any manner. Then $a$ fixes $A$ pointwise, and hence so does any $G$-conjugate of $a$, and so does any product of $G$-conjugates: that is, $\langle a^G\rangle$ fixes $A$ pointwise. On the other hand, if $g\in G$ acts non-trivially on $A$, then so does $ag$, and $ag\in\langle a,G\rangle\setminus G$. So these two semigroups are not equal, and $G$ is not normalizing. Regarding (ii) suppose that $G$ is imprimitive and let $B$ be a non-trivial $G$-invariant partition of $\{1,\ldots,n\}$. Choose a set $S$ of representatives for the $B$-classes, and let $a$ be the map which takes every point to the unique point of $S$ in the same $B$-class. Then $a$ fixes all $B$-classes (in the sense that it maps any $B$-class into itself), and hence so does any $G$-conjugate of $a$, and so does any product of $G$-conjugates. On the other hand, the transitivity of $G$ implies that there exists $g\in G$ that does not fix all $B$-classes, so that neither does the element $ag\in\langle a,G\rangle\setminus G$. As before, it follows that $G$ is not normalizing. Now we are ready to prove the main lemma of this section. But before that we introduce some terminology and results. For natural numbers $i,j{\leqslant}n$ with $i{\leqslant}j$, a group $G{\leqslant}{\mathcal{S}_{n}}$ is said to be $(i,j)$-homogeneous if for every $i$-set $I$ contained in $[n]$ and for every $j$-set $J$ contained in $[n]$, there exists $g\in G$ such that $Ig\subseteq J$. This notion is linked to homogeneity since an $(i,i)$-homogeneous group is an $i$-homogeneous (or $i$-set transitive) group in the usual sense. The goal of next lemma is to prove that a normalizing group is $(k-1,k)$-homogeneous, for all $k$ such that $1{\leqslant}k{\leqslant}\lfloor \frac{n+1}{2}\rfloor$. But before stating our next lemma we state here two results about $(k-1,k)$-homogeneous groups. (We denote the dihedral group of order $2p$ by $D(2p)$.) (See [@ArCa12]) \[thkk-1\] If $n{\geqslant}1$ and $2{\leqslant}k{\leqslant}\lfloor \frac{n+1}{2} \rfloor$ is fixed, then the following are equivalent: 1. $G$ is a $(k-1,k)$-homogeneous subgroup of ${\mathcal{S}_{n}}$; 2. $G$ is $(k-1)$-homogeneous or $G$ is one of the following groups 1. $n=5$ and $G\cong C_5$ or $D(25),$ $k=3$; 2. $n=7$ and $G\cong{\mbox{\rm AGL}}(1,7)$, with $k=4$; 3. $n=9$ and $G\cong{\mbox{\rm ASL}}(2,3)$ or ${\mbox{\rm AGL}}(2,3)$, with $k=5$. These groups admit an analogue of the Livingstone–Wagner [@lw] result about homogeneous groups. (See [@ArCa12]) \[corkk-1\] Let $n{\geqslant}1$, let $3{\leqslant}k{\leqslant}\lfloor \frac{n+1}{2}\rfloor$ be fixed, and let $G{\leqslant}{\mathcal{S}_{n}}$ be a $(k-1,k)$-homogeneous group. Then $G$ is a $(k-2,k-1)$-homogeneous group, except when $n=9$ and $G\cong{\mbox{\rm ASL}}(2,3)$ or ${\mbox{\rm AGL}}(2,3)$, with $k=5$. Now we state and prove the main lemma in this section. Let $G{\leqslant}{\mathcal{S}_{n}}$ be a normalizing group such that $n{\geqslant}10$. Then, for all $k$ such that $2{\leqslant}k{\leqslant}\lfloor \frac{n+1}{2}\rfloor$, the group $G$ is $(k-1,k)$-homogeneous. Suppose that $G$ fails to have the $(k-1,k)$-homogenous property, for some $k< \lfloor\frac{n+1}{2}\rfloor$. Then it follows that $G$ fails to be $(m-1,m)$-homogeneous, for $m= \lfloor\frac{n+1}{2}\rfloor$, that is, there exist two sets, $I$ and $J$, such that $Ig\not\subseteq J$, for all $g\in G$. Without loss of generality (since we can replace $G$ by some appropriate $g^{{-1}}Gg{\leqslant}{\mathcal{S}_{n}}$) we can assume that $I=\{1,\ldots,m-1\}$, $J=\{a_{1},\ldots,a_{m}\}$ and hence there is no $g\in G $ such that $$\{1,\ldots ,m-1\}g\subseteq \{a_{1},\ldots, a_{m}\}.$$ Now pick $a\in {\mathcal{T}_{n}}$ such that $$a=\left(\begin{array}{ccccccc} \{1\}&\ldots &\{m-1\} &[n]\setminus\{1,\ldots,m-1\}\\ a_{1} &\ldots&a_{m-1} &a_{m} \end{array}\right).$$ Observe that (for all $g\in G$) we have ${\operatorname{rank}}(aga)<{\operatorname{rank}}(a)$, because there is no set in the orbit of $\{a_{1},\ldots,a_{m}\}$ that contains $\{1,\dots ,m-1\}$; therefore there is only one chance for $G$ to normalize $a$: $$\begin{aligned} \label{obs} (\forall g \in G)(\exists h\in G)\ ag = h^{-1}ah. \end{aligned}$$ On the other hand, $$\|\{a_{1},\ldots,a_{m}\}\cap \{1,\dots , m-1\}\|=r,$$ implies that ${\operatorname{rank}}(a^{2})=r+1$, and hence ${\operatorname{rank}}((h^{-1}ah)^{2})=r+1$ as well. Now we have two situations: either there exists a constant $c$ such that for all $g\in G$ we have $$\|\{a_{1},\ldots,a_{m}\}g\cap \{1,\dots , m-1\}\|=c,$$ or not. We start by the second case. We are going to build a map $ah\in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$ and pick a permutation $h^{-1}g\in G$ such that $(ah)h^{-1}g$ is not normalized by $G$. By assumption there exists $g\in G$ such that $$\|\{a_{1},\ldots,a_{m}\}g\cap \{1,\ldots ,m-1\}\|=c$$ and there exists $h\in G$ such that $$\|\{a_{1},\ldots,a_{m}\}h\cap \{1,\ldots ,m-1\}\|=d<c.$$ Then, by the observation above, ${\operatorname{rank}}((ah)^{2})=d+1$ and so the rank of any one of its conjugates is also $d+1$: for all $h_{1 }\in G$ we have ${\operatorname{rank}}((h^{-1}_{1}(ah)h_{1})^{2})=d+1$. On the other hand, ${\operatorname{rank}}((ah\cdot h^{-1}g)^{2})=c+1(>d+1)$ so that $$(\forall h_{1}\in G) ah\cdot h^{-1}g \neq h^{-1}_{1}(ah)h_{1}$$ and hence by (\[obs\]) $$ah\cdot h^{-1}g \not\in \langle (ah)^{h_{1}}\mid h_{1}\in G\rangle ,$$ a contradiction. It is proved that if the size of the following intersection $$\|\{a_{1},\ldots,a_{m}\}g\cap \{1,\dots , m-1\}\|$$ varies with $g\in G$, then it is possible to build a map that is not normalized by $G$. Now we turn to the first possibility, namely, exists a constant $c$ such that, for all $g\in G$, we have $$\|\{a_{1},\ldots,a_{m}\}g\cap \{1,\dots , m-1\}\|=c.$$ First observe that if $c=1$, then $m(m-1)=n$, which holds only when $n=6$ (see Lemma \[enormous\] and recall that $m=\lfloor\frac{n+1}{2}\rfloor$). Since $n{\geqslant}10$ we have $c{\geqslant}2$. As $ \|\{a_{1},\ldots,a_{m}\}g\cap \{1,\dots , m-1\}\|=c $, for all $g\in G$, it follows that (for $g=1$) we have $\|\{a_{1},\ldots,a_{m}\}\cap \{1,\dots , m-1\}\|=c.$ Without loss of generality (in order to increase the readability of the map $a$ below), we will assume that $a_{i}=i$, for $i=1,\ldots ,c$. Now, as $G$ is transitive, pick $g\in G$ such that $1g=2$, and suppose there exists $h\in G$ such that $ag=a^{h}$, with $$a=\left(\begin{array}{ccccccc} \{1\}&\ldots &\{c\}&\{c+1\}&\ldots &\{m-1\} &[n]\setminus\{1,\ldots,m-1\}\\ 1 &\ldots &c&a_{c+1} &\ldots &a_{m-1}&a_{m} \end{array}\right),$$ $$ag=\left(\begin{array}{ccccccc} \{1\}&\ldots &\{c\}&\{c+1\}&\ldots &\{m-1\} &[n]\setminus\{1,\ldots,m-1\}\\ 1g=2 &\ldots &cg&a_{c+1}g &\ldots &a_{m-1}g&a_{m}g \end{array}\right)$$ and $$a^{h}=\left(\begin{array}{ccccccc} \{1\}h&\ldots &\{c\}h&\{c+1\}h&\ldots &\{m-1\}h &[n]\setminus\{1,\ldots,m-1\}h\\ 1h &\ldots &ch&a_{c+1}h &\ldots &a_{m-1}h&a_{m}h \end{array}\right).$$ In $ag$, $2$ is not a fixed point and $\|2(ag)^{{-1}}\|=1$. Therefore $2$ is not a fixed point of $a^{h}$ and $\|2(a^{h})^{{-1}}\|=1$. As the possible non-fixed points of $a^{h}$ with singleton inverse image (under $a^{h}$) are contained in $\{a_{c+1}h, \ldots , a_{m-1}h\}$, it follows there must be an element $a_{j}\in \{a_{c+1}, \ldots ,a_{m-1}\}$ such that $a_{j}h=2$. But this means that $h$ does not permute $\{1,\ldots,m-1\}$ and hence $$\{\{1\},\dots,\{m-1\}\}h\neq \{\{1\},\ldots, \{m-1\}\}$$ yielding that the kernel of $a^{h}$ and $ag$ are different, a contradiction. It is proved that if $G$ fails to be $(k-1,k)$-homogeneous, for some $k$ such that $1{\leqslant}k{\leqslant}\lfloor \frac{n+1}{2} \rfloor$, then $G$ is not normalizing. The result follows. We have now everything needed in order to prove Theorem \[main\] regarding the groups of degree at least $10$. In fact, if $G$ is normalizing, then $G$ is $(k-1,k)$-homogenous for all $k$ such that $1< k{\leqslant}\lfloor \frac{n+1}{2} \rfloor$ and hence the group (of degree at least $10$) is $(k-1)$-homogeneous (by Theorem \[thkk-1\]). A primitive group (of degree $n$) is proper if it does not contain the alternating group of degree $n$. Therefore, if $n=10$, then a proper primitive normalizing group must be $(k=\lfloor \frac{n-1}{2} \rfloor=4)$-homogenous, but there are no such groups of degree $10$. For $n=11$, a proper primitive normalizing group must be $(k=\lfloor \frac{n-1}{2} \rfloor=5)$-homogenous, but there are no such groups of degree $11$. If $n=12$, then the group must be $(k=\lfloor\frac{n-1}{2} \rfloor=5)$-homogenous, whose unique example (of degree $12$) is $M_{12}$. However $M_{12}$, as the group of permutations of $\{1,\ldots ,12\}$ generated by the following permutations $$\begin{array}{lll} (1\ 2\ 3)(4\ 5\ 6)(7\ 8\ 9),& (2\ 4\ 3\ 7)(5\ 6\ 9\ 8),& (2\ 9\ 3\ 5)(4\ 6\ 7\ 8),\ \\ (1\ 10)(4\ 7)(5\ 6)(8\ 9),& (4\ 8)(5\ 9)(6\ 7)(10\ 11),& (4\ 7)(5\ 8)(6\ 9)(11\ 12), \end{array}$$ fails to normalize the following map: $$a=\left(\begin{array}{cccccc} \{1\}&\{2\}&\{3\}&\{4\}&\{5,6\}&\{7,\ldots,12\}\\ 1 &2&3&4&5&6 \end{array}\right).$$ In fact, it is easily checked (using GAP [@GAP]) that no element of $M_{12}$ maps $\{1,\ldots,6\}$ to a section for the kernel of this map $a$. So, by Lemma \[tiny\], we only have to check whether, for every $g\in M_{12}$, there exists $h\in M_{12}$ such that $ag=h^{-1}ah$. This fails for $g=(132)(465)(798)$. For $n>12$, the group must be $(k=\lfloor\frac{n-1}{2} \rfloor{\geqslant}6)$-homogenous, but for $k{\geqslant}6$ there are no proper primitive $k$-homogeneous groups [@dixon Theorem 9.4B, p. 289]. Therefore the unique groups that can be normalizing are the trivial group, the symmetric and alternating groups, and some primitive groups of degree at most $9$. In the next section we explain how we used [GAP]{} [@GAP], [orb]{} [@orb] and [Citrus]{} [@citrus], to check these groups of small degree. That the symmetric and the alternating groups are normalizing is already well known. ([@lmm Theorem 5.2])\[sym\] The groups ${\mathcal{S}_{n}}$ and ${\mathcal{A}_{n}}$ are normalizing. Computational considerations ============================ In this section we describe the computational methods used to find the normalizing groups of degree at most $9$. Regarding primitive groups of degree at most $3$ they contain the alternating group and the result follows by Theorem \[sym\]. Therefore, from now on we assume that $4{\leqslant}n{\leqslant}9$. We know that a normalizing group $G{\leqslant}\mathcal{S}_{n}$ is primitive and $(k-1,k)$-homogeneous for all $k{\leqslant}\lfloor\frac{n+1}{2}\rfloor$. By Theorem \[thkk-1\] we have two situations: 1. $G$ is $(\lfloor \frac{n-1}{2}\rfloor)$-homogeneous and hence (by inspection of the GAP library of primitive groups) is one of the groups below: $\ $ Degree $G$ -------- ----------------------------------------------------------------------------------------------------------------------------------------- 5 ${\mbox{\rm AGL}}(1,5)$ 6 ${\mbox{\rm PSL}}(2,5)$, ${\mbox{\rm PGL}}(2,5)$ 8 ${\mbox{\rm AGL}}(1,8)$, ${\mbox{\rm A}\Gamma {\rm L}}(1,8)$, ${\mbox{\rm ASL}}(3,2)$, ${\mbox{\rm PSL}}(2,7)$, ${\mbox{\rm PGL}}(2,7)$ 9 ${\mbox{\rm PSL}}(2,8)$, ${\mbox{\rm P}\Gamma {\rm L}}(2,8)$ $\ $ 2. or $G$ is one of the groups in Theorem \[thkk-1\] ($C_{5}$ and $D(25)$ of degree $5$; ${\mbox{\rm AGL}}(1,7)$ of degree $7$; ${\mbox{\rm ASL}}(2,3)$ and ${\mbox{\rm AGL}}(2,3)$ of degree 9). To check that a group $G \le {\mathcal{S}_{n}}$ is $a$-normalizing for some $a \in {\mathcal{T}_{n}}\setminus{\mathcal{S}_{n}}$ it is enough to check that $aG \subseteq \langle g^{-1}ag \mid g\in G\rangle$, since the latter is closed under conjugation with elements from $G$. So we only have to enumerate the $G$-orbit of $a$ with right multiplication as action and check membership in the semigroup $\langle g^{-1}ag \mid g \in G\rangle$ for all its elements. This is essentially achieved by the following -commands using the packages (see [@orb]) and (see [@citrus]): gap> o := Orb(G,a,OnRight);; Enumerate(o);; gap> o2 := Orb(G,a,OnPoints);; Enumerate(o2);; gap> s := Semigroup(o2);; gap> ForAll(o,x->x in s); true However, for the larger examples on $9$ points checking this for all $a \in {\mathcal{T}_{n}}\setminus {\mathcal{S}_{n}}$ would have taken too long. Fortunately, this was not necessary, since if $G$ is $a$-normalizing, then it is of course $a^g$-normalizing for all $g \in G$. So we only have to check this property for representatives of the $G$-orbits on ${\mathcal{T}_{n}}\setminus{\mathcal{S}_{n}}$ under the conjugation action. To compute a set of representatives we first implemented an explicit bijection of ${\mathcal{T}_{n}}$ to the set $\{ i \in \mathbb{N} \mid 1 \le i \le n^n \}$. Then we organised a bitmap of length $n^n$ and enumerated all conjugation $G$-orbits in ${\mathcal{T}_{n}}$, crossing off the transformations we had already encountered in the bitmap. Having the representatives as actual transformations then allowed us to perform the test explained above. A slight speedup was achieved by actually verifying a stronger condition, namely that $aG$ is a subset of the $\mathcal{R}$-class of $a$ in the semigroup $\langle a^g \mid g \in G\rangle$, which turned out to be the case whenever $G$ was normalizing. Testing membership in the $\mathcal{R}$-class of $a$ in the transformation semigroup $S:=\langle a^g \mid g \in G\rangle$ can be done by computing the strong orbit of the image of $a$ under the action of $S$ and the permutation group induced by the elements of $S$ that stabilise the image of $a$ setwise; as described in [@Linton1998aa]. This method is implemented in the [Citrus]{} package [@citrus] for [GAP]{}. For degree $5$, only ${\mbox{\rm AGL}}(1,5)$ is normalizing, since the group $C_{5}$ fails to normalize the map $$a=\left(\begin{array}{ccc} \{1,2,5\}&\{3\}&\{4\}\\ 1 &3&4 \end{array}\right),$$ and the group $D(25)$ fails to normalize the map $$a=\left(\begin{array}{ccc} \{1,2,3\}&\{4\}&\{5\}\\ 1 &3&2 \end{array}\right).$$ For degree $6$, both groups ${\mbox{\rm PSL}}(2,5)$ and ${\mbox{\rm PGL}}(2,5)$ are normalizing. For degree $7$, we only had to check ${\mbox{\rm AGL}}(1,7)$, which fails to normalize the map $$a=\left(\begin{array}{ccc} \{1,\ldots,5\}&\{6\}&\{7\}\\ 1 &2&3 \end{array}\right).$$ For degree $8$, all three groups ${\mbox{\rm AGL}}(1,8)$, ${\mbox{\rm A}\Gamma {\rm L}}(1,8)$ and ${\mbox{\rm ASL}}(3,2)$ fail to normalize the map $$a=\left(\begin{array}{cccc} \{1,\ldots,5\}&\{6\}&\{7\}&\{8\}\\ 1 &2&3&4 \end{array}\right),$$ the group ${\mbox{\rm PSL}}(2,7)$ fails to normalize the map $$a=\left(\begin{array}{cccc} \{1,\ldots,5\}&\{6\}&\{7\}&\{8\}\\ 1 &2&3&5 \end{array}\right),$$ and finally the group ${\mbox{\rm PGL}}(2,7)$ fails to normalize the map $$a=\left(\begin{array}{cccc} \{1,\ldots,5\}&\{6\}&\{7\}&\{8\}\\ 1 &2&4&7 \end{array}\right).$$ For degree $9$, the two groups ${\mbox{\rm PSL}}(2,8)$ and ${\mbox{\rm P}\Gamma {\rm L}}(2,8)$ are normalizing, whereas both groups ${\mbox{\rm ASL}}(2,3)$ and ${\mbox{\rm ASL}}(2,3)$ fail to normalize the map $$a=\left(\begin{array}{cccccc} \{1,8\}&\{2,3,7\}&\{4\}&\{5\}&\{6\}&\{9\}\\ 7 &8&6&9&4&5 \end{array}\right).$$ These computational results complete the proof of our main Theorem \[main\]. Problems ======== Regarding this paper, the main problem that has to be tackled now should be the classification of the $k$-normalizing groups. Let $k$ be a fixed number such that $1<k<\lfloor \frac{n+1}{2}\rfloor$. Classify the $k$-normalizing groups, that is, classify the groups that satisfy $\langle a,G\rangle\setminus G=\langle a^{g}\mid g\in G\rangle$, for every rank $k$ map. To solve this problem is necessary to use the results of [@ArCa12], but that will be just a starting point since many delicate considerations will certainly be required. The theorems and problems in this paper admit linear versions that are interesting for experts in groups and semigroups, but also to experts in linear algebra and matrix theory. For the linear case, we already know that any singular matrix with any group containing the special linear group is normalizing [@ArSi1; @ArSi2] (see also the related papers [@Gr; @Pa; @Ra]). Classify the linear groups $G{\leqslant}GL(n,q)$ that, together with any singular linear transformation $a$, satisfy $$\langle a,G\rangle \setminus G = \langle h^{-1}a{h}\mid h\in G\rangle.$$ A necessary step to solve the previous problem is to solve the following. \[11\] Classify the groups $G{\leqslant}GL(n,q)$ such that for all rank $k$ (for a given $k$) singular matrix $a$ we have that ${\operatorname{rank}}(aga)={\operatorname{rank}}(a)$, for some $g\in G$. To handle this problem it is useful to keep in mind the following results. Kantor [@kantor:inc] proved that if a subgroup of ${\mbox{\rm P}\Gamma {\rm L}}(d,q)$ acts transitively on $k$-dimensional subspaces, then it acts transitively on $l$-dimensional subspaces for all $l\le k$ such that $k+l\le n$; in [@kantor:line], he showed that subgroups transitive on $2$-dimensional subspaces are $2$-transitive on the $1$-dimensional subspaces with the single exception of a subgroup of ${\mbox{\rm PGL}}(5,2)$ of order $31\cdot5$; and, with the second author [@cameron-kantor], he showed that such groups must contain ${\mbox{\rm PSL}}(d,q)$ with the single exception of the alternating group $A_7$ inside ${\mbox{\rm PGL}}(4,2)\cong A_8$. Also Hering [@He74; @He85] and Liebeck [@Li86] classified the subgroups of ${\mbox{\rm PGL}}(d,p)$ which are transitive on $1$-spaces. (See also [@kantor:inc; @kantor:line].) Solve analogues of the results (and problems) in this paper for independence algebras (for definitions and fundamental results see [@ArEdGi; @arfo; @cameronSz; @gould]). Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to express their gratitude to the referee for a very careful review and for suggestions that prompted a much simplified paper. The first author was partially supported by FCT through the following projects: PEst-OE/MAT/UI1043/2011, Strategic Project of Centro de Álgebra da Universidade de Lisboa; and PTDC/MAT/101993/2008, Project Computations in groups and semigroups . The second author is grateful to the Center of Algebra of the University of Lisbon for supporting a visit to the Centre in which some of this research was done. [10]{} J. Araújo and Peter J. Cameron. J. Araújo, M. Edmundo and S. Givant. J. Araújo and J. Fountain. Proceedings of the Workshop on Semigroups and Languages (Lisbon 2002), World Scientific, (2004), 54–67 J. Araújo, J. D. Mitchell and C. Schneider. J. Araújo and F.C. Silva. J. Araújo and F.C. Silva. B. J. Birch, R. G. Burns, Sheila Oates Macdonald and Peter M. Neumann. Peter J. Cameron. Peter J. Cameron and William M. Kantor. P. J. Cameron and C. Szabó, , John D. Dixon and Brian Mortimer. The GAP Group. V. Gould, L. Grunenfelder, M. Omladič, H. Radjavi, A. Sourour. C. Hering. C. Hering. Donald G. Higman. John M. Howie. William M. Kantor, William M. Kantor. William M. Kantor. William M. Kantor. I. Levi, D. B. McAlister, and R. B. McFadden. I. Levi and R. B. McFadden. I. Levi. M.W. Liebeck. S. A. Linton, G. Pfeiffer, E. F. Robertson, and N. Ru[š]{}kuc. S. A. Linton, G. Pfeiffer, E. F. Robertson, and N. Ru[š]{}kuc. Donald Livingstone and Ascher Wagner. Donald B. McAlister. James D. Mitchell. The -package . Version 0.6.\ `http://www-groups.mcs.st-andrews.ac.uk/ jamesm/citrus/index.html`, 2012. Jürgen Müller, Max Neunhöffer and Felix Noeske. The -package . Version 4.3.\ `http://www-groups.mcs.st-and.ac.uk/ neunhoef/Computer/Software/Gap/orb.html`,\ 2012. C.S. Pazzis. K.P.S.B. Rao.
--- abstract: 'Based on Density Functional Theory and Density Functional Perturbation Theory we have studied the thermodynamical and vibrational properties of Mg(AlH$_4$)$_2$. The crystal structure recently proposed on the basis of x-ray powder diffraction data has been confirmed theoretically by the comparison of the experimental and theoretical IR and Raman spectra. The main discrepancy regards the position of the hydrogen atoms which makes the theoretical AlH$_4$ tetrahedra more symmetric than the experimental ones. The calculated thermodynamical decomposition temperature is also in good agreement with experimental result.' author: - 'E. Spanò and M. Bernasconi' title: 'Ab-initio study of the vibrational properties of Mg(AlH$_4$)$_2$' --- epsf Introduction ============ In the search for suitable materials for reversible hydrogen storage the class of metal alanates has attracted much attention in recent years due to their high hydrogen content, but primarly because of the possibility to accelerate the kinetics of hydrogen uptake by doping [@bodga1; @bogda2]. Sodium alanate, the prototypical material of this class, has long been known. Although its thermodynamics decomposition temperature with hydrogen release is relatively low, the kinetics of the dehydrogenation/rehydrogenation reactions is very slow. The interest in this material for hydrogen storage increased dramatically in 1997 when Bogdanovic [et al.]{} [@bodga1] demonstrated that reversible storage can be achieved at moderate temperature and hydrogen partial pressure by the addition of catalysts, most notably titanium. This discovery boosted an intense activity on the study of other members of the alkali metal alanates as well [@LiAl; @KAl]. More recently, magnesium alanate Mg(AlH$_4$)$_2$, as a representative of the alkali earth metal alanates, has also been considered as a possible materials for hydrogen storage [@ficth0; @ficth1; @ficth2; @ficth3]. Mg(AlH$_4$)$_2$ decomposes readly in the temperature range 110 $^o$C - 200 $^o$C according to the reaction $$Mg(AlH_4)_2 \rightarrow MgH_2 + 2 Al + 3 H_2 \label{reaction}$$ which would correspond to a maximum reversible hydrogen content of 7 wt$\%$. Although the reversibility of reaction \[reaction\] has not been demonstrated yet, the easy of the decomposition reaction even in the absence of doping, suggests that, in analogy with NaAlH$_4$, the insertion of suitable catalyst might enhance the kinetics of hydrogen uptake. The structure of Mg(AlH$_4$)$_2$ has been recently resolved from X-ray powder diffraction pattern aided by quantum-chemical calculations on cluster models which allowed to tentatively assign also the position of hydrogen atoms, not detectable by X-ray diffraction [@ficth3]. In this work, we present an ab-initio study of the thermodynamical and vibrational properties of Mg(AlH$_4$)$_2$ aiming at providing a better estimate of the decomposition temperature from calculations on periodic models and at confirming the structure inferred experimentally from the comparison of theoretical and experimental IR and Raman spectra. Computational details ===================== Calculations are performed within the framework of Density Functional Theory (DFT) with a gradient corrected exchange and correlation energy functional [@blyp], as implemented in the codes PWSCF and PHONONS [@pwscf]. Norm conserving pseudopotentials [@TM] and plane wave expansion of Kohn-Sham (KS) orbitals up to a kinetic cutoff of 40 Ry have been used. Non linear core corrections are included in the pseudopotential of magnesium [@nlcc]. Brillouin Zone (BZ) intergration has been performed over Monkhorst-Pack (MP) [@MP] 6x6x6, 4x4x4 and 16x16x16 meshes for Mg(AlH$_4$)$_2$, MgH$_2$ and metallic Al, respectively. Hermite-Gaussian smearing [@meth] of order one with a linewidth of 0.01 Ry has been used in the reference calculations on metallic Al. Equilibrium geometries have been obtained by optimizing the internal and lattice structural parameters at several volumes and fitting the energy versus volume data with a Murnaghan function [@murna]. Residual anisotropy in the stress tensor at the optimized lattice parameters at each volume is below 0.6 kbar. Infrared and Raman spectra are obtained from effective charges, dielectric susceptibilities and phonons at the $\Gamma$ point within density functional perturbation theory [@dfpt]. Since Mg(AlH$_4$)$_2$ is an uniaxial crystal, the infrared absorption spectrum depends on the polarization of the trasmitted electromagnetic wave with respect to the optical axis. Relevant formula for the calculation of the IR and Raman spectra for a polycristalline sample to be compared with experimental data are given in section II. Results ======= Structure and energetics ------------------------ The structure of Mg(AlH$_4$)$_2$ has been recently assigned to a CdI$_2$-like layered crystal from X-ray powder diffraction data [@ficth3]. Rietveld refinement assigned the space group $P{\bar 3}m1$ ($D_{3d}^3$) and lattice parameter $a$=$b$=5.199 $\rm\AA$ and $c$=5.858 $\rm\AA$. The position of the four independent atoms in the unit cell are given in Table I. In the AlH$_4$ tetrahedron there are two Al-H bond lengths: the length of the Al-H1 bond with non-bridging H1 hydrogen aligned with the $c$ axis and the length of Al-H2 bond with the H2 atom bridging between Al and Mg. -------- --------------- ----------------- --------------- Mg(1a) 0 0 0 Al(2d) 1/3 2/3 0.7 (0.7053) H1(2d) 1/3 2/3 0.45 (0.4349) H2(6i) 0.16 (0.1678) -0.16 (-0.1678) 0.81 (0.8111) -------- --------------- ----------------- --------------- : Experimental and theoretical (in parenthesis) positions (in crystal units) of the four independent atoms at the experimental equilibrium lattice parameters (space group $P{\bar 3}m1$, $a$=5.199 $\rm\AA$, $c$=5.858 $\rm\AA$). The Wyckoff notation is used. If not reported the theoretical positions coincide with the experimental ones by symmetry. H1 and H2 are non-bridging and bridging hydrogen atoms, respectively. \[position\] A picture of the crystal structure is given in Fig. \[structure\]. It can be seen as a stacking along the $c$ axis of AlH$_4$-Mg-AlH$_4$ neutral trilayers. = 5. truecm = 5. truecm The theoretical equilibrium lattice parameters obtained from the Murnaghan equation of state are $a$=5.229 $\rm\AA$ (exp. 5.199) and $c$= 6.238 $\rm\AA$ (exp. 5.858). In the geometry optimization we have also relaxed the constrained of $P{\bar 3}m1$ symmetry, but we have always recovered the experimental space group. The misfit in the $c$ axis ( 6 $\%$) is larger than usual in DFT-based calculations. As already pointed out in previous cluster calculations on Mg(AlH$_4$)$_2$ [@ficth3] the trilayers stacked along the $c$ axis are neutral and not chemically bonded each other. Thus van der Waals interactions have been proposed to play an important role in interlayer cohesion. Since van der Waals interations are not present in current approximations to the exchange and correlation energy functional, the interlayer bonding is underestimated which implies an expansion of the $c$ axis with respect to the experimental data. The energy difference between the relaxed configurations at the experimental and theoretical equilibrium lattice parameters is nevertheless small ( 38 meV per formula unit). The theoretical atomic positions at the experimental lattice parameters are given in Table I. The theoretical geometry of the AlH$_4$ tetrahedron is closer to an ideal tetrahedron than the experimental geometry. In fact the theoretical length of the two Al-H bonds are Al-H1= 1.584 $\rm\AA$ (exp. 1.46) and Al-H2= 1.614 $\rm\AA$ (exp. 1.69). The Al-H bond lengths do not change sizably (within 0.01 $\rm\AA$) by changing the lattice parameters from the experimental values to the theoretical equilibrium values. The electronic band structure along the high symmetry directions of the Irreducible Brillouin Zone is reported in Fig. \[bands\]. The analysis of the projection of the KS states on the atomic orbitals at the $\Gamma$ point shows that the top of the valence band is mainly formed by 3p states of aluminum and 1s state of hydrogen while the bottom of the conduction band is mainly formed by 3s states of magnesium. The band gap is thus a charge transfer excitation. [ = 5. truecm]{} By neglecting at first the vibrational contribution to the entropy of the solids, the thermodynamical decomposition temperature $T_D$ of reaction \[reaction\] can be estimated as $$T_D = \frac{\Delta E + P\Delta V}{3 (S_{{\rm H}_2}-R)} \label{TD}$$ where $\Delta V$ is the difference in volume of the solid reactants and products of reaction \[reaction\], $P$ is the pressure, $\Delta E$ is the energy difference of reactants and products in \[reaction\], $S_{{\rm H}_2}$ is the entropy per mole of gaseus H$_2$ and the gas constant $R$ in the denominator comes from the PV term of gaseus H$_2$. $\Delta V$ and $\Delta E$ are obtained from the ab-initio calculations. For MgH$_2$ the theoretical equilibrium structural parameters are $a$= 4.470 $\rm\AA$ (exp. 4.501 [@MgH2]), $c$= 2.943 $\rm\AA$ (exp. 3.010 [@MgH2]) and the 4$f$ position (in Wyckoff notation of the $P4_2/mnm$ space group) of the independent H atom in the unit cell is (0.3046,0.3046,0) (exp. (0.3040,0.3040,0) [@MgH2]). The theoretical equilibrium lattice parameter of metallic Al is $a$= 4.065 $\rm\AA$ (exp. 4.050 $\rm\AA$ [@ash]). From the total energies at equilibrium we obtain $\Delta E$= 144 kJ/mol (48 kJ/mol per H$_2$ molecule) which becomes $\Delta E$= 139.5 kJ/mol (46.5 kJ/mol per H$_2$ molecule) by including the zero point energy of the optical phonons (at the $\Gamma$ point only) and of the acoustic bands within a Debye model [@notadebye]. This result is in good agreement with previous estimate from the extrapolation of cluster calculations (123 kJ/mol [@ficth3], 41 kJ/mol per H$_2$ molecule). The $P\Delta V$ in Eq. \[TD\] is negligible (4.4 J/mol). By assuming $S_{{\rm H}_2}$= 130 J/mol K$^{-1}$ at 300 K [@SH] and atmospheric pressure, we obtain $T_D$= 111 $^o$C. The predicted T$_D$ shifts by 2 $^o$C by including the temperature dependence of $S_{{\rm H}_2}$ as given by the Sackur-Tetrode expression [@atkins], the vibrational contribution to the free energy of the solids and the translational and rotational energy of gaseous H$_2$ [@notadebye]. The theoretical $T_D$ is comparable with the experimental decomposition temperature which falls in the range 110 $^o$C - 200 $^o$C which implies that kinetics effect do not affect dramatically the decomposition reaction of Mg(AlH$_4$)$_2$. Vibrational properties ---------------------- Phonons at the $\Gamma$-point can be classified according to the irreducible representations of the $D_{3d}$ point group of Mg(AlH$_4$)$_2$ as $\Gamma$= 4 A$_{1g}$ + 2 A$_{2g}$ + 5 E$_{g}$ + 5 A$_{2u}$ + 6 E$_{u}$. A$_{1g}$ and E$_{g}$ modes are Raman active while A$_{2u}$ and E$_{u}$ modes are IR active. One A$_{2u}$ and one E$_{u}$ mode are uniform translational modes. The IR active modes display a dipole moment which couple to the inner macroscopic longitudinal field which shifts the LO phonon frequencies via the non-analytic contribution to the dynamical matrix [@dfpt] $$D^{NA}_{\alpha,\beta}(\kappa,\kappa')=\frac{4 \pi}{V_o} \frac{Z_{\alpha,\alpha'}(\kappa)q_{\alpha'} Z_{\beta,\beta'}(\kappa')q_{\beta'} } {{\bf q} \cdot \tens{\varepsilon}^{\infty} \cdot {\bf q}}, \label{macro}$$ where $\tens{Z}$ and $\tens{\varepsilon}^{\infty}$ are the effective charges and electronic dielectric tensors, $V_o$ is the unit cell volume and [q]{} is the phononic wavevector. The effective charge tensor for the three independent atoms in the unit cell (cfr. Table \[position\]) are $$Z(Mg)=\left(\begin{array}{ccc} 2.180 & & \\ & 2.180 & \\ & & 1.974 \\ \end{array} \right), \ \ \ \label{ZM}$$ $$Z(Al)=\left(\begin{array}{ccc} 1.934 & & \\ & 1.934 & \\ & & 1.678\\ \end{array} \right), \ \ \ \label{ZA}$$ $$Z(H1)=\left(\begin{array}{ccc} -0.627 & & \\ & -0.627 & \\ & & -0.641 \\ \end{array} \right), \ \ \ \label{H1}$$ $$Z(H2)=\left(\begin{array}{ccc} -0.564 & & \\ & -1.029 & 0.248 \\ & 0.289 & -0.672 \\ \end{array} \right), \ \ \ \label{H2}$$ The electronic dielectric tensor is $$\tens{\varepsilon}^{\infty}=\left(\begin{array}{ccc} 2.894 & & \\ & 2.894 & \\ & & 2.781 \\ \end{array} \right)\ \ \ , \label{epsi81}$$ The calculated phonon frequencies at the $\Gamma$ point, neglecting the contribution of the longitudinal macroscopic field (Eq. \[macro\]), are reported in Table II for the equilibrium geometry at the experimental lattice parameters. The phonon frequencies at the theoretical lattice parameters differ at most by 10 cm$^{-1}$ with respect to those reported in Table II. -------------- -------------------- ------- -------- -------- -- Modes Energy (cm$^{-1}$) f$_j$ a (c) b (d) E$_{g}$ (1) 87 0.976 0.578 A$_{2g}$ (1) 169 A$_{1g}$ (1) 232 2.337 1.210 E$_{u}$ (1) 282 3.162 E$_{g}$ (2) 298 A$_{2u}$ (1) 302 0.103 A$_{2g}$ (2) 355 E$_{u}$ (2) 360 0.758 E$_{u}$ (3) 620 3.896 A$_{2u}$ (2) 663 0.784 E$_{u}$ (4) 716 0.064 E$_{g}$ (3) 742 -0.457 3.653 E$_{g}$ (4) 758 2.746 0.023 A$_{1g}$ (2) 812 3.970 0.681 A$_{1g}$ (3) 1845 8.768 17.540 A$_{2u}$ (3) 1850 0.072 E$_{g}$ (5) 1852 5.202 0.339 E$_{u}$ (5) 1905 0.597 A$_{2u}$ (4) 2013 0.040 A$_{1g}$ (4) 2077 12.270 5.160 -------------- -------------------- ------- -------- -------- -- : Theoretical phonon frequencies at the $\Gamma$ point, oscillator strengths ($f_j$ in Eq. \[epsiperp\]) of IR $u$-modes and coefficients of the Raman tensor of the Raman active mode, $a$, $b$ for A$_{1g}$ and $c$, $d$ for E$_g$ modes (in unit of 10$^{-3}$ $V_o$= 0.2743 $\rm\AA^3$, see section III B.2). The contribution of the inner longitudinal macroscopic field is not included (LO-TO splitting) (see Fig. \[disp\] for the displacement patterns). \[phonon\] For a uniaxial crystal like Mg(AlH$_4$)$_2$, the macroscopic field contribution to the dynamical matrix (Eq. \[macro\]) introduces an angular dispersion of the phonons at the $\Gamma$ point, i.e. the limit of the phononic bands $\omega({\bf q})$ for ${\bf q} \rightarrow 0$ depends on the angle $\theta$ formed by [q]{} with the optical axis. The angular dispersion of the A$_{2u}$ and E$_{u}$ modes due to the macroscopic field is reported in Fig. \[adisp\]. = 6 truecm ### IR spectrum Tha absorption coefficient of an uniaxial crystal depends on the the polarization of the trasmitted light with respect to the optical axis. The optical properties of the crystal can be obtained from the two dielectric functions $\epsilon_{\perp}(\omega)$ and $\epsilon_{\parallel}(\omega)$ which represent the response of the crystal to electromagnetic wave with electric field perpendicular ([E]{} $\perp$ [c]{}) and parallel ([E]{} $\parallel$ [c]{}) to the optical axis, respectively. The trasmitted electromagnetic wave at a generic wavevector ${\bf q}$ forming an angle $\theta$ with the optical axis would split in an ordinary wave with electric field perpendicular to the optical axis and in an extraordinary wave with electric field lying in the plane formed by the optical axis and ${\bf q}$. The dielectric function $\epsilon_{\perp}(\omega)$, independent on $\theta$, describes the response to the ordinary waves while the dielectric function $\epsilon_{\theta}(\omega)$ for the extraordinary wave is $\theta$-dependent and is given by $$\epsilon_{\theta}(\omega)=\frac{\epsilon_{\perp}(\omega)\epsilon_{\parallel}(\omega)} {\epsilon_{\perp}(\omega)sin^2\theta + \epsilon_{\parallel}(\omega)cos^2\theta}. \label{extra}$$ The extraordinary wave coincides with the ordinary wave for $\theta=0$, i.e. propagation along the optical axis. $\epsilon_{\perp}(\omega)$ and $\epsilon_{\parallel}(\omega)$ can be obtained from ab-initio phonons, effective charges and electronic dielectric tensor as $$\begin{aligned} \epsilon_{\perp}(\omega) & = &\epsilon^{\infty}_{\perp} + \frac{4 \pi}{V_o}\sum_{j=1}^{\nu} \| \sum_{\kappa =1}^N \tens{Z} \cdot \frac{{\bf e}(j,\kappa)}{\sqrt{M_{\kappa}}}\|^2 \frac{1}{\omega^2_j -\omega^2} \nonumber \\ & = & \epsilon^{\infty}_{\perp} + \sum_{j=1}^{\nu} \frac{f_j \omega^2_j}{\omega^2_j -\omega^2} \label{epsiperp}\end{aligned}$$ $$\epsilon_{\parallel}(\omega)=\epsilon^{\infty}_{\parallel} + \frac{4 \pi}{V_o}\sum_{j=1}^{\mu} \| \\ \sum_{\kappa =1}^N \tens{Z} \cdot \frac{{\bf e}(j,\kappa)}{\sqrt{M_{\kappa}}}\|^2 \frac{1}{\omega^2_j -\omega^2}, \label{epsipar}$$ where $\nu$ and $\mu$ are the number of E$_u$ and A$_{2u}$ TO modes, respectively. The phonons entering in Eq. \[epsiperp\] (\[epsipar\]) have wavevector [q]{} parallel (perpendicular) to the optical axis. The notation $\parallel$ and $\perp$ refers to the orientation with respect to the optical axis of the dipole moment of the phonon which coincides with that of the electric field of the trasmitted wave it couples to. The sum over $\kappa$ run over the $N$ atoms in the unit cell with mass $M_{\kappa}$. ${\bf e}(j,\kappa)$ and $\omega_j$ are the eigenstates and eigenvalues of the dynamical matrix at the $\Gamma$ point, without the contribution of the macroscopic field which has no effect on the purely TO modes. The absorption coefficient for the ordinary wave is given by $$\begin{aligned} \alpha_{\perp}(\omega) & = & \frac{\omega}{nc} {\rm Im} \epsilon_{\perp}(\omega+i\gamma, \gamma \rightarrow 0) \nonumber \\ & = & \frac{2 \pi^2}{V_onc}\sum_{j=1}^{\nu} \| \sum_{\kappa =1}^N \tens{Z} \cdot \frac{{\bf e}(j,\kappa)}{\sqrt{M_{\kappa}}}\|^2 \delta(\omega-\omega_j), \label{absorb} \end{aligned}$$ $c$ is the velocity of light in vacuum and $n$ is the refractive index. The absorption coefficient for the extraordinary waves $\alpha_{\parallel}(\omega)$ with [E]{} $\parallel$ [c]{} (and [q]{} $\perp$ [c]{}) is given by the analougous expression by changing $\nu$ with $\mu$ in the sum over phonons in Eq. \[absorb\]. The $\delta$-functions in Eq. 12 are approximated by Lorenztian functions as $$\delta(\omega-\omega_j)=\frac{4}{\pi}\frac{ \omega^2 \gamma}{(\omega^2-\omega_j^2)^2 + 4 \gamma^2\omega^2}. \label{lorenz}$$ The functions $\alpha_{\perp}(\omega)$ and $\alpha_{\parallel}(\omega)$ are shown in Fig. \[abs\]. The phonons around 700-800 cm$^{-1}$ are bending modes of the Al-H bonds. The highest frequency modes in the range 1800-2000 cm$^{-1}$ are stretching modes of the Al-H bonds. In particular the A$_{2u}$(3) mode is a stretching of the Al-H$_1$ apical bond. The other modes are stretching of the Al-H$_2$ bridging bonds. The displacement pattern of the IR and Raman active modes are shown in Fig. \[disp\]. The modes below 360 cm$^{-1}$ (cfr. Table \[phonon\]) are lattice modes involving a rigid motion of the teatrahedra. = 9. truecm = 5. truecm Since the experimental data are available only for a polycrystalline sample [@ficth2] an angle-averaged absorption coefficient is needed to compared theoretical and experimental data. For a generic [q]{} forming an angle $\theta$ with the optical axis the absorption coefficient for the extraordinary wave can be obtained from the imaginary part of $\epsilon_{\theta}$ (Eq. \[extra\]). For non-polarized light the total absorption coeffient can be equivalently expressed as $$\alpha_{\theta}(\omega) = \frac{2 \pi^2}{V_onc}\sum_{j=1}^{\nu + \mu} \| \sum_{\kappa =1}^N {\bf \hat{q}} \wedge \tens{Z} \cdot \frac{{\bf \tilde{e}}(j,\kappa)}{\sqrt{M_{\kappa}}}\|^2 \delta(\omega-\tilde{\omega}_j), \label{abstheta}$$ where ${\bf \tilde{e}}(j,\kappa)$ and $\tilde{\omega}_j$ are eigenstates and eigenvalues of the full dynamical matrix including the non-analytic term (Eq. \[macro\]) which mixes E$_u$ and A$_{2u}$ modes. The angle-averaged absorption coefficient has been obtained from Eq. \[abstheta\] as $$\alpha_{ave}= \sum_n sin(\theta_n)\alpha_{\theta_n}, \label{absave}$$ where the sum runs over ten angles equally spaced in the range 0-$\pi$. The resulting theoretical absorption coefficient for a polycrystalline sample is compared with the experimental IR spectrum in Fig. \[absexp\]\ = 5 truecm Good agreement with experimental data is obtained with a Lorentzian broadening of $\gamma$= 40 cm$^{-1}$. (Fig. \[absexp\]), but for the experimental shoulder around 800 cm$^{-1}$ which is absent in the theoretical spectra. This misfit might be either due to an underestimation of the A$_{2u}$(2) mode which is very close to the strongest IR mode E$_{2u}$(3) (cf. Fig. \[abs\]) or to a large inhomogeneous broadening of the experimental spectra. In fact, the line-width of the experimental peak are very large and might be partially due to residual solvent adducts which are released only after dehydrogenation [@ficth2]. ### Raman spectrum The differential cross section for Raman scattering (Stokes) in non-resonant conditions is given by the following expression [@cardona; @bruesh] (for a unit volume of scattering sample) $$\frac{d \sigma}{d \Omega d \omega} = \sum_{j} \frac{\omega_S^4 }{c^4} \| {\bf e_S} \cdot \tens{R}^j \cdot {\bf e_L}\|^2 (n(\omega)+1)\delta(\omega-\omega_j), \label{raman}$$ where $n(\omega)$ is the Bose factor, $\omega_S$ is the frequency of the scattered light, ${\bf e_S}$ and ${\bf e_L}$ are the polarization vectors of the scattered and incident light, respectively. The Raman tensor $\tens{R}^j$ associated with the $j$-th phonon is given by $$R_{\alpha,\beta}^j = \sqrt{\frac{V_o \hbar}{2 \omega_j}} \sum_{\kappa=1 }^N \frac{\partial \chi_{\alpha,\beta}^{\infty}}{\partial {\bf r}(\kappa)} \cdot \frac{{\bf e}(j,\kappa)}{\sqrt{M_{\kappa}}}, \label{ramanT}$$ where $V_o$ is the unit cell volume (274.25 $\rm\AA^3$) [@ficth3], ${\bf r}(\kappa)$ is the position of atom $\kappa$-th and $\tens{\chi}^{\infty}=(\tens{\varepsilon}^{\infty}-{\bf \delta})/4\pi$ is the electronic susceptibility. The inner longitudinal macroscopic electric field has no effect on $g$-modes. The tensor $\tens{R}^j$ is computed from $\tens{\chi}^{\infty}$ by finite differences, by moving the atoms along the phonon displacement pattern with maximum displacement of 0.002 $\rm\AA$. The Raman tensor (Eq. \[raman\]) for the Raman-active irreducible representations has the form $$A_{1g} \Rightarrow \left [ \begin{array}{ccc} a & . & . \\ . & a & . \\ . & . & b \\ \end{array} \right ] \label{a1g}$$ $$E_g, 1 \Rightarrow \left [ \begin{array}{ccc} c & . & . \\ . & - c & d \\ . & d & . \\ \end{array} \right ] \label{eg1}$$ $$E_g, 2 \Rightarrow \left [ \begin{array}{ccc} . & -c & -d \\ -c & . & . \\ -d & . & . \\ \end{array} \right ] \label{eg2}$$ The coefficients $a$,$b$,$c$ and $d$ calculated from first principles as outlined above are given for each mode in Table I. The experimental Raman spectrum [@ficth2] is available only for non-polarized light and backscattering geometry on a polycristalline sample. To compare with experimental data, Eq. \[raman\] must be integrated over the solid angle by summing over all possible polarization vectors ${\bf e_S}$ and ${\bf e_L}$ consistent with the backscattering geometry. The total cross section for unpolarized light in backscattering geometry is obtained from Eq. \[raman\] with the substitution $$\begin{aligned} 4 &(&R_{xx}^2+R_{yy}^2+R_{zz}^2) +7(R_{xy}^2+R_{xz}^2+R_{yz}^2)+ \nonumber \\ &(&R_{xx}R_{yy}+R_{xx}R_{zz}+R_{zz}R_{yy}) \rightarrow \| {\bf e_S} \cdot \tens{R}^j \cdot {\bf e_L}\|^2 \nonumber\\\end{aligned}$$ The resulting theoretical Raman spectrum is compared with experimental data in Fig. \[ramanexp\]. = 5. truecm The agreement is good, but for the highest frequency peak which is too high in energy and in intensity with respect to experiments. The displacement pattern of the Raman active intra-tetrahedra modes are sketched in Fig. \[disp\]. The strongest Raman peak is due to the A$_{1g}$ stretching mode of the Al-H$_2$ bond at 1845 cm$^{-1}$. Conclusions =========== Based on Density Functional Theory we have optimized the structure of Mg(AlH$_4$)$_2$. Due to the lack of van der Waals forces within the current approximations to the energy functional, the interlayer spacing between the neutral AlH$_4$-Mg-AlH$_4$ sheets stacked along the $c$ axis is largely overestimated (6 $\%$) in our calculations. Conversely, by fixing the lattice parameters to the experimental ones, the optimization of the internal structure provides a geometry in fair agreement with that inferred experimentally from x-ray powder diffraction data. The main discrepancy regards the position of the hydrogen atoms (which however can not be detectd accurately from x-ray powder diffraction) resulting in Al-H bond lengths which differ up to 0.1 $\rm\AA$ from the exprimental ones. As a consequence, the AlH$_4$ tetrahedra are much more symmetric in the theoretical geometry than in that proposed experimentally. However, the IR and Raman spectra calculated within density functional perturbation theory are in good agreement with the experimental spectra which supports the correctness of the crystal structure emerged from the ab-initio calculations. Aknowledgments ============== We gratefully thank G. Benedek, V. Boffa, G. Dai, V. Formaggio and S. Serra for discussion and information. This work is partially supported by the INFM Parallel Computing Initiative. [169]{} B. Bogdanović and M. J. Schwickardi, J. Alloys Compd. [253]{}, 1 (1997). B. Bogdanović and G. Sandrock, Material Research Bulletin, [27]{}, 712 (2002). J. Chen, N. Kuriyama, Q. Xu, H. T. Takeshita, and T. Sakai, J. Phys. Chem. B [105]{}, 11214 (2001). H. Morioka, K, Kazizaki, S.-C. Chung, and A. Yamada, J. Alloys Compd. [353]{}, 310 (2003). M. Fichtner and O. Furh, J. Alloys Compd. [345]{}, 286 (2002). M. Fichtner, O. Furh, and O. Kircher, J. Alloys Compd. [356]{}, 418 (2003). M. Fichtner, O. Furh, O. Kircher, and O. Rubner, Mat. 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The zero-point energy per unit cell of the acoustic bands within the Debye model is given by $E_o=9/8k_B\Theta_D= (3^7/2^{11}\pi)^{1/3} h \bar{c} V_o^{-1/3}$ (Ref. [@ash]), where $\Theta_D$ is the Debye temperature, $V_o$ the unit cell volume and $\bar{c}$ is the phase velocity of the acoustic branches (1/3 $v_{LA}$ + 2/3 $v_{TA}$) averaged over all directions. For Al $\Theta_D$= 394 K. For MgH$_2$ $\bar{c} \approx$ 4.5 km/s, from the phonons calculation in Ref. [@phonMgh2] (by assuming isotropic phase velocities). For Mg(AlH$_4$)$_2$, $\bar{c} \approx$ $v_{Z}/3$ + $v_{L}2/3$, where $v_{Z}$= 3.75 km/s and $v_{L}$= 4.38 km/s are the phase velocities (averaged over the TA and LA branches) along the $\Gamma$Z and $\Gamma$L directions of the BZ, calculated within dendisty functional pertirbation theory. The zero point energy per unit cell of the optical modes of Mg(AlH$_4$)$_2$, MgH$_2$ and H$_2$ are 162.4, 73.2, and 26.2 kJ/mol, respectively. The contribution of the vibrational modes to the free energy at finite temperature has been computed by considering only the optical phonons at the $\Gamma$-point and the acoustic phonons within the Debye approximation as given above. P.W. Atkins, [sl Physical Chemistry]{}, 5th Edition (Oxford University Press, Oxford, 1994), p. C11. Ref. [@SH], p. 686. , edited by M. Cardona and G. Güntherodt (Springer-Verlag, Berlin, 1982). P. Brüesch, [Phonons: Theory and Experiments II]{} (Springer-Verlag, Berlin, 1986). J. Zak, [Irreducible Representations of Space Groups]{} (Benjamin, New York, 1969). N. W. Ashcroft and N. D. Mermin, [Solid State Physics]{} (Holt-Saunders International Edition, Philadelphia, 1976). H. G. Schimmel, M. R. Johnson, g. J. Kearey, A. J. Ramirez-Cuesta, J. Huot, and F. M. Mulder, Mat. Sci. Eng. B [108]{}, 38 (2004).
--- abstract: 'Prosody affects the naturalness and intelligibility of speech. However, automatic prosody prediction from text for Chinese speech synthesis is still a great challenge and the traditional conditional random fields (CRF) based method always heavily relies on feature engineering. In this paper, we propose to use neural networks to predict prosodic boundary labels directly from Chinese characters without any feature engineering. Experimental results show that stacking feed-forward and bidirectional long short-term memory (BLSTM) recurrent network layers achieves superior performance over the CRF-based method. The embedding features learned from raw text further enhance the performance.' address: \| $^1$School of Computer Science, Northwestern Polytechnical University, Xi’an, China\ $^2$School of Software and Microelectronics, Northwestern Polytechnical University, Xi’an, China\ [{cding, lxie, jyan, wnzhang, yangliu}@nwpu-aslp.org]{} bibliography: - 'refs.bib' title: 'Automatic Prosody Prediction for Chinese Speech Synthesis using BLSTM-RNN and Embedding Features' --- automatic prosody prediction, speech synthesis, neural network, BLSTM, embedding features Introduction ============ Prosody refers to the rhythm, stress and intonation of speech, including variations in duration, loudness and pitch. It is well known that speech prosody plays an important perceptual role in human speech communication [@qian2010automatic]. Specifically, perception of prosodic boundaries is essential for listeners. In Chinese speech synthesis systems, typical prosody boundary labels consist of prosodic word (PW), prosodic phrase (PPH) and intonational phrase (IPH), which construct a three-layer prosody structure tree [@sun2009chinese], as shown in Fig. \[fig:prosody-structure\]. The leaf nodes of tree structure are lexical words that can be derived from a lexical-based word segmentation module. Whether the prosody labels are properly predicted will directly affect the naturalness and intelligibility of the synthesized speech. Previous studies have investigated a great number of features, their relevance to prosody generation in speech production and various prosodic modeling methods. Some syntactic cues like part-of-speech (POS), syllable identity, syllable stress and their contextual counterparts are commonly used for prosody boundary prediction [@jeon2009automatic; @rangarajan2007exploiting; @koehn2000improving]. Many statistical methods have been investigated to model speech prosody, including classification and regression tree [@chu2001locating], hidden Markov model [@nie2003automatic], maximum entropy model [@li2004chinese] and conditional random fields (CRF) [@levow2008automatic]. To our knowledge, the best reported results were achieved with CRF due to its ability of relaxing strong model independence assumption and solving the label bias problem [@qian2010automatic; @LaffertyCRF]. Despite years of research, it is still a great challenge to predict correct prosodic labels from unrestricted text for a text-to-speech (TTS) system. Obviously, there are two major drawbacks of the CRF-based prosody prediction in Chinese speech synthesis. First, it heavily relies on the performances of Chinese word segmentation (CWS) and POS tagging [@sheng2002learning]. Second, the particle size and the inevitable segmentation errors in CWS have negative effects on the subsequent prosodic boundary prediction task. Moreover, the choice of effective features, from a broad set of feature templates, is critical to the success of such systems [@zheng2013deep]. Much of the effort goes into feature engineering, which is notoriously labor-intensive, mainly based on the experience of an annotator. ![Three-layer prosody structure tree in Chinese.](figure/prosody-structure.pdf){width="70"} \[fig:prosody-structure\] Recently, deep neural networks (DNN) have been increasingly investigated in order to minimize the effort of feature engineering in sequential labeling tasks. Zheng et al. [@zheng2013deep] applied neural networks to CWS and POS tagging and proposed a perceptron-style algorithm to speed up the training process with negligible loss in performance. Pei et al. [@pei2014maxmargin] proposed a max-margin tensor neural network for CWS to model interactions between tags and context characters by exploiting tag embeddings and tensor-based transformation. These researches have proved that DNN is able to achieve similar or even superior performance over CRF-based method with minimal feature engineering in sequential labeling tasks. Therefore, it is promising to apply DNN architectures to automatic prosody prediction. However, we notice that the neural networks used in previous researches are feed-forward structures that keep the assumption of sample independence and provide only limited context modeling ability by operating on a fixed-size window of input samples. Instead, bidirectional recurrent neural networks (BRNN) are able to incorporate contextual information from both past and future inputs [@schuster1997bidirectional]. Specifically, BRNN with long short-term memory (LSTM) cells, namely BLSTM-RNN, has become a popular model [@hochreiter1997long]. In this paper, we address the prosodic boundary prediction problem using neural networks. There are three main contributions. (1) We propose a neural network approach to predict prosody labels directly from Chinese characters without any feature engineering. (2) We show that superior performance is achieved by stacking feed-forward and bidirectional long short-term memory (BLSTM) recurrent layers. (3) We leverage a large raw text corpus to obtain useful character embedding features. Both objective and subjective evaluations show that the proposed architecture achieves superior performance over the CRF-based method and the embedding features further enhance the performance. The Proposed Approach ===================== Just like CWS and POS tagging, automatic prosody prediction can be treated as a sequential labeling task that assigns boundary labels to characters of an input sentence. In order to make the prediction models less dependent on the feature engineering, we choose to use a variant of the neural network architecture proposed by [@bengio2003neural] for probabilistic language model. This architecture was subsequently used for CWS and POS tagging [@zheng2013deep]. As shown in Fig. \[fig:NNarchitecture\], the architecture takes raw text as input and maps each Chinese character into a basic feature vector. The following layers are two types of neural networks, FFNN and BLSTM-RNN, used to discover multiple levels of feature representations from the basic feature vectors. The output layer is a graph over which tag inference is achieved by the Viterbi algorithm. ![The neural network architecture for prosodic boundary prediction. In tag inference, B, NB and O denote boundary, non-boundary and others (e.g., punctuation), respectively.](figure/NNarchitecture.pdf){width="70"} \[fig:NNarchitecture\] Feature Vectors --------------- The characters fed into network are transformed into feature vectors by a mapping operation. Typically, a character dictionary $D$ of size $\|D\|$ is extracted from the training set and unknown characters are mapped to a special symbol that is not used elsewhere. Each Chinese character can be typically represented by a one-hot vector, the size of which is $\|D\|$, and all dimensions are marked as $0$ except the location of the character in $D$, which is marked as $1$. However, the one-hot representation, with high dimensions, fails to model the semantic similarity between the ideographic characters. In contrast, the distributed representation or embedding feature, in form of a low dimensional continuous-valued vector learned using neural networks from raw text in a fully unsupervised way, is assumed to carry important syntactic and semantic information [@mikolov2013distributed] [@mikolov2013linguistic]. Recently, Mansur et al. [@mansur2013feature] have shown superior performance in Chinese word segmentation by the use of embedding features based on a neural language model [@bengio2003neural]. Besides [@bengio2003neural], Mikolov et al. [@mikolov2013distributed] proposed a faster skip-gram model called word2vec[^1]. As our preliminary experiments do not show much performance difference among various embedding features, we simply choose word2vec in this study because it can be trained much faster. Network Structures and Training ------------------------------- Two types of neural networks are investigated in this paper: FFNN and BLSTM-RNN. FFNN, trained with a back-propagation learning algorithm [@horikawa1992fuzzy], is widely used in many practical applications. In a typical FFNN, every unit in a layer is connected with all the units in the previous layer, which takes in the output of the previous layer and computes a new set of non-linear activations for next layer. However, the assumption of sample independence brings in only limited context modeling ability. Researchers have proposed RNN to solve the limitation of FFNN. However, conventional RNN is only able to make use of previous context information. This is not accurate in modeling speech prosody that is highly related with both past and future contexts. Instead, bidirectional RNN can access both the preceding and succeeding input contexts with two separate hidden layers, which are then fed to the same output layer. The activation function $\mathcal{H}$ of RNN is usually a sigmoid or hyperbolic tangent function, which often causes the gradient vanishing problem that prevents RNN from modeling the long-span relations in sequence features. An LSTM architecture, which uses purpose-built memory cells to store information, can overcome this problem and model longer contexts. Fig. \[fig:LSTMcell\] illustrates a single LSTM memory cell. For LSTM, $\mathcal{H}$ is implemented by the following functions: $$\begin{aligned} & i_t = \sigma(W_{xi}x_t + W_{hi}h_{t-1} + W_{ci}c_{t-1} + b_i) \\\end{aligned}$$ $$\begin{aligned} & f_t = \sigma(W_{xf}x_t + W_{hf}h_{t-1} + W_{cf}c_{t-1} + b_f) \\ & c_t = f_tc_{t-1} + i_ttanh(W_{xc}x_t + W_{hc}h_{t-1} + b_c) \\ & o_t = \sigma(W_{xo}x_t + W_{ho}h_{t-1} + W_{co}c_t + b_o) \\ & h_t = o_ttanh(c_t)\end{aligned}$$ where $x = (x_1, x_2, ... x_t ..., x_T)$ is the input feature sequence, $\sigma$ is the logistic function, and $i$, $f$, $o$ and $c$ are the input gate, forget gate, output gate and cell memory, respectively. $W$ is the weight matrix and the subscript indicates it is the matrix between two different gates. BLSTM-RNN is a combination of LSTM and BRNN. Deep bidirectional LSTM-RNN can be established by stacking multiple BLSTM-RNN hidden layers on top of each other. The output sequence of one layer is used as the input sequence of the next layer. The hidden state sequences, $h^n$, consist of forward and backward sequences $\overrightarrow{h}^n$ and $\overleftarrow{h}^n$, iteratively computed from $n = 1$ to $N$ and $t = 1$ to $T$ as follows: $$\begin{aligned} & \overrightarrow{h}_t^n = \mathcal{H}(W_{\overrightarrow{h}^{n-1}\overrightarrow{h}^n}\overrightarrow{h}_t^{n-1} + W_{\overrightarrow{h}^n\overrightarrow{h}^n}\overrightarrow{h}_{t - 1}^n + b_{\overrightarrow{h}}^n), \\ & \overleftarrow{h}_t^n = \mathcal{H}(W_{\overleftarrow{h}^{n-1}\overleftarrow{h}^n}\overleftarrow{h}_t^{n-1} + W_{\overleftarrow{h}^n\overleftarrow{h}^n}\overleftarrow{h}_{t - 1}^n + b_{\overleftarrow{h}}^n), \\ & y_t = W_{\overrightarrow{h}^Ny}\overrightarrow{h}_t^N + W_{\overleftarrow{h}^Ny}\overleftarrow{h}_t^N + b_y.\end{aligned}$$ where $y = (y_1, y_2, ... y_t ..., y_T)$ is the output prosodic boundary sequence. In our study, the feed-forward layers are trained with typical backpropagation (BP) algorithm and the back-propagation through time (BPTT) method is used for training of BLSTM layers. BPTT is applied to both forward and backward hidden nodes and back-propagates layer by layer. The weight gradients are computed over the entire utterance [@williams1995gradient]. The neural networks can be trained effectively in a layer-wised training manner, which makes it convenient to stack different types of neural network layers on top of each other to form a deep architecture. The deep architecture is able to build up progressively higher level representations of the input data, which is a crucial factor of the recent success of hybrid systems [@graves2013hybrid]. ![Long short-term memory cell.](figure/LSTMcell.pdf){width="80"} \[fig:LSTMcell\] Tag Inference ------------- To model the tag dependency and infer the tag sequence globally, given a set of tags $G=\{B, NB, O\}$, a transition score $S_{ab}$ is introduced for jumping from tag $a \in G$ to tag $b \in G$. For the input character sequence of a sentence $c_{[1:T]}$ with a tag sequence $tag_{[1:T]}$, a sentence-level score is then given by the sum of transition and network scores [@zheng2013deep; @collobert2011natural]: $$\begin{aligned} & l(c_{[1:T]},tag_{[1:T]},\theta) = \sum_{t=1}^{T}(S_{tag_{t-1}tag_t} + f_\theta(tag_t\|c_t))\end{aligned}$$ where $f_\theta(tag_t\|c_t))$ indicates the score output for $tag_t$ at the $t$-th character by the networks. Given a sentence $c_{[1:T]}$, we can find the best tag path $tag^_{[1:T]}$ by maximizing the sentence score: $$\begin{aligned} & tag^_{[1:T]} = \arg\max_{\forall l_{[1:T]}} l(c_{[1:T]}, tag_{[1:T]}, \theta).\end{aligned}$$ The Viterbi algorithm can be used for tag inference. The description above shows that it is easy to stack feature vectors, neural networks and tag inference together. Thus, the proposed architecture can be trained in a layer-wised fashion. Experiments =========== Totally 48210 sentences randomly selected from People’s Daily were used in our experiments. Prosodic boundaries (PW, PPH and IPH) were labelled by professional annotators with corresponding speech and labeling consistency is ensured. Word segmentation and POS tagging were carried out by a front-end preprocessing tool. The accuracy of word segmentation is 97% and the accuracy of POS tagging is 96%. The corpus was partitioned into three parts: a training set with 43390 utterances, a validation set with 2410 utterances for parameter tuning and a testing set with another 2410 utterances. A character dictionary $D$ of size 4030 was extracted from the training set. A large set of raw texts was also collected from People’s Daily for unsupervised embedding feature learning. All texts were preprocessed with text normalization. In the experiments, PW, PPH and IPH were predicted separately. That is to say, three separate neural network models were trained independently for PW, PPH and IPH using the CURRENNT toolkit [@weninger2014introducing]. Each character in a sentence was assigned to one of the following three boundary tags: B for a prosodic boundary, NB for a non-boundary, and O for other symbols such as punctuation. Precision (P), recall (R) and F-score (F) were calculated as standard objective evaluation criteria. A CRF-based prosodic boundary prediction approach was used as baseline and boundary prediction (B, NB and O) was operated at word level. Atomic features in the CRF approach include word identity, POS tags, the length of word and the predicted tag from the previous boundary level. A linear statistical model was applied to optimize the feature templates. Parameters grid search was adopted to achieve the best performance of the CRF model. The CRF++ toolkit[^2] was used for the CRF-based prosodic boundary prediction. The baseline results are shown in Table \[tab:resultCRF\]. Boundary P (%) R (%) F (%) ---------- ------- ------- ------- PW 95.34 96.73 96.03 PPH 83.41 83.68 83.06 IPH 84.85 73.39 78.71 : The results of CRF-based prosody prediction.[]{data-label="tab:resultCRF"} [\|c\|c\|]{} Topology & ------------------------------ B, BB, BBB, BBBB FFB, FBF, BFF, FBB, BFB, BBF ------------------------------ : Different network configurations in the experiments. \ Num of nodes & 32, 64, 128, 256\ \[tab:networkcnf\] We investigated the performance of neural network architecture with different topologies, as described in Table \[tab:networkcnf\], where F and B denote a feed-forward layer and a BLSTM layer, respectively. The number of the nodes were kept the same for all hidden layers in every tested network architecture. Specifically, the network input is an $M$-dimensional feature vector, where $M$=4030 for the PW prediction and $M$=4031 for the PPH and IPH prediction [^3]. The network output corresponds to the three boundary tags (B, NB and O). All networks were trained with a momentum of 0.9, a learning rate of 1e-3 for PW and 1e-4 for PPH and IPH. BPTT was performed using stochastic gradient descent (SGD) with 32 parallel sentences. The training stops if no lower error on the validation set can be achieved within the last 10 epochs. The best performances for different prosodic boundary levels are shown in Table \[tab:fbb\]. We interestingly discover that the best performances at different levels are all obtained with a topology of FBB. When we compare Table 3 with the CRF-baseline Table 1, we find that the proposed neural network approach achieves competitive performance at the PW level and significant improvements at the PPH and IPH levels. We also studied the effectiveness of the character embedding features. Different sizes of unsupervised training data (400M, 800M, 1200M, 1600M and 2000M text) and embedding feature sizes (100, 200, 300 and 400) were tested. The best network architectures, as shown in Table 3, were used in the experiments. Please note that the dimension of feature vector is greatly reduced as compared with the one-hot representation. The results shown in Table \[tab:embedding\] indicate that the embedding features can further improve the performance of automatic prosodic boundary prediction. Boundary P (%) R (%) F (%) TP / Num of nodes ---------- ------- ------- ------- ------------------- PW 96.02 96.69 96.35 FBB / 32 PPH 82.50 86.75 84.57 FBB / 128 IPH 84.06 79.33 81.63 FBB / 64 : The best performance of each level and the corresponding network topology (TP).[]{data-label="tab:fbb"} Boundary P (%) R (%) F (%) Embedding feature size ---------- ------- ------- ------- ------------------------ PW 96.27 96.91 96.59 300 PPH 82.89 87.13 84.96 400 IPH 84.81 79.88 82.27 100 : The results of neural network architecture with embedding features and the corresponding feature size. \[tab:embedding\] We further conducted an A/B preference test on the naturalness of the synthesized speech. A set of 100 sentences were randomly selected from the test set and the prosodic boundary labels were achieved by: - CRF-based model in Table \[tab:resultCRF\]; - NN with one-hot representation in Table \[tab:fbb\]; - NN with embedding features in Table \[tab:embedding\]. We carried out two sessions of comparative evaluations: (1) vs (2) and (2) vs (3). A set of 20 sentence pairs of each session was randomly selected from the 100 pairs with different prosody prediction results and speech was generated through a typical HMM-based TTS system. A group of 10 subjects were asked to choose which one was better in terms of the naturalness of synthesis speech. The percentage preference is shown in Figure \[fig:sub\]. We can clearly see that the NN architecture with one-hot representation can achieve better naturalness of synthesized speech as compared with CRF, while the use of embedding features further improves the natrualness. ![The percentage preference of A/B test.[]{data-label="fig:sub"}](figure/sub.pdf){width="90"} Conclusion and Future Work ========================== In this paper, we propose to use neural network architectures to predict prosodic boundary labels directly from Chinese characters without feature engineering. We show that superior performance is achieved by stacking feed-forward and bidirectional long short-term memory (BLSTM) recurrent layers. We obtain useful character embedding features from raw text. Both objective and subjective evaluations show that the proposed neural network approch achieves superior performance over the CRF-based approach and the use of embedding features can further boost the performance. For future work, it is promising to predict PW, PPH and IPH labels in a unified neural network and n-gram character embedding features can be further investigated. Acknowledgements ================ This work was supported by the National Natural Science Foundation of China (61175018 and 61571363). [^1]: https://code.google.com/p/word2vec/ [^2]: http://taku910.github.io/crfpp/ [^3]: The predicted tag from the previous level was used as a feature.
--- abstract: 'We are interested in time series of the form $y_{n} = x_{n} + \xi_{n}$ where $\{ x_{n}\}$ is generated by a chaotic dynamical system and where $\xi_{n}$ models observational noise. Using concentration inequalities, we derive fluctuation bounds for the auto-covariance function, the empirical measure, the kernel density estimator and the correlation dimension evaluated along $y_{0}, \ldots, y_{n}$, for all $n$. The chaotic systems we consider include for instance the Hénon attractor for Benedicks-Carleson parameters.' address: - 'CPhT, CNRS-École Polytechnique, 91128 Palaiseau Cedex, France' - 'Email address: `[email protected]`' author: - Cesar Maldonado bibliography: - 'bib.bib' title: Fluctuation bounds for chaos plus noise in dynamical systems --- Introduction ============ Practically all experimental data is corrupted by noise, whence the importance of modeling dynamical systems perturbed by some kind of noise. In the literature one finds two principal models of noise. On one hand, the dynamical noise model in which the noise term evolves within the dynamics (see for instance [@Arn] and references therein). And on the other hand, the so-called observational noise model, in which the perturbation is supposed to be generated by the observation process (measurement). In this paper we focus on the latter model of noise. Suppose that we are given a finite ‘sample’ $y_{0},\ldots,y_{n-1}$ of a discrete ergodic dynamical system perturbed by observational noise. Consider a general observable $K(y_{0},\ldots,y_{n-1})$. We are interested in estimating the fluctuations of $K$ and its convergence properties as $n$ grows. Our main tool is concentration inequalities. Roughly speaking, concentration inequalities allow to systematically quantify the probability of deviation of an observable from its expect value, requiring that the observable is smooth enough. The systems for which concentration inequalities are available must have some degree of hyperbolicity. Indeed, in [@ChGo], the authors prove that the class of non-uniformly hyperbolic maps modeled by Young towers satisfy concentration inequalities. They are either exponential or polynomial depending on the tail of the corresponding return-times. Concentration inequalities is a recent topic in the study of fluctuations of observables in dynamical systems. The reader can consult [@Ch] for a panorama. The article is organized as follows. In section 2, we give some general definitions concerning observational noise and concentration inequalities. We give some typical examples of systems perturbed by observational noise. In section 3, we prove our main theorem, namely, concentration inequalities for observationally perturbed systems (observed systems). As a consequence, we obtain estimates on the deviation of any separately Lipschitz observable $K(y_{0},\ldots,y_{n-1})$ from its expected value. Section 4 is devoted to some applications. We derive a bound for the deviation of the estimator of the auto-covariance function in the observed system. We provide an estimate of the convergence in probability of the observed empirical measure. We study the $L^{1}$ convergence of the kernel density estimator for a observed system. We also give a result on the variance of an estimator of the correlation dimension in the observed system. The observables we consider here were studied in [@ChCS] and [@ChGo] for dynamical systems without observational noise. Generalities ============ Dynamical systems as stochastic process --------------------------------------- We consider a dynamical system $(X,T,\mu)$ where $(X,d)$ is a compact metric space and $\mu$ is a $T$-invariant probability measure. In practice, $X$ is a compact subset of $\rr^{n}$. One may interpret the orbits $(x, Tx, \ldots)$ as realizations of the stationary stochastic process defined by $X_{n}(x) = T^{n}x$. The finite-dimensional marginals of this process are the measures $\mu_{n}$ given by $$\label{marginal} d\mu_{n}(x_{0},\ldots,x_{n-1}) = d\mu(x_{0})\prod_{i=1}^{n-1}\delta_{x_{i}=Tx_{i-1}}.$$ Therefore, the stochasticity comes only from the initial condition. When the system is sufficiently mixing, one may expect that the iterate $T^{k}x$ is more or less independent of $x$ if $k$ is large enough. Observational noise ------------------- The noise process is modeled as bounded random variables $\xi_{n}$ defined on a probability space $(\Omega, \mathcal{B},P)$ and assuming values in $X$. Without loss of generality, we can assume that the random variables $\xi_{n}$ are centered, i.e. have expectation equal to 0. In most cases, the noise is small and it is convenient to represent it by the random variables $\varepsilon\xi_{i}$ where $\varepsilon>0$ is the amplitude of the noise and $\xi_{i}$ is of order one. We introduce the following definition. For every $i\in\nn\cup\{0\}$ (or $i\in\zz$ if the map $T$ is invertible), we say that the sequence of points $\{y_{i}\}$ given by $$%\label{observ-noise} y_{i} := T^{i}x + \varepsilon\xi_{i},$$ is a trajectory of the dynamical system $(X,T,\mu)$ perturbed by the observational noise $(\xi_{n})$ with amplitude $\varepsilon>0$. Hereafter we refer to it simply as the observed system. Next, we make the following assumptions on the noise. Standing assumption on noise: 1. $(\xi_{n})$ is independent of $X_{0}$ and $\lVert\xi_{n}\rVert\leq1$; 2. The random variables $\xi_{i}$ are independent. As we shall see, the $\xi_{i}$ need not be independent, although it is a natural assumption in practice. We notice that, under the same assumption on the noise, the authors of [@LN] give a consistent algorithm for recovering the unperturbed time series from the sequence $\{y_{i}\}$. They assume that the process $(X_{n})$ is generated by a sufficiently chaotic dynamical system. The merit of Lalley and Nobel ([@LN]) is that a few assumptions are made (compare with Kantz-Schreiber’s or Abarbanel’s books [@KS; @Aba]). In contrast, for the case of unbounded noise (e.g. Gaussian) and if the system present strongly homoclinic pairs of points, then with positive probability it is impossible to recover the initial condition of the true trajectory even observing an infinite sequence with noise (see also [@LN]). Examples -------- \[ex1\] Consider Smale’s solenoid map, $T_{S}:\rr^{3}\to\rr^{3}$ which maps the torus into itself: $$T_{S}\left(\phi,u,v\right) = \left( 2\phi \ \ \mod\ 2\pi, \beta u+\alpha\cos(\phi),\beta v+\alpha\sin(\phi) \right),$$ where $0<\beta<1/2$ and $\beta<\alpha<1/2$. Let the random variables $\xi_{i}$ be uniformly distributed on the solid sphere of radius one. For every vector $x = (\phi,u,v)$ in the torus, the observed system is given by $y_{i} = T_{S}(x_{i}) + \varepsilon \xi_{i}$, for some fixed $\varepsilon>0$. Take $\mathbb{S}^{1}$ (the unit circle) as state space. Let us fix an increasing sequence $a_{0}<a_{1}<\cdots<a_{k}=a_{0}$, and consider for each interval $(a_{j},a_{j+1})$ ($0\leq j\leq k-1$) a monotone map $T_{j}:(a_{j},a_{j+1}) \to \mathbb{S}^{1}$. The map $T$ on $\mathbb{S}^{1}$ is given by $T(x)=T_{j}(x)$ if $x\in(a_{j},a_{j+1})$. It is well known that when the map $T$ is uniformly expanding, it admits an absolutely continuous invariant measure $\mu$. It is unique under some mixing assumptions. Let $P$ be the uniform distribution on $\mathbb{S}^{1}$. The observed sequence is $y_{i} = T^{i}(x) +\varepsilon \xi_{i}$. ![Top: Simulation of the Lozi map for the parameters a=1.7 and b=0.5. Bottom: A simulation of the observed Lozi map with observational noise whose magnitude is bounded by $\varepsilon = 0.06$. ](LoziNoise.pdf){width="60.00000%"} The Lozi map $T_{L}:\rr^{2}\to\rr^{2}$ is given by $$T_{L}(u,v) = \left( 1-a\lvert u\rvert + v ,bu \right), \hspace{1cm} (u,v)\in\rr^{2}.$$ For $a=1.7$ and $b=0.5$ one observes numerically a strange attractor. In [@CL] the authors constructed a SRB measure $\mu$ for this map. It is also included in Young’s framework [@Y1]. Now, as state space of the random variables we take $B_{1}(0)$, the ball centered at zero with radius one. Consider the uniform probability distribution on $B_{1}(0)$. Let us denote by $x$ the vector $(u,v)$ and let $\varepsilon>0$, so, the observed system is given by $y_{i} = T_{L}^{i}x +\varepsilon \xi_{i}$. \[ex4\] Consider the Hénon map $T_{H}:\rr^{2}\to\rr^{2}$ defined as $$T_{H}(u,v) = \left( 1-au^{2} + v , bu \right), \hspace{1cm} (u,v)\in\rr^{2}.$$ Where $0<a<2$ and $b>0$ are some real parameters. The state space of the random variables is again $B_{1}(0)$ with the uniform distribution on it. Let be $x=(u,v)$, then the observed system is given by $y_{i} = T_{H}^{i}x +\varepsilon \xi_{i}$. It is known that there exists a set of parameters $(a,b)$ of positive Lebesgue measure for which the map $T_{H}$ has a topologically transitive attractor $\Lambda$, furthermore there exists a set $\Delta\subset\rr^{2}$ with $\Leb(\Delta)>0$ such that for all $(a,b)\in\Delta$ the map $T_{H}$ admits a unique SRB measure supported on $\Lambda$ ([@BY]). ![Top: A simulation of the Hénon map for the classical parameters a=1.4 and b=0.3. Bottom: Simulation of the observed Hénon map with observational noise whose magnitude is uniformly bounded by $\varepsilon=0.04$.](HenonNoise.pdf){width="70.00000%"} \[ex5\] The Manneville-Pomeau map is an example of an expansive map, except for a point where the slope is equal to 1 (neutral fixed point). Consider $X=[0,1]$, and for the sake of definiteness take $$T_{\alpha}(x) = \begin{cases} x+2^{\alpha}x^{1+\alpha} & \mbox{ if }\ x\in[0,1/2)\\ 2x-1 & \mbox{ if }\ x\in[1/2,1), \end{cases}$$ where $\alpha\in(0,1)$ is a parameter. It is well known that there exists an absolutely continuous invariant probability measure $d\mu(x)=h(x)dx$ and $h(x)\sim x^{-\alpha}$ when $x\to0$. The observed sequence is defined by $y_{i}= T^{i}_{\alpha}(x) +\varepsilon \xi_{i}$. The random variables $\xi_{i}$ are uniformly distributed in $X$. One identifies the $[0,1]$ with the unit circle to avoid leaks. Concentration inequalities -------------------------- Let $X$ be a metric space. For any function of $n$ variables $K: X^{n}\to\rr$, and for each $j$, $0\leq j \leq n-1$, let $$\Lip_{j}(K) := \sup_{x_{0},\ldots,x_{n-1}}\sup_{x_{j}\neq x'_{j}}\frac{\lvert K(x_{0},\ldots,x_{j},\ldots,x_{n-1})-K(x_{0},\ldots,x'_{j},\ldots,x_{n-1}) \rvert}{ d(x_{j},x'_{j})}.$$ We say that $K$ is separately Lipschitz if, for all $0\leq j \leq n-1$, $\Lip_{j}(K)$ is finite. Now, we may state the following definition. The stochastic process $(Y_{n})$ taking values on $X$ satisfies an exponential concentration inequality if there exists a constant $C>0$ such that, for any separately Lipschitz function $K$ of $n$ variables, one has $$\label{exp-ineq} \ee\left( e^{K(Y_{0},\ldots,Y_{n-1}) - \ee(K(Y_{0},\ldots,Y_{n-1}))}\right)\leq e^{C\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}}.$$ Notice that the constant $C$ depends only on $T$, but neither on $K$ nor on $n$. A weaker inequality is given by the following definition. The stochastic process $(Y_{n})$ taking values on $X$ satisfies a polynomial concentration inequality with moment $q\geq2$ if there exists a constant $C_{q}>0$ such that, for any separately Lipschitz function $K$ of $n$ variables, one has $$\label{poly-ineq} \ee\left( \lvert K(Y_{0},\ldots,Y_{n-1}) - \ee(K(Y_{0},\ldots,Y_{n-1}))\rvert^{q} \right)\leq C_{q}\left(\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}\right)^{q/2}.$$ As in the previous definition the constant $C_{q}$ does not depend neither on $K$ nor on $n$. When $q=2$, we have a bound for the variance of $K(Y_{0},\ldots,Y_{n})$. \[remark-iid\] If $(Y_{n})$ is a bounded i.i.d. process then it satisfies (see e.g. [@Led]). It also satisfies for all $q\geq2$, see e.g. [@BBL] for more details. These concentration inequalities allow us to obtain estimates on the deviation probabilities of the observable $K$ from its expected value. If the process $(Y_{n})$ satisfies the exponential concentration inequality then for all $t>0$ and for all $n\geq1$, $$\label{exp-prob-deviation} \pp\left\{\lvert K(Y_{0},\ldots,Y_{n-1}) - \ee(K(Y_{0},\ldots,Y_{n-1})) \rvert>t\right\} \leq 2 e^{\frac{-t^{2}}{4C\sum_{j=0}^{n-1}\Lip_{j}(K)^{2} }}.$$ If the process satisfies the polynomial concentration inequality for some $q\geq2$, then we have that for all $t>0$ and for all $n\geq1$, $$\label{poly-prob-deviation} \pp\left\{\lvert K(Y_{0},\ldots,Y_{n-1})-\ee(K(Y_{0},\ldots,Y_{n-1})) \rvert >t\right\} \leq \frac{C_{q}}{t^{q}}\Big(\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}\Big)^{q/2}.$$ The inequality follows from the basic inequality $\pp(Z>t) \leq e^{-\lambda t}\ee(e^{\lambda Z})$ with $\lambda >0$ applied to $Z = K(Y_{0},\ldots,Y_{n-1})-\ee(K(Y_{0},\ldots,Y_{n-1}))$, using the exponential concentration inequality and optimizing over $\lambda$. The inequality follows easily from and the Markov inequality (see [@Ch] for details). It has been proven that a dynamical system modeled by a Young tower with exponential tails satisfies the exponential concentration inequality [@ChGo]. The systems in the examples from \[ex1\] to \[ex4\] are included in that framework. The example \[ex5\] satisfies the polynomial concentration inequality with moment $q<\frac{2}{\alpha}-2$ for $\alpha\in(0,1/2)$, which is the parameter of the map (see [@ChGo] for full details). Main theorem & corollary ======================== Let us first introduce some notations. We recall that $P$ is the common distribution of the random variables $\xi_{i}$. The expected value with respect to a measure $\nu$ is denoted by $\ee_{\nu}$. Recall the expression for the measure $\mu_{n}$. Hence in particular $$\begin{aligned} \ee_{\mu_{n}}(K) =& \int\cdots\int K(x_{0},\ldots,x_{n-1})\dd\mu_{n}(x_{0},\ldots,x_{n-1})\\ =& \int K(x,\ldots, T^{n-1}x)\dd\mu(x).\end{aligned}$$ Next, we denote by $\mup$ the product of the measures $\mu_{n}$ and $P^{n}$, where $P^{n}$ stands for $P\otimes\cdots\otimes P$ ($n$ times). The expected value of $K(y_{0}, \ldots,y_{n-1})$ is denoted by $$\ee_{\mup}(K) := \int K(x + \varepsilon\xi_{0},\ldots, T^{n-1}x+\varepsilon\xi_{n-1})\dd\mu(x)\dd{P}(\xi_{0})\cdots\dd{P}(\xi_{n-1}).$$ Our main result is the following. \[PerturbedIneq\] If the original system $(X,T,\mu)$ satisfies the exponential inequality , then the observed system satisfies an exponential concentration inequality. For any $n\geq1$, it is given by $$\label{exp-ineq-pert} \ee_{\mup}\left( e^{K(y_{0},\ldots,y_{n-1})-\ee_{\mup}(K(y_{0},\ldots,y_{n-1}))}\right)\leq e^{D(1+\varepsilon^{2})\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}},$$ Furthermore, if the system $(X,T,\mu)$ satisfies the polynomial concentration inequality with moment $q\geq2$, then the observed system satisfies a polynomial concentration inequality with the same moment. For any $n\geq1$, it is given by $$\label{poly-ineq-pert} \begin{split} \ee_{\mup}\left( \left\lvert K(y_{0},\ldots,y_{n-1})-\ee_{\mup}(K(y_{0},\ldots,y_{n-1}))\right\rvert^{q} \right) \leq D_{q}(1+\varepsilon)^{q}\Big(\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}\Big)^{q/2}. \end{split}$$ Observe that one recovers the corresponding concentration inequalities for the original dynamical system when $\varepsilon$ vanishes. Our proof works provided the noise process satisfies a concentration inequality (see Remark \[remark-iid\]). We have stated the result in the particular case of i.i.d. noise because it is reasonable to model the observational perturbations in this manner. Nevertheless, one can slightly modify the proof to get the result valid for correlated perturbations. First let us fix the noise $\{\xi_{j}\}$ and let $\overline{\xi} := (\xi_{0}, \xi_{1}, \ldots,\xi_{n-1})$. Introduce the auxiliary observable $$\Kxi(x_{0},\ldots,x_{n-1}) := K(x_{0} + \varepsilon\xi_{0}, \ldots,x_{n-1}+\varepsilon\xi_{n-1}).$$ Since the noise is fixed, it is easy to see that $\Lip_{j}(\Kxi)=\Lip_{j}(K)$ for all $j$. Notice that $\Kxi(x,\ldots, T^{n-1}x)= K(x+\varepsilon\xi_{0},\ldots, T^{n-1}x+\varepsilon\xi_{n-1}) = K(y_{0},\ldots,y_{n-1})$. Next we define the observable $F(\xi_{0}, \ldots, \xi_{n-1})$ of $n$ variables on the noise, as follows, $$F(\xi_{0}, \ldots, \xi_{n-1}):= \ee_{\mu_{n}}(\Kxi(x,\ldots, T^{n-1}x)).$$ Observe that, $\Lip_{j}(F) \leq \varepsilon\Lip_{j}(K)$. Now we prove inequality . Observe that is equivalent to prove the inequality for $$%\begin{split} %\ee_{\mup}\left(e^{K(y_{0},\ldots,y_{n-1})-\ee_{\mup}(K(y_{0},\ldots,y_{n-1}))} \right) = \ee_{\mup}\left(e^{\Kxi(x,\ldots,T^{n-1}x)-\ee_{\mup}(\Kxi(x,\ldots,T^{n-1}x))} \right). %\end{split}$$ Adding and subtracting $\ee_{\mup}(\Kxi(x, \ldots, T^{n-1}x))$ and using the independence between the noise and the dynamical system, we obtain that the expression above is equal to $$\ee_{\mu_{n}}\left(e^{\Kxi(x, \ldots,T^{n-1}x)- \ee_{\mu_{n}}(\Kxi(x,\ldots,T^{n-1}x))} \right)\ee_{P^{n}}\left(e^{F(\xi_{0}, \ldots, \xi_{n-1})-\ee_{P^{n}}(F(\xi_{0}, \ldots, \xi_{n-1}))} \right).$$ Since in particular, i.i.d. bounded processes satisfy the exponential concentration inequality (see Remark \[remark-iid\] above), we may apply to the dynamical system and the noise, yielding $$\begin{split} \ee_{\mu_{n}}\left(e^{\Kxi(x, \ldots,T^{n-1}x)- \ee_{\mu_{n}}(\Kxi(x,\ldots,T^{n-1}x))} \right)\ee_{P^{n}}\left(e^{F(\xi_{0}, \ldots, \xi_{n-1})-\ee_{P^{n}}(F(\xi_{0}, \ldots, \xi_{n-1}))} \right)\\ \leq e^{C\sum_{j=0}^{n-1}\Lip_{j}(\Kxi)^{2}}e^{C'\varepsilon^{2}\sum_{j=0}^{n-1}\Lip_{j}(F)^{2}} \leq e^{D(1+\varepsilon^{2})\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}}, \end{split}$$ where $D := \max\{C,C'\}$. Next, we prove inequality similarly. We use the binomial expansion after the triangle inequality with $\ee_{\mu_{n}}(\Kxi(x,\ldots,T^{n-1}x))$. Using the independence between the noise and the dynamics, we get $$\label{sep} \begin{split} \ee_{\mup}( \lvert K(y_{0},\ldots, y_{n-1}) - \ee_{\mup}(K(y_{0},\ldots,y_{n-1}))\rvert^{q}) \hspace{4cm}\\ \leq\sum_{p=0}^{q}\Big(\begin{array}{c} q\\p \end{array}\Big)\ee_{\mu_{n}}(\lvert \Kxi(x,\ldots,T^{n-1}x) - \ee_{\mu_{n}}(\Kxi(x,\ldots,T^{n-1}x))\rvert^{p})\times\\ \ee_{P^{n}}\left(\lvert F(\xi_{0},\ldots,\xi_{n-1})-\ee_{P^{n}}(F(\xi_{0},\ldots,\xi_{n-1}))\rvert^{q-p} \right). \end{split}$$ We proceed carefully using the polynomial concentration inequality. The terms corresponding to $p=1$ and $p=q-1$ have to be treated separately. For the rest we obtain the bound $$%\begin{split} %\EmuP( \lvert K(y_{0},\ldots, y_{n-1}) - \EmuP(K(y_{0},\ldots,y_{n-1}))\rvert^{q}) \hspace{4cm}\\ %\leq \sum_{\substack{ p=0\\ p\neq1,q-1}}^{q} \binom{q}{p} C_{p}\Big(\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}\Big)^{p/2}\times C'_{q-p}\Big(\varepsilon^{2}\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}\Big)^{\frac{q-p}{2}}. %+\\ %q\cdot\Emu\left(\lvert \Kxi(x,\ldots,T^{n-1}x)-\Emu(\Kxi(x,\ldots,T^{n-1}x))\rvert\right)\cdot C'_{q-1}\Big(\varepsilon^{2}\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}\Big)^{\frac{q-1}{2}}+\\ %q\cdot C_{q-1}\Big(\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}\Big)^{\frac{q-1}{2}}\cdot\EP\left(\lvert F(\xi_{0},\ldots,\xi_{n-1}-\EP(F(\xi_{0},\ldots,\xi_{n-1}))\rvert\right). %\end{split}$$ For the case $p=1$, we apply Cauchy-Schwarz inequality and for $q=2$ to get $$\ee_{\mu_{n}}\left(\lvert \Kxi(x,\ldots,T^{n-1}x)-\ee_{\mu_{n}}(\Kxi(x,\ldots,T^{n-1}x))\rvert\right)\leq \sqrt{C_{2}}\Big(\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}\Big)^{1/2}.$$ If $q=2$, we proceed in the same way for the second factor in the right hand side of . The case $p=q-1$ is treated similarly. Finally, putting this together and choosing adequately the constant $D_{q}$ we obtain the desired bound. Next we obtain an estimate of deviation probability of the observable $K$ from its expected value. If the system $(X,T,\mu)$ satisfies the exponential concentration inequality, then for the observed system $\{y_{i}\}$, for every $t>0$ and for any $n\geq1$ we have, $$\label{exp-deviation-pert} \mup\big( \lvert K(y_{0},\ldots,y_{n-1})-\ee_{\mup}(K)\rvert \geq t \big)\leq 2\exp\left(\frac{-t^{2}}{4D(1+\varepsilon^{2})\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}}\right).$$ If the system $(X,T,\mu)$ satisfies the polynomial concentration inequality with moment $q\geq2$, then the observed system satisfies for every $t>0$ and for any $n\geq1$, $$\label{poly-deviation-pert} \begin{split} \mup\big(\lvert K(y_{0},\ldots,y_{n-1}) - \ee_{\mup}(K)\rvert >t\big) \leq \frac{D_{q}}{t^{q}}(1+\varepsilon)^{q}\left(\sum_{j=0}^{n-1}\Lip_{j}(K)^{2}\right)^{q/2}. \end{split}$$ The proof is straightforward and left to the reader. Applications ============ Dynamical systems ----------------- Concentration inequalities are available for the class of non-uniformly hyperbolic dynamical systems modeled by Young towers ([@ChGo]). Actually, systems with exponential tails satisfy an exponential concentration inequality and if the tails are polynomial then the system satisfies a polynomial concentration inequality. The examples given in section 2 are included in that class of dynamical systems. We refer the interested reader to [@Y1] and [@Y2] for more details on systems modeled by Young towers. Here we consider dynamical systems satisfying either the exponential or the polynomial concentration inequality. We apply our result of concentration in the setting of observed systems to empirical estimators of the auto-covariance function, the empirical measure, the kernel density estimator and the correlation dimension. Auto-covariance function ------------------------ Consider the dynamical system $(X,T,\mu)$ and a square integrable observable $f:X\to\rr$. Assume that $f$ is such that $\int f\dd\mu=0$. We remind that the auto-covariance function of $f$ is given by $$\mathrm{Cov}(k):= \int f(x)f(T^{k}x)\dd\mu(x).$$ In practice, one has a finite number of iterates of some $\mu$-typical initial condition $x$, thus, what we may easily obtain from the data is the empirical estimator of the auto-covariance function: $$\Cemp(k) := \frac{1}{n}\sum_{i=0}^{n-1}f(T^{i}x)f(T^{i+k}x).$$ From Birkhoff’s ergodic theorem it follows that $\mathrm{Cov}(k)=\lim_{n\to\infty}\Cemp(k)$ $\mu$-almost surely. Observe that the expected value of the estimator $\Cemp(k)$ is exactly $\mathrm{Cov}(k)$. The following result gives us a priori theoretical bounds to the fluctuations of the estimator $\Cemp$ around $\mathrm{Cov}$ for every $n$. This result can be found in [@ChGo], here we include it for the sake of completeness. \[Prop-Concen-Cov\] Let $\mathrm{Cov}(k)$ and $\Cemp(k)$ be defined as above. If the dynamical system $(X,T,\mu)$ satisfies the exponential concentration inequality then for all $t>0$ and any integer $n\geq1$ we have $$\mu\left( \left\lvert \Cemp(k) - \mathrm{Cov}(k) \right\rvert > t \right) \leq 2\exp\left(\frac{-t^{2}}{16Ca_{f}^{2} }\left(\frac{n^{2}}{n+k}\right) \right),$$ where $a_{f} = \Lip(f)\lVert{f}\rVert_{\infty}$ and $C$ is the constant appearing in . If the system satisfies the polynomial concentration inequality with moment $q\geq2$, then for all $t>0$ and any integer $n\geq1$ we have $$\mu\left( \left\lvert \Cemp(k)-\mathrm{Cov}(k)\right\rvert>t \right)\leq C_{q}\left(\frac{2a_{f}}{t}\right)^{q}\left(\frac{n+k}{n^{2}}\right)^{q/2},$$ where $C_{q}$ is the constant appearing in . Consider the following observable of $n+k$ variables, $$K(z_{0},\ldots,z_{n+k-1}):= \frac{1}{n}\sum_{i=0}^{n-1}f(z_{i})f(z_{i+k}).$$ In order to estimate the Lipschitz constant of $K$, consider $0\leq l\leq n+k-1$ and replace the value $z_{l}$ with $z'_{l}$. Note that the absolute value of the difference between $K(z_{0},\ldots,z_{l},\ldots,z_{n+k-1})$ and $K(z_{0},\ldots,z'_{l},\ldots,z_{n+k-1})$ is less than or equal to $$\frac{1}{n}\left\lvert f(z_{l-k})f(z_{l}) + f(z_{l})f(z_{l+k})-f(z_{l-k})f(z'_{l})-f(z'_{l})f(z_{l+k})\right\rvert,$$ and so for every index $l$, we have that $$\Lip_{l}(K) \leq \sup_{z_{0},\ldots,z_{n+k-1}}\sup_{z_{l}\neq z'_{l}} \frac{1}{n}\frac{\lvert(f(z_{l})-f(z'_{l}))(f(z_{l-k})+f(z_{l+k})) \rvert}{d(z_{l}, z'_{l})}\leq \frac{2}{n}\Lip(f)\lVert f\rVert_{\infty}.$$ Next, if the exponential inequality holds, we use to obtain $$\begin{aligned} \mu\left(\Cemp(k)-\mathrm{Cov}(k)>t\right) \leq& \exp\left(\frac{-t^{2}}{16C\Lip(f)^{2}\lVert f\rVert^{2}_{\infty}}\left( \frac{n^{2}}{n+k}\right) \right).\end{aligned}$$ Applying similarly the inequality to the function $-K$, we get the result by a union bound. The polynomial case follows from inequality . ### Auto-covariance function for observed systems Let us consider the observed orbit $y_{0},\ldots,y_{n-1}$. Define the observed empirical estimator of the auto-covariance function as follows $$\label{cov-funct-pert} \Cpert(k) := \frac{1}{n}\sum_{i=0}^{n-1}f(y_{i})f(y_{i+k}).$$ We are interested in quantifying the influence of noise on the correlation. We provide a bound on the probability of the deviation of the observed empirical estimator from the covariance function. Let $\Cpert(k)$ be given by . If the dynamical system $(X,T,\mu)$ satisfies the exponential inequality then for all $t>0$ and for any integer $n\geq1$ we have $$\begin{split} \mup\left(\left\lvert \Cpert(k) -\mathrm{Cov}(k)\right\rvert > t + 2a_{f}\varepsilon \right)\leq 2\exp\left( \frac{-t^{2}}{64Da_{f}^{2}(1+\varepsilon^{2})}\left(\frac{n^{2}}{n+k}\right)\right)\\ + 2\exp\left(\frac{-t^{2}}{16Ca_{f}^{2} }\left(\frac{n^{2}}{n+k}\right)\right), \end{split}$$ where $a_{f}=\Lip(f)\lVert f\rVert_{\infty}$, $C$ and $D$ are the constants appearing in and respectively. If the system satisfies the polynomial inequality with moment $q\geq2$, then for all $t>0$ and any integer $n\geq1$ we have $$\mup\left( \left\lvert \Cpert(k) - \mathrm{Cov}(k)\right\rvert >t + 2a_{f}\varepsilon \right)\leq \left(2^{q}D_{q}(1+\varepsilon)^{q}+C_{q}\right)\left(\frac{2a_{f}}{t}\right)^{q}\left(\frac{n+k}{n^{2}}\right)^{q/2},$$ where $C_{q}$ and $D_{q}$ are the constants appearing in and respectively. To prove this assertion we will use an estimate of $$\mup\left( \left\lvert \Cpert(k) -\Cemp(k) \right\rvert > t + \ee_{\mup}\left(\left\lvert\Cpert(k) -\Cemp(k)\right\rvert\right) \right).$$ First let us write $x_{i}:=T^{i}x$, and observe that by adding and subtracting $f(x_{i}+\varepsilon\xi_{i})f(x_{i+k})$, the quantity $\lvert\Cpert(k) -\Cemp(k)\rvert$ is less than or equal to $$\frac{1}{n}\sum_{i=0}^{n-1}\left\lvert f(x_{i}+\varepsilon\xi_{i})[f(x_{i+k}+\varepsilon\xi_{i+k})-f(x_{i+k})] +[f(x_{i}+\varepsilon\xi_{i})-f(x_{i})]f(x_{i+k})\right\rvert ,$$ which leads us to the following estimate, $$\label{expect-CpertCemp} \ee_{\mup}\left(\left\lvert\Cpert(k) - \Cemp(k)\right\rvert\right) \leq 2\varepsilon\Lip(f)\lVert f\rVert_{\infty}.$$ For a given realization of the noise $\{ e_{i}\}$, consider the following observable of $n+k$ variables $$K(z_{0},\ldots,z_{n+k-1}) := \frac{1}{n}\sum_{i=0}^{n-1}\left( f(z_{i}+\varepsilon e_{i})f(z_{i+k}+\varepsilon e_{i+k}) - f(z_{i})f(z_{i+k})\right).$$ For every $0\leq l\leq n-1$, one can easily obtain that $$\Lip_{l}(K)\leq \frac{4}{n}\Lip(f)\lVert f\rVert_{\infty}.$$ In the exponential case, from the inequality and the bound on the expected value of $K$, we obtain that $$\mup\left(\left\lvert \Cpert(k)-\Cemp(k)\right\rvert> t +2\varepsilon a_{f}\right)\leq 2\exp\left( \frac{-t^{2}}{64Da_{f}^{2}(1+\varepsilon^{2})}\left(\frac{n^{2}}{n+k}\right) \right).$$ Using proposition \[Prop-Concen-Cov\], a union bound and an adequate rescaling, we get the result. In order to prove the polynomial inequality, proceed similarly applying . Empirical measure ----------------- The empirical measure of a sample $x_{0},\ldots,x_{n-1}$ is given by $$\E := \frac{1}{n}\sum_{i=0}^{n-1}\delta_{x_{i}},$$ where $\delta_{x}$ denotes the Dirac measure at $x$. If the given sample $x_{0},\ldots,x_{n-1}$ is the sequence $x,\ldots,T^{n-1}x$ for a $\mu$-typical $x\in X$, then from Birkhoff’s ergodic theorem it follows that the sequence of random measures $\{\E\}$ converges weakly to the $T$-invariant measure $\mu$, almost surely. Consider the observed itinerary $y_{0}, \ldots,y_{n-1}$ and define the observed empirical measure by $$\Epert := \frac{1}{n}\sum_{i=0}^{n-1}\delta_{y_{i}}.$$ Observe that this measure is well defined on $X$. Again Birkhoff’s ergodic theorem implies that almost surely $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}g(y_{i}) = \int\int g(x+\xi)\dd\mu(x)\dd{P}(\xi),$$ for every continuous function $g$. More precisely, this convergence holds for a set of $\mu$-measure one of initial conditions for the dynamical system $(X,T)$ and a set of measure one of noise realizations $(\xi_{i})$ with respect to the product measure $P^{\nn}$. We want to estimate the speed of convergence of the observed empirical measure. For that purpose, we chose the Kantorovich distance on the set of probability measures, which is defined by $$\kappa(\mu,\nu) := \sup_{g\in\mathcal{L}} \int g \dd\mu - \int g \dd\nu,$$ where $\mu$ and $\nu$ are two probability measures on $X$ and $\mathcal{L}$ denotes the space of all real-valued Lipschitz functions on $X$ with Lipschitz constant at most one. Now, we study the fluctuations of the Kantorovich distance of the observed empirical measure to the measure $\mu$, around its expected value. The statement is the following. \[Fluctua-Emp-Meas\] If the system $(X,T,\mu)$ satisfies the exponential concentration inequality , then for all $t>0$ and any integer $n\geq1$, $$\mup\left( \kappa(\Epert,\mu)>t +\ee_{\mup}\big(\kappa(\Epert,\mu)\big)\right)\leq e^{-\frac{t^{2}n}{4D(1+\varepsilon^{2})}}.$$ If the system satisfies the polynomial concentration inequality with moment $q\geq2$, then for all $t>0$ and any integer $n\geq1$, $$\mup\left( \kappa(\Epert,\mu) >t + \ee_{\mup}\big(\kappa(\Epert,\mu)\big) \right)\leq \frac{D_{q}(1+\varepsilon)^{q}}{{t}^{q}}\frac{1}{n^{q/2}}.$$ Using the following separately Lipschitz function of $n$ variables, $$K(z_{0},\ldots,z_{n-1}) := \sup_{g\in\mathcal{L}}\left[\frac{1}{n}\sum_{i=0}^{n-1}g(z_{i})-\int g \dd\mu\right].$$ It is easy to check that $\Lip_{j}(K)\leq \frac{1}{n}$, for every $j=0,\ldots,n-1$. The proposition follows from the concentration inequalities and . We are not able to obtain a sufficiently good estimate of $\ee_{\mup}\left(\kappa(\Epert,\mu)\right)$ in dimension larger than one, thus in the following we restrict ourselves to systems with $X\subset\rr$. \[Bound-Expect\] Let $(X,T,\mu)$ be a dynamical system with $X\subset\rr$. If there exists a constant $c>0$ such that for every Lipschitz function $f$, the auto-covariance function $\mathrm{Cov}_{f}(k)$ satisfies that $\sum_{k=1}^{\infty}\lvert \mathrm{Cov}_{f}(k)\rvert\leq{c}\lVert{f}\rVert_{\Lip}^{2}$, then there exists a constant $B$ such that for all $n\geq1$ $$\ee_{\mu_{n}}\left(\kappa(\E,\mu)\right) \leq \frac{B}{n^{1/4}}.$$ The proof of the preceding lemma is found in [@ChCS Section 5]. It relies in the fact that in dimension one, it is possible to rewrite the Kantorovich distance using distribution functions. Then by an adequate Lipschitz approximation of the distribution function, the estimate bound follows from the summability condition on the auto-covariance function. As a consequence of proposition \[Fluctua-Emp-Meas\] and the previous lemma, we obtain the following result. Assume that the system $(X,T,\mu)$ satisfies the assumptions of lemma \[Bound-Expect\]. Let $\Epert$ be the observed empirical measure. If the system satisfies the exponential inequality then for all $t>0$ and for all $n\geq1$ we have that $$\mup\left( \kappa(\Epert,\mu)> \frac{t +B}{n^{1/4}}+\varepsilon \right)\leq e^{-\frac{t^{2}\sqrt{n}}{4D(1+\varepsilon^{2})}}.$$ If the system satisfies the polynomial inequality with moment $q\geq2$, then for all $t>0$ and for all $n\geq1$ we obtain $$\mup\left( \kappa(\Epert,\mu)> \frac{t+B}{n^{1/4}}+ \varepsilon \right)\leq \frac{D_{q}(1+\varepsilon)^{q}}{t^{q}}\frac{1}{n^{q/4}}.$$ Clearly $ \ee_{\mup}(\kappa(\Epert,\mu))\leq \ee_{\mup}(\kappa(\Epert,\E)) + \ee_{\mup}(\kappa(\E,\mu))$. A straightforward estimation yields $$\begin{aligned} \ee_{\mup}(\kappa(\Epert,\E)) \leq&\int \sup_{g\in\mathcal{L}}\left[ \frac{1}{n}\sum_{i=0}^{n-1} \Lip(g)\varepsilon\lVert \xi_{i}\rVert \right] \dd\mup \ \leq \ \varepsilon.\end{aligned}$$ We obviously have $\ee_{\mup}\left(\kappa(\E,\mu)\right)=\ee_{\mu_{n}}\left(\kappa(\E,\mu)\right)$. Using the exponential estimate of proposition \[Fluctua-Emp-Meas\] and lemma \[Bound-Expect\] we obtain, for any $t>0$, $$\mup\left( \kappa(\Epert,\mu)\geq t+\varepsilon +\frac{B}{n^{1/4}} \right)\leq \exp\left(\frac{-t^{2}n}{4D(1+\varepsilon^{2})} \right).$$ Rescaling adequately we get the result. For the polynomial case, one uses the polynomial estimate of proposition \[Fluctua-Emp-Meas\]. Kernel density estimator for one-dimensional maps ------------------------------------------------- In this section we consider the system $(X,T,\mu)$ where $X$ is a bounded subset of $\rr$. We assume the measure $\mu$ to be absolutely continuous with density $h$. For a given trajectory of a randomly chosen initial condition $x$ (according to $\mu$), the empirical density estimator is defined by, $$\h(x; s) :=\frac{1}{n\alpha_{n}}\sum_{j=0}^{n-1}\psi\left(\frac{s-T^{j}x}{\alpha_{n}}\right),$$ where $\alpha_{n}\to0$ and $n\alpha_{n}\to\infty$ as $n$ diverges. The kernel $\psi$ is a bounded and non-negative Lipschitz function with bounded support and it satisfies $\int\psi(s)\dd{s}=1$. We shall use the following hypothesis. \[hypo-density\] The probability density $h$ satisfies $$\int\lvert h(s) - h(s-\sigma)\rvert \dd s\leq C'\lvert \sigma \rvert^{\beta}$$ for some constants $C'>0$ and $\beta>0$ and for every $\sigma\in\rr$. This assumption is indeed valid for maps on the interval satisfying the axioms of Young towers with exponential tails (see [@ChCS Appendix C]). For convenience, we present the following result on the $L^{1}$ convergence of the density estimator ([@ChGo]). Let $\psi$ be a kernel defined as above. If the system $(X,T,\mu)$ satisfies the exponential concentration inequality and the hypothesis \[hypo-density\], then there exist a constant $C_{\psi}>0$ such that for any integer $n\geq1$ and every $t>C_{\psi}\left(\alpha_{n}^{\beta}+\frac{1}{\sqrt{n}\alpha_{n}^{2}}\right)$, we have $$\mu\left( \int\left\lvert \h(x;s)-h(s)\right\rvert \dd{s} >t\right)\leq e^{-\frac{n\alpha_{n}^{4}t^{2}}{4C\Lip(\psi)^{2}}}.$$ Under the same conditions above, if the system satisfies the polynomial concentration inequality for some $q\geq2$, then for any integer $n\geq1$ and every $t>C_{\psi}\left(\alpha_{n}^{\beta}+\frac{1}{\sqrt{n}\alpha_{n}^{2}}\right)$, we obtain, $$\mu\left( \int\left\lvert \h(x;s)-h(s)\right\rvert \dd{s} >t\right)\leq \frac{C_{q}}{t^{q}}\left(\frac{\Lip(\psi)}{\sqrt{n}\alpha_{n}^{2}} \right)^{q}.$$ The parameter $\beta$ is the same constant appearing in the hypothesis \[hypo-density\]. For the proof of this statement see [@ChGo] or Theorem 6.1 in [@ChCS]. ### Kernel density estimator for observed maps on the circle In order to avoid ‘leaking’ problems, now we assume $X=\mathbb{S}^{1}$. Given the observed sequence $\{y_{j}\}$, let us define the observed empirical density estimator by $$\hpert(y_{0},\ldots,y_{n-1};s) : = \frac{1}{n\alpha_{n}}\sum_{j=0}^{n-1}\psi\left(\frac{s-y_{j}}{\alpha_{n}}\right).$$ Our result is the following. If $(X,T,\mu)$ satisfies the hypothesis \[hypo-density\] and the exponential concentration inequality, then there exists a constant $C_{\psi}>0$ such that, for all $t>C_{\psi}\left(\alpha_{n}^{\beta} + \frac{1}{\sqrt{n}\alpha_{n}^{2}}\right)$ and for any integer $n\geq1$, $$\mup\left( \int\left\lvert \hpert(y_{0},\ldots,y_{n-1};s) - h(s)\right\rvert \dd{s}> t + \Lip(\psi)\frac{\varepsilon}{\alpha_{n}^{2}}\right) \leq \exp\left(-\frac{n\alpha_{n}^{4}t^{2}}{R(1+\varepsilon^{2})}\right),$$ where $R:=4D\Lip(\psi)^{2}$. If the system satisfies the hypothesis \[hypo-density\] and the polynomial concentration inequality, then for all $t>C_{\psi}\left(\alpha_{n}^{\beta} + \frac{1}{\sqrt{n}\alpha_{n}^{2}}\right)$ and for any integer $n\geq1$, we have $$\mup\left( \int\left\lvert \hpert(y_{0},\ldots,y_{n-1};s) - h(s)\right\rvert \dd{s}> t + \Lip(\psi)\frac{\varepsilon}{\alpha_{n}^{2}}\right) \leq D_{q}\left( \frac{(1+\varepsilon)\Lip(\psi)}{t\sqrt{n}\alpha_{n}^{2}}\right)^{q}.$$ The parameter $\beta$ is the same constant appearing as in the hypothesis \[hypo-density\]. Consider the following observable of $n$ variables, $$K(z_{0},\ldots,z_{n-1}) := \int\Big\lvert \frac{1}{n\alpha_{n}}\sum_{j=0}^{n-1}\psi\left( \frac{s-z_{j}}{\alpha_{n}}\right) - h(s)\Big\rvert \dd{s}.$$ It is straightforward to obtain that $\Lip_{l}(K) \leq\frac{\Lip(\psi)}{n\alpha_{n}^{2}}$, for every $l=0,\ldots, n-1$. Next, we need to give an upper bound for the expected value of the observable $K$, first $$\begin{aligned} \ee_{\mup}(K) \leq& \int \Big(\int \Big\lvert \frac{1}{n\alpha_{n}} \sum_{j=0}^{n-1}\left[\psi\Big(\frac{s-y_{j}}{\alpha_{n}}\Big)- \psi\left(\frac{s-x_{j}}{\alpha_{n}}\right)\right]\Big\rvert \dd{s}\Big)\dd\mup \\ & \hspace{1.3cm}+ \int\Big(\int \Big\lvert \frac{1}{n\alpha_{n}}\sum_{j=0}^{n-1}\psi\Big(\frac{s-x_{j}}{\alpha_{n}}\Big)-h(s)\Big\rvert \dd{s}\Big)\dd\mu_{n}.\end{aligned}$$ Subsequently we proceed on each part. For the first one we get $$\begin{split} \int \Big(\int \Big\lvert \frac{1}{n\alpha_{n}} \sum_{j=0}^{n-1}\left[\psi\Big(\frac{s-y_{j}}{\alpha_{n}}\Big)-\psi\Big(\frac{s-x_{j}}{\alpha_{n}}\Big)\right]\Big\rvert \dd{s}\Big)d\mup\\ \leq \int \Big(\frac{1}{n\alpha_{n}}\sum_{j=0}^{n-1}\frac{\Lip(\psi)\varepsilon}{\alpha_{n}}\Big) \dd\mup &\ \leq \ \Lip(\psi)\frac{\varepsilon}{\alpha^{2}_{n}}. \end{split}$$ For the second part, there exist some constant $C_{\psi}$ such that $$\int\Big(\int \Big\lvert \frac{1}{n\alpha_{n}}\sum_{j=0}^{n-1}\psi\Big(\frac{s-x_{j}}{\alpha_{n}}\Big)-h(s)\Big\rvert \dd{s}\Big)\dd\mu_{n} \leq C_{\psi}\left(\alpha_{n}^{\beta} + \frac{1}{\sqrt{n}\alpha_{n}^{2}}\right).$$ The proof of this statement is found in [@ChCS Section 6]. We finish the proof applying and , respectively. Correlation dimension --------------------- The correlation dimension $d_{c} = d_{c}(\mu)$ of the measure $\mu$ is defined by $$d_{c} = \lim_{r\searrow0}\frac{\log{\int\mu(B_{r}(x))\dd\mu(x)}}{\log{r}},$$ provided the limit exists. We denote by $\C(r)$ the spatial correlation integral which is defined by $$\C(r) = \int\mu(B_{r}(x))\dd\mu(x).$$ As empirical estimator of $\C(r)$ we choose the following function of $n$ variables $$K_{n,r}(x_{0},\ldots,x_{n-1}) := \frac{1}{n^{2}}\sum_{i\neq j}H(r-d(x_{i},x_{j})),$$ where $H$ is the Heaviside function. It has been proved (see e.g. [@Ser]) that $$\C(r) = \lim_{n\to\infty}K_{n,r}(x,\ldots, T^{n-1}x),$$ $\mu$-almost surely at the continuity points of $\C(r)$. Next, given a $\mu$-typical initial condition, let us consider the observed sequence $y_{0},\ldots,y_{n-1}$, and define the estimator of $\C(r)$ for observed systems, as follows $$\K(y_{0},\ldots,y_{n-1}) := \frac{1}{n^{2}}\sum_{i\neq j}H(r-d(y_{i},y_{j})).$$ Since $\K(y_{0},\ldots,y_{n-1})$ is not a Lipschitz function we cannot apply directly concentration inequalities. The usual trick is to replace $H$ by a Lipschitz continuous function $\phi$ and then define the new estimator $$\label{estim-pert-Corr} \K^{\phi}(y_{0},\ldots,y_{n-1}):=\frac{1}{n^{2}}\sum_{i\neq j}\phi\left(1-\frac{d(y_{i},y_{j})}{r}\right).$$ The result of this section is the following estimate on the variance of the estimator $\K^{\phi}$. Let $\phi$ be a Lipschitz continuous function. Consider the observed trajectory $y_{0},\ldots,y_{n-1}$ and the function $\K^{\phi}(y_{0},\ldots,y_{n-1})$ given by . If the system $(X,T,\mu)$ satisfies the polynomial concentration inequality with $q=2$, then for any integer $n\geq1$, $$\mathrm{Var}(\K^{\phi})\leq D_{2}\Lip(\phi)^{2}(1+\varepsilon)^{2}\frac{1}{r^{2}n},$$ where $\mathrm{Var}(Y) := \ee(Y^{2}) - \ee(Y)^{2}$ is the variance of $Y$. The proof follows the lines of section 4 in [@ChCS], and by applying the inequality with $q=2$ and noticing that $\Lip_{l}(\K^{\phi}) \leq \frac{\Lip(\phi)}{rn}$ for every $l=0,\ldots,n-1$.
--- abstract: 'Glitches are sudden spin-up events that punctuate the steady spin down of pulsars and are thought to be due to the presence of a superfluid component within neutron stars. The precise glitch mechanism and its trigger, however, remain unknown. The size of glitches is a key diagnostic for models of the underlying physics. While the largest glitches have long been taken into account by theoretical models, it has always been assumed that the minimum size lay below the detectability limit of the measurements. In this paper we define general glitch detectability limits and use them on 29 years of daily observations of the Crab pulsar, carried out at Jodrell Bank Observatory. We find that all glitches lie well above the detectability limits and by using an automated method to search for small events we are able to uncover the full glitch size distribution, with no biases. Contrary to the prediction of most models, the distribution presents a rapid decrease of the number of glitches below $\sim0.05\,\mu$Hz. This substantial minimum size indicates that a glitch must involve the motion of at least several billion superfluid vortices and provides an extra observable which can greatly help the identification of the trigger mechanism. Our study also shows that glitches are clearly separated from all the other rotation irregularities. This supports the idea that the origin of glitches is different to that of timing noise, which comprises the unmodelled random fluctuations in the rotation rates of pulsars.' author: - \| C.M. Espinoza,$^{1, 2}$[^1] D. Antonopoulou,$^{3}$ B.W. Stappers,$^{1}$ A. Watts$^{3}$ and A.G. Lyne$^{1}$\ $^{1}$Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK.\ $^{2}$Instituto de Astrofísica, Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile.\ $^{3}$Astronomical Institute Anton Pannekoek, University of Amsterdam, Postbus 94249, 1090GE Amsterdam, The Netherlands.\ bibliography: - 'journals.bib' - 'modrefs.bib' - 'psrrefs.bib' - 'crossrefs.bib' - 'new1.bib' title: Neutron star glitches have a substantial minimum size --- \[firstpage\] pulsars: general – pulsars: individual: PSR B0531+21 – stars: neutron Introduction ============ Neutron stars are the highly-magnetised and rapidly-rotating remnants of the collapse of the cores of once more-massive stars. Having masses of approximately $1.4\,\rm{M}_\odot$ and radii of about $12$km, the high densities of neutron stars indicate a structure of a crystalline-like crust and a superfluid interior [@bpp69; @NS1]. Their large and steady moments of inertia mean that they have extremely stable rotational frequencies, which slowly decrease as energy is lost through electromagnetic radiation and acceleration of particles in their magnetospheres. However, this regular spin-down is occasionally interrupted by sudden spin-up events, known as glitches [@rm69; @elsk11]. The exact mechanism responsible for glitches is not fully understood but it is thought to involve a sudden transfer of angular momentum from a more rapidly rotating superfluid component to the rest of the star [@ai75]. This component resides in regions of the interior where neutron vortices, which carry the angular momentum of the superfluid, are impeded in moving by pinning on crustal nuclei or on superconducting vortices in the core (or on both). Since a superfluid in such conditions cannot slow down by outwards motion and expulsion of vortices, the superfluid component will retain a higher rotational frequency as the rest of the star slows down. A glitch occurs when vortices are suddenly unpinned and free to move outwards, allowing for a rapid exchange of angular momentum and the observed spin-up of the crust. Catastrophic unpinning of vortices is expected once the velocity lag between the two components exceeds a maximum threshold, above which the pinning force can no longer sustain the hydrodynamic lift force exerted on the pinned vortices by the ambient superfluid. It has also been shown [@gl09; @agh13] that, beyond some critical lag, a two-stream instability might develop and trigger the unpinning. If in such events the lag is completely relaxed (or partially relaxed by a fixed amount) then the interglitch time interval corresponds to the time it takes for the system to reach the critical threshold again, driven by the nearly constant external torque. Models relying on such a build-up and depletion of the superfluid angular momentum reservoir have been successfully used to explain the regular, similar glitches of some young pulsars [@accp93; @pizz11; @hps12]. However this simple picture cannot account for the wide range of glitch sizes and waiting times between glitches seen in most pulsars. Glitch sizes in rotational frequency can range over four orders of magnitude in individual pulsars and appear to follow a power-law distribution [@mpw08]. This favours scale-invariant models of the dynamics of individual vortices in the presence of a pinning potential, such as the vortex avalanche model [@wm08] and the coherent noise model [@mw09]. Alternative models involve non-superfluid mechanisms that can act as unpinning triggers before the critical lag is reached, such as crustquakes [@rud69; @bp71] or heating episodes [@le96]. The crustquake-induced glitch model has been particularly favoured for the Crab pulsar as it may explain the persistent changes in slow-down rate observed after some of its glitches [@girp77; @accp94] and could possibly lead to a power-law distribution of event sizes, similar to earthquakes. A second type of irregularity is often seen in the rotational behaviour of pulsars, namely timing noise. Thought to be partially caused by torque variations driven by two or more magnetospheric states [@lhk+10], it manifests as a continuous and erratic wandering of the rotation rate around the predictions of a simple slow-down model. While glitches are rapid and sporadic events in rotation rate, timing noise appears as a slow and continuous process. Owing to observational limitations such as infrequent and irregular sampling and the presence of timing noise, the detection of glitches is an uncertain process. Moreover, the signature of timing noise in the data can be confused with glitches, so that the lower end of a glitch-size distribution is possibly contaminated by spurious detections. Knowledge of this distribution is essential for any glitch theory. The largest glitches are easily detected and can be used to constrain the minimum superfluid moment of inertia that can act as an angular momentum reservoir [@aghe12; @chamel13]. The biases involved and the question of whether there is a minimum glitch size have not been addressed; so far, the smallest possible glitch has been assumed to lie below our detection limits. In this paper we study the glitch detection capabilities of the current detection methods and define limits depending upon the intrinsic pulsar rotational stability, observing cadence and sensitivity. We apply these definitions to an extensive set of observations of the Crab pulsar and, by using an automated glitch detector, uncover the full glitch size distribution and show that there is a minimum glitch size. Limits on glitch detection ========================== To assess the level of completeness of the existing glitch samples, we quantify simple observational limits on glitch detection, applicable for a given pulsar and observing setup. The first step towards this is establishing a working definition of what constitutes a glitch. Traditionally, glitches are identified by visual inspection of the pulsar’s timing residuals, which are defined as the phase differences between measurements and the predictions of a model for the rotation. To put this on a more formal footing, we define a glitch as an event characterised by a sudden, discrete positive change in rotational frequency ($\Delta\nu$) and a discrete negative or null change in frequency spin-down rate ($\Delta\dot{\nu}$). These two sudden changes together make glitches distinguishable from timing noise [@elsk11; @lhk+10]. The timing residuals will be flat if the model describes the rotation of the pulsar well. For such a model, the timing residuals after a glitch at $t=t_g$ will follow a quadratic signature given by $$\label{gresids} \phi_g=-\Delta\nu(t-t_g)-\Delta\dot{\nu}\frac{(t-t_g)^2}{2} \quad ; \quad(t>t_g)\,.$$ The frequency change $\Delta\nu>0$ produces a linear drift of the post-glitch residuals towards negative values, with the slope being the magnitude of the frequency step. The effect of a change $\Delta\dot{\nu}<0$ is a parabolic signature which lifts the residuals towards positive values. Therefore a glitch with a large, negative change in spin-down rate will produce positive residuals rising quadratically soon after the glitch (Fig. \[figS1\]). ![Example of a glitch signature in the timing residuals. The residuals are with respect to a model which describes well the rotation before the glitch ($t<0$ on the plot). This is a simulated glitch in the Crab pulsar’s rotation, with $\Delta\nu=0.01$$\mu$Hz and $\Delta\dot{\nu}=-3.0\times 10^{-15}$Hzs$^{-1}$.[]{data-label="figS1"}](glitexamp.pdf){width="84mm"} Based upon these facts, we can define simple limits that describe our ability to detect glitches in the timing residuals. If the observing cadence is not very frequent, it is possible that no observations occur before the post-glitch residuals rise above the extrapolation of the line defined by the pre-glitch ones (Fig. \[figS1\]). This effect will primarily mask glitches with small $\Delta\nu$ and large $\|\Delta\dot{\nu}\|$. Requiring at least one observation before the rise of the residuals defines a minimum $\Delta\nu$ that can be detected (Eq. \[limits\]). If $\Delta\nu$ became smaller, the dip would become shallower and in the case that it is undetectably small, the event is unlikely to be recognised as a glitch and might appear as timing noise. To ensure detection, the maximum negative departure of the residuals ought to be larger than both the root mean square (RMS) of the timing residuals prior to the glitch and the typical error of the TOAs. Therefore, a detectable glitch is a rapid event in which the effects of $\Delta\nu$ are recognisable over the effects of $\Delta\dot{\nu}$ and the limiting detectable value of $\Delta\nu$ depends on the observation cadence (one observation every $\Delta T$ days) and the largest of either the sensitivity of the observations or the typical dispersion of the timing residuals in rotational phase, $\sigma_\phi$, as $$\label{limits} \Delta\nu_\textrm{\scriptsize \,lim}=\rm{max} \left\{ \begin{array}{l} \Delta T\|\Delta\dot{\nu}\|/2 \\ \\ \sqrt{2\,\sigma_\phi\|\Delta\dot{\nu}\|} \end{array} \right . \quad \quad (\textrm{for}\quad \Delta\dot{\nu}<0).$$ For simplicity and because of our particular focus on small events, any exponential recovery of the frequency, often observed after glitches, is not considered here. Nonetheless, we constructed several detectability curves, with decaying components and timescales similar to those observed in the Crab pulsar, and verified that our conclusions are not altered if exponential recoveries are present. These limits are consistent with the glitch samples of several pulsars, hence we believe they offer a realistic way to assess glitch detectability as it is commonly carried out. The Crab pulsar glitches {#Crab} ======================== The Crab pulsar (PSR B0531+21; PSR J0534+2200) is the central source of the Crab Nebula and a young neutron star widely studied since it was first observed in 1968. The rotation of the Crab pulsar has been monitored almost every day for the last 29 years with the 42-ft radio telescope operating at $610$MHz at the Jodrell Bank Observatory (JBO) in the UK [@lps88; @lps93]. This offers an ideal dataset to test the completeness of the glitch sample because of its rapid cadence, good sensitivity and low dispersion of the timing residuals. Observations ------------ The product of each observation was the time of arrival (TOA) of one pulse at the observatory, corrected to the solar system barycenter. The dataset comprises $8862$ TOAs starting in January 1984. There is one TOA per day in general and two TOAs per day during some periods of time. In addition, towards the beginning of the dataset, there are some isolated cases in which groups of TOAs are separated by up to $5$ days. Finally, there are also a few gaps with no observations, generally no larger than $\sim20$days, when the telescope or observing hardware were unavailable due to maintenance. The TOAs generally have errors of less than $0.001$ rotation, with more than $75$% having uncertainties less than $0.0004$ rotation. For groups of $20$ TOAs, which cover $20$ days on average, the timing residuals with respect to a simple slow-down model with two frequency derivatives typically give a dispersion similar to the TOA uncertainties (hence $\sigma_\phi\sim 0.0004$ rotations). Detection limits and the sample of detected glitches ---------------------------------------------------- To study the glitch size distribution of the Crab pulsar we need a complete list of glitches for the time interval defined by the 42-ft dataset, and their main parameters $(\Delta\nu,\Delta\dot{\nu})$. As described above, we classify events as glitches based on the assumption that a glitch is a sudden, unresolved change in spin frequency, implying clearly defined features in the timing residuals. We use the events included in the JBO online glitch catalogue[^2], which correspond to all the events published by [@elsk11] plus one new glitch that occurred on MJD$55875.5$ [@ejb+11]. The event on MJD$\sim50489$, originally reported by [@wbl01], was rejected because of its anomalous characteristics, already described by them. No other glitches have been reported for this time-span by other authors [e.g. @wbl01; @wwty12] and we confirmed this by visually inspecting the timing residuals for all our dataset. Our final list contains 20 glitches, with parameters covering the ranges $0.05\leq\left(\Delta\nu/\mu\textrm{Hz}\right)\leq6.37$ and $45\leq\left(\|\Delta\dot{\nu}\|/10^{-15}\textrm{Hz\,s}^{-1} \right)\leq2302$. Here we use the glitch sizes reported by @elsk11 [@ejb+11]. We note the clear presence of four other glitches prior to the start of the 42-ft observations. However, the available data for that period is highly inhomogeneous and contains large gaps with no observations, making it difficult to define single detectability limits and complicating the use of the glitch detector (see below). Hence, in order to work with a set of glitches that we know is statistically complete, we have not included them in our sample. Using $\Delta T=1$day and $\sigma_\phi=0.0004$ in Eq. \[limits\] we find that all glitches (including the four early ones) show a clear separation from the detectability limits (Fig. \[fig1\]). Thus, at least for intermediate and large glitches, we are uncovering the true $\Delta\nu$–$\Delta\dot{\nu}$ distribution, with no biases. The glitch detector {#detector} =================== To confirm that we have identified every glitch in the data, especially small ones which may be missed by standard techniques, we developed an automated glitch detector to find and measure every timing signature that might be regarded as a glitch. The detector assumes that a glitch occurred after every observation and attempts to measure its size, producing an output of glitch candidates (GCs) whenever $\Delta\nu>0$ and $\Delta\dot{\nu}\leq0$ are detected. Method ------ The detector’s technique is based on the fact that timing residuals, in the presence of timing noise or glitches, quickly depart from the best fit model of previous data, resulting in deviations from a mean of zero as newer observations are included and the model is not updated. Below we describe the method step by step, optimised for the JBO dataset for the Crab pulsar, described above. Different parameters should be used for different datasets. A fit for $\nu$, $\dot{\nu}$ and $\ddot{\nu}$ is performed over a set of $20$ TOAs using the timing software [psrtime]{} and [tempo2]{} [@hem06], following standard techniques. To test for a glitch occurring after the last TOA in a set, the timing residuals of the following $10$ TOAs, relative to that model, are fitted with a quadratic function of the form of Eq. (\[gresids\]) and separately with just the linear term in that equation. The latter is to test the case $\Delta\dot{\nu}=0$. The fit with the smallest reduced $\chi^2$ is selected. When the quadratic fit is selected, an event is characterised as a GC only if the reduced $\chi^2$ is less than $15$, the quadratic part of the fit is negative ($\Delta\dot{\nu}<0$) and if the minimum of the fitted curve is at least $2.5$ times the dispersion of the timing residuals of the $20$ TOAs below zero. This last condition ensures a positive $\Delta\nu$ and a solid detection of its magnitude. If the linear fit is selected, a new GC is created only if the reduced $\chi^2$ is less than $15$ and the slope of the fit is negative, indicating $\Delta\nu>0$. In this case the GC has a null or undetectable $\Delta\dot{\nu}$. The conclusions of our analysis are not dependent on the choice of the maximum allowed reduced $\chi^2$ threshold[^3]. The chosen value of $15$ is high enough to avoid missing signatures that one might regard as a glitch. The next step is to move the analysis forward by one TOA to define a new set of $20$ TOAs and test for a glitch occurring after this new TOA. By doing this over the whole dataset, the dataset is explored for glitches after every single observation (with the exception of the first $19$ TOAs and the last 10 TOAs). This method, however, causes some events to be detected multiple times. This happens because the effects of $\Delta\nu$ and $\Delta\dot{\nu}$ may be detectable not only in the set of TOAs starting immediately after the event but also in some of the neighbouring trials. Close inspection of the results shows that detections typically cluster in groups of $2$–$5$ trials, separated by no more than 2 days, and that clusters are typically $20$ to $30$ days apart. To remove the repeated detections and produce a final list of GCs we select from each cluster of candidates the detection with the largest $\Delta\nu$ value. This is a conservative choice which makes the final list of GCs a representation of the maximum possible activity present in the data. Also, this choice follows the experience gained from the detection of previously known glitches (section \[output\]). Results ======= We ran the detector over the 42-ft dataset, using the data from January 1984 to February 2013. The detector found all but one of the known glitches in this time-span as well as a large number of GCs. The output of the glitch detector {#output} --------------------------------- ![Previously known glitches (diamonds, from @elsk11 [@ejb+11]), glitch candidates (GCs) and anti-glitch candidates (AGCs). The top panel shows their distribution in the $\|\Delta\nu\|$–$\|\Delta\dot{\nu}\|$ plane. The straight line with the smallest slope represents the detection limit expected (Eq. \[limits\]) for a cadence of $\Delta T=1$day. The one with the largest slope represents the detection limit expected for residuals of $\sigma_\phi=0.0004$. Our observations are not sensitive to glitches in the shaded areas above these lines. The middle panel shows the $\|\Delta\nu\|$ values of those candidates with undetectable $\|\Delta\dot{\nu}\|$. The lower panel shows histograms for the $\|\Delta\nu\|$ values of the known glitches (filled grey), GCs (thick black) and AGCs (thin black). The inset shows a zoom in the region $\|\Delta\nu\|>0.02\,\mu$Hz.[]{data-label="fig1"}](df0df1.pdf){width="8.5cm"} ![image](mjddf0.pdf){width="17.7cm"} The only previously known glitch that was not found by the detector occurred on MJD$\sim52146.8$, only $63$ days after the previous glitch. It was not labelled as a possible glitch because none of the fits, neither the quadratic nor the linear, gave a reduced $\chi^2$ less than $15$, one of the conditions to create a GC. The smallest reduced $\chi^2$ among the fits around this glitch was $18$. We attribute these poor fits to the influence of the recovery from the previous glitch. The sizes $\Delta\nu$ and $\Delta\dot{\nu}$ that the detector measured for the known glitches are in good agreement with the values published by [@elsk11]. Nevertheless, some differences can be found among the $\Delta\nu$ measurements. As discussed above, because of the way the detector works, the effects of every glitch were detected in more than one set of TOAs. The $\Delta\nu$ value coming from the set of TOAs offering the best fit (smallest reduced $\chi^2$) is always smaller than the published value, which is obtained by standard timing techniques. In addition, this set of TOAs is normally the one starting one to three TOAs after the glitch epoch. On the other hand, the glitch sizes obtained when testing at the correct glitch epoch are typically the largest and the most similar to the published values, though the fits have larger reduced $\chi^2$ values. These effects are likely caused by unmodelled rapid exponential recoveries and were taken into account when selecting one candidate from a group of several candidates in the overall search. The uncertainties of the GC sizes are the square root of the variances of the parameters, given by the Levenberg-Marquardt algorithm used to fit the data, multiplied by the square root of the reduced $\chi^2$ of the fit. We reviewed the output of the glitch detector with the aim of producing a clean list of GCs. First, we removed from the original list all those GCs related to known glitches. Then we kept only one candidate (the one having the largest $\Delta\nu$) per event, as mentioned in the description of the method (section \[method\]). Next, we visually inspected the timing residuals for all GCs having $\Delta\nu\geq0.02$$\mu$Hz and eliminated three which involved large data gaps or with timing residuals clearly contaminated by glitch recoveries. We also examined the possibility that some GCs in this $\Delta\nu$ range could be caused by rapid changes in the electron density towards the Crab pulsar [@lps93], which strongly affects the travel time of the pulsar emission at these low frequencies, introducing signatures in the data which can mimic a glitch. To do so, we used observations taken at higher frequencies (mostly at $1400$MHz, with the Lovell telescope) and removed a further three GCs that were clearly caused by this effect. However, the cadence of the Lovell observations is not as rapid as that of the 42-ft observations and we were unable to confirm some other possible cases of such non-achromatic events. We inspected the timing residuals of the largest remaining GCs ($\|\Delta\nu\|\geq0.02\,\mu$Hz) and found their signatures to be indistinguishable from timing noise, though we acknowledge that discrimination between small glitches and timing noise is difficult. Nonetheless, in many cases no sharp transitions, typical of the known glitches, are observed at the GC epochs and the residuals are consistent with a smooth connection with the pre-GC-epoch residuals. Our final list contains $381$ GCs. They are homogeneously distributed over the entire time-span and are clustered as a population in $\Delta\nu$–$\|\Delta\dot{\nu}\|$ space (Figs. \[fig1\], \[fig2\]). The vast majority of them exhibit $\Delta\nu$ steps that are smaller than all previously detected glitches, leaving a gap between the $\Delta\nu$ distributions of real glitches and GCs which would be hard to populate with undetected events. Search for anti-glitches ------------------------ Given the distinct properties of the GC population, it is possible that the glitch-like signatures found by the detector are a component of the Crab pulsar’s timing noise. To test this idea and explore the noise nature of these irregularities, we performed a search for events with the opposite signature to a glitch, i.e. anti-glitch candidates (AGCs) with $\Delta\nu<0$ and $\Delta\dot{\nu}\geq0$, which are subject to the equivalent detection constraints as the normal glitches. After removing repeated detections and $10$ events caused by glitch recoveries, gaps with no data and non-achromatic events (see above), we obtain $383$ AGCs. They are also separate from the glitch population and show very similar characteristics to the GCs (Figs. \[fig1\], \[fig2\]). Discussion ========== GCs and AGCs: glitches or timing noise? --------------------------------------- Using the Kolmogorov–Smirnov (K–S) test we can compare the $\|\Delta\nu\|$ distributions for GCs and AGCs, which are found to be statistically consistent with coming from the same parental distribution (with a K–S statistic of $D=0.037$ and $p_{KS}(D)=0.96$, thus a probability of only $\sim4\%$ for a false null hypothesis). The $\|\Delta\nu\|$ distributions for GCs and AGCs can be well described by lognormal distributions, with probability density function (PDF) of the form $$\label{LG} p(\|\Delta\nu\|)=\frac{1}{\sqrt{2\pi}\sigma(\|\Delta\nu\|-\theta)}\exp{\left[-\left(\frac{\ln(\|\Delta\nu\|-\theta)-\mu}{4\sigma}\right)^2\right]}$$ for $\|\Delta\nu\|>\theta$, which gives $p_{KS}=0.85$ and $0.92$ respectively. However, this result is only indicative since the lower ends of these distributions are not well probed by the observations (Figs. \[fig1\], \[fig2\]). We also compared the $\|\Delta\nu\|$–$\|\Delta\dot{\nu}\|$ distributions of GCs and AGCs using a 2-dimensional K–S test [@ptvf92]. The test gives $D_\mathrm{2D}=0.084$, implying a probability of $\sim40\%$ that they come from the same distribution. This relatively low probability is likely to be produced by differences in the $\|\Delta\dot{\nu}\|$ distributions between GCs and AGCs, since a K–S test over these two gives $p_{KS}=0.55$, considerably smaller than the one for $\|\Delta\nu\|$. Neither a power-law nor a lognormal distribution can describe well the joint $\Delta\nu$ distribution of the $20$ glitches plus all the GCs, with $p_{KS}<10^{-4}$. A power-law with a lower cut-off at $\Delta\nu\sim0.01\,\mu$Hz, to account for the incompleteness of the sample at small sizes, gave a similarly poor fit. Although it is possible that some of the GCs correspond to real glitches, we interpret all the above results as confirmation that the GCs and AGCs are generated by a symmetric noise process and that no new glitches have been found. This timing noise component produces a continuous departure from a simple slow-down trend with variations that can be characterised by changes of $\|\Delta\nu\|\leq 0.03\,\mu$Hz and $\|\Delta \dot{\nu}\|\leq 200\times 10^{-15}$Hzs$^{-1}$. The glitch size distribution ---------------------------- Having established that the 20 glitches form the complete sample of glitches the Crab pulsar has had in the last 29 years, we can address their statistical properties. To determine the best-fit exponent $\alpha$ for a power-law PDF of the form $$\label{PL} p(\Delta\nu)=C{\Delta\nu}^{-\alpha} \quad ,$$ with $\Delta\nu_{\rm{min}}\leq\Delta\nu\leq\Delta\nu_{\rm{max}}$ and $C=(1-\alpha)(\Delta\nu_{\rm{max}}^{1-\alpha}-\Delta\nu_{\rm{min}}^{1-\alpha})^{-1}$, we use the maximum-likelihood estimator method. Setting $\Delta\nu_{\rm{min}}=0.05\,\mu$Hz and $\Delta\nu_{\rm{max}}=6.37\,\mu$Hz, the values for the smallest and largest glitches observed respectively, we obtain $\alpha=1.36\,(+0.15,-0.14)$ (Fig. \[figS2\]). The value of the exponent does not depend strongly on the choice of limits, as long as these are a few times smaller or larger than the observed ones. To assess the goodness of the fit, we calculate the K–S statistic, $D=0.1$, and its probability value $p_{KS}(D)=0.9$, which corresponds to a $10\%$ probability that our null hypothesis (that the data follow the PDF described by Eq. (\[PL\])) is false. Thus our results confirm that the Crab glitch $\Delta\nu$ distribution is consistent with a power-law, a description motivated by theoretical models. If this power-law continued below $\Delta\nu_{\rm{min}}$, we would expect to have detected more than 10 glitches with $0.02\,\mu$Hz $<\Delta\nu<\Delta\nu_{\rm{min}}$ in the searched data, and the gap between glitches and GCs (in Figs. \[fig1\] and \[fig2\]) should have been populated. Thus we observe a rapid fall-off of the power-law for $\Delta\nu<\Delta\nu_{\rm{min}}$. However, the small sample size makes it impossible to exclude other distributions. For example, the same K–S probability is obtained for a lognormal distribution (Eq. \[LG\]) with parameters $\mu=-1.79$, $\sigma=1.9$ and $\theta=0.049\,\mu$Hz, whose probability density function also quickly vanishes for $\Delta\nu<\Delta\nu_{\rm{min}}$. The same conclusions hold if the four glitches from before the start of this dataset are included in the sample. ![The cumulative distribution function of the observed glitch sizes, $\rm{s}$, and the corresponding power-law fit (solid line) given by Eq. \[PL\], with $0.05\leq \Delta\nu\,(\mu$Hz$) \leq 6.37$ and $\alpha=1.36$. The dashed line corresponds to a lognormal fit (Eq. \[LG\]) with $\mu=-1.79$, $\sigma=1.9$ and $\theta=0.049\,\mu$Hz.[]{data-label="figS2"}](cdf.pdf){width="8.4cm"} Further confirmation of the rare occurrence of small glitches comes from the study of the $\|\Delta\nu\|$–$\|\Delta\dot{\nu}\|$ distribution. Having shown that the latter is not affected by observational biases, the correlation between $\|\Delta\nu\|$ and $\|\Delta\dot{\nu}\|$ (apparent in Fig. \[fig1\]) is confirmed to be a robust feature. While $\Delta\nu$ measurements are very accurate, the acquired values of $\Delta\dot{\nu}$ are less certain and depend upon the method used to determine them, leading sometimes to large discrepancies. For this work we consistently calculated the glitch parameters for all 20 glitches, using the technique described in [@elsk11]. Using those measurements, the Spearman’s rank correlation coefficient between $\Delta\nu$ and $\|\Delta\dot{\nu}\|$ is $\rm{rs}=0.776$ with $p(\rm{rs})=6\times10^{-5}$, which indicates a strong correlation. We note that the correlation becomes stronger if the four early glitches are included. Given this relationship, any additional glitches would occupy a region of the $\|\Delta\nu\|$–$\|\Delta\dot{\nu}\|$ space well probed by the observations. Therefore, we observe a rapid decrease of the probability for glitch sizes below $\Delta\nu_{\rm{min}}$, which cannot be ascribed to incompleteness of the sample and hence indicates the existence of a minimum glitch size for the Crab pulsar. Implications for theoretical models =================================== Such a limit for the smallest glitch size is challenging to our current understanding of glitches and has the potential to constrain the proposed mechanisms. Some simple considerations can be used to get a rough order of magnitude estimate for the number of neutron superfluid vortices that need to unpin to produce the smallest Crab glitch. Each superfluid vortex carries a quantum of circulation $\kappa=h/2m_n\sim2\times10^{-3}$cm$^2$s$^{-1}$. Neglecting differential rotation of the superfluid (and entrainment), its total circulation at distance $r$ from the rotational axis will be $\Gamma=\oint v_s\cdot dl=N_v(r)\kappa=2\Omega_s(r) A$, where $N_v(r)$ is the number of vortices in the enclosed area $A$ and $\Omega_s$ is the superfluid angular velocity. Using $r=10^6$cm, the total number of vortices for the Crab pulsar is of the order $N_v\sim6\times10^{17}$. Conservation of the total angular momentum implies that the angular velocity change of the superfluid, $\delta\Omega_s$, relates to the observed glitch size $\Delta\nu$ by $\delta\Omega_s=2\pi\Delta\nu I_c/I_s$, where $I_c$ is the moment of inertia of the coupled component and $I_s$ is the superfluid moment of inertia that participates in the glitch. Using a typical value of $I_c/I_s\sim10^2$ and $\Delta\nu=\Delta\nu_{\rm{min}}=0.05\,\mu$Hz, the total number of vortices must be reduced by $\delta N_v\sim10^{11}$. The actual change in the superfluid angular momentum $L_s$ depends on the number of vortices that unpinned, the location and size of the region where this happened and the distance travelled by those vortices before they repin. For a more rigorous estimate, the change in $L_s$ can be approximated by $\Delta L_s\sim \hat{\rho}\kappa\xi R^3\delta N_v$, where $\hat\rho$ is the average density of the region involved, $R$ is the stellar radius and $\xi$ is the fraction of $R$ that unpinned vortices travel [@wm13]. For a typical value of $I_c\sim10^{45}$gcm$^2$ for the moment of inertia of the coupled component, the smallest glitch observed translates to an angular momentum change of $\Delta L_c=2\pi I_c \Delta\nu_{\rm{min}}=3\times10^{38}$gcm$^2$s$^{-1}$. Conservation of angular momentum leads to $\xi\delta N_v\simeq1.5\times10^9$ if one assumes typical values for the base of the crust, like $R=10$km and $\rho=10^{14}$gcm$^{-3}$. Vortices are expected to repin after encountering a few available pinning sites, however as a conservative order of magnitude estimate we assume they cover a distance comparable to the thickness of the crust ($1.5$km) and take $\xi\leq0.15$, which means that at least $10^{10}$ vortices must unpin in a glitch with $\Delta\nu=\Delta\nu_{\rm{min}}$. Therefore the observed minimum glitch size, which is well above that expected for single-vortex unpinning events, implies the existence of a smaller characteristic length-scale which sets the lower cut-off for the range of the scale-invariant behaviour. The vortex avalanche model is based on the notion of self-organised criticality [SOC, @btw87], applications of which can be found for example in earthquake dynamics [@Her02] or superconducting flux-tube avalanches [@wwam06]. SOC occurs without the need of fine tuning of parameters, in several dynamical systems consisting of many interacting elements (the superfluid vortices in the case of a neutron star) which, under the act of an external slow driving force (the spin-down of the star), self-organise in a critical stationary state with no characteristic spatiotemporal scale. A small perturbation in such systems can trigger an avalanche of any size. Thus in the glitch avalanche model of @wm08 vortex density is assumed to be greatly inhomogeneous and many metastable reservoirs of pinned vortices are formed, which relax independently giving rise to the observed spin-ups. Since such a system has no preferred scale the resulting glitch magnitudes follow a power-law distribution. This behaviour should however continue down to events involving the unpinning of only a few vortices, which is orders of magnitude below the observed cut-off. The coherent noise model [@ns96] is a different, non critical mechanism which produces scale-free dynamics, even in the absence of interaction between the system’s elements. In such systems a global stress is imposed to all elements coherently, to which they respond if it exceeds their individual unpinning threshold, giving rise to avalanches of various sizes. Both threshold levels (for each element) and stress strength are randomly chosen from respective probability distribution functions. The elements with thresholds smaller than the applied stress will participate in an avalanche and then be re-assigned new threshold values. New thresholds must always be assigned to a few elements, even when no avalanche is triggered, otherwise such a system will stagnate. A possible mechanism for this process in superfluids is the thermally activated unpinning of vortices [@mw09], while the global Magnus force acts as the coherent stress. The model predicts a minimum for the glitch magnitude, which represents the thermal creep only events, present even if all thresholds lie above the applied stress strength. But it also predicts an excess (with respect to the resulting power-law) of such small glitches, in contradiction to what we observe for the Crab pulsar. The lack of this overabundance of small events requires a broad distribution for the pinning potentials. @mw09 studied the top-hat distribution and applied their model to the Crab pulsar. They found that the half-width of the distribution should be comparable to the mean pinning strength. Even when such a broad distribution of pinning energies is introduced, independent unpinning of vortices as a random Poisson process of variable rate proves insufficient to produce scale-invariant glitches [@wm13], indicating that the interaction between vortices and collective unpinning (a domino-like process) must be taken into account. The most prominent mechanism for collective unpinning is the proximity effect, in which a moving vortex triggers the unpinning of its neighbours. However such a mechanism requires extreme fine tuning, since power-law size distributions occur only if this effect is neither too weak (where thermal creep dominates) nor too strong (which always leads to large, system-spanning, avalanches) [@wm13]. Another process which could lead to scale-invariant glitches are crustquakes [@Morl96]. Stresses develop in the solid crust of a neutron star because of the change in its equilibrium oblateness as the spin decreases, but also due to the interaction of the crustal lattice with the magnetic field and superfluid vortices in the interior. If the crust cannot readjust plastically it will do so abruptly when the breaking strain $\epsilon_{\rm{cr}}=\sigma_{\rm{cr}}/\mu$ is exceeded (where $\sigma_{\rm{cr}}$ is the critical stress and $\mu$ the mean modulus). This will result in both a spin-up (due to the moment of inertia decrease) and in a reaction of the superfluid [@accp96; @rzc98], which is evident in the post-glitch relaxation. The maximum fractional moment of inertia change associated with the $\Delta\nu_{\rm{min}}$ glitch is $\|\Delta I\|/I\leq 10^{-9}$; we note here that the glitch size can be significantly boosted by the crustquake induced unpinning of vortices [@ll99; @Eich10]. Elastic stress on the crust due to change of the equilibrium oblateness builds up because of the almost-constant secular $\dot{\nu}$. Therefore the critical stress $\sigma_{\rm{cr}}$ will be reached in regular time intervals if all stress is relieved in each crustquake, and the total energy released will be $\Delta E_{\rm{el}}\propto\epsilon^2_{\rm{cr}}$. If the stress is only partially relaxed then the energy released will depend on the stress drop $\Delta\sigma$, and the time interval to the next crustquake will depend on the size of the preceding one. The latter correlation is observed for the glitches in PSR J0537-6910, which have been interpreted as crustquakes [@mmw+06]. For the Crab pulsar however, the lack of any such trends in our glitch sample indicates a more complicated picture. Conclusions {#fin} =========== We have quantified our current glitch detection capabilities and, after a meticulous search for small glitches, we have shown that in the case of the Crab pulsar all glitches in this dataset have already been detected. The full glitch size distribution exhibits an under-abundance of small glitches and implies a lower cut-off at $\Delta\nu\sim0.05\,\mu$Hz. The existence of such a minimum glitch size implies a threshold-dominated process as their trigger, which still needs to be identified. Besides the occasional glitches, we have detected a continuous presence of timing noise having a well defined maximum amplitude, which can be described by step changes $\|\Delta\nu\|\leq 0.03\,\mu$Hz and $\|\Delta \dot{\nu}\|\leq 200\times 10^{-15}$Hzs$^{-1}$. The distinct properties of this noise component compared to the glitches imply that timing noise cannot be attributed solely to unresolved small glitches produced by the exact same mechanism. Acknowledgments {#acknowledgments .unnumbered} =============== Pulsar research at JBCA is supported by a Consolidated Grant from the UK Science and Technology Facilities Council (STFC). C.M.E. acknowledges the support from STFC and FONDECYT (postdoctorado 3130512). D.A. and A.L.W. acknowledge support from an NWO Vidi Grant (PI Watts). \[lastpage\] [^1]: E-mail: [email protected] [^2]: <http://www.jb.man.ac.uk/pulsar/glitches.html> [^3]: Changing this threshold to 20, for example, we obtained 12 new GCs, homogeneously distributed across the frequency range of GCs. There are no effects on the statistical results described in later sections.
--- abstract: 'We investigate the value of the near-infrared imaging from upcoming surveys for constraining the ellipticities of galaxies. We select galaxies between $0.5 \le z < 3$ that are brighter than expected [Euclid]{} sensitivity limits from the GOODS-S and N fields in CANDELS. The co-added CANDELS/HST V+I and J+H images are degraded in resolution and sensitivity to simulate [Euclid]{}-quality optical and near-infrared (NIR) images. We then run GALFIT on these simulated images and find that optical and NIR provide similar performance in measuring galaxy ellipticities at redshifts $0.5 \le z < 3$. At $z>1.0$, the NIR-selected source density is higher by a factor of 1.4 and therefore the standard error in NIR-derived ellipticities is about 30% smaller, implying a more precise ellipticity measurement. The good performance of the NIR is mainly because galaxies have an intrinsically smoother light distribution in the NIR bands than in the optical, the latter tracing the clumpy star-forming regions. In addition, the NIR bands have a higher surface brightness per pixel than the optical images, while being less affected by dust attenuation. Despite the worse spatial sampling and resolution of [Euclid]{} NIR compared to optical, the NIR approach yields equivalent or more precise galaxy ellipticity measurements. If systematics that affect shape such as dithering strategy and point spread function undersampling can be mitigated, inclusion of the NIR can improve galaxy ellipticity measurements over all redshifts. This is particularly important for upcoming weak lensing surveys, such as with [Euclid]{} and WFIRST.' author: - Bomee Lee - 'Ranga-Ram Chary' - 'Edward L. Wright' title: 'Galaxy Ellipticity Measurements in the Near-Infrared for Weak Lensing' --- =1 Introduction ============ Weak gravitational lensing (WL) is a slight deflection of light rays from distant galaxies when they propagate through the tidal gravitational field of intervening large scale structure. The amplitude of the WL distortion can be used to map dark matter and measure dark energy by statistically quantifying the shear distortions encoded in the observed shapes of background galaxies, namely galaxy ellipticities [e.g. @kai95; @bar01]. The ellipticities of galaxies are typically distorted only about a per cent by WL [@tro15], so the WL signal in individual galaxies is challenging to detect. WL measurements thus rely on averaging over a very large sample to obtain the distortions and sufficiently unbiased estimates of galaxy shapes, which in turn require a correction for the impact of the point spread function (PSF) of the telescope. In that sense, WL observations demand high quality images because it requires a large number density of resolved galaxies and high signal-to-noise ratio (SNR), while minimizing the PSF corrections and related systematic uncertainties, with well-sampled PSFs [@mas13; @sch18]. [Euclid]{} [@lau11] is a survey mission designed to understand the expansion and growth history of the Universe, and is scheduled to launch in the next decade. [Euclid]{} will image 15,000deg$^{2}$ of sky in one broad optical band VIS spanning 550 to 920nm, and three additional near-infrared (NIR) bands ($Y$, $J$, and $H$). [Euclid]{} will detect cosmic shear with VIS by measuring ellipticities of $\sim30$ resolved galaxies per arcmin$^2$ with a resolution better than 0.18$\arcsec$ (PSF FWHM) with 0.1$\arcsec$ pixels. The near-infrared bands will primarily be used to derive photometric redshifts for the weak lensing sample, in conjunction with ground-based observations at visible wavelengths. The [Euclid]{} wide survey is expected to provide WL galaxy shape measurements for 1.5 billion galaxies with space-quality resolution. To measure WL through surveys, one should measure galaxy ellipticities and its uncertainty, including systematics, very accurately. In particular, it is necessary to measure the shapes of typically faint and small, distant galaxies with high-SNR observations. In this work, we demonstrate that NIR bands result in a comparable or more precise galaxy ellipticity measurement compared to optical bands for WL studies despite their worse spatial resolution (0.3$\arcsec$ compared to 0.18$\arcsec$) and pixel sampling (0.3$\arcsec$ vs. 0.1$\arcsec$ pixel scale). There are several advantages to using NIR bands [@tun17]; first, NIR wavelengths sample the rest-frame optical light, which traces the older stellar population (hence the bulk of stellar mass) and is less affected by dust extinction. The VIS band covers the rest-frame UV and blue wavelengths, which predominantly traces emission from star-forming regions [@dic00]. In particular, the shapes of galaxies as seen in the rest-frame UV are more clumped and irregularly distributed than older stellar populations. The second advantage is that galaxies in the NIR bands have an intrinsically smoother light distribution resulting in a lower shape noise than in the optical [@sch18]. Third, NIR images of galaxies have a higher surface brightness with more than nine times the number of source photons per pixel, based on a calculation using images in this study; this is at least partly due to the relative importance of the bulge compared to the disk as a function of wavelength. Finally, we find that the NIR bands are sensitive to a larger number density of distant galaxies than the VIS band (see Section 2). In this paper, we study the shapes of the galaxy sample expected from [Euclid]{}-quality imaging and forecast how we can improve the shape measurement by using co-added NIR images[^1]. To do that, we select galaxies from [HST]{}/CANDELS (Cosmic Assembly Near-infrared Extragalactic Legacy Survey; [@gro11]; [@koe11]) observations satisfying the [Euclid]{} sensitivity limits and simulate [Euclid]{}-resolution images. The structure of this paper as follows. The sample selection using CANDELS data is introduced in Section 2. We describe the procedure of simulating [Euclid]{}-quality optical and NIR images from [HST]{} images in Section 3. In Section 4, we explain how GALFIT (Peng et al. 2010) is used to measure the ellipticity of galaxies after accounting for the PSF, and compare the ellipticities obtained from GALFIT in simulated-[Euclid]{} and CANDELS images. Finally, we present our conclusions in Section 5. Initial Sample Selection ======================== We select a sample of galaxies at optical and near-infrared (NIR) wavelengths from the HST/CANDELS survey that closely resembles the [Euclid]{} weak lensing (WL) sample. Among the five CANDELS fields, we use the GOODS-S and GOODS-N fields which include the CANDELS Deep survey and covers about 340 $arcmin^{2}$ in V (0.606$\rm \mu m$), I (0.814$\rm \mu m$), J (1.25$\rm \mu m$), and H (1.6$\rm \mu m$). These fields are several magnitudes deeper than the [Euclid]{} survey. We use CANDELS photometric redshifts measured for all galaxies by [@dah13], unless spectroscopic redshifts are available. For the WL shape measurement, the [Euclid]{} survey will detect galaxies in a broad optical R+i+z band (VIS: 0.55–0.92$\rm \mu m$) down to 24.5 mag (10 $\sigma$). It will use three additional NIR bands (Y, J, H in the range of 0.92–2.0 $\mu m$) reaching AB mag 24 (5 $\sigma$) in each. To achieve the required dark energy figure of merit through weak lensing, the surface density of resolved galaxies needs to be at least 30 $arcmin^{-2}$ [Euclid Red book; @lau11]. We start by replicating the [Euclid]{} expected sensitivity selection on the CANDELS catalogs. We find that I $< $24.5 AB mag results in about 30 galaxies per $arcmin^{2}$ with a mean redshift of $\sim 0.9$, which is consistent with the [Euclid]{} requirement. Applying the [Euclid]{} $H<24$ mag selection on the CANDELS NIR sample, results in a mean $z\sim1.1$ with about 37 galaxies per $arcmin^{2}$. The redshift distribution of each sample selection is shown in Figure \[fig:histz\]. One clear advantage of the NIR is at $z>1$, where the NIR bands select many more galaxies than the optical. This suppresses the shape noise induced by the intrinsic ellipticities of distant galaxies if the individual ellipticity uncertainties were similar to that in the optical; we assess the veracity of this in the following sections. In this study, we specifically use galaxies at a redshift range of $0.5\le z < 3$ to compare ellipticities estimated from optical and NIR images. ![Number density of galaxies at a redshift range, $0.5\le z < 3.0$, in the GOODS-S and GOODS-N fields. We compare number density of galaxies in redshift bins of width $\delta z=0.1$ of different galaxy samples, selected using expected magnitude depths of the [Euclid]{} survey. The red histogram represents a NIR selection of $H< 24$mag while the blue histogram is for ${I} < 24.5$mag. We also sub-select galaxies having $H<23.5$mag based on the quality of their ellipticity fits and present their redshift distribution with the purple histogram (see a further explanation about this selection in Section 4.3). At $z>1.0$, the NIR yields a higher surface density of galaxies than in the optical. The median (mean) redshift is 0.95(1.1), 1.1(1.3), and 1.0(1.2) for the optical, NIR, and $H<23.5$ selected sample, respectively, within a redshift range of $0.5\le z < 3.0$. []{data-label="fig:histz"}](fig1_new.png){width="4.5in"} [Euclid]{} images made from CANDELS/[HST]{} images ====================================================== ![Postage stamps of five galaxies in the GOODS-S field at different redshifts. Each image covers an area of 4$\arcsec\times 4\arcsec$. From left to right, CANDELS I, H, simulated [Euclid]{} V+I, and J+H images, GALFIT model fit to the Euclid VI image and residual, model fit to the [Euclid]{} JH image and residual. Redshift, AB magnitude of I and H bands for each galaxy are given in the top (red texts) and the modulus of the galaxy ellipticity calculated from the GALFIT results using respective images are given in the bottom (blue texts). []{data-label="fig:post"}](fig2_new.png){width="7in"} We simulate [Euclid]{} VIS and NIR images using CANDELS/[HST]{} V (0.606$\rm \mu m$), I (0.814$\rm \mu m$), J (1.25$\rm \mu m$), and H (1.6$\rm \mu m$). Each of the [Euclid]{} VIS, Y, J and H bands will have four images taken per unit area of sky with 0.1$\arcsec$ pixel scale in the VIS and 0.3$\arcsec$ pixel scale in the NIR bands [@lau11]. For the VIS images, we combine the CANDELS V and I band images which span the bandwidth of the [Euclid]{} VIS band, 0.55$\rm \mu m$–0.92$\rm \mu m$. In both the VIS and NIR bands, there will be significant correlated noise if coadded images are made on finer pixel scale with just four frames. By combining the $J$ and $H$ band images, we can both increase the signal to noise in the NIR and drizzle on a factor of two oversampled pixel scale with a point-kernel (e.g. CANDELS WIDE survey; [@gro11], HST/ACS COSMOS; [@rho07]), thereby minimizing the impact of correlated noise. We therefore combine the CANDELS $J$ and $H$ band images ([Euclid]{} $J$: 1.16–1.58$\rm \mu m$, $H$: 1.52–2.04$\rm \mu m$). We estimate that the [Euclid]{} survey strategy results in a median of between 10 and 11 valid frames per pixel when combining all three bands $Y$, $J$ and $H$. However, the PSF undersampling in the shortest wavelength band and color gradients across such a wide wavelength range may introduce other systematics. The impact on the undersampled PSF as a result of the drizzling and the [Euclid]{} dither strategy is beyond the scope of this work and is currently being investigated. Furthermore, since the CANDELS $Y$ band imaging does not cover the entire GOODS-S and -N fields, we avoid including the $Y-$band in this analysis. The step-by-step procedure to simulate the [Euclid]{}-quality images is outlined below: 1. Produce cutouts of science and noise images ($rms$) for each galaxy from the large [HST]{}/CANDELS V, I, J, and H mosaics. 2. Combine V and I or J and H by weighting each pixel according to the weight map (inverse variance), i.e. $f_{comb} = (f_1w_1+f_2w_2)/(w_1+w_2)$, where $f_{1,2}$ and $w_{1,2}$ are the pixel values of science image and weight map, and $w_{1,2} \sim 1/rms_{1,2}^2 $. Because each pixel value has noise associated with it, and the noise is somewhat heterogeneous due to the observing strategy, an inverse variance weighting is the optimal approach to combine images and increase the signal-to-noise of co-added images [@koe11]. 3. Re-bin the pixel scale from 0.06$\arcsec$ of CANDELS to 0.1$\arcsec$ (V+I) or 0.15$\arcsec$ (J+H). By using co-added J+H images, which will double the number of images, the co-added NIR images can be drizzled onto a 0.15$\arcsec$ pixel scale, half of the original NIR pixel scale. This is challenging to do for VIS, since only four frames will be taken. 4. Smooth the combined images with a Gaussian kernel to correct for the difference in PSF FWHM between [HST]{} and [Euclid]{} (for optical, 0.1$\arcsec$ vs. 0.18$\arcsec$; for NIR, 0.18$\arcsec$ vs. 0.3$\arcsec$). 5. Make noise maps for the V+I and J+H following a random Gaussian distribution with 1$\sigma$ measured from the quoted sensitivity of [Euclid]{} VIS and NIR images. For the VIS images, the sensitivity is 24.5 mag at 10 $\sigma$ (estimated from an extended source with a 0.3$\arcsec$ radius; [@cro12]). For the NIR images, the sensitivity is 24 mag at 5 $\sigma$ in each band (measured from a point source), respectively. By combining the $J$ and $H$ bands, the effective sensitivity is therefore 24 AB mag at 7$\sigma$. 6. Add the noise maps to the images to obtain the simulated [Euclid]{} VI(V+I) and JH(J+H) images. Since CANDELS’ background noise is negligible (more than 50 times smaller than that of [Euclid]{}), we do not remove the noise in the CANDELS images before adding [Euclid]{} noise maps. A few galaxies in the sample (1% and 3% in JH and VI, respectively) are not observed in J and V bands because the CANDELS coverage of the field at different bands varies slightly. Thus, after excluding these sources, we have 7,248 galaxies for VI and 9,887 galaxies for JH. For illustrative purposes, the images of 5 galaxies in the CANDELS I and H bands, the simulated [Euclid]{} VI and JH images, and their GALFIT fits are shown in Figure \[fig:post\]. Ellipticity measurements using GALFIT ===================================== Masking sources --------------- Weak lensing measurements, due to the small signal, typically rely on averages over a large number of galaxies. As a result, they usually require aggressive masks of samples to correct systematic effects. In particular, due to the sensitivity limit and spatial resolution, we find that [Euclid]{} will suffer from blending of galaxies with nearby objects and non-detections which the higher spatial resolution of the VIS band may be able to reveal. As demonstrated in Figure \[fig:post\], the spatial resolution and SNR of simulated Euclid images (3rd and 4th columns) unsurprisingly appear to be significantly worse than CANDELS images (1st and 2nd). We therefore run a source detection algorithm, SExtractor [@ber96], on the VI and JH cutouts in Section 3 and remove sources for which the photometry as measured by the SEXtractor, AUTO MAG, deviates from the expected magnitude (I or H band magnitude from CANDELS photometry catalog) by more than two times the uncertainty in the difference between derived and expected magnitudes. In addition, we exclude galaxies from the original sample which are now offset by more than 0.7$\arcsec$ (VI) and 0.75$\arcsec$ (JH) relative to the original positions because it implies that the detection in the [Euclid]{} simulated image is either noise or affected by source confusion. After masking out about 13.3% and 11.4% sources from Section 3, we have 6,283 and 8,762 galaxies for VI and JH respectively. At the expected sensitivity limit, we find that we are about 80% complete at $24.5$AB mag in VI and $24.0$AB mag in JH. GALFIT ------ We measure the ellipticities for the galaxies in the sample using GALFIT [@pen02]. GALFIT fits a Sérsic law to the surface brightness profile measured within elliptical isophotes of a galaxy. Importantly, GALFIT includes the PSF in the fitting process. The accounting for the PSF is crucial in WL because the smoothing from the PSF make galaxies appear rounder than they actually are [@hol09] and significantly biases the ellipticity measurements. The appropriate PSF model is essential for the accuracy of the ellipticity estimation. It is inappropriate to derive a PSF for galaxies from the stars because stars typically have a Rayleigh-Jeans spectrum across the bandpass while galaxies are significant redder, implying a broader intrinsic PSF. We therefore construct a model PSF for the CANDELS using the TinyTim software package [@kri95] for the ACS I band and WFC3 H band by assuming a flat galaxy spectrum ($f_{\nu} \sim constant$). They are then re-sampled to the CANDELS pixels scale, 0.06$\arcsec$. For [Euclid]{}, we re-sample I and H Tinytim PSF to the [Euclid]{} pixel scales, 0.1$\arcsec$ (VI) and 0.15$\arcsec$ (JH) and, subsequently, smooth with a Gaussian smoothing kernel to correct for the difference in PSF FWHM between CANDELS and [Euclid]{}. We let GALFIT fit the images with central position, magnitude, half-light radius (R$_{e}$) measured along the major axis, Sérsic index, axis ratio (q = semi-minor axis/semi-major axis), and position angle as free parameters. The SExtractor measurements are used to feed GALFIT with initial guesses for these parameters. In each image cutout, neighboring objects detected from the SExtractor are fit simultaneously or masked out if they are less than 2 magnitudes fainter than the target galaxy. Any fit resulting in problems (i.e., axis ratio errors $>$1.0) or non-existent results (fits crashed) are excluded. According to experiments undertaken by [@vdw12], about 60% of galaxies have a good fit (GALFIT flag $=$ 0) from CANDELS GOODS-S GALFIT catalog, which can be used as reliable measures of ellipticity. We find a slightly higher percentage, 4,737 galaxies (65.4%) for VI and 6,449 galaxies (65.2%) for JH, after excluding all problematic galaxies as discussed in Section 4.1 and bad fits. Figure \[fig:post\] illustrates best-fit GALFIT model images and residuals images showing the difference between model and original image which are dominated by noise. The absolute value of the ellipticity (see the definition in Section 4.3) computed using the GALFIT results for those galaxies are given in the bottom of I, H, VI, and JH images. Although GALFIT is one of the most popular fitting tools for measuring galaxy shapes, we lose a large amount of our sample due to unreliable fits. Furthermore, it is known that the GALFIT is not suitable to fit small, faint galaxies, mainly high redshift galaxies [@vdw12; @sif15]. This is mostly due to the number of parameters that GALFIT tries to fit for, which results in unreliable fits in the low signal to noise ratio regime [@Jee]. The widely used shear measurement algorithms uniquely developed for WL, such as the KSB (Kaiser, Squires & Broadhurst) algorithm [@kai95; @hoe98] and $\it{lens}$fit [@mil07; @kit08], might provide better performance in measuring the ellipticity of galaxies. However, [@sif15] compared the KSB results for bright cluster galaxies to GALFIT shapes and showed that the ellipticities measured by both methods are generally consistent. A detailed assessment of the accuracy of ellipticity measurements from different techniques is beyond of the scope of this paper and we use GALFIT for our main goal of comparing the ellipticities estimated from CANDELS to [Euclid]{}-quality images. Comparison of ellipticities between CANDELS and Euclid-quality images --------------------------------------------------------------------- Typically, only galaxies with a size comparable or larger than the PSF have a well measured shape; so the shape of the smallest galaxies becomes ill-defined. Also, as the SNR decreases, the ability to measure galaxy shapes decreases since the imaging data is only sensitive to the highest surface brightness regions of the galaxy. Therefore, the galaxy samples for shape measurement require a lower limit to the SNR of about 10, and the radius of the galaxy larger than 1.25 times the PSF FWHM (Euclid Red book: @lau11). In order to satisfy those requirements, we restrict the galaxy sample in the NIR band to have H $< 23.5$mag ($\sim$ SNR $>11$ for J+H), and a half-light radius ($R_{e}$) measured from GALFIT on the simulated [Euclid]{} JH image of larger than half of the NIR PSF FWHM of [Euclid]{} (R$_{e} > $ 0.15$\arcsec$). For the optical, we select galaxies having I $< 24.5$mag and SNR(I) $>10$ with a size limit, $R_{e}$ measured from VI images $> 0.1\arcsec$. This is about half the FWHM of the PSF in the VIS band. In Figure \[fig:re\], we show the distribution of $R_{e}$ as a function of a redshift for the optical and NIR sample with $I<24.5$mag (left) and $H<23.5$mag (right), respectively. About 2.2% of optical and 9.8% of NIR selected galaxies have $R_{e} < 0.1\arcsec$ and $R_{e} < 0.15\arcsec$, respectively. As a final sample for analyzing ellipticities, we use 4,634 and 4,770 galaxies for the optical and NIR respectively. In Table 2, the number densities of the galaxy samples in the optical and NIR are listed at five different redshift bins. The relatively high SNR cut of H$< 23.5$mag results in a similar total number density of galaxies with the VI band as also shown in Figure \[fig:histz\], but still translates to a higher number density by a factor of 1.4 at $z >1$ (6.9 vs. 5.1 arcmin$^{-2}$). In order to study galaxy shapes, the complex galaxy ellipticity is typically used in weak lensing studies [@mil13; @sch15]. Using axis ratio (q) and position angle estimated from GALFIT, we compute the complex galaxy ellipticity ($e$) of our final sample which is defined as $$e = e_{1} + \mathrm{i}e_{2} = \|e\|\mathrm{e}^{2\mathrm{i}\phi}, \label{eq:ell}$$ where, the modulus of the ellipticity ($\|e\|$) is defined as (1-q)/(1+q) and $\phi$ corresponds to the position angle of the major axis. We then compare complex galaxy ellipticities ($e$) measured from Euclid-quality images with the CANDELS values. Here, we consider CANDELS-measured ellipticity as the original ellipticity of an observed galaxy because of the much larger depth and better resolution compared to [Euclid]{} data. Through this comparison, we can investigate the robustness of the galaxy shape measurements on the simulated [Euclid]{} images. In Figure \[fig:complex\], we plot differences in ellipticities between CANDELS and [Euclid]{}-quality data, $\Delta e_{\alpha}$ = ($e_{\alpha}$ of Euclid-quality – $e_{\alpha}$ of CANDELS), as a function of $e_{\alpha}$ of CANDELS for both ellipticity components, $\alpha=1,2$, in the range of $-1<e_{\alpha}<1$. Most galaxies scatter systematically around $\Delta e_{\alpha} = 0$ for both optical (purple) and NIR (green) with a median $\sim 0$ at all redshift ranges considered in this study. The uncertainty in $\Delta e$ quantifies how well the galaxy shapes are recovered with Euclid-quality images. The lowest redshift bin has a measured scatter ($\sigma$, the standard deviation of $\Delta e_{\alpha}$) in the NIR which is a factor of 1.2 larger than that in the VI-band. However, the measured scatter in the NIR is similar to that in the VI at all other redshifts, $z>0.7$. Overall, we find that the ellipticities of individual galaxies can be measured with a similar scatter from the Euclid VIS- and NIR-like images. There is a weak trend that the scatter of $\Delta e_{\alpha} $ increases with redshifts in both selections. Galaxies are fainter and smaller at higher redshift; so, the limited [Euclid]{} spatial resolution and sensitivity will result in a larger scatter in the measured shape. In particular, at higher redshifts, galaxies with larger ellipticities (in absolute values) tend to have larger discrepancies (see diagonal trends at $z>1.3$). This is likely because highly elongated galaxies in CANDELS appear to be rounder, and with less-constrained position angles at [Euclid]{}-quality resolution. This trend appears to be a bit stronger for the NIR high-z sample due to the pixelization in the JH data. In Figure \[fig:ell\], we compare the modulus of the ellipticity, $\|e\|$ from Equation \[eq:ell\], derived using CANDELS I band to the simulated [Euclid]{} VI and CANDELS H band to the simulated [Euclid]{} JH. The ellipticities derived from the simulated [Euclid]{} images are correlated very well with the ellipticities derived from CANDELS images with a median of $\Delta \|e\|$ = ($\|e\|$ of Euclid-quality – $\|e\|$ of CANDELS) $\sim 0$ for both optical and NIR imaging. At $z>1.0$, NIR yields a similar to lower scatter than the optical, while the scatter in VI-derived ellipticities is significantly smaller at $z<0.7$. This trend of the uncertainty in $\Delta \it{e}$ is more obvious in Figure \[fig:sigma\]–a). We compare the standard error (SE) of $\Delta \|e\|$ ($=\sigma(\Delta \|e\| / \sqrt{N}$) at each redshift bin for JH (blue) and VI (red). At $z<1.0$, the standard error of JH is 1.2–1.1 times larger than one of VI. But, the trend reverses at $z>1.0$ so that the standard error of JH is significantly smaller than VI by a factor of 1.5–1.3. As shown in Figure \[fig:ell\], the measured $\sigma(\Delta \|e\|)$ of galaxies in JH is very similar with that of VI over the redshift range considered here; thus, the higher number density of galaxies at $z>1.0$ drives a lower standard error of ellipticity differences in JH. In Figure \[fig:sigma\]–b, we show the median fractional error of ellipticity, which is defined as (median of $\frac{\Delta \|e\|}{\|e\|[CANDELS]}$)$/\sqrt{N}$, as a function of redshift. The small values indicate that the derived ellipticity with $Euclid$-quality JH and VI imaging is very close to the true value from CANDELS on average. We find that the performance of the JH band in galaxy ellipticity measurements is comparable to the VI at all redshifts despite the significantly worse spatial resolution of JH. This result is very similar to that derived by @tun17 who found that for a 1.2m class telescope, the $K_{s}-$band yields an ellipticity measurement error which is a factor of $\sim$3 smaller than in an $R-$band selected catalog while the $J-$band is a factor of $1.5-2.5$ worse than the $K_{s}-$selected catalog. This is also consistent with the results of @sch18 who found that ground-based $K_{s}$ imaging with a PSF FWHM$\sim$0.35$\arcsec$ yields an ellipticity dispersion for $z\gtrapprox$ 1.4 galaxies, which is 0.76 times that of optically-selected galaxy samples with single-orbit [HST]{} imaging. A comparison between ellipticities derived from the simulated [Euclid]{} JH data and CANDELS I-band data indicates a correlation; however, the I-band ellipticities are larger than that in the NIR and the scatter is larger than shown in Figure \[fig:ell\]. This is likely because the $I-$band ellipticities are dominated by disk light while the NIR ellipticities are tracing a combination of disk and more-compact bulge light. Thus, if systematics arising from PSF under-sampling and dither strategy on the NIR images can be accounted for in future work, the shape noise can be minimized by including ellipticity measurements from the NIR bands. Conclusion: Precise Ellipticity Measurements in the NIR ======================================================= We investigate galaxy ellipticities in simulated [Euclid]{}-quality optical (VIS) and near-infrared (NIR) images constructed from [HST]{}/CANDELS co-added V+I (VI) and J+H (JH) images. We select galaxies in CANDELS GOODS-S and -N fields (covering about 340 arcmin$^2$) with photometry in I and H bands at similar depths as the planned [Euclid]{} survey. In this study, we specifically use galaxies at a redshift range of $0.5\le z<3$. After applying a SNR $\gtrapprox 10$ cut, the total number density of galaxies at $0.5\le z<3$ is comparable in the NIR and optical; however, the NIR bands select 1.4 times higher number density of galaxies relative to the optical selection at $z>1.0$, which enable us to reduce the statistical uncertainties in the shape measurements of distant galaxies. By co-adding [Euclid]{}-quality J and H-band images, which double the number of frames and the exposure time, we can generate images at 0.15$\arcsec$ pixel scale (half of the original [Euclid]{} NIR detector pixel scale) and achieve $S/N \gtrapprox 7$ at $H<24$mag and $\gtrapprox11$ at $H<23.5$mag. Using GALFIT, we measure ellipticities of galaxies in CANDELS I and H band images with 0.06$\arcsec$ pixel scale and the simulated [Euclid]{} VI and JH images with 0.1$\arcsec$ and 0.15$\arcsec$ pixel scale, respectively. We then compare ellipticities between CANDELS and simulated [Euclid]{}-quality images while considering [HST]{}/CANDELS ellipticity as the original ellipticity of a galaxy due to its superior depth and resolution. A comparison between ellipticities derived from CANDELS and [Euclid]{}-quality VIS and NIR imaging shows that both wavelength ranges provide similar performance in measuring galaxy ellipticities at all redshifts included in this study despite the worse spatial resolution and pixel sampling of the NIR imaging. When combined with the higher source density in the NIR selection, we find that the standard error in NIR-derived ellipticities is about 30% smaller than the optical bands at $z>1.0$, which implies a more precise ellipticity measurement than in the optical alone. Since the VIS and NIR galaxy shape measurements with [Euclid]{} have different fractional contributions of the bulge and disk, a combination of the two can improve the precision with which galaxy ellipticities are measured. The next step that is required before the NIR data can be used for WL studies is to assess how the drizzling affects both the undersampled telescope PSF, and the correlated noise [see e.g. @rho07]. However, even though the FWHM of the [Euclid]{} NIR imaging does not quite reach [HST]{} or WFIRST resolution, the NIR data provides a major advantage for weak lensing measurements compared to optical ground-based observations that typically achieve a PSF FWHM$\sim 0.6\arcsec-0.7\arcsec$ in good seeing conditions [e.g. @kui15; @man18]; while the latter provides good sensitivity to the weak lensing signal with a median redshift of the sample of z $\sim 0.85$, about half the galaxy sample will be unresolved due to the small size of galaxies, as shown in Figure \[fig:re\], implying a higher statistical uncertainty in their ellipticities. In conclusion, by using co-added J+H band [Euclid]{}–quality images, we show that the galaxy sample selected at NIR wavelengths yields a more precise ellipticity measurement, especially at high redshifts. This suggests that a careful evaluation of NIR shape systematics for future weak gravitational lensing surveys, such as with [Euclid]{} and WFIRST, should be undertaken. This work is partly funded by NASA/[Euclid]{} grant 1484822 and is based on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. The authors thank the anonymous referee for very useful comments that helped to improve the presentation of the paper. We also thank Lance Miller, Stefanie Wachter, Peter Schneider, Henk Hoekstra and Jason Rhodes for thoughtful comments which improved this manuscript. Bartelmann,M. & Schneider, P. 2001, Phys. Rep., 340, 291 Bertin, E. & Arnouts, S. 1996, A&AS, 117, 393 Dahlen, T., Bahram, M., Faber, S. M., Ferguson, H.C., et al. 2013, , 775, 93 Dickinson, M. 2000, Philos. Trans. R. Soc. London A, 358, 2001 Grogin, N. A., Kocevski, D. 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--- abstract: 'The $\nu_l n \rightarrow l^- p$ QE reaction on the A-target is used as a signal event or/and to reconstruct the neutrino energy, using two-body kinematics. Competition of another processes could lead to misidentification of the arriving neutrinos, being important the fake events coming from the CC1$\pi$ background. A precise knowledge of cross sections is a prerequisite in order to make simulations in event generators to substract the fake ones from the QE countings, and in this contribution we analyze the different nuclear effects on the CC1$\pi$ channel. Our calculations also can be extended for the NC case.' author: - 'A. Mariano$^\dag$ and C. Barbero$^\dag$' date: '$^\dag$Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata - IFLP (Conicet), C.C. 67, La Plata, Argentina' title: 2p2h effects on the weak pion production cross section --- $\nu_\mu \rightarrow \nu_x$ disappearance experiments uses $\nu_\mu n \rightarrow \mu^- p$ CCQE to detect neutrinos and reconstruct its energy. $E_\nu$ determination could be wrong for a fraction of CC1$\pi^+$ background events (20%) $ \nu_\mu p \rightarrow \mu^- p \pi^+$, that can mimic a CCQE one if the pion is absorbed and/or not detected. These processes take place into the target nucleus and nuclear effects as smearing (S) of the reconstructed energy by the momentum distribution ($n_A$) of the target binding (B) nucleons, should be taken into account. In addition final state interactions (FSI) of the emerging hadrons generate energy lost,change of direction,charge transfer or multiple nucleon knock out(np-nh). Finally meson exchange currents (MEC) processes lead to additional contributions to one-body current generated. In what follows we concentrate on the 2p2h+1$\pi$ contributions to the pion production cross section, and compare with the 1p1h+1$\pi$ one already analyzed previously [@Lalakulich2012]. The 2p2h+1$\pi$ amplitude is depicted in Figure 1 and the corresponding differential cross section reads (${\cal N}^2(\mbk)\equiv { f\over (2\pi)^3E(\mbk)}, f=1/2( M)$ for bosons(fermions)) ![Amplitude for the $\nu N \rightarrow l N' N_1 N_2 \pi$ ($2p2h + 1\pi$) process.[]{data-label="oneloop"}](2p2h_6.eps){width="9.cm"} &&d\_[A]{}\^[2p2h+1]{} = [n\_A()(1-n\_A(\_1))\|\_- \_A\|2 E()]{} [1 2]{}\_[m\_,m\_N,m\_l,m\_[N\_1]{}]{}\ &&\ &&\|W\^(N\_1 m\_[N\_1]{},l m\_l, ’, N m\_N,m\_) J\^l\_(l m\_l, m\_)\|\^2\ &&(2)\^4 \^4(N\_1+’+l--N) d\^3 l [N]{}\^2() d\^3 N\_1 [N]{}\^2(\_1) d\^3N [N]{}(),being $N,\pi,l,\nu \equiv (E(\mbN,\mbpi,\mbl,\mbnu), \ \mbN,\mbpi,\mbl,\mbnu)$, $E(\mbk)=\sqrt{\mbk^2+M^2}$, $m\equiv$ spin, $\S$ symmetrization factor, and T $\pi' N'\rightarrow \pi N_2$ rescattering is simplified replacing &&[\[ \]]{} 2(’\^2 - M\_\^2 - ((’))),as shown in Figure 2. We work out this self-energy $\Pi(\pi')$ in the $\Delta-h$ approach, taking into account that the $\Delta$ width will account the final pion-nucleon additional state through the absorptive evaluation of the pion-nucleon $\Delta$-self energy. Figure 3 shows results for the total CC$1\pi^+$ cross section and the gradual effect of the B, S and FSI within the 1p1h+1${\pi}$ configuration space. Also the results for B+S+FSI in the 2p2h+1${\pi}$ one and full 1p1h+2p2h+1${\pi}$ one are shown. They are also shown for the differential cross section in Figure 4.B effects are considered within the Relativistic Hartree approximation (RHA) of QHD I [@Serot86], for N and $\Delta$ using universal couplings. $n_A$ is obtained from a perturbative approach in nuclear matter within a 2p2h+4p4h configuration space[@Mariano96]. FSI on nucleons is taken (Toy model !) through the RHA effective fields also for final N, while for pions we use the Eikonal approach[@fsi]. ![Simplification to calculate $\pi' N'(h)$ rescattering.[]{data-label="oneloop"}](2p2h_7.eps){width="13.5cm"} ![Total 1$\pi^+$ cross section compared with MiniBooNE data (see[@Lalakulich2012]).[]{data-label="oneloop"}](2p2h_1.eps){width="10.8cm"} ![1$\pi^+$ production differential cross section.[]{data-label="oneloop"}](2p2h_2.eps){width="10.cm"} We conclude that 2p2h contribution is important and comparable to the 1p1h one. In the $\pi^0$ channel results are not so good in reproducing the data and nonresonant contributions and charge exchange terms should be included in the rescattering amplitude. Finally, MEC should be included at the same time that 2p2h contributions in order to have a more real estimation. Acknowledgments: The assistance of A.M. to the Nuint 2012 was supported partially by the CONICET under the PIP 0349. O. Lalakulich and U. Mosel, arXiv:1210.4717 \[nucl-th\] (2012). B.D. Serot and J.D. Walecka, [Adv. Nucl. Phys.]{}[16]{},(1986) 1. A. Mariano, F. Krmpotić and A.F.R de Toledo Piza, [Phys. Rev.]{} [C53]{},(1996)1. C. Barbero, A. Mariano, and S. B. Duarte, [Phys. Rev. C]{}, [82]{} (2010) 067305.
--- abstract: 'In this paper, we develop a fast imaging technique for small anomalies located in homogeneous media from $S-$parameter data measured at dipole antennas. Based on the representation of $S-$parameters when an anomaly exists, we design a direct sampling method (DSM) for imaging an anomaly and establishing a relationship between the indicator function of DSM and an infinite series of Bessel functions of integer order. Simulation results using synthetic data at $f = 1 $GHz of angular frequency are illustrated to support the identified structure of the indicator function.' address: 'Department of Information Security, Cryptology, and Mathematics, Kookmin University, Seoul, 02707, Korea' author: - 'Won-Kwang Park' bibliography: - '../References.bib' title: 'Direct sampling method for anomaly imaging from $S-$parameter' --- Direct sampling method ,$S-$parameter ,Bessel functions ,simulation results Introduction ============ In this study, we consider an inverse scattering problem that determines the locations of small anomalies in a homogeneous background using $S-$parameter measurements. This study has been motivated by microwave tomography for small-target imaging, such as in the case of tumors during the early stages of breast cancer. Because of the intrinsic ill-posedness and nonlinearity of inverse scattering problems, this problem is very hard to solve; however, it is still an interesting research topic because of its relevance in human life. Many researchers have focused on various imaging techniques that are mostly based on Newton-type iteration-based techniques [@CZBN Table II]. However, the success of Newton-type based techniques is highly dependent on the initial guess, which must be close to the unknown targets. Furthermore, Newton-type based techniques have various limitations such as large computational costs, local minimizer problem, difficulty in imaging multiple anomalies, and selecting appropriate regularization. Because of this reason, developing a fast imaging technique for obtaining a good initial guess is highly required. Recently, various non-iterative techniques have been investigated, e.g., MUltiple SIgnal Classification (MUSIC) algorithm, linear sampling method, topological derivative strategy, and Kirchhoff/subspace migrations. A brief description of such techniques can be found in [@AGKPS; @AK2; @K2; @P-SUB3; @SZ]. Direct sampling method (DSM) is another non-iterative technique for imaging unknown targets. Unlike the non-iterative techniques mentioned above, DSM requires either one or a small number of fields with incident directions [@IJZ1; @IJZ2; @LZ]. Furthermore, this is a considerably effective and stable algorithm. In a recent study [@PKLS], the MUSIC algorithm was designed for imaging small and extended anomalies; however, DSM has not yet been designed and used to identify unknown anomalies from measured $S-$parameter data. To address this issue, we design a DSM from $S-$parameter data collected by a small number of dipole antennas to identify the outline shape anomaly with different conductivity and relative permittivity compared to the background medium and a significantly smaller diameter than the wavelength. To investigate the feasibility of the designed DSM, we establish a relationship between the indicator function of DSM and an infinite series of Bessel functions of integer order. Subsequently, we present the simulation results that confirm the established relationship using synthetic data generated by the CST STUDIO SUITE. The remainder of this paper is organized as follows. In Section \[sec:2\], we briefly introduce the DSM for imaging anomalies from $S-$parameter data. Subsequently, in Section \[sec:3\], we present simulation results for the synthetic data generated at $f = 1 $GHz of angular frequency, which is followed by a brief conclusion in Section \[sec:4\]. Preliminaries {#sec:2} ============= In this section, we briefly survey the three-dimensional forward problem in which an anomaly $\mathrm{D}$ with a smooth boundary ${\partial}\mathrm{D}$ is surrounded by $N-$different dipole antennas. For simplicity, we assume that $\mathrm{D}$ is a small ball with radius $\rho$, which is located at ${\mathbf{r}}_\mathrm{D}$ such that $$\mathrm{D}={\mathbf{r}}_\mathrm{D}+\rho\mathbf{B},$$ where $\mathbf{B}$ denotes a simply connected domain. We denote ${\mathbf{r}}_{\mathrm{\tiny TX}}$ as the location of the transmitter, ${\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}$ as the location of the $n-$th receiver, and $\Gamma$ as the set of receivers. $$\Gamma=\{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}:n=1,2,\cdots,N\quad\mbox{with}\quad\|{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}\|=R\}.$$ Throughout this paper, for every material and anomaly to be non-magnetic, they are classified on the basis of the value of their relative dielectic permittivity and electrical conductivity at a given angular frequency $\omega=2\pi f$. To reflect this, we set the magnetic permeability to be constant at every location such that $\mu({\mathbf{r}})\equiv\mu=4\cdot10^{?7}\pi$, and we denote ${\varepsilon}_\mathrm{B}$ and $\sigma_\mathrm{B}$ as the background relative permittivity and conductivity, respectively. By analogy, ${\varepsilon}_\mathrm{D}$ and $\sigma_\mathrm{D}$ are respectively those of $\mathrm{D}$. Then, we introduce piecewise constant relative permittivity ${\varepsilon}({\mathbf{r}})$ and conductivity $\sigma({\mathbf{r}})$, $${\varepsilon}({\mathbf{r}})=\left\{\begin{array}{rcl} {\varepsilon}_\mathrm{D} & \mbox{if} & {\mathbf{r}}\in\mathrm{D},\\ {\varepsilon}_\mathrm{B} & \mbox{if} & {\mathbf{r}}\in\mathbb{R}^3\backslash\overline{\mathrm{D}}, \end{array} \right. \quad\mbox{and}\quad \sigma({\mathbf{r}})=\left\{\begin{array}{rcl} \sigma_\mathrm{D} & \mbox{if} & {\mathbf{r}}\in\mathrm{D},\\ \sigma_\mathrm{B} & \mbox{if} & {\mathbf{r}}\in\mathbb{R}^3\backslash\overline{\mathrm{D}}, \end{array} \right.$$ respectively. Using this, we can define the background wavenumber $k$ as $$k=\omega^2\mu\left({\varepsilon}_\mathrm{B}+i\frac{\sigma_\mathrm{B}}{\omega}\right)=\frac{2\pi}{\lambda},$$ where $\lambda$ denotes the wavelength such that $\rho<\lambda/2$. Let ${\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}})$ be the incident electric field in a homogeneous medium because of a point current density at ${\mathbf{r}}_{\mathrm{\tiny TX}}$. Then, based on the Maxwell equation, ${\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}})$ satisfies $$\nabla\times{\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}})=-i\omega\mu{\mathbf{H}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}})\quad\mbox{and}\quad\nabla\times{\mathbf{H}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}})=(\sigma_\mathrm{B}+i\omega{\varepsilon}_\mathrm{B}){\mathbf{E}}_{\mbox{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}})$$ Analogously, let ${\mathbf{E}}_{\mathrm{\tiny tot}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})$ be the total field in the existence of $\mathrm{D}$ measured at ${\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}$. Then, ${\mathbf{E}}_{\mathrm{\tiny tot}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})$ satisfies $$\nabla\times{\mathbf{E}}_{\mathrm{\tiny tot}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})=-i\omega\mu{\mathbf{H}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})\quad\mbox{and}\quad\nabla\times{\mathbf{H}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})=(\sigma({\mathbf{r}})+i\omega{\varepsilon}({\mathbf{r}})){\mathbf{E}}_{\mathrm{\tiny tot}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})$$ with transmission condition on the boundary ${\partial}\mathrm{D}$ and the open boundary condition: $$\lim_{\|{\mathbf{r}}\|\to\infty}{\mathbf{r}}\bigg(\nabla\times{\mathbf{E}}_{\mathrm{\tiny tot}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})-ik\frac{{\mathbf{r}}}{\|{\mathbf{r}}\|}\times{\mathbf{E}}_{\mathrm{\tiny tot}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})\bigg)=0.$$ Let $\mathrm{S}(n)$ be the $S-$parameter, which is the ratio of the reflected waves at the $n-$th receiver ${\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}$ to the incident waves at the transmitter ${\mathbf{r}}_{\mathrm{\tiny TX}}$. Herein, $\mathrm{S}_{\mathrm{\tiny scat}}(n)$ denotes the scattered field $S-$parameter, which is obtained by subtracting the $S-$parameters from the total and incident fields. Based on [@HSM2], $\mathrm{S}_{\mathrm{\tiny scat}}(n)$ because of the existence of an anomaly, $\mathrm{D}$ can be represented as follows. This representation plays a key role in the DSM that will be designed in the next section. $$\label{Formula-S} \mathrm{S}_{\mathrm{\tiny scat}}(n)=\frac{ik^2}{4\omega\mu}\int_\mathrm{D}\chi({\mathbf{r}}){\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}}){\mathbf{E}}_{\mathrm{\tiny tot}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})d{\mathbf{r}},\quad\chi({\mathbf{r}})=\frac{{\varepsilon}({\mathbf{r}})-{\varepsilon}_\mathrm{B}}{{\varepsilon}_\mathrm{B}}+i\frac{\sigma({\mathbf{r}})-\sigma_\mathrm{B}}{\omega\sigma_\mathrm{B}}.$$ Indicator function of direct sampling method: introduction and analysis {#sec:3} ======================================================================= In this section, we design an imaging algorithm based on the DSM, which uses the collected $S-$parameters $\mathrm{S}_{\mathrm{\tiny scat}}(n)$ such that $\mathbb{S}={\left\{\mathrm{S}_{\mathrm{\tiny scat}}(n):n=1,2,\cdots,N\right\}}$. Because we assumed that $\mathrm{D}$ is a small ball such that $\rho<\lambda/2$, using the Born approximation, $\mathrm{S}_{\mathrm{\tiny scat}}(n)$ of (\[Formula-S\]) can be approximated as follows: $$\label{Formula-S2} \mathrm{S}_{\mathrm{\tiny scat}}(n)\approx\rho^3\frac{ik^2}{4\omega\mu}\chi({\mathbf{r}}_\mathrm{D}){\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}}_\mathrm{D}){\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_\mathrm{D},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}).$$ Based on this approximation, the imaging algorithm based on the DSM can be introduced as follows; for a search point ${\mathbf{r}}\in\Omega$, the indicator function of DSM is expressed as follows: $$\label{ImagingFunction} \mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}}):=\frac{\|\langle \mathrm{S}_{\mathrm{\tiny scat}}(n),{\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})\rangle_{L^2(\Gamma)}\|}{\|\|\mathrm{S}_{\mathrm{\tiny scat}}(n)\|\|_{L^2(\Gamma)}\|\|{\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})\|\|_{L^2(\Gamma)}},$$ where $\Omega$ is a search domain, $$\langle \mathbf{F}_1(n),\mathbf{F}_2(n)\rangle_{L^2(\Gamma)}:=\sum_{n=1}^{N}\mathbf{F}_1(n)\overline{\mathbf{F}}_2(n),\quad\mbox{and}\quad\|\|\mathbf{F}\|\|_{L^2(\Gamma)}=\left(\langle \mathbf{F},\mathbf{F}\rangle_{L^2(\Gamma)}\right)^{1/2}.$$ Then, $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ has a peak magnitude of $1$ at ${\mathbf{r}}={\mathbf{r}}_\mathrm{D}$ and a small magnitude at ${\mathbf{r}}\ne{\mathbf{r}}_\mathrm{D}$ so that the shape of anomaly $\mathrm{D}$ can be easily identified. Following [@IJZ2; @LZ], the structure of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ can be represented as follows: $$\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})\approx\|J_0(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)\|,$$ where $J_m$ is the Bessel function of the first kind of order $m$. However, this does not explain the complete phenomena that were illustrated in the simulation results in the next section; thus, further analysis is required. Through careful analysis, we can identify the structure of the indicator function as follows: \[StructureDSM\] Assume that the total number of antennas $N$ is small, a sufficiently large wavenumber $k$ and search point ${\mathbf{r}}\in\Omega$ satisfy $k\|{\mathbf{r}}-{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}\|\gg0.25$. Let ${\boldsymbol{\theta}}_n={{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}/\|{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}\|=(\cos\theta_n,\sin\theta_n)$ and ${\mathbf{r}}-{\mathbf{r}}_\mathrm{D}=\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|(\cos\phi_\mathrm{D},\sin\phi_\mathrm{D})$. Then, if ${\mathbf{r}}$ is far from ${\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}$, $$\label{ImagingFunction} \mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})=\frac{\|\Phi({\mathbf{r}})\|}{\displaystyle\max_{{\mathbf{r}}\in\Omega}\|\Phi({\mathbf{r}})\|},$$ where $$\label{Structure} \Phi({\mathbf{r}})=J_0(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)+\frac{1}{N}\sum_{n=1}^{N}\sum_{m\in\mathbb{Z}^\backslash{\left\{0\right\}}}i^m J_m(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)e^{im(\theta_n-\phi_\mathrm{D})}.$$ Here, $\mathbb{Z}^=\mathbb{Z}\cup{\left\{-\infty,\infty\right\}}$ and $J_m$ denotes the Bessel function of integer order $m$ of the first kind. Because $k\|{\mathbf{r}}-{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}\|\gg0.25$, applying (\[Formula-S2\]) and the asymptotic form of the Hankel function $$H_0^{(2)}(k\|{\mathbf{r}}-{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}\|)=\frac{1+i}{4\sqrt{k\pi}}\frac{e^{ik\|{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}\|}}{\sqrt{\|{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}\|}}e^{-ik{\boldsymbol{\theta}}_n\cdot{\mathbf{r}}}+o\left(\frac{1}{\sqrt{\|{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}\|}}\right),$$ we can observe that $$\begin{aligned} \langle \mathrm{S}_{\mathrm{\tiny scat}}(n),{\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})\rangle_{L^2(\Gamma)}&\approx\sum_{n=1}^{N}\rho^3\frac{ik^2}{4\omega\mu}\chi({\mathbf{r}}_\mathrm{D}){\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}}_\mathrm{D}){\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_\mathrm{D},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})\overline{{\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})}\\ &=\frac{ik\rho^3}{32\omega\mu\pi}\chi({\mathbf{r}}_\mathrm{D}){\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}}_\mathrm{D})\sum_{n=1}^{N}\frac{1}{\|{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}\|}e^{ik{\boldsymbol{\theta}}_n\cdot({\mathbf{r}}-{\mathbf{r}}_\mathrm{D})}.\end{aligned}$$ Because $\|{{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}}\|=R$, ${\boldsymbol{\theta}}_n\cdot({\mathbf{r}}-{\mathbf{r}}_\mathrm{D})=\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|(\cos(\theta_n-\phi_\mathrm{D}),\sin(\theta_n-\phi_\mathrm{D}))$, and the following Jacobi-Anger expansion holds uniformly, $$e^{ix\cos\theta}=J_0(x)+\sum_{m\in\mathbb{Z}^\backslash{\left\{0\right\}}}i^m J_m(x)e^{im\theta},$$ we can derive $$\begin{aligned} \sum_{n=1}^{N}e^{ik{\boldsymbol{\theta}}_n\cdot({\mathbf{r}}-{\mathbf{r}}_\mathrm{D})}&=\sum_{n=1}^{N}\left(J_0(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)+\sum_{m\in\mathbb{Z}^\backslash{\left\{0\right\}}}i^m J_m(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)e^{im(\theta_n-\phi_\mathrm{D})}\right)\\ &=NJ_0(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)+\sum_{n=1}^{N}\sum_{m\in\mathbb{Z}^\backslash{\left\{0\right\}}}i^m J_m(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)e^{im(\theta_n-\phi_\mathrm{D})}.\end{aligned}$$ Thus, we arrive at $$\begin{gathered} \langle \mathrm{S}_{\mathrm{\tiny scat}}(n),{\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})\rangle_{L^2(\Gamma)}\approx\frac{ik\rho^3}{32R\omega\mu\pi}\chi({\mathbf{r}}_\mathrm{D}){\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}}_\mathrm{D})\sum_{n=1}^{N}e^{ik{\boldsymbol{\theta}}_n\cdot({\mathbf{r}}-{\mathbf{r}}_\mathrm{D})}\\ =\frac{iNk\rho^3}{32R\omega\mu\pi}\chi({\mathbf{r}}_\mathrm{D}){\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}}_{\mathrm{\tiny TX}},{\mathbf{r}}_\mathrm{D})\left(J_0(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)+\frac{1}{N}\sum_{n=1}^{N}\sum_{m\in\mathbb{Z}^\backslash{\left\{0\right\}}}i^m J_m(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)e^{im(\theta_n-\phi_\mathrm{D})}\right).\end{gathered}$$ Using this, we apply H[ö]{}lder’s inequality $$\|\langle \mathrm{S}_{\mathrm{\tiny scat}}(n),{\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})\rangle_{L^2(\Gamma)}\|\leq\|\|\mathrm{S}_{\mathrm{\tiny scat}}(n)\|\|_{L^2(\Gamma)}\|\|{\mathbf{E}}_{\mathrm{\tiny inc}}({\mathbf{r}},{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)})\|\|_{L^2(\Gamma)},$$ to obtain (\[Structure\]). This completes the proof. \[Remark\]Based on the result of Theorem \[StructureDSM\], we examine some properties of the DSM. 1. Because $J_0(0) = 1$ and $J_m(0) = 0$ for all $m = 1, 2, \cdots,$ we can observe that $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})\approx1$ at ${\mathbf{r}}= {\mathbf{r}}_\mathrm{D}\in\mathrm{D}$. This is the theoretical reason for which the location of $\mathrm{D}$ can be imaged using the DSM. 2. The imaging performance is highly dependent on the value of $k$ and $N$, i.e., to accurately detect the location of $\mathrm{D}$, the value of $N$ must be sufficiently large. This is the theoretical reasoning for increasing the total number of antennas to guarantee good imaging results. 3. If the value of $N$ is not sufficiently large, the right-hand side of (\[Structure\]) $$\sum_{m\in\mathbb{Z}^\backslash{\left\{0\right\}}}i^m J_m(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)e^{im(\theta_n-\phi_\mathrm{D})}$$ will deteriorate the imaging performance by generating large numbers of artifacts. 4. If $N$ is sufficiently large, the effect of the deteriorating term becomes negligible and $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ becomes $$\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})\approx\|J_0(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)\|.$$ This result is same as the one derived in [@LZ]. 5. \[P5\] If the radius of $\mathrm{D}$ is larger than $\lambda$, then it is impossible to apply Born approximation (\[Formula-S\]). This means that the designed DSM cannot be applied to the imaging of extended targets. If multiple small anomalies $\mathrm{D}_l$, $l=1,2,\cdots,L$, whose radii, permittivities, and conductivities are $\rho_l$, ${\varepsilon}_l$, and $\sigma_l$, respectively, exist $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ can be represented as $$\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})=\frac{\|\Phi({\mathbf{r}})\|}{\displaystyle\max_{{\mathbf{r}}\in\Omega}\|\Phi({\mathbf{r}})\|},$$ where $$\Phi({\mathbf{r}})=\sum_{l=1}^{L}\rho_l^3\left(\frac{{\varepsilon}_l-{\varepsilon}_\mathrm{B}}{{\varepsilon}_\mathrm{B}}+i\frac{\sigma_l-\sigma_\mathrm{B}}{\omega\sigma_\mathrm{B}}\right)\left(J_0(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)+\frac{1}{N}\sum_{n=1}^{N}\sum_{m\in\mathbb{Z}^\backslash{\left\{0\right\}}}i^m J_m(k\|{\mathbf{r}}-{\mathbf{r}}_\mathrm{D}\|)e^{im(\theta_n-\phi_\mathrm{D})}\right).$$ Based on this structure, we can observe that the imaging performance of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ is highly dependent on the values of permittivity, conductivity, size of anomalies, and the total number of dipole antennas $N$. This means that if the permittivity, conductivity, or the size of one anomaly is significantly larger than that of the others, the shape of the anomaly can be identified via the map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$. Otherwise, it will be difficult to identify the shape of the anomaly via the map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$. Simulation results {#sec:4} ================== In this section, simulation results are presented to demonstrate the effectiveness of DSM and to support the mathematical structure derived in Theorem \[StructureDSM\]. For this purpose, $N = 16$ dipole antennas were used with an applied frequency of $f = 1 $GHz. For the transducer and receivers, we set $${\mathbf{r}}_{\mathrm{\tiny TX}}=0.09\mbox{m}\left(\cos\frac{3\pi}{2},\sin\frac{3\pi}{2}\right)\quad\mbox{and}\quad{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}=0.09\mbox{m}\left(\cos\theta_n,\sin\theta_n\right),\quad\theta_n=\frac{3\pi}{2}-\frac{2\pi(n-1)}{N}.$$ Hence, $R=\|{\mathbf{r}}_{\mathrm{\tiny RX}}^{(n)}\|= 0.09 $m. The $S-$parameters $\mathrm{S}_{\mathrm{\tiny scat}}(n)$ for $n = 1,2,\cdots,N$ were generated using the CST STUDIO SUITE. The relative permittivity and conductivity of the background were set to ${\varepsilon}_\mathrm{B} = 20$ and $\sigma_\mathrm{B} = 0.2 $S/m, respectively, the search domain $\Omega$ was set to be an interior of a circle with radius $0.085 $m centered at the origin, i.e., $\Omega={\left\{{\mathbf{r}}:\|{\mathbf{r}}\|\leq0.085\mbox{m}\right\}}$, and the step size of ${\mathbf{r}}$ to be of the order of $0.002 $m. \[Example1\] In this result, we consider the imaging of small anomalies. For this, we placed an anomaly at $(0.01\mbox{m},0.03\mbox{m})$ with a radius, relative permittivity, and conductivity of $\rho = 0.01 $m, ${\varepsilon}_\mathrm{D} = 55$, and $\sigma_\mathrm{D} = 1.2 $S/m, respectively. Figure \[Small\] shows the test configuration with the anomaly and the map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ with an identified location of $D$. Based on these results, we detected almost the exact location of the anomaly by considering that ${\mathbf{r}}$ satisfies $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})\approx1$. Furthermore, because of the presence of the infinite series of Bessel functions in (\[Structure\]), the appearance of artifacts was found to be quite different from the usual form shown in [@IJZ1; @LZ]. ![\[Small\]Test configuration (left), map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ (center) and the identified location of $\mathrm{D}$ (right).](1.eps "fig:"){width="32.50000%"} ![\[Small\]Test configuration (left), map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ (center) and the identified location of $\mathrm{D}$ (right).](2.eps "fig:"){width="32.50000%"} ![\[Small\]Test configuration (left), map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ (center) and the identified location of $\mathrm{D}$ (right).](3.eps "fig:"){width="32.50000%"} \[Example2\] To examine \[P5\] of Remark \[Remark\], we consider the imaging of extended anomalies. For this, we placed an anomaly at $(0.01\mbox{m},0.02\mbox{m})$ with a radius, relative permittivity, and conductivity of $\rho=0.05 $m, ${\varepsilon}_\mathrm{D} = 15$, and $\sigma_\mathrm{D}=0.5 $S/m, respectively. Figure \[Large\] shows the test configuration with the anomaly and a map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$. Based on these results, compared to the imaging of small anomalies in Example \[Example1\], it is impossible to recognize the shape of the anomaly. This result shows the limitation of DSM and that an improvement is necessary. ![\[Large\]Test configuration (left), map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ (center) and the ${\partial}\mathrm{D}$ (right).](4.eps "fig:"){width="32.50000%"} ![\[Large\]Test configuration (left), map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ (center) and the ${\partial}\mathrm{D}$ (right).](5.eps "fig:"){width="32.50000%"} ![\[Large\]Test configuration (left), map of $\mathfrak{F}_{\mathrm{DSM}}({\mathbf{r}})$ (center) and the ${\partial}\mathrm{D}$ (right).](6.eps "fig:"){width="32.50000%"} Conclusion {#sec:5} ========== We designed and employed DSM for fast imaging of small anomalies from $S-$parameter values. By considering the relationship between the indicator function and an infinite series of Bessel functions of integer order, certain properties of the DSM were examined. Based on the simulation results with synthetic data, we concluded that DSM is an effective algorithm for detecting small anomalies. Thus, we anticipate its development for its use in real-world applications such as breast cancer detection in biomedical imaging. Acknowledgement {#acknowledgement .unnumbered} =============== The author is acknowledge to Kwang-Jae Lee and Seong-Ho Son for providing $S-$parameter data from CST STUDIO SUITE. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2017R1D1A1A09000547).
--- abstract: 'The emergence of self-replication and information transmission in life’s origin remains unexplained despite extensive research on the topic. A hypothesis explaining the transition from a simple organic world to a complex RNA world is offered here based on physical factors in hydrothermal vent systems. An interdisciplinary approach is taken using techniques from thermodynamics, fluid dynamics, oceanography, statistical mechanics, and stochastic processes to examine nucleic acid dynamics and kinetics in a hydrothermal vent from first principles. Analyses are carried out using both analytic and computational methods and confirm the plausibility of a reaction involving the PCR-like assembly of ribonucleotides. The proposal is put into perspective with established theories on the origin of life and more generally the onset of order and information transmission in prebiotic systems. A biomimicry application of this hypothetical process to PCR technology is suggested and its viability is evaluated in a rigorous logical analysis. Optimal temperature curves begin to be established using Monte Carlo simulation, variational calculus, and Fourier analysis. The converse argument is also made but qualitatively, asserting that the success of such a modification to PCR would in turn reconfirm the biological theory.' author: - \| Stan Palasek\ Princeton University date: May 2013 title: \| Primordial RNA Replication\ and Applications in PCR Technology --- Introduction ============ The mutual dependence of proteins and nucleic acids in their replication leads to a catch-22 in the origin of life: the first organism would require enzymes to replicate its genome, but these enzymes cannot be synthesized without the nucleic acids that code for them. [@weaver p. 223-4] This paradox was first resolved with Thomas Cech’s discovery of self-splicing RNA in the protozoan Tetrahymena thermophila. [@kruger] The existence of such a “ribozyme” that can both carry genetic information and perform catalysis removes the necessity of an event during which proteins and nucleic acids spontaneously emerged. Thus, the idea that an RNA world preceded the modern DNA-RNA world, once speculation by Crick and his colleagues, [@crick; @orgel] had reached the mainstream as the “RNA world hypothesis.” [@gilbert] According to Cech, an RNA world transitioned to a ribonucleoprotein (RNP) world of protein-bound nucleotides which were favorable for their greater versatility in catalysis. A final transition led to the current DNA-RNA world along with the last universal common ancestor (LUCA) to take advantage of DNA’s greater resistance to hydrolysis. [@cech] The remaining problem is the origin of the ribozymes themselves. By the early 1990s, leading molecular biologists rejected “the myth of a self-replicating RNA molecule that arose de novo from a soup of random polynucleotides.” [@joyceorgel] Biophysicists of the last decade seem to have ignored this submission and went on to evaluate the plausibility of spontaneous assembly in the contexts of thermodynamics, statistical mechanics, and information theory. (e.g. [@baaske; @chen7; @obermayer; @chen12; @krammer]) This paper will proceed in the spirit of the cited physical approaches, particularly Obermayer 2011 where the objective is to use mathematics and computer simulation to identify qualitatively new phenomena. The premise here, as it was there, is that information stored in an RNA sequence is lost upon hydrolytic cleavage so somehow a lasting memory must have emerged, likely in the form of a replicator. It is imprudent to be overly concerned with the numerical details due to our ignorance of the chemical and thermal factors of ancient oceans. [@pagani; @pinti] Also in the manner of the cited literature, the origin of the nucleotide monomers themselves will not be addressed as this is a different problem entirely (see for instance [@powner] and the assumptions made in [@obermayer; @robertson]). Nucleotide Flow in a Hydrothermal Vent ====================================== Hydrothermal gradients ---------------------- Let the local energy per unit distance at a height $z$ above a hydrothermal vent be $U(z)$. The generic scalar transport equation in steady state becomes Poisson’s equation [@shubin p. 14], $\alpha\nabla^2U+P(\textbf{r})=0$ where $\alpha$ is the thermal diffusivity and $P$ is the instantaneous power per unit distance. The hydrothermal vent is treated as a point source with $P_{vent}=2P_V\delta(z)$ where $\delta(z)$ is the Dirac delta function evaluated at a height $z$ above the vent, requiring the factor of two so that the differential power integrates to $P_V$ on $\mathbb{R}^+$. With just the constant influx of hydrothermal energy, the system would never reach equilibrium. Let $U_0$ be the total thermal energy in steady state. Suppose that there is dissipation per unit distance proportional to the local energy density, so $P_{lost}=UP_V/U_0$ in such a way that the net power is 0. Poisson’s equation in the vertical direction takes the form of the screened Poisson equation [@fetter pp. 312-313] with $\lambda=\sqrt{P_V/\alpha U_0}$ and $f(z)=(2P_V/\alpha)\delta(z)$. $$\label{heat} \alpha\frac{d^2U}{dz^2}+2P_V\delta(z)-P_V\frac{U}{U_0}=0$$ A solution is desired for which $\int_0^{\infty}U(z)dz$ both converges and equals $U_0$. These two constraints, though not independent, uniquely solve the initial value problem, giving an exponential function with decay constant $\lambda$. Exploiting the linearity between temperature and and average thermal energy and defining the ambient ($z=\infty$) and vent ($z=0$) temperatures as $T_0$ and $T_V$ respectively yields the following temperature distribution. $$\label{expTemp} T(z)=T_0+(T_V-T_0)\exp\left(-\sqrt{\frac{P_V}{U_0\alpha}}z\right)$$ A new model for thermophoresis ------------------------------ Thermophoresis is the tendency of particles to move against temperature gradients. Its mechanism is up for debate and has been modeled primarily as an entropic phenomenon across only small gradients. [@duhr] Here a model is derived based on the principle that more collisions with a test particle occur from the direction of higher temperature. Let the test particle occupy a van der Waals region $\mathcal{R}$ with a surface $\partial\mathcal{R}$ and volume $V$. By the ideal gas law, the ratio of the pressure orthogonal to the surface to the temperature is $k_BC$, where $k_B$ is Boltzmann’s constant and $C$ is the local concentration of the solvent. Integration over the surface provides the total resulting force. $$\label{surfaceInt} \textbf{F}\hspace{-.02in}_{thermal}=-\oiint_{\partial\mathcal{R}}k_BCT(x,y,z)\,d\hspace{-.02in}\textbf{A}$$ It is shown[^1] that this surface integral can be converted to an integral over the volume. The limit is taken as the molecule’s size approaches zero and the temperature gradient becomes approximately uniform throughout the region. $$\label{netF} \eqref{surfaceInt}=-k_BC\lim_{\\|\mathcal{R}\\|\rightarrow 0}\iiint_{\mathcal{R}}\nabla TdV=-k_BCV\nabla T$$ The force field due to thermophoresis is therefore conservative. This is equivalent to Archimedes’ law with pressure replaced by the most probable (mode) ratio of thermal energy to volume. The constant is perhaps related to the Soret coefficient [@cussler pp. 522-3] of classical thermophoresis which has dimensions of inverse temperature rather than distance per temperature. Temperature oscillation ----------------------- The oscillations of interest are in the vertical direction over the hydrothermal vent, so consider in a single dimension superposed with constant acceleration of gravity $g$: $md^2z/dt^2=-k_BCV\partial_zT-mg$. Multiplying through by $dz/dt$ and integrating with respect to $dt$ explicitly gives the kinetic energy and gravitational potential energy as a function of temperature. Adding the work done by thermophoresis (integrating ) yields the constant Hamiltonian $H=k_BCV(T_V-T_0)$. When oscillations are small and the right hand side of the trajectory equation is approximated with a first-order Taylor polynomial, each nucleotide behaves as a harmonic oscillator with angular frequency $\omega$. $$\label{traj} x(t)=\left(\sqrt{\frac{U_0\alpha}{P_V}}-\frac{g}{\omega^2}\right)\left(1-\cos\omega t\right)\hspace{.4in}\omega^2=\frac{k_BCVP_V}{\alpha mU_0}(T_V-T_0).$$ Figure 1 shows temperature as a function of time as a result of of these oscillations without the harmonic approximation. The shaded blue curve is the theoretical deterministic trajectory from the nonlinear equation obtained numerically with a Runge-Kutta method [@hamming pp. 413-4]. The scattered data depict four runs of the process taking into account not only thermophoresis and gravity, but also buoyancy, diffusion, and the temperature dependence of the transport coefficients [@cussler p. 114]. The existence of thermal cycling is critical to the rest of this paper. ![Temperature oscillations from the perspective of a particle in a hydrothermal vent according to both deterministically (shaded blue) and stochastically (scattered).]("2012_Trajectories".PNG) The Quasi-Polymerase Chain Reaction Hypothesis ============================================== In the previous section it was shown that particles in a hydrothermal vent system experience temperature oscillations. Here it is proposed that these oscillations could mirror the thermal cycling that is the fundamental basis of the polymerase chain reaction (PCR). A “quasi-polymerase chain reaction” (quasi-PCR) would consist of two primary stages: denaturation and rehybridization. The former stage takes place most favorably near the vent so that the hydrogen bonds united the complementary RNA strands may be destroyed. The latter stage must take place far from the vent at the ambient ocean temperature so that loci that have undergone rehybridization—a slow process when uncatalyzed—are not prematurely broken by rogue thermal fluctuations. It is essential to provide long spans of time at this low temperature. The oscillations shown in Figure 1 are merely to illustrate their existence and are too wide and too frequent to result from a hydrothermal vent that creates a temperature gradient potentially larger than 400$^\circ$C. [@haase] Hybridized molecules will actually stay at cooler temperatures longer than this model predicts by virtue of their size: the diffusion coefficient is inversely proportional to the particle’s mass (also by the Stokes-Einstein equation [@cussler eodem loc.]), fluid density and with it buoyancy increase as the water temperature decreases, and acceleration due to gravity is intensive. This proposal will be examined in more detail with mathematics and in computer simulation in subsection 5.3. It should be emphasized that the author is not suggesting that a bona fide PCR could progress in a hydrothermal vent. Indeed, it would be contrary to the motivating RNA world hypothesis to assume that polymerase (the P in PCR) enzymes would have yet been in existence in the same vent system. Without enzymes, the primers that delineate the nucleic acid fragment to be replicated are not needed and neither is an annealing step. It is acknowledged that the reaction’s specificity and speed are compromised by removing these elements but it will be argued that only a simplified process is needed to lead in to an RNA world. The plausibility of the reaction without biological catalysts will be addressed in section 6. Analytic Modeling ================= A description of an analytic model for the process described in the previous section will be presented before the Monte Carlo simulation as its consequences will be needed throughout the rest of the paper. Quasi-PCR as a branching process -------------------------------- A method for examining the rate of polynucleotide branching in continuous time with discrete molecules will be developed. Let $f_n(t)$ be the probability mass function for the number of molecules ($n$) as a function of time ($t$). Branching of a given molecule occurs at a time-dependent rate $R(t)$, meaning the probability it branches (replicates) on an interval $[t,t+\Delta t]$ as $\Delta t\rightarrow0$ is $R(t)\Delta t$. Using a finite difference $\Delta t$, $f_n(t)$ can be expressed recursively as $$\label{rec} f_n(t+\Delta t)=(n-1)f_{n-1}(t)R(t)\Delta t+f_n(t)\left[1-n R(t)\Delta t\right]\textnormal{ for }n=1,2,3,\ldots$$ meaning the probability of being at $n$ on a given time step is the probability of being at $n-1$ on the previous time step and making a successful transition plus the probability of being at $n$ on the current time step and not making a successful transition. The probability a transition occurs is proportional to the number of loci involved.[^2] In the continuum limit, $\left[f(t+\Delta t)-f(t)\right]/\Delta t$ becomes $d\!f_n(t)/dt$. $$\label{diff} \frac{d\!f_n(t)}{dt}=R(t)\left[(n-1)f_{n-1}(t)-nf_{n}(t)\right]=-R(t)[n\Delta(f_n(t))+f_{n-1}(t)]\textnormal{ for }n=1,2,3,\ldots$$ adopting the notation of difference equations in the second representation. This system of countably infinite differential equations or, equivalently, a combined differential equation and recursive relation, has initial conditions $f_{0}(t)=0$ for all $t$ meaning that a system from non-zero intial conditions can never have zero particles and $f_n(0)=\delta_{n-1}$ for all $n$ where $\delta$ is the Kronecker delta meaning at $t=0$ the system is certain to contain only one molecule.[^3] The general solution of the first-order linear equation is known, [@santos] allowing the representation of $f_n$ as an explicit function $f_{n-1}$. $$\label{branchonce} f_n(t_n)=(n-1)\int_0^{t_n}\exp\left(-n\int_{t_{n-1}}^{t_n}R(k)dk\right)f_{n-1}(t_{n-1})R(t_{n-1})dt_{n-1}$$ During process of the calculation, $t_n$ will be used as the argument of the function $f_n$ to disambiguate between the infinite independent variables.[^4] Each $f_k$ can be replaced iteratively by an expression in terms of $f_{k-1}$ using until $f_1$ is reached. Recalling that $f_0(t)=0$ and $f_1(0)=1$, yields $f_1(t)=\exp\left(-\int_0^tR(k)dk\right)$. $$(n-1)!\int_0^{t_n}\cdots\int_0^{t_2}\exp\left(-\sum_{i=1}^{n-1}\int_{t_i}^{t_{i+1}}(i+1)R(k)dk\right)\exp\left(-n\int_0^{t_1}R(k)dk\right)\prod_{i=1}^{n-1}R(t_i)dt_1\cdots dt_{n-1}$$ The integrals within the exponential do not simply combine because each one has a different coefficient. They can instead be combined into $n-1$ integrals with the same coefficients by combining the first through the $(n-1)$st, including the separate contribution from $f_1$, into a single integral over $[0,t_n]$; then the second to the $(n-1)$th into one on $[t_1,t_n]$; etc. Now the integrand can be concisely written as $\prod_{i=1}^{n-1}\exp\left(-\int_{t_i}^{t_n}R(k)dk\right)R(t_i)dt_i$ and the substitution $u_i=-\int_{t_i}^{t_n}R(k)dk$ can be made leaving only $\exp(u_1+\cdots+u_{n-1})du_1\cdots du_2$. From here it is found and can be easily verified by induction or by substitution into that $f_n(t_n)=\exp\left(-\int_0^{t_n}R(k)dk\right)\left[1-\exp\left(-\int_0^{t_n}R(k)dk\right)\right]^{n-1}$. Finally, we revert to $t$ without the subscript and rescale such that $f_n(0)=\delta_{n-n_0}$ where $n_0$ is the initial quantity of molecules. $$\label{branch} f_n(t)=\exp\left(-n_0\int_0^tR(k)dk\right)\left[1-\exp\left(-n_0\int_0^tR(k)dk\right)\right]^{n-n_0}$$ Recalling that $f$ is a probability mass function over the random variable $n$, if there were not a logical flaw, would be properly normalized.[^5] The distribution is further confirmed in Figure 2 in which the narrowness of the confidence interval (red) is a consequence of the very little deviation from the mean scaled by the square root of the sample size. Taking the sum from $n=n_0$ ad infinitum, the exponential component is independent of $n$ and factors out of the sum, while all that remains is the sum of $(1-x)^n$ as $n$ goes from 0 to infinity. This is $1/x$ by the formula for a geometric series and happens to be the precise inverse of the exponential in . Thus it is normalized. To compute the expectation of the random variable $n$ as a function of time, compute the sum of $nf_n(t)$. Some manipulation gives $$\label{mean} \langle n\rangle=n_0-1+\exp\left(n_0\int_0^tR(k)dk\right)$$ where the angular brackets denote the expectation operator. It can be seen that is identical to the probability mass function for the shifted geometric distribution [@turin p. 68] shifted by $n_0$ rather than the conventional 1 and with $\lambda=\exp\left(-\int_0^tR(k)dk\right)$. This equivalence is only formal as nowhere in the formulation is there a notion of a Bernoulli process. ![The dark line on the left is the number of molecules as a function of time predicted by when $R(t)=1+\sin(3t)$, surrounded by the 95% confidence interval for the mean based on a sample of $10^4$ of such processes with $\Delta t=0.01$. The predicted and observed correspond closely. $R(t)$ is depicted below on the same time scale. To the right is the distribution of the number of molecules at the final time apposed with the geometric distribution given by .]("newpicture".PNG) Computing $R(t)$ ---------------- $R(t)$, the rate at which complete replications occur for a polynucleotide of length $m$, is (the probability that $m-1$ loci are hybridized at time $t$) $\times$ (the rate of hybridization of the final $m$th locus). The former probability is given simply by the binomial distribution, $m(1-p_0)p_0^{m-1}$ where $p_0$ is the probability that a given locus is hybridized. Under several circumstances this can be approximated by $mp_0^m$, including when $p_0\approx1/2\Rightarrow p_0\approx1-p_0$, when $l$ is large and edge effects are negligible, and when partial degradation of a fully replicated strand is infrequent. The latter justification can be put quantitatively as $p=(\textnormal{probability completely replicated})=p_0^m$ (assuming independence) such that when the reverse reaction is improbable $d\!p/dt=mp_0^{m-1}d\!p_0/dt$ and $d\!p_0/dt$ behaves as the rate of hybridization of the $m$th locus. One could then easily take into account the reverse reaction with $R-m\times(\textnormal{denaturation rate})=d\!p/dt$. $p_0$ has two multiplicative components that are assumed to be independent: the probability the original double strand has been denatured and the probability it is in a hybridized state given it has been denatured. In order to make this problem tractable, we assume that the change in thermal energy is slow compared to the molecular and submolecular processes being examined here. The slowness that follows from this kind of adiabadicity would confine the system to a state infinitesimally close to equilibrium.[^6] This allows many of the processes considered here to be accurately treated as Poisson (memoryless). First we compute the steady-state probability a locus is in a denatured state. This must be significantly lower for longer molecules since many loci will be quickly denatured and remain idle while the remainder of the molecule is being hybridized. Let $r_+$ and $r_-$ be the respective hybridization and denaturation rates, borrowing from the notation of [@obermayer]. The probability the Poisson event representing denaturation has occurred as of time $t$ from the beginning of a quasi-PCR cycle is $1-e^{-r_-t}$. [@turin p. 400] On average, the length of a cycle is the inverse of the turnover rate ($1/R$). The probability of a randomly chosen point in time being within the time interval $[t,t+dt]$ into the current cycle is $dt/(\textnormal{average cycle period})=Rdt$. Therefore, the expected probability of denaturation is strangely given by the integral of an already cumulative distribution, $\sum_{t\in\textnormal{period}}P[\textnormal{denatured~at~}t] \times P[t\textnormal{~into~the~cycle}]=\int_0^{1/R}\left(1-e^{-r_-t}\right)Rdt=1-\frac{R}{r_-}\left(1-e^{-r_-/R}\right)$. The memorylessness of hybridization-dehybridization that can be assumed in quasistatic equilibrium implies, finding the steady state of the simple two-state Markov chain, that the probability of being in a hybridized state is $1/(1+r_-/r_+)$. These results combined finally yield $p_0$ (again, assuming the independence of the probabilities). $$\label{p0} p_0=\frac{1-\frac{R}{r_-}\left(1-e^{-r_-/R}\right)}{1+r_-/r_+}\sim\frac{1}{2R\left(r_+^{-1}+r_-^{-1}\right)}=\frac{r_{\textnormal{harmonic mean}}}{2R}\textnormal{ for non-negligible replication } (R\gg r_-).$$ Solving this with $R-mr_-=mp_0^{m-1}d\!p_0/dt$ gives $R$ but is dependent on which rates are considered invariant with time. Specifics will not be given here in the context of quasi-PCR as these mathematics do not become useful until biotechnology applications are discussed toward the end of the paper where the kinetics fundamentally change. Monte Carlo Simulation of Quasi-PCR =================================== Due to the system’s complexity, it is not practical to rely entirely on analytic methods. The implementation of a Monte Carlo simulation of the process outlined in Section 3 is detailed here. Implementing the hydrothermal gradient -------------------------------------- The temperature gradient is of magnitude $\frac{\partial T}{\partial z}=-\lambda(T_V-T_0)\exp\left(\lambda\,z\right)$ where $\lambda$ is again $\sqrt{\frac{P_V}{U_0\alpha}}$ and seems to act as a linear approximation for the exponential temperature gradient. The ratio $P_V/U_0$ is the ratio of the vent’s power to the ocean’s total potential energy relative to the ambient energy on the order of $k_BT_0$. It can thus be thought of as a turnover rate for energy in a vent system. $U_0$ can be estimated based on the temperature at the base of the vent. Direct solution of shows that the linear energy density at the base is $\sqrt{\frac{P_VU_0}{\alpha}}$. In a single dimension, the average kinetic energy of a single particle relative to the energy at the ambient temperature is $\frac{1}{2}\,k_B(T_V-T_0)$. Letting $\rho$ be the linear density of particles, we can equate the two energy densities, solve for $U_0$, and rewrite $\lambda=2P_V/[\alpha\rho k_B(T_V-T_0)]$. The remaining parameters must be empirical. Little et al. examined hydrothermal vent flow in the East Pacific Rise, finding power $P_V=3.7\pm0.8$ MW. [@little] Thermal conductivity values for water at various high temperatures and pressures were established at the Sixth International Conference on the Properties of Steam. [@sengers] Vent temperatures and pressures on the Mid-Atlantic Ridge fall on or above seawater’s critical point of 407$^\circ$ C and 29.8MPa. [@kos] Implementing thermophoresis --------------------------- To compute the particle dynamics as modeled by , the van der Waals volumes of nucleotides and water are computed using Zhao’s method. [@zhao] Averaging over pyrimidine- and purine-derived bases and assuming Chargaff’s rules apply [@weaver p. 140], the constant factor in can be obtained with dimensions of specific (intensive) entropy. Rearranging the equation gives an acceleration due to thermophoresis $\textbf{a}=\left(421.5\frac{\textnormal{J}}{\textnormal{kg\,K}}\right)\nabla T$. In the limit of long polynucleotide length, molecular mass becomes proportional to van der Waals volume and the thermophoretic acceleration becomes independent of the length. Implementing diffusion ---------------------- The dynamics are modeled as a finite difference process with time step $\Delta t$ where the state of a particle at time $t$ is determined by its position $z_t$ and velocity $v_t$. On each step, the position is updated nondeterministically to $z_{t+\Delta t}=z_t+v_t\,\Delta t$. Then velocity is updated[^7] to $v_{t+\Delta t}=v_t+a(z_{t+\Delta t},l)\,\Delta t$ where $a$ is the acceleration that is explicitly dependent on both position and polymer length. Particles diffuse in a Gaussian distribution centered about the average drift of the substance in bulk with variance $2Dt$ where $D$ is the diffusion coefficient. [@cussler p. 37] To achieve this, when computing $x_t$ at each step, a Gaussian random variable is added with variance $2D\Delta t$. Since the difference is finite and large thermal fluctuations can occur, it is necessary to reset the particle at $z=0$ if an anomaly puts it in an unphysical location. The diffusion coefficient must vary dramatically with the temperature and molecule size. The standard Stokes-Einstein relation cited earlier will not be appropriate because it is restricted to spherical particles. Instead we will resort to the most general Einstein relation $D=\mu k_BT$ where $\mu$ is the particle’s mobility. [@einstein] In the case of spherical particles and low Reynolds number the mobility would be replaced by the reciprocal drag coefficient which is given simply by Stoke’s law as $\textit{drag}=6\pi\eta R$ where $\eta$ is the dynamic viscosity of the fluid and $R$ is the sphere’s radius. However it would be naïve to assume that $R$ scales with molecule size without going back to first principles. Ideally, one would begin with the Navier-Stokes equations and use a cylindrical coordinate system (where long molecules can be neatly represented) rather then a spherical one (which is most convenient computationally because the stream function can be decomposed into radial and polar components). This formulation is still in progress. Instead, consider the fact that precisely one-third of the drag force is due to pressure and the rest is the result of shear stress. [@mit] The magnitude of a pressure gradient is dependent on the vertical size of the molecule. This size can be modeled as the difference between the maximum and minimum heights reached in a random walk in continuous three-dimensional space with fixed step size (neglecting that a molecule is self-avoiding). A Monte Carlo simulation shows that this range has a least-squares fit of $l_Bl\,^{0.516}$ ($R^2>0.9999$, best fit in the asymptotic limit of large $l$ as shown in Figure 3) where $l_B$ is the length of a single segment. The viscosity, on the other hand, is a consequence of stress between the fluid and the molecule which must be proportional to the surface area. Nucleic acids with their negatively charged phosphate groups will conform to maximize this surface area, making it proportional to the molecule length. Weighting the dependencies on $l^{1/2}$ and $l$ based on their contributions to the total drag gives $\textit{drag}=2\pi\eta R(\hspace{-.02in}\sqrt{l}+2l)$ where $R$ is the radius of a single nucleotide approximated by a sphere. In the monomeric case, this formula correctly reduces to the Stokes-Einstein equation. ![Average sizes of folded nucleotides modeled with a random walk, computing the size as a function of length up to length 1000 sampling 1000 conformations. The unit for distance is the length of a monomer. The log-log scale makes clear that the linear dimension asymptotically approaches precisely $\sqrt{m}$. This is confirmed in the residual plot in which the magnitude of the sampling error is correctly on the order of the square root of the observed value.]("sqrt".PNG) Implementing bond kinetics -------------------------- Maxwell-Boltzmann statistics describe the probability that a given particle is in a state with energy $E$ as proportional to $\exp\left(-\frac{E}{k_BT}\right)$. [@gibbs] Then one can integrate to find a quantity proportional to the probability the energy exceeds $E$. For small $T$, this mirrors the simplest form of the Arrhenius equation where $E$ is the activation energy. For large $T$, however, the integrated form is more accurate as now the reaction rate diverges linearly with $T$. $$\textnormal{reaction rate}=rk_BT\exp\left(-\frac{E}{k_BT}\right)=-rE+rk_BT+O\left(\frac{1}{T}\right)$$ where $r$ is a constant with dimensions of inverse action.[^8] This constant is dependent upon the concentration of the reactants (free nucleotides) so it cannot be known with any certainty for primordial kinetics. Nonetheless, it should be kept in mind that the purpose of this study is to determine the plausibility of phenomena so it will be sufficient to give a range of $r$ values for which the proposed reaction can progress. In the Monte Carlo simulation, the probability of a reaction occurring successfully will be $\textnormal{rate}\times\Delta t$ such that in the limit as $\Delta t\rightarrow0$ the average waiting time for an event is $1/\textnormal{rate}$ provided the rate remains constant (a Poisson process). Simulation results ------------------ Figure 4 depicts the progression of this process using the ensemble of estimated constants that was described throughout Section 5. ![RNA molecules vs. time on a log scale for a quasi-polymerase chain reaction modeled as Sections 4 and 5 describe it (blue). In purple is the same trial but removing the particles’ mobility that is critical to quasi-PCR. To the right is the quasi-PCR efficacy after 100 seconds as a function of the rate constant that depends on physical factors around a prebiotic hydrothermal vent. Since this must be empirical, a range of plausible values is given as promised in subsection 5.4.]("newpicture2".PNG) It is seen that at the constant vent temperature there is virtually no replication; after these 3000 seconds, the probability a molecule has been replicated is $\sim2\%$, averaged over 1000 runs of the process. Because by the end the true quasi-PCR simulation generates $\sim100$ independent polynucleotides that each become an effectively parallel computation, there was no need to examine many separate systems. The in silico power of the proposed quasi-PCR establishes the reaction as a potential replication mechanism that deserves further examination.[^9] Discussion: Quasi-PCR ===================== Mentioned in the introduction is ribozyme-discoverer Thomas Cech’s view on the progression from complex organic molecules to an RNA world to an RNP world to the modern world as it is expressed in his 2011 paper. [@cech] He cites a “likely self-replicating systems that preceded RNA” and describes a potential replicating molecule that would be related to RNA. The molecular biology literature appears to be caught up in the specific chemistry that could have allowed repeated ligation of a polynucleotide. Here it is shown that the thermodynamics of the system could provide sufficient impetus for even reactions of high energy barrier to occur. As for pre-enzymatic reactivity, Baaske et al. [@baaske] describe how deep-sea thermal processes induce extreme accumulation of particles in vent pores, creating ideal settings for reactions. These successes show that the focus henceforth needs to be on the general physical factors that could have produced the inexplicable emergence of complexity. Indeed, the second plot in Figure 4 shows that a quasi-PCR can progress at a reasonable pace with the hybridization rate varying over four orders of magnitude. This suggests that the mechanism presented thus far can provide a more general framework for rapid polymerization in hydrothermal vent systems, irrespective of the detailed chemistry. On an even more fundamental level, this reaction is analogous to the classic Urey-Miller experiment [@miller] as they both involve systematically shocking small molecular components in hopes that several will exceed some high activation energy to ultimately yield a more ordered product.[^10] The mechanism presented here is particularly effective because it makes use of the large energy influx from a super-powerful 1300 horsepower hydrothermal vent. Compare the directness and minimal stochasticity of this hypothesis to that of Obermayer et al. [@obermayer] which has the same purpose but depends on a subtle probabilistic tendency for hydrolytic cleavage to occur at unhybridized loci. [@usher] Though this did effectively induce a selective pressure for more complex RNA conformations allowing the investigators to see recurring sequence as well as structural motifs, there was no observation of strong exponential information replication akin to the one established in this paper. Applications: PCR Technology ============================ Nonlinear PCR ------------- It should be noted a priori that two arguments will be given and because one is the converse of the other, they must be thought of as independent assertions to avoid circularity. A fundamental difference between true PCR as it is used in the laboratory and quasi-PCR is the nature of the thermal cycling: while the former is carried out in very discrete stages of denaturation, annealing, and elongation, the latter involves the periodic and continuous temperature oscillations shown in Figure 1. [l]{}[0.5]{} ![image](bijective.png){width="53.00000%"} The inevitable consequence of discrete cycling is that each stage must be maintained long enough so that the expected proportion of the molecules that are successful in whatever process is associated with that stage is greater than a certain near-unity threshold value. One familiar with mathematical optimization might find it absurd that the ideal temperature function out of all functions that map positive reals to positive reals would be a step function. Let “nonlinear PCR” be a PCR reaction whose temperature function was carefully extremized. The applicability of such a reaction will be supported by an argument of biomimicry of quasi-PCR. Conditions for biomimicry of quasi-PCR -------------------------------------- “Quasi-PCR” and “RNA” will refer respectively to the thermodynamics of the quasi-PCR reaction as they are outlined in subsection 2.3 and to the extant genetic molecules. The underlying argument is as follows. Let $T$ denote the set of all possible thermal cycling phenomena and $I$ the set of all possible information transmitters. Thus we can say $\textnormal{quasi-PCR}\in T$ and $\textnormal{RNA}\in I$. Also let $f:T\rightarrow I$ and $I\rightarrow T$ meaning that given the argument of $f$ occurred, the output of $f$ is the member of the codomain that results in the most effective prebiotic replication. For the moment we take as a premise the quasi-PCR hypothesis: RNA came about in a PCR-like process with Figure 1’s nonlinear temperature cycles. This would suggest that the hereditary molecules present today were those which most prevailed in quasi-PCR, ie. $f(\textnormal{quasi-PCR})=\textnormal{RNA}$. In order to justify biomimicry, the converse $f(\textnormal{RNA})$ must be determined. It will be shown that this equals quasi-PCR as one might expect only when bijectivity between $T$ and $I$ is assumed. Suppose $t_k$ and $i_k$ for $k=1,2,3,\dots$ are distinct members of $T$ and $I$ respectively excluding quasi-PCR and RNA. It was already argued that $\textnormal{quasi-PCR}\rightarrow\textnormal{RNA}$. $f(\textnormal{RNA})$ may take on one of two values: quasi-PCR (this is the bijective case), or something distinct represented by $t_1$. From here $t_1$ can map back to RNA (this is the $\textnormal RNA\leftrightarrow t_1$ case) or to a distinct $i_1$. Now $i_1$ cannot map back to quasi-PCR because that implies that a reaction between quasi-PCR and $i_1$ would be more effective than a reaction between $i_1$ and $t_1$ which must be more effective than $t_1$ with RNA (otherwise $t_1$ would map back to RNA but $i_1$ is distinct) which must be more effective than quasi-PCR with RNA (otherwise RNA would map back to quasi-PCR which was already the bijective case). Since $i_1$ cannot map to quasi-PCR, it must either map to $t_1$ (this is the $t_1\leftrightarrow i_1$ case) or to a distinct $t_2$. This method can be continued ad infinitum and three types of structures will be left: bijectivities with inverse ($\textnormal{quasi-PCR}\leftrightarrow\textnormal{RNA}$ and $t_k\leftrightarrow i_k$ cases), chains of $t_1\rightarrow i_1\rightarrow t_2\rightarrow i_2\rightarrow\cdots$ interrupted by bijectivities, and chains that continue indefinitely. In practice, however, arbitrarily long chains cannot exist as at each link the replication must become more rapid but there must be a practical upper limit on its speed. With the two possible structures that remain, if bijectivity between the $T$ and $I$ can be shown, then $f$ must be its own inverse function establishing that $f(\textnormal{RNA})=\textnormal{quasi-PCR}$. Mapping $I$ to $T$ ------------------ The previous paragraph concluded that upon the premise of the quasi-PCR hypothesis, biomimicry of quasi-PCR is valid if and only if the following statement is accurate: If A is the thermal system that best replicates genetic molecule B, then B is the genetic molecule best replicated by A. Here we develop a method to uniquely map information carriers to the thermodynamic system that would most encourage replication. There is no need to proceed de novo as most of the requisite mathematics were already formulated in this paper in the analysis of quasi-PCR. We wish to maximize the expected number of molecules that will be left after replication which is given by . This is equivalent to extremizing the functional $\int R(t)dt=\int\left(\frac{d}{dt}p_0^m+mr_-\right)dt$ which is a variational problem. $p_0$ is given in . The degradation rate $r_-$ will be the function subject to optimization since it can be explicitly expressed in terms of the temperature (see 5.4). The hybridization rate $r_+$ will be treated as a constant since we are now operating under the conditions of enzyme-catalyzed modern PCR. Since the inner function has no explicit time dependence, we can use Gelfand and Fomin’s result that the Euler-Lagrange equation reduces to [@gelfand p. 19] $$\label{awful} R-\frac{dr_-}{dt}\frac{\partial R}{\partial r_-'}=C\hspace{.08in}\textnormal{where}\hspace{.08in}R=\frac{d}{dt}\left[\frac{1-\frac{R}{r_-}\left(1-e^{-r_-/R}\right)}{1+r_-/r_+}\right]^m+mr_-\textnormal{ from }\eqref{p0}$$ and $C$ is the constant obtained from reducing the order of the equation. The author is currently working to approximate an analytic solution to while preserving its critical features.[^11] Until then, preliminary Monte Carlo simulations for several temperature functions can be presented. Figure 6 depicts the average factor by which the number of molecules has been amplified for constant, square-wave, and quasi-PCR temperature functions. Note that simulations were only carried out for polynucleotides of length five and the results are expected to become even more dramatic for longer chains. The initial drop is due to the initial single molecule consistently undergoing denaturation among all the trials. It is the recovery from this depletion that occurs at differential rates among the paradigms. ![Mean number of complete replications in simulated PCR with constant temperature $\sim 100^\circ C$ (blue), square wave temperature oscillations (purple), and quasi-PCR oscillations (gold). Averages are taken over 50 trials beginning from a single molecule. The latter method appears best, but analysis is by no means complete as there is no obvious way to put the paradigms on equal footing for a comparison when they are so fundamentally different.]("2013_PCR_Monte_Carlo_7".png) Variational calculus over periodic functions -------------------------------------------- It is a concern that the optimal temperature function according to may not be characteristic of a chain reaction—that is, it may not be periodic. A method is developed here to force periodicity when extremizing functionals that depend on the function and first time derivative. Let the functional operating on temperature $T$ be $\mathcal{L}=\mathcal{L}\{T,T'\}$. For simplicity and since it is reasonable to start the chain reaction at the peak temperature, we will assume $T$ is an even function with angular frequency $\omega$. Using a Fourier cosine series we can rewrite the functional as $$\mathcal{L}=\mathcal{L}\left\{\frac{a_0}{2}+\sum_{i=1}^{\infty}a_i\cos(i\omega t),-\omega\sum_{i=1}^{\infty}ia_i\sin(i\omega t)\right\}.$$ Now this is an optimization of a countably infinite number of variables so we can resort to methods of traditional rather than variational calculus. In other words, all the derivatives of $\mathcal{L}$ with respect to the coefficients must identically vanish when the coefficients are optimal. Using the multivariable chain rule, when $j=1,2,3,\ldots$, $$\frac{\partial\mathcal{L}}{\partial a_j}=\frac{\partial\mathcal{L}}{\partial T}\frac{\partial T}{\partial a_j}+\frac{\partial\mathcal{L}}{\partial T'}\frac{\partial T'}{\partial a_j}=\cos(j\omega t)\frac{\partial\mathcal{L}}{\partial T}-\omega j\sin(j\omega t)\frac{\partial\mathcal{L}}{\partial T'}=0.$$ After rearrangement, the implicit function theorem allows the ratio of the partial derivatives to be reduced. $$\frac{\partial\mathcal{L}/\partial T}{\partial\mathcal{L}/\partial T'}=-\left(\frac{\partial T'}{\partial T}\right)_{\!\mathcal{L}}=\omega j\tan(j\omega t)\hspace{.1in}\textnormal{for }j=1,2,3,\ldots$$ where the subscript means $\mathcal{L}$ is being held constant as $T$ and its derivative are allowed to vary. Note the minus sign that is not a typo but is required by the implicit function theorem. Computation is likely easier using the less attractive $\partial_T\mathcal{L}/\partial_{T'}\mathcal{L}$. The optimization was done for the symbol temperature but of course it would generalize to $r_-$ when applied to . $R(t)$ for general chain reaction processes ------------------------------------------- In a scenario more general than PCR, consider two processes $A$ and $B$ which occur in turn with respective temperature-dependent rates of $R_A(T)$ and $R_B(T)$. Let $s_0$ and $s_1$ be, respectively, the initial state and the state after process $A$ has occurred but before $B$. $B$ then is the transition from $s_1$ to $s_2$, making the reaction cyclic. Letting $\textbf{P}(t)$ be the probability vector over the system’s states, the evolution has a simple master equation. $$\label{master} \frac{d\textbf{P}}{dt}=\left(\begin{smallmatrix} -R_A & R_B \\ R_A & -R_B \end{smallmatrix}\right)\textbf{P}$$ Given that all components begin in $s_0$ and that $R(t)=R_2(t)\times P[\textnormal{final state}]$ as described in subsection 4.2, the first-order linear system yields $$R(t)=R_B(t)\int_0^tR_A(\tau)\exp\left(-\int_\tau^t(R_A(k)+R_B(k))dk\right)d\tau$$ where the explicit temperature dependence of $R_A$ and $R_B$ are omitted. Again, the functional to be maximized is the integral of $R(t)$ from 0 to a constant so as to extremize . Applications support theory --------------------------- Here the independent converse argument mentioned in this section’s introduction will be briefly described. The biomimicry proposal was premised upon the validity of the quasi-PCR hypothesis. Though evidence can be gathered for quasi-PCR, one can of course never be certain of what actually occurred in prebiotic times. Therefore one can never be certain whether nonlinear PCR is actually mimicking anything that ever occurred in biology. What one can plausibly be certain of is that our current biochemistry is such that it replicates very effectively when subjected to the thermal conditions of subsection 2.3. This then would serve as evidence that quasi-PCR was responsible for the emergence of lasting genetic memory. Figure 7 shows the relationship between these two arguments and highlights some of the peculiarities of the proposals made here. While the previous subsections made the unusual claim that hypothetical primordial biology applies to modern technology, this section is perhaps more obscure in its assertion that inferences about prebiotic systems can be made based on observations in biotechnology. This all stems from the unorthodox nature of the biomimicry presented here which is mimicking the physical environment in which the biological system is immersed, not the biology itself. ![Given either the validity quasi-PCR or the favorability nonlinear PCR, the other is suggested. If the assumptions for both arguments are not carefully formulated they may appear circular.](cycle.PNG) Conclusion ========== A simulation of nucleotide flow, ligation, and cleavage encompassing all physical factors thought to be in play in prebiotic hydrothermal vents was conducted. Vent thermodynamics were shown to induce oscillations of temperature from each particle’s perspective which result in a polymerase chain reaction-like process (quasi-PCR). Values for the rate constant for hydrogen bond degradation could not be obtained from the literature because a more precise denaturation model was derived in this paper that yields accurate kinetics for the large range of temperatures over which quasi-PCR operates. Nonetheless, it was shown that the reaction can progress effectively over several orders of magnitude of this constant. Thus one can be confident that reactions like the quasi-PCR could proceed in prebiotic systems. However, due to the nature of studying the origin of life, one can never be certain if this was truly the reaction that provided the critical transition from inert organic compounds to the self-replicating information-storing pioneer molecules of the RNA world. This specificity is not needed for the results of this paper to be significant because, as discussed in section 6, quasi-PCR provides a very general framework for spontaneous polymerization that is fundamentally analogous to both the Urey-Miller experiment and diversity through radiation-induced mutation. The fact the quasi-PCR can occur, regardless of whether it did motivates the conclusion that this instrumental transition in the origin of life without which the RNA world hypothesis has no credibility is thermodynamically possible. The biotechnology application was an unexpected implication of what began as a pursuit in pure biology. The initial computer simulations appear promising, but as discussed earlier, the “experiments” are were not yet carefully controlled. The most definitive conclusion on nonlinear PCR will be reached once the variational calculus approach (in progress) is completed. [99]{} Weaver, R. F., and Hedrick, P. W. Genetics, Wm. C. 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Sci. USA 73, 1149-1153. Schrödinger, E. What is Life?, Cambridge University Press: Cambridge 1992. Gelfand, I. M. and Fomin, S. V. Calculus of Variations, Dover Publications: New York, NY 1963. Bender, A. E. An Introduction to Mathematical Modeling, Wiley: New York, NY 1978. Wolfram, S. A New Kind of Science, Wolfram Media: Champaign, IL 2002. [^1]: Assume $\mathcal{R}$ is compact and has a piecewise smooth boundary. Applying the divergence theorem to a vector field $\textbf{F}(x,y,z)=T(x,y,z)\,\textbf{c}$ where $\textbf{c}$ is a constant vector eventually gives $\textbf{c}\cdot\left(\oiint_{\partial\mathcal{R}}Td\textbf{A}-\iiint_R\nabla TdV\right)=0$ for all $\textbf{c}$. [^2]: The probability is actually given by $1-(1-r\Delta t)^n$ which, when $\Delta t\approx0$, has a Taylor approximation of $nr\Delta t$. [^3]: Other initial conditions can be accommodated simply by scaling the final probability mass function. [^4]: All but $t_n$ turn out to be dummy variables anyway. [^5]: A normalized result would be encouraging as nowhere in the derivation was it explicitly forced. [^6]: These assumptions are in line with those of the quasistatic field approximation that is likewise useful in electrical engineering. [@bansal p. 31] [^7]: Casual experimentation shows that when using a computationally-economical large time step, evaluating acceleration at $z_{t+\Delta t}$ rather than $z_t$ produces trajectories that diverge slower from those that are in the ideal continuum limit. [^8]: According to the IUPAC, using a power of temperature in the pre-exponential factor is not novel. [@iupac] In their model however, $T$ is arbitrarily scaled and raised to any real power, probably resulting in overfitting of empirical data when they should really seek a more fundamental model. [^9]: The results section is brief due not to a lack of data collection and analysis but to an effort to emphasize the key finding: the possibility of progression of a quasi-polymerase chain reaction. Thorough analysis and discussion will be given in the next section. [^10]: Radiation-induced mutation provides yet another parallel of this process. Brief quantized excitations upset everything so that the system may land in a more favorable state. Why this is crucial when discussing the nature of life was discussed by Schrödinger. [@schrod pp. 42-45] [^11]: Necessary and sufficient simplification is apparently not an uncommon dilemma, as it is the subject of a preface by Bender [@bender pp. 2-4] and a whole book by Wolfram [@wolfram expressed concisely on p. 1025 in the context of biological evolution].
--- abstract: 'The Casimir and Casimir-Polder interactions are investigated in a stack of equally spaced graphene layers. The optical response of the individual graphene is taken into account using gauge invariant components of the polarization tensor extended to the whole complex frequency plane. The planar symmetry for the electromagnetic boundary conditions is further used to obtain explicit forms for the Casimir energy stored in the stack and the Casimir-Polder energy between an atom above the stack. Our calculations show that these fluctuation induced interactions experience strong thermal effects due to the graphene Dirac-like energy spectrum. The spatial dispersion and temperature dependence in the optical response are also found to be important for enhancing the interactions especially at smaller separations. Analytical expressions for low and high temperature limits and their comparison with corresponding expressions for an infinitely conducting planar stack are further used to expand our understanding of Casimir and Casimir-Polder energies in Dirac materials. Our results may be useful to experimentalists as new ways to probe thermal effects at the nanoscale in such universal interactions.' address: - '$^1$ Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, 09210-170 Santo André, SP, Brazil' - '$^2$ Institute of Physics, Kazan Federal University, Kremlevskaya 18, Kazan, 420008, Russia' - '$^3$ Department of Physics, University of South Florida, Tampa, Florida 33620, USA' author: - 'Nail Khusnutdinov$^{1,2}$, Rashid Kashapov$^2$ and Lilia M. Woods$^3$' title: 'Thermal Casimir and Casimir-Polder interactions in $N$ parallel 2D Dirac materials' --- Introduction ============ Long-ranged dispersive forces originating from zero-point vacuum fluctuations exist between any types of objects. Specifically, the presence of boundaries and/or molecular structures modifies the electromagnetic boundary conditions, which in turn gives rise to such fluctuation induced forces including Casimir and Casimir-Polder interactions [@Casimir:1948:otabtpcp; @Casimir:1948:tiorotlvdwf; @Lifshitz:1956:ttomafbs]. Dispersive forces dominate in inert materials and they are related to important phenomena, such as stability of composites, adhesion, and sticktion in tiny instruments and biological matter among others [@Woods:2016:mpocavdwi; @Klimchitskaya:2009:tcfbrmeat]. Understanding the underlying mechanisms is important not only for the fundamental science of these ubiquitous forces, but also for control in the laboratory. The Casimir-Polder interaction, in particular, is especially strong in near-field interference setups, where distortion of the wave fronts due to different atoms or surfaces can lead to significant changes [@Nimmrichter:2008:tonfmwibtea; @Gerlich:2007:akifhpm]. Trapping atoms near surfaces of Bose-Einstein condensates are also sensitive to Casimir-Polder forces [@Leanhardt:2003:becnams; @Druzhinina:2002:eooqrithel]. The Casimir force, on the other hand, can limit the operation of nano/micro electro-mechanical systems or it can be used as a driving force for actuators [@Chan:2001:qmaomsbtcf; @Buks:2001:saeatceims; @Esquivel-Sirvent:2010:vdwtibemf]. Recently, special attention has been devoted to long-ranged dispersive forces involving carbon materials, such as graphene, carbon nanotubes and nanoribbons [@Drosdoff:2014:qatdfatgn; @Popescu:2011:cdcni; @Bordag:2006:ltffgaswcnvdwaci; @Sarabadani:2011:mbeitvdwibgl; @Gomez-Santos:2009:tvdwibgl]. The Dirac-like electronic structure and the unique optical properties of these materials have resulted in new functionalities in terms of magnitude, sign, and characteristic distance dependences, which are not found in Casimir interactions in standard materials [@Woods:2016:mpocavdwi; @Klimchitskaya:2009:tcfbrmeat]. Moreover, it has been shown that the Hall phase diagram in the graphene family, composed of graphene, silicene, germanene, and stanene, accessible via external fields results in Casimir force phase transitions with even greater range of tunability [@Rodriguez-Lopez:2017:cfptitgf]. With experiments being performed at room temperature, the impact of thermal fluctuations on Casimir-Polder/Casimir phenomena is of special interest. It appears that the massless charge excitations in 2D graphene materials have strong effects on dispersive interactions when temperature is taken into account. Specifically, due to the much smaller Fermi velocity compared to the speed of light, the onset of thermal fluctuations involving 2D Dirac-like materials begins at much smaller separations as compared to typical metals and dielectrics [@Sushkov:2011:oottcf; @Bimonte:2017:htotgteitcffgs]. The remarkable manifestation of thermal fluctuations in graphene materials is intertwined with their electronic structure and optical response properties. A successful model for the graphene response properties is the polarization tensor [@Bordag:2015:qftdftrog; @Klimchitskaya:2016:copgtautpt]. This approach is based on the graphene Dirac Hamiltonian and it takes into account temperature and spatial dispersion through the wave vector. This polarization tensor description is complementary to the recently presented optical conductivity tensor for the entire graphene family based on the linear response Kubo formalism [@Rodriguez-Lopez:2018:noritptitgf]. The reflection coefficients of graphene have been expressed in terms of the polarization tensor and have been utilized in the Lifshitz approach to calculate Casimir interactions involving systems with a single graphene layer [@Bordag:2015:qftdftrog]. It has been demonstrated that the strong thermal effects at small separations originate not only from the zero Matsubara dominance, but also from the polarization tensor dependence on temperature, frequency, and wave vector [@Klimchitskaya:2015:oolteitcibtgs]. These strong thermal effects cannot be captured by other, simpler models for the graphene response properties, such as a constant conductivity model [@Kuzmenko:2008:uocog], which reflects the small frequency range (less that $3$ eV) and is thus suitable for Casimir interactions at large distances [@Drosdoff:2010:cfags; @Fialkovsky:2012:frig]. In this work, we consider the Casimir interaction energy stored in a stack of $N$ graphene planes and the Casimir-Polder interaction between an atom and the layered graphene stack. Due to the planar symmetry of the system, the response of each graphene can be represented as two decoupled conductivities, which are expressed in terms of the gauge invariant components of the polarization tensor extended to the whole complex frequency plane. This enables the numerical evaluation of the different interactions as a function of separation, temperature, mass gaps, and chemical potential. Results obtained via the constant conductivity show how the two models for the response compare in different regions. Numerical and analytical results for a stack of infinitely conducting planes are also shown as a reference and further comparisons. This work is especially beneficial as it provides a unified approach for Casimir and Casimir-Polder interactions in configurations with finite number of infinitely thin layers by taking into account the most sophisticated and complete analytical representation of the graphene response. Studying more than one graphene layers and tuning the chemical potential are especially interesting from experimental point of view, since this can be done by changing $N$ and varying the external electrostatic potential. Our results offer new opportunities for probing Casimir interactions and micro- and nanomechanical device applications. Optical response ================ The Casimir and Casimir-Polder interactions are determined by the electromagnetic modes supported by the system. For planar systems, which are of interest here, these can be separated into transverse magnetic () and transverse electric () contributions. The optical response of the individual graphenes in the multilayered system is also a key ingredient for the interactions. A reliable approach here is to utilize the polarization tensor approach, which enables taking into account the electronic structure of graphene as well as frequency, temperature, and chemical potential [@Bordag:2009:cibapcagdbtdm; @Fialkovsky:2011:ftcefg; @Bordag:2016:ecefdg; @Bordag:2017:eecefdg]. The components of the polarization tensor, $\Pi_{lj}$, can be related to the components of the optical conductivity tensor $\sigma_{lj}$ as $$\sigma_{lj} = \frac{\Pi_{lj}}{i\omega}.$$ Due to gauge invariance, the polarization tensor has only two independent components, $\Pi_{00}$ and $\Pi_{tr} = \Pi^{\mu\nu} g_{\mu\nu} = \Pi_{00} - \Pi_{11} - \Pi_{22}$. The explicit expressions for $\Pi_{00}$ and $\Pi_{tr}$ were previously obtained in Ref. [@Bordag:2009:cibapcagdbtdm; @Fialkovsky:2011:ftcefg; @Bordag:2016:ecefdg; @Bordag:2017:eecefdg]. It turns out that the graphene optical conductivity tensor entering into the reflection coefficients for the interactions calculations can be decoupled into components related to the gauge invariant quantities $\Pi_{00}$ and $\Pi_{tr}$. Specifically, using imaginary frequencies $\lambda = i \omega$ the normalized to the graphene universal conductivity $\sigma_{gr} = e^2/4$, the $\overline{\sigma}_{\textsf{\scriptsize tm}}$ and $\overline{\sigma}_{\textsf{\scriptsize te}}$ components, corresponding to the TM and TE modes, are expressed as $$\overline{\sigma}_{\textsf{\scriptsize tm}}= \frac{4\lambda}{e^2k^2} \Pi_{00}, \overline{\sigma}_{\textsf{\scriptsize te}}= \frac{4}{e^2\lambda}\left(\Pi_{tr} -\frac{\lambda^2+k^{2}}{k^2} \Pi_{00}\right),$$ where the trace of the $\overline{\bm{\sigma}}$ tensor is $\Tr \overline{\bm{\sigma}} = \frac{4}{e^{2} \lambda} \left(\Pi_{tr} - \Pi_{00}\right) = \frac{4}{e^{2} \lambda} \Pi^k_k$. Here and throughout the paper all relations are given in $\hbar = c = k_B =1$ units. The expressions for the two types of conductivities can further be rewritten as $$\begin{aligned} \overline{\sigma}_{\textsf{\scriptsize tm}}&=& \frac{4m\lambda }{\pi(\lambda^2 + v_F^2 k^2)}\left(1 + \frac{\lambda^2 + v_F^2 k^2 - 4m^{2}}{2m\sqrt{\lambda^2 + v_F^2 k^2}} \arctan \frac{\sqrt{\lambda^2 + v_F^2 k^2}}{2m}\right) + \Delta\overline{\sigma}_{\textsf{\scriptsize tm}},{\nonumber \\}\overline{\sigma}_{\textsf{\scriptsize te}}&=& \frac{4m}{\pi \lambda} \left(1 + \frac{\lambda^2 + v_F^2 k^2 - 4m^{2}}{2m\sqrt{\lambda^2 + v_F^2 k^2}} \arctan \frac{\sqrt{\lambda^2 + v_F^2 k^2}}{2m} \right) + \Delta\overline{\sigma}_{\textsf{\scriptsize te}},{}\label{eq:T0}\\ \Delta\overline{\sigma}_{\textsf{\scriptsize tm}}&=& \frac{8}{\pi} \Re \int_m^\infty dz \frac{q (q^2 + v_F^2 k^2+ 4m^2) - \lambda r }{r(r + q \lambda)} \left(\frac{1}{e^{\frac{z + \mu}{T}} +1} + \frac{1}{e^{\frac{z - \mu}{T}} +1}\right),{\nonumber \\}\Delta\overline{\sigma}_{\textsf{\scriptsize te}}&=& \frac{8}{\pi \lambda} \Re \int_m^\infty dz \frac{(4m^2+q^2) (\lambda^2 q^2+v_F^2k^2q^2 + 4 m^2 v_F^2 k^2) - \lambda^2 q^2(\lambda^2 +v_F^2 k^2)}{r(\lambda^2q^2 +v_F^2 k^2 q^2+ 4m^2 k^2 v_F^2 + q \lambda r)} \left(\frac{1}{e^{\frac{z + \mu}{T}} +1} + \frac{1}{e^{\frac{z - \mu}{T}} +1}\right), \label{eq:SigmaTot}\end{aligned}$$ where $r = \sqrt{(\lambda^{2}+v_F^2 k^2)^2 (q^2 + k^2 v_F^2) + 4m^2 k^2 v_F^2}$ and $q=\lambda-2i z$. One notes that the first terms in Eqs. do not contain temperature or chemical potential. These $T$ and $\mu$ dependences are found in the $\Delta\overline{\sigma}_{{\textsf{\scriptsize te}}, {\textsf{\scriptsize tm}}}$ terms. The and optical response expressions in Eqs. , take into account the temporal dispersion due to the frequency, spatial dispersion due to the wave vector ${\bf k}$, temperature $T$, chemical potential $\mu$, and finite mass gap $m$. Note that the ${\bf k}$-dependence in the longitudinal (TM) and transverse (TE) conductivity components are different. In the limit of $k=0$ with $m=\mu=0$ we recover the isotropic and homogeneous case $$\overline{\sigma}_{\textsf{\scriptsize tm}}= \overline{\sigma}_{\textsf{\scriptsize te}}= \frac{8T\ln 2}{\pi \lambda } + \frac{2}{\pi}\int_0^\infty\frac{\tanh(\lambda x/4T)}{x^2+1}dx$$ as obtained by others (see Ref. [@Gusynin:2007:mocig; @Falkovsky:2007:stdogc]). Further taking $T\rightarrow 0$ we also recover the well known universal conductivity value $\overline{\sigma}_{\textsf{\scriptsize tm}}= \overline{\sigma}_{\textsf{\scriptsize te}}= 1\ (\sigma_{\textsf{\scriptsize tm}}= \sigma_{\textsf{\scriptsize te}}= \sigma_{gr} = e^2/4)$. Let us also study the $T\rightarrow 0$ limit in the case when $m$ and $\mu$ are different from zero with spatial dispersion taken into account. Using Eqs. , we find that when $\mu < m$ the $\Delta\overline{\sigma}_{\textsf{\scriptsize tm}}, \Delta\overline{\sigma}_{\textsf{\scriptsize te}}$ terms are zero. When $\mu > m$, however, all contributions to $\overline{\sigma}_{\textsf{\scriptsize tm}}, \overline{\sigma}_{\textsf{\scriptsize te}}$ in Eq. are nonzero. ![The normalized to $\sigma_{gr} = e^2/4$ graphene conductivity components difference $ \overline{\sigma}_{{\textsf{\scriptsize tm}},{\textsf{\scriptsize te}}} - \Delta\overline{\sigma}_{{\textsf{\scriptsize tm}},{\textsf{\scriptsize te}}}$ for a mass gap $m=0$ (left panel) and $m=0.1$ eV (right panel) and different values of the wave vector $k$. The line-color legend is the same in both panels. []{data-label="fig:cnd1"}](f1){width="9.8truecm"} To further analyze the different characteristic dependences in the and optical response, in Fig.\[fig:cnd1\] we show how the normalized to $\sigma_{gr}$ temperature independent component difference $(\overline{\sigma}_{{\textsf{\scriptsize tm}},{\textsf{\scriptsize te}}} - \Delta\overline{\sigma}_{{\textsf{\scriptsize tm}},{\textsf{\scriptsize te}}})/\sigma_{gr}$ evolves as a function of the imaginary frequency $\lambda= i \omega$ for different $m$ and $k$ values. Fig. \[fig:cnd1\] shows that the low frequency response is dominated by the contributions which diverge as $\lambda\rightarrow 0$. The wave vector, however, enhances the role of the modes for gapeless graphene as evident from the left panel, where more pronounced region of nonlinear behavior is seen when compared with the $m=0.1$ $eV$ case. As $\lambda$ increases $(\overline{\sigma}_{{\textsf{\scriptsize tm}},{\textsf{\scriptsize te}}} - \Delta\overline{\sigma}_{{\textsf{\scriptsize tm}},{\textsf{\scriptsize te}}})/\sigma_{gr}$ approaches $1$, which corresponds to the universal graphene conductivity. One notes that this limit is reached much slower for gapped graphene when compared with the $m=0$ case. The effects of temperature are shown in Fig.\[fig:cnd2\] for a graphene with nonzero chemical potential and mass gaps for different values of the wave vector. One finds that temperature affects primarily the low frequency regime, where the contribution is small while the part has a large finite value. The effect in vs disparity is more pronounced for smaller temperature and wave vector values. ![The ratio $\delta \overline{\sigma}_{{\textsf{\scriptsize te}},{\textsf{\scriptsize tm}}} = \Delta \overline{\sigma}_{{\textsf{\scriptsize te}},{\textsf{\scriptsize tm}}}/(\overline{\sigma}_{{\textsf{\scriptsize tm}},{\textsf{\scriptsize te}}} - \Delta\overline{\sigma}_{{\textsf{\scriptsize tm}},{\textsf{\scriptsize te}}})$ at $T = 30$ K (left panel) and $T = 300$ K (right panel) at different values of the wave vector $k$. Here $\mu = 0.1$ eV and $m=0.1$ eV. The line-color legend is the same for both panels. []{data-label="fig:cnd2"}](f2){width="9.8truecm"} Method of Calculations ====================== The system under consideration here consists of $N$ equally spaced infinitely thin layers along the $z$ axis such that each layer extends in the $x-y$ plane as shown in Fig. \[fig:stack\]. We are interested in the Casimir energy stored in this stack of planes. For the Casimir-Polder interaction initially we take that the half space above the top layer and including the atom is occupied by a dielectric medium specified with a dielectric function $\epsilon(\omega)$, which is then rarefied [@Lifshitz:1956:ttomafbs]. As a result, $\epsilon(\omega)=1+4\pi L\alpha(\omega)$, where $L$ is the number of atoms making up the medium and their atomic polarizability is $\alpha(\omega)$. In the limit of $L\rightarrow 0$, one obtains the Casimir-Polder interaction between one atom and the stack of planes, as shown in Fig. \[fig:stack\]. We find that the Casimir (${\textsf{\scriptsize C}}$) free energy per unit area and the Casimir-Polder (${\textsf{\scriptsize CP}}$) free energy can be given as $${\mathcal{F}}^{{\textsf{\scriptsize C}},({\textsf{\scriptsize CP}})} = {\mathcal{F}}_{\textsf{\scriptsize tm}}^{{\textsf{\scriptsize C}},({\textsf{\scriptsize CP}})} + {\mathcal{F}}_{\textsf{\scriptsize te}}^{{\textsf{\scriptsize C}},({\textsf{\scriptsize CP}})}.$$ The and contributions are obtained explicitly by using the electromagnetic boundary conditions for the systems in Fig. \[fig:stack\] and summing the zero-point energy excitations [@Khusnutdinov:2016:cpefasocp; @Khusnutdinov:2015:cefasocp; @Kashapov:2016:tcefpls] $$\begin{aligned} \mathcal{F}_{\textsf{\scriptsize tm}}^{\textsf{\scriptsize C}}&=& \frac{T }{4\pi^2}\sum_{n=0}^\infty{}' \int d^2 k_\perp \ln \Psi_N\left( \frac{\eta_n^{\textsf{\scriptsize tm}}\kappa_n}{\xi_n} \right),{\nonumber \\}\mathcal{F}_{\textsf{\scriptsize te}}^{\textsf{\scriptsize C}}&=& \frac{T }{4\pi^2}\sum_{n=0}^\infty{}' \int d^2 k_\perp \ln \Psi_N\left( \frac{\eta_n^{\textsf{\scriptsize te}}\xi_n}{\kappa_n} \right),\label{eq:FCa}\\ \mathcal{F}^{{\textsf{\scriptsize CP}}}_{\textsf{\scriptsize tm}}&=& \frac{T }{2\pi}\sum_{n=0}^\infty{}'\int d^2 k_\perp \alpha_n \Phi_N\left( \frac{\eta_n^{\textsf{\scriptsize tm}}\kappa_n}{\xi_n} \right)\left(\frac{\xi_n^2}{\kappa_n^2} -2\right),{\nonumber \\}\mathcal{F}^{{\textsf{\scriptsize CP}}}_{\textsf{\scriptsize te}}&=& \frac{ T }{2\pi}\sum_{n=0}^\infty{}' \int d^2 k_\perp \alpha_n \Phi_N\left( \frac{\eta_n^{\textsf{\scriptsize te}}\xi_n}{\kappa_n} \right)\left(-\frac{\xi_n^2}{\kappa_n^2}\right),\label{eq:Fa}\end{aligned}$$ where the following auxiliary functions are defined $$\begin{aligned} \Psi_N(t) &=& \frac{e^{-d \kappa_n (N-1)}}{(1+t)^N}\frac{1}{f(t)^{N-1}} \frac{1-f(t)^{2N}}{1-f(t)^2} {\nonumber \\}&\times& \left(1 + t - e^{-d \kappa_n }f(t) \frac{1-f(t)^{2(N-1)}}{1-f(t)^{2N}}\right),{\nonumber \\}\Phi_N(t) &=& \frac{t z e^{-2a \kappa_n }}{1 + t - e^{-d \kappa_n }f(t) \frac{1-f(t)^{2(N-1)}}{1-f(t)^{2N}}}. \end{aligned}$$ Here $\kappa_n = \sqrt{k_\perp^2 + \xi_n^2 }$ and $\alpha_n = \alpha(\xi_n), \eta_n^{{\textsf{\scriptsize te}},{\textsf{\scriptsize tm}}} = 2\pi \sigma_{{\textsf{\scriptsize te}},{\textsf{\scriptsize tm}}} (\xi_n)$ with $\xi_n = 2\pi n T$ being the Matsubara frequencies. Also, we have defined $f(t) = \sqrt{(\cosh d \kappa_n + t \sinh d\kappa_n)^2-1}+ \cosh d\kappa_n + t \sinh d\kappa_n$. The prime in each summation denotes that the zero term is multiplied by $1/2$. ![Schematic representation of infinitely thin layers equally spaced by a distance $d$ along the $z$-direction. The Casimir-Polder interaction is calculated for an atom placed at a distance $a$ above the top layer, while the Casimir energy for the stack only is calculated using the shown coordinate system. []{data-label="fig:stack"}](f3){width="4.3truecm"} Eqs. and constitute the main framework of calculating the Casimir and Casimir-Polder interactions in a multilayered graphene system. The separation into and contributions achieved in the optical response of an individual graphene is an important factor in the interaction forces being written as a sum of such parts. Let us note that these expressions are quite general as they take into account the distance between the layers, the finite number of layers, and temperature. Additionally, these results also include the spatial dispersion via the 2D wave vector in the graphene conductivity tensor. Potentially, Eqs. and can be applied to other multilayered materials characterized by different optical conductivity properties. Results and Discussion ====================== Before considering the graphene multilayers, we investigate the limiting case of a stack composed of infinitely conducting planes. We find it convenient to recast Eqs. and in a form using Poisson’s formula $\sum_{n=-\infty}^{\infty}\phi(n)=4\pi \sum_{l=0}^{\infty}\int_{0}^{\infty}\phi(s)\cos(2\pi ls)ds$ (see Ref. [@Bordag:2009:ACE] for details). Let us note that the $T=0$ limiting case, found by substituting $T\sum'^\infty_n \to \int_0^{\infty}\frac{d\omega}{2\pi}$ corresponds to the $l=0$ term in the Poisson’s expressions for the energies, as reported in Ref. [@Khusnutdinov:2016:cpefasocp; @Khusnutdinov:2015:cefasocp]. Using the Poisson’s summation formula, the Casimir interaction in the stack of infinitely conducting layers is obtained essentially as the energy between two planes with $\sigma\rightarrow \infty$ written as $$\begin{aligned} \mathcal{F}^{\textsf{\scriptsize C}}&=&\frac{N-1}{\pi^2 d^3}\sum_{l=0}^\infty{}' \int_{0}^\infty y^2 dy \int_0^1 dx \cos \left(\frac{y x l}{dT}\right) {\nonumber \\}&\times& \ln \left(1 - e^{-2y} \right).\end{aligned}$$ The above expression enables us to find the small and large temperature limits, $$\begin{aligned} \left. \mathcal{F}^{\textsf{\scriptsize C}}\right\|_{T\to 0} &=& (N-1)\mathcal{E}^{\textsf{\scriptsize C}}_0 \left\{1 + \frac{45\zeta_R(3)}{\pi^6}\left(2\pi T d\right)^3 \right\},{\nonumber \\}\left. \mathcal{F}^{\textsf{\scriptsize C}}\right\|_{T\to \infty} &=& - \frac{\zeta_R(3) T}{8\pi d^2}, \label{eq:Cid}\end{aligned}$$ where $\zeta_R(3)$ is Riemann zeta function and $\mathcal{E}^{\textsf{\scriptsize C}}_0 = - \pi^2 /720 d^3$ denotes the quantum mechanical ($T=0$) energy for two infinitely conducting planes multiplied by $(N-1)$. Eqs. show that the low $T$ correction to the standard quantum mechanical interaction between the perfectly conducting planes is $\sim T^3$. The high $T$ limit is consistent with the classical thermal fluctuations results for the $n=0$ Matsubara frequency. ![(a) The Casimir energy stored in a stack of infinitely conducting planes and normalized to $\mathcal{E}^{{\textsf{\scriptsize C}}}_0$ for different interplane distances. (b) The Casimir-Polder energy normalized to $\mathcal{E}_{0}^{{\textsf{\scriptsize CP}}}$ between a hydrogen atom and a stack of planes for different atom-stack distances.[]{data-label="fig:CandCPstack"}](f4a "fig:"){width="8truecm"} ![(a) The Casimir energy stored in a stack of infinitely conducting planes and normalized to $\mathcal{E}^{{\textsf{\scriptsize C}}}_0$ for different interplane distances. (b) The Casimir-Polder energy normalized to $\mathcal{E}_{0}^{{\textsf{\scriptsize CP}}}$ between a hydrogen atom and a stack of planes for different atom-stack distances.[]{data-label="fig:CandCPstack"}](f4b "fig:"){width="8truecm"} ![image](f5a){width="5.8truecm"}![image](f5b){width="5.8truecm"}![image](f5c){width="5.8truecm"} From the Poisson’s formula we also find that the Casimir-Polder energy in the $\sigma\rightarrow \infty$ case becomes $$\begin{aligned} \mathcal{F}^{\textsf{\scriptsize CP}}&=&-\frac{1}{2\pi a^4}\sum_{l=0}^\infty{}' \int_{0}^\infty dy (2y^2 + 2y + 1) e^{-2y} {\nonumber \\}&\times&\cos \left(\frac{yl}{a T}\right) \alpha\left(\frac{y}{a}\right).\end{aligned}$$ For large $a$ separations the atomic polarizability becomes $\alpha(y/a)\rightarrow \alpha(0)$, where $\alpha(0)$ is taken at zero frequency. In this case, the integration over $y$ and the summation over $l$ can be performed explicitly, $$\mathcal{F}^{{\textsf{\scriptsize CP}}}\hspace{-1ex} = \frac{\mathcal{E}^{{\textsf{\scriptsize CP}}}_0 \chi}{3} \left( \coth \chi + \chi {\mathrm{csch}}^2 \chi + \chi^2 \coth \chi {\mathrm{csch}}^2 \chi \right),\label{eq:CPid}$$ where $\chi= 2\pi a T$ and $\mathcal{E}^{\textsf{\scriptsize CP}}_0 =-3\alpha(0)/8\pi a^4$ corresponds to the quantum mechanical Casimir-Polder energy between an atom and an infinitely conducting plane. Eq. is in agreement with [@Bezerra:2008:Ltaiatqr] and it shows that when the atom is far away from the stack, the interaction is determined by the closest to it layer. It is now easy to see that in the limits of small and high temperatures, the Casimir-Polder energy becomes $$\begin{aligned} \left.\mathcal{F}^{\textsf{\scriptsize CP}}\right\|_{T\to 0} &=& \mathcal{E}^{\textsf{\scriptsize CP}}_{0} \left\{1 - \frac{1}{135}\left(2 \pi T a\right)^4 \right\},{\nonumber \\}\left.\mathcal{F}^{\textsf{\scriptsize CP}}\right\|_{T\to \infty} &=& \frac{3\alpha (0)}{4a^3} T.\label{eq:idealCP}\end{aligned}$$ These expressions show that the low $T$ correction to the quantum mechanical result for the energy is $\sim T^4$, which is different that the Casimir-Polder case in Eqs. . Similarly to the Casimir interaction, the high temperature limit for the Casimir-Polder energy corresponds to the $n=0$ Matsubara term from Eq. describing the thermal fluctuation contribution. Results for the calculated energies involving infinitely conducting planes are shown in Fig. \[fig:CandCPstack\]. The left panel indicates that the stored Casimir energy for small separations is not significantly affected by temperature. However, as $d$ is increased, the energy begins to deviate from $\mathcal{E}^{\textsf{\scriptsize C}}_{0}$, which characterizes the $T=0$ interaction between infinitely conducting objects. The deviations for smaller $T$ appear as $d$ becomes larger as can be seen from the $d=100$ and $300$ nm distances. Similar trends are found for the Casimir-Polder interaction. This behavior is in agreement with the low temperature approximations given by Eq. and , where correction to the Casimir energy is $\sim T^3$ and Casimir-Polder energy is $\sim (- T^4)$. Let us now consider the Casimir and Casimir-Polder interactions involving a stack of graphene layers, as specified in Fig. \[fig:stack\]. Asymptotic low and high temperature expansions are found for the Casimir and Casimir-Polder interactions when the response is taken to be described by the constant graphene universal conductivity, $$\begin{aligned} \left. \mathcal{F}^{\textsf{\scriptsize C}}\right\|_{T\to 0} &=& \mathcal{F}^{\textsf{\scriptsize C}}_{T=0} \left\{1 + B(\sigma_{gr}, N) \left(2\pi T d\right)^2 \right\},{\nonumber \\}\left. \mathcal{F}^{\textsf{\scriptsize C}}\right\|_{T\to \infty} &=& - \frac{\zeta_R(3)}{8\pi d^2} T, \label{eq:Cideal}{}\\ \left.\mathcal{F}^{\textsf{\scriptsize CP}}\right\|_{T\to 0} &=& \mathcal{F}^{\textsf{\scriptsize CP}}_{T= 0} \left\{1 + A(\sigma_{gr}, N)\left(2\pi T a\right)^2 \right\},{\nonumber \\}\left.\mathcal{F}^{\textsf{\scriptsize CP}}\right\|_{T\to \infty} &=& \frac{3\alpha (0)}{4a^3} T.\label{eq:CPn}\end{aligned}$$ Here $\mathcal{F}^{\textsf{\scriptsize C}}_{T=0}$ and $\mathcal{F}^{\textsf{\scriptsize CP}}_{T= 0}$ are the Casimir and Casimir-Polder energies, respectively, in the quantum mechanical limit where the Matsubara frequency summation is transformed into an integral ($T\sum_{n=0}^\infty{}' \rightarrow \int\frac{d\omega}{\pi}$) in Eqs. , and $T=0$ in the optical response in Eq. , [@Khusnutdinov:2016:cpefasocp; @Khusnutdinov:2015:cefasocp]. Also, $A$ and $B$ are non-trivial functions of $\sigma_{gr}$ and the number of graphene planes in the stack, but they are temperature independent ($A$ and $B$ are not given explicitly here). It appears that the low $T$ behavior is rather different than the low $T$ behavior of a stack of infinitely conducting planes. Specifically, these corrections to the Casimir and Casimir-Polder interactions are $\sim T^2$ and they are positive. For the infinitely conducting planes the low-$T$ Casimir correction is $\sim T^3$ and it is positive, but the low-$T$ Casimir-Polder correction is $\sim T^{4}$ and it is negative. In general, analytical formulas for the $A$ and $B$ parameters are not possible, however for a single graphene with a constant $\sigma_{gr}$ conductivity, we find that $A = 1/18\eta^{\textsf{\scriptsize tm}}$. The high $T$ limit is determined by the $n=0$ Matsubara mode in Eqs. , . We note that for the constant conductivity response model the arguments of the $\Psi_N$ and $\Phi_N$ functions tend to infinity for the mode, while they vanish for the mode. As a result the high $T$ interaction is determined by the mode giving Eqs. and . When the graphene is described via the polarization tensor, the arguments of the $\Psi_N$ and $\Phi_N$ functions are constant for the zero Matsubara frequency mode. In the $T\rightarrow \infty$ limit, one arrives at the expected classical thermal fluctuations results. ![image](f6a){width="5.8truecm"}![image](f6b){width="5.8truecm"}![image](f6c){width="5.8truecm"} In Fig.\[fig:CasfT1\] we show numerically calculated results for the Casimir energy stored in a stack of graphene planes by using the constant conductivity and polarization tensor approaches. Panel a) shows that the two models have different results for $T<100$ K for the chosen distances, however, they yield practically the same results for higher temperatures. This means that the spatial dispersion and frequency dependence taken via the polarization tensor are not important for the Casimir energy in this case. The particular optical response model has a pronounced role when the chemical potential is varied, as depicted in panel b). When $\mu \leq m$ the energy is independent upon the chemical potential, although the polarization tensor model gives smaller values as compared to the ones with $\sigma_{gr}$. As $\mu$ is increased, the intraband transitions (taken into account via the polarization tensor, but not present in $\sigma_{gr}$) begin to dominate and the energy obtained with polarization tensor starts to increase with $\mu$, while the energy obtained with $\sigma_{gr}$ stays constant. Finally, panel c) shows how the interaction depends on the number of graphenes in the stack. For small $N$, the energy is linear with $N$ and it approaches a constant value as the number of planes is increased. The two models for the response essentially give the same result for higher $T$. For smaller temperatures, however, the constant conductivity model underestimates the Casimir energy. The Casimir-Polder interaction is also investigated numerically by taking a Cs atom above the stack as an example. Fig. \[fig:CasPfT1\](a) shows that there are deviations between the constant conductivity and polarization tensor models at smaller temperatures and larger $a$ separations, similar to the situation in Fig. \[fig:CasfT1\](a). The behavior of the energy of the Cs atom/graphene stack system as a function of the chemical potential (Fig. \[fig:CasPfT1\] (b)) and number of planes in the stack ( Fig. \[fig:CasPfT1\](c)) is also similar as the one of corresponding Casimir interactions (Fig. \[fig:CasfT1\](b), (c)). Conclusions =========== In this work, a unified description of the Casimir and Casimir-Polder interactions involving a stack of $N$ infinitely thin equally spaced parallel layers is presented. This formalism is applied to graphene and infinitely conducting stacks. Using the planar symmetry separations between TM and TE contributions in the corresponding energies are found. The optical response is key in the interactions and in the graphene case, we consider two optical conductivity models, which involve the constant universal conductivity and the polarization tensor extended over the entire complex frequency plane. Considering various dependences upon separations, chemical potential, and number of planes in the stack and comparing with results for infinitely conducting layers, we show that thermal effects have a unique role in Casimir-like interactions in Dirac-like materials. Thermal fluctuations become strong and even dominant at separations that are much smaller than the typical $\mu m$ scale for standard materials. Our results further indicate that the polarization tensor model for the response is necessary in order to quantify the temperature effects in graphene properly. This study may be useful to experimentalists as new ways to probe classical thermal fluctuations in electromagnetic interactions at the nanoscale. The Casimir-Polder energy for two different models of conductivity namely, constant conductivity and conductivity calculated in framework of polarization tensor approach and for atom Cs is shown in Fig. \[fig:CasPfT1\]. All parameters for Cs maybe found in Ref. [@Khusnutdinov:2016:cpefasocp]. References {#references .unnumbered} ==========
--- abstract: 'We investigate a superconducting circuit consisting of multiple capacitively-coupled charge qubits. The collective Rabi oscillation of qubits is numerically studied in detail by imitating environmental fluctuations according to the experimental measurement. For the quantum circuit composed of identical qubits, the energy relaxation of the system strongly depends on the interqubit coupling strength. As the qubit-qubit interaction is increased, the system’s relaxation rate is enhanced firstly and then significantly reduced. In contrast, the inevitable inhomogeneity caused by the nonideal fabrication always accelerates the collective energy relaxation of the system and weakens the interqubit correlation. However, such an inhomogeneous quantum circuit is an interesting test bed for studying the effect of the system inhomogeneity in quantum many-body simulation.' address: - '$^{1}$Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore' - '$^{2}$Institute of Advanced Studies, Nanyang Technological University, 60 Nanyang View, Singapore 639673, Singapore' - '$^{3}$National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616, Singapore' - '$^{4}$MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore' - '$^{5}$Division of Physics and Applied Physics, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore' author: - 'Deshui Yu$^{1}$, Leong Chuan Kwek$^{1,2,3,4}$, & Rainer Dumke$^{1,5}$' title: 'Relaxation of Rabi Dynamics in a Superconducting Multiple-Qubit Circuit' --- Introduction ============ Owing to the fascinating properties such as flexibility, tunability, scalability, and strong interaction with electromagnetic fields, superconducting Josephson-junction circuits provide an outstanding platform for quantum information processing (QIP) [@Science:Devoret2013; @RPP:Wendin2017], quantum simulation of many-body physics [@NatCommun:Barends2015; @PhysRevA.95.042330; @SciRep:Lamata2017], and exploring the fundamentals of quantum electrodynamics in and beyond the ultrastrong-coupling regime [@NatPhys:Niemczyk2010; @NatPhys:Yoshihara2017]. Additionally, hybridizing these solid-state devices with the atoms may enable the information transfer between macroscopic and microscopic quantum systems [@PRA:Yu2016-1; @SciRep:Yu2016; @PRA:Yu2016-2; @QST:Yu2017; @PRA:Yu2017; @ProcSPIE:Hufnagel2017], where the superconducting circuits play the role of rapid processor while the atoms act as the long-term memory. Nonetheless, the strong coupling to the environmental noise significantly limits the energy-relaxation ($T_{1}$) and dephasing ($T_{2}$) times of superconducting circuits [@PRL:Astafiev2004; @PRL:Yoshihara2006; @PRA:Koch2007]. Recently, there has been focus on large-scale QIP [@PRL:Song2017; @npj:Lu2017]. Several network schemes have already been demonstrated in experiments: one-dimensional spin chain with the nearest-neighbor interaction [@Nature:Kelly2015; @Nature:Barends2016], two-dimensional lattice with quantum-bus-linked qubits [@NatCommun:Corcoles2015; @NatCommun:Riste2015], multiple artificial atoms interacting with the same resonator [@PRL:Song2017], and many cavities coupled to a superconducting qubit [@NewJPhys:Yang2016]. However, only little attention has been paid to the quantum circuit composed of nearly identical superconducting qubits, where an arbitrary individual directly interacts with others and, in addition, all qubits are biased by the same voltage, current, or magnetic bias and are exposed to the same fluctuation source. Studying such a multiple-qubit architecture is of importance to superconducting QIP network. For one thing, the nearest- or next-nearest-neighbour-coupling approximation, which is commonly employed in scalable superconducting schemes [@NewJPhys:Wallquist2005; @PRB:Storcz2005; @PRB:Richer2016], does not always hold the truth in realistic systems. For another, transferring the quantum information from one processor to another over a long distance, which relies on the long-range interqubit coupling, is essential for cluster quantum computing [@RepMathPhys:Nielsen2006] and quantum algorithms [@Nature:DiCarlo2009]. Moreover, in a large-scale network composed of strong or ultrastrong interacting qubits, the energy relaxation and dephasing of one qubit strongly influence the dynamics of others and this fluctuation can be rapidly boosted and affect more qubits. Such a collective dissipation may significantly degrade the information transfer fidelity between two remotely separated qubits. However, to our best knowledge, this collective dissipation of a large ensemble of directly-coupled superconducting qubits has not been explored before. Here we investigate the collective relaxation effect in a multiple-qubit circuit, where all charge qubits are biased by the same voltage source, capacitively coupled via linking all islands together, and influenced by the same noise source. The system’s Rabi oscillation is numerically studied in detail. It is shown that for the homogeneous system with identical qubits, the interqubit coupling strongly modifies the collective relaxation of quantum circuit. In contrast, the nonideal-fabrication-induced extra inhomogeneity always enhances the system’s relaxation rate and also provides a platform for studying many-body localization. Results ======= Physical model -------------- ![Multiple-qubit scheme. $N$ charge qubits are biased by the voltage source $V_{g}$ via the identical gate capacitors $C_{g}=300$ aF. In this work, we set $N=10$. All Cooper-pair boxes are linked together via the identical coupling capacitors $C_{c}$. The inhomogeneity of the system only arises from the nonidentical Josephson junctions. $C_{j,k}$ and $E_{j,k}$ with $k=1,\ldots,N$ denote the self-capacitances and Josephson energies of different Josephson junctions. The corresponding mean values are $C_{j}=30$ aF and $E_{J}=2\pi\hbar\times3$ GHz.[]{data-label="Fig1"}](Fig1 "fig:"){width="12.0cm"}\ We consider $N$ capacitively-coupled single-Josephson-junction charge qubits as shown in figure \[Fig1\]. A voltage source $V_{g}$ biases the array of superconducting islands via the identical gate capacitors $C_{g}$. All Cooper-pair boxes are linked together via the identical coupling capacitors $C_{c}$, where we have applied the fact that the practical inhomogeneities of capacitors $C_{g}$ and capacitors $C_{c}$ are much smaller than that of Josephson junctions. For the $k$-th ($k=1,\ldots,N$) charge qubit, we define the self-capacitance of Josephson junction, the total capacitance, the charging energy, and the Josephson energy as $C_{j,k}$, $C_{\Sigma,k}=C_{g}+C_{j,k}+C_{c}$, $E_{C,k}=\frac{(2e)^{2}}{2C_{\Sigma,k}}$, and $E_{J,k}$, respectively. The total excess charge $Q_{k}$ of the $k$-th box distributes on capacitor plates of $C_{g}$, $C_{j,k}$, and $C_{c}$ that are involved in the Cooper-pair box. The corresponding charges are defined as $Q_{g,k}$, $Q_{j,k}$, and $Q_{c,k}$, respectively, and we have $$Q_{k}=Q_{g,k}+Q_{j,k}+Q_{c,k}.$$ According to Kirchhoff’s circuit laws and charge conservation, one obtains $$\begin{aligned} &&\textstyle\frac{Q_{g,1}}{C_{g}}-\frac{Q_{j,1}}{C_{j,1}}=\frac{Q_{g,2}}{C_{g}}-\frac{Q_{j,2}}{C_{j,2}}=\ldots=-V_{g},\\ &&\textstyle\frac{Q_{c,1}}{C_{c}}-\frac{Q_{j,1}}{C_{j,1}}=\frac{Q_{c,2}}{C_{c}}-\frac{Q_{j,2}}{C_{j,2}}=\ldots,\\ &&Q_{c,1}+Q_{c,2}+\ldots+Q_{c,N}=0.\end{aligned}$$ The total charging energy of the system is then given by $$\begin{aligned} \nonumber E_{ch}&=&\textstyle\sum_{k}\left(\frac{Q^{2}_{g,k}}{2C_{g}}+\frac{Q^{2}_{j,k}}{2C_{j,k}}+\frac{Q^{2}_{c,k}}{2C_{c}}+V_{g}Q_{g,k}\right)\\ &=&\textstyle\sum_{k}E_{C,k}\left(N_{k}-N_{g}\right)^{2}+\left[\sum_{k}\frac{E_{C,k}}{\sqrt{NV}}\left(N_{k}-N_{g}\right)\right]^{2},\end{aligned}$$ where $N_{k}=-\frac{Q_{k}}{(2e)}$ denotes the number of excess Cooper pairs in the $k$-th box and $N_{g}=\frac{C_{g}V_{g}}{(2e)}$ is the gate-charge bias. We have also defined $$V=\textstyle\frac{(2e)^{2}}{2C_{c}}-\frac{1}{N}\textstyle\sum_{k}E_{C,k},$$ which is positive and denotes the difference between the electrostatic energy of a Cooper pair in $C_{c}$ and the mean value of charging energies of different Cooper-pair boxes. Adding the tunneling energies of Cooper pairs into $E_{ch}$ [@Book:Dittrich1998], the system’s Hamiltonian is derived as $H=H_{0}+H_{1}$, where $$\begin{aligned} H_{0}&=&\textstyle{\sum_{k}\left[\frac{E_{C,k}}{2}(1-2N_{g})\sigma^{(k)}_{z}-\frac{E_{J,k}}{2}\sigma^{(k)}_{x}\right]},\label{H1a}\\ H_{1}&=&\textstyle{\left[\sum_{k}\frac{E_{C,k}}{2\sqrt{NV}}(\sigma^{(k)}_{z}+1-2N_{g})\right]^{2}},\label{H1b}\end{aligned}$$ in the two-state approximation. The $x$- and $z$-components of the Pauli operator are given by $\sigma^{(k)}_{x}=(\|0\rangle\langle1\|)_{k}+(\|0\rangle\langle1\|)_{k}$ and $\sigma^{(k)}_{z}=(\|1\rangle\langle1\|)_{k}-(\|0\rangle\langle0\|)_{k}$. $\|0\rangle_{k}$ and $\|1\rangle_{k}$ represent the absence and presence of a single excess Cooper pair in the $k$-th island. The multiple-qubit system operates in the charging limit of $E_{C,k}\gg E_{J,k}$. $H_{0}$ gives the total energy of free charge qubits while $H_{1}$ corresponds to the interqubit interaction energy. It is seen that besides a constant, the qubit-qubit coupling leads to the linear terms of $(1-2N_{g})\sigma^{(k)}_{z}$, which may be involved into $H_{0}$, and the quadratic $zz$-interaction terms of $\sigma^{(k_{1})}_{z}\sigma^{(k_{2})}_{z}$ that are diagonal in the charge number basis. In the limit of $C_{c}\sim0$, we have $V\rightarrow\infty$, for which $H_{1}\sim0$ and charge qubits become independent with each other. As $C_{c}$ is increased, $E_{C,k=1,\ldots,N}$ go down. In this work, we restrict $C_{c}$ within the range satisfying the charging limit condition. Due to the assumption of identical gate and coupling capacitors, the system’s inhomogeneity completely comes from the nonidentical Josephson junctions. Homogeneous Circuit ------------------- We first consider the homogeneous system, where all Josephson junctions are identical, i.e., $C_{j,k}=C_{j}$, $E_{C,k}=E_{C}$, and $E_{J,k}=E_{J}$. It is easy to obtain $\frac{V}{E_{C}}=\frac{C_{g}+C_{j}}{C_{c}}$ and $H$ can be simplified as $$\label{H2} \textstyle\frac{H}{N}=\frac{E^{2}_{C}}{4V}\left[\frac{J_{z}}{N}+\Xi(V,N_{g})\right]^{2}-\frac{E_{J}}{2}\frac{J_{x}}{N},$$ by defining the collective operators $J_{z}=\sum_{k}\sigma^{(k)}_{z}$ and $J_{x}=\sum_{k}\sigma^{(k)}_{x}$ and the function $$\textstyle\Xi(V,N_{g})=\left(1+\frac{V}{E_{C}}\right)(1-2N_{g}).$$ Equation (\[H2\]) illustrates that multiple qubits behave in the same way if they are initialized in the same state. We define the dimensionless parameter $\eta=\frac{E^{2}_{C}}{E_{J}V}$ to measure the interqubit coupling strength. ### Ground state of nondissipative system ![(a) Ground-state expectation value $\frac{\langle\beta\|J_{z}\|\beta\rangle}{N}$ of nondissipative homogeneous circuit as a function of $\eta$ and $N_{g}$. Solid lines: $\Xi(V,N_{g})=\pm1$. The varying range of $\eta$ ensures the charging limit of $\frac{E_{C}}{E_{J}}\gg1$. (b) Dependence of expectation value $\frac{\langle\beta\|J_{x}\|\beta\rangle}{N}$ on $\eta$ and $N_{g}$.[]{data-label="Fig2"}](Fig2 "fig:"){width="12.0cm"}\ We are interested in the ground state of the nondissipative system in the limit of $N\rightarrow\infty$. The Holstein-Primakoff transformation [@PRA:Yu2014], $$\begin{aligned} \textstyle\frac{J_{z}}{2}&=&b^{\dag}b-\frac{N}{2},\\ \textstyle\frac{J_{x}}{2}&=&\sqrt{N-b^{\dag}b}b+b^{\dag}\sqrt{N-b^{\dag}b},\end{aligned}$$ is employed to map the multiple-qubit system onto a bosonic mode. The annihilation $b$ and creation $b^{\dag}$ operators fulfill the bosonic commutation relation $[b,b^{\dag}]=1$. Then, we introduce the macroscopic displacements, i.e., $b\rightarrow b+\sqrt{N}\beta$ and $b^{\dag}\rightarrow b^{\dag}+\sqrt{N}\beta^{\ast}$, to this bosonic mode and obtain a displaced Hamiltonian $${\cal{H}}=e^{-\sqrt{N}(\beta b^{\dag}-\beta^{\ast}b)}He^{\sqrt{N}(\beta b^{\dag}-\beta^{\ast}b)}.$$ ${\cal{H}}$ is further expanded as ordered power series in $b$ and $b^{\dag}$. In the second-order approximation, choosing $\beta$ such that the linear terms associated with $b$ and $b^{\dag}$ vanish leads to the ground state $\|\beta\rangle$ and the corresponding energy $$\begin{aligned} \nonumber\textstyle{\frac{{\cal{E}}}{N}}&\equiv&\textstyle{\frac{\langle\beta\|{\cal{H}}\|\beta\rangle}{N}}\\ &=&\textstyle\frac{E^{2}_{C}}{4V}\left[2\beta^{2}-1+\Xi(V,N_{g})\right]^{2}-2E_{J}\sqrt{1-\beta^{2}}\beta.\end{aligned}$$ Finally, we find that $\beta$ is real and determined by $$\textstyle\frac{E^{2}_{C}}{V}\left[2\beta^{2}-1+\Xi(V,N_{g})\right]=E_{J}\frac{1-2\beta^{2}}{\beta\sqrt{1-\beta^{2}}},$$ from which multiple solutions may be derived and the accepted one should minimize ${\cal{E}}$. We focus on the observable operator $J_{z}$. Figure \[Fig2\](a) shows the ground-state expectation value $\frac{\langle\beta\|J_{z}\|\beta\rangle}{N}=(2\beta^{2}-1)$ vs. $\eta$ and $N_{g}$. One can see that the large interqubit coupling ($\eta\gg1$) opens a wide intermediate regime, where $\frac{\langle\beta\|J_{z}\|\beta\rangle}{N}$ strongly relies on the gate-charge bias $N_{g}$. In contrast, for $\eta\ll1$ the system returns to an ensemble of independent qubits, and $\frac{\langle\beta\|J_{z}\|\beta\rangle}{N}\approx1$ ($-1$) when $N_{g}>\frac{1}{2}$ ($<\frac{1}{2}$). We should note that, unlike the Dicke model of a single cavity mode interacting with an ensemble of two-level atoms [@PRL:Emary2003; @PRE:Emary2003], both first and second derivatives of ${\cal{E}}$ with respect to $\beta$ are continuous, indicating the absence of phase transitions in our system. In the limit of $E_{J}\sim0$, one obtains the simple solution, $\frac{\langle\beta\|J_{z}\|\beta\rangle}{N}=-\Xi(V,N_{g})$ and ${\cal{E}}=0$. Due to the condition of $-1\leq\frac{\langle\beta\|J_{z}\|\beta\rangle}{N}\leq1$, the ground-state diagram may be divided into three regimes: $\frac{\langle\beta\|J_{z}\|\beta\rangle}{N}=-1$ and $\|\beta=0\rangle$ (all qubits are in $\|0\rangle$), $\frac{\langle\beta\|J_{z}\|\beta\rangle}{N}=-\Xi(V,N_{g})$, and $\frac{\langle\beta\|J_{z}\|\beta\rangle}{N}=1$ and $\|\beta=1\rangle$ (all qubits are in $\|1\rangle$), and the boundaries are given by $\Xi(V,N_{g})=\pm1$ \[see figure \[Fig2\](a)\]. We also displays the expectation value $\frac{\langle\beta\|J_{x}\|\beta\rangle}{N}$ in figure \[Fig2\](b), where $\frac{\langle\beta\|J_{x}\|\beta\rangle}{N}$ always maximizes at the sweet spot $N_{g}=\frac{1}{2}$ due to the resonant driving. ### Dissipative system In the practical operation, the many-qubit system is unavoidably interfered by environmental fluctuations. It has been experimentally demonstrated that the dominant noise sources in a single charge qubit include the high-frequency Ohmic dissipation from the DC voltage source and the low-frequency $1/f$ noise induced by background charge fluctuations [@PRL:Astafiev2004; @PRL:Nakamura2012; @Pashkin2009]. The former mainly determines the relaxation time $T_{1}$ of the charge qubit, i.e., the characteristic time scale of the damped qubit Rabi oscillation, while the latter primarily affects the dephasing time $T_{2}$, i.e., the characteristic time scale of the damped qubit Ramsey/spin echo oscillation. According to [@Pashkin2009; @Makhlin2003], the whole noise may be mapped onto the gate-charge bias $N_{g}$, i. e., the voltage source, in the mathematical treatment. This is still valid in this multi-qubit scheme. Since all islands are electrostatically biased by the same voltage source, different qubits are subjected to the same Ohmic noise whose spectrum is proportional to the noise frequency $f$. In addition, all islands are strongly coupled in the system. When environmental fluctuations interrupt the dynamic of one Cooper-pair box, the local voltage of the corresponding gate capacitor is disturbed. This voltage fluctuation may be mapped onto the voltage source and it further influences the dynamics of other boxes in the same way, leading to the energy relaxation of other charge qubits. Thus, one may map all local voltage fluctuations occurring in different Cooper-pair boxes on to the single voltage source. Moreover, in this work we mainly focus on the qubit Rabi oscillation, whose damping time is $T_{1}$, and the low-frequency $1/f$ noise in the circuit hardly influences $T_{1}$. Therefore, for these reasons it is valid to model environmental fluctuations by the fluctuation of gate-charge bias (voltage source), i.e., all charge qubits suffer the same noise. This differs from the many-body system composed of excited-state atoms that spontaneously decay independently. ![(a) Power spectral density $S_{\delta N_{g}}(f)$ of gate-charge fluctuations $\delta N_{g}(t)$. Solid (dash) line corresponds to the numerical (analytical) result. (b) Single-charge-qubit Rabi and Ramsey oscillations with the system parameters of $C_{g}$, $C_{j}$, and $E_{J}$ same to figure (\[Fig1\]) and $E_{C}=78E_{J}$ for $C_{c}=0$. For the Rabi oscillation, the qubit is initialized in $\|1\rangle$ with $N_{g0}=\frac{1}{2}$. For performing Ramsey fringes, the qubit is initially prepared in $\|1\rangle$ and $N_{g0}$ is set at $\frac{1}{2}$ during two $\frac{\pi}{2}$-pulses while $N_{g0}=0$ in the free evolution period. Solid curves: numerical results. Dash lines: decay-envelope fittings. In the Ramsey oscillation, the detailed behavior around the characteristic time sale $T_{2}$, which is surrounded by the rectangular frame, is zoomed in, and the oscillation frequency is given by $E_{\uparrow\downarrow}=\sqrt{E^{2}_{C}(1-2N_{g})^{2}+E^{2}_{J}}$. For all curves of ensemble average $\langle j_{z}(t)\rangle_{E}$, the size of trajectory ensemble is $10^{5}$.[]{data-label="Fig3"}](Fig3 "fig:"){width="13.0cm"}\ We rewrite the gate-charge bias as $$N_{g}(t)=N_{g0}+\delta N_{g}(t),$$ i.e., a constant value $N_{g0}$ plus a fluctuating term $\delta N_{g}(t)$. The noise spectral density $$S_{\delta N_{g}}(f)=\textstyle\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\delta N_{g}(t+\tau)\delta N_{g}(t)e^{-i2\pi f\tau}dtd\tau,$$ is given by $$S_{\delta N_{g}}(f)=\textstyle\frac{\pi R\hbar C^{2}_{g}}{e^{2}}f+\frac{\alpha}{2\pi f},$$ where the typical impedance of the voltage-source circuit is $R=50$ $\Omega$ and $\alpha=5.0\times10^{-7}$ [@APL:Zimmerli1992; @JAP:Verbrugh1995; @IEEE:Wolf1997]. Accordingly to $S_{\delta N_{g}}(f)$, one may numerically generate $\delta N_{g}(t)$ \[see figure \[Fig3\](a)\]. We choose $\{\|n_{1}\rangle\otimes\cdots\otimes\|n_{N}\rangle;n_{k=1,\ldots,N}=0,1\}$ to span the Hilbert space. Using the Schrödinger equation, $$\textstyle i\hbar\frac{d}{dt}\psi(t)=H\psi(t),$$ one can simulate the system’s state $\psi(t)$ for a given initial state $\psi(0)$, resulting in the trajectory $$j_{z}(t)=\langle\psi(t)\|J_{z}\|\psi(t)\rangle/N.$$ Repeating the simulation with the same initial condition leads to the ensemble mean observation $\langle j_{z}(t)\rangle_{E}$. Here we use $\langle\ldots\rangle_{E}$ to denote the ensemble average. As an example, figure \[Fig3\](b) depicts the Rabi and Ramsey oscillations of a single charge qubit, from which the decoherence times $T_{1,2}$ may be extracted. In the following we only focus on the collective Rabi oscillation of multiple-qubit circuit, where the system is initialized in $\|1\rangle_{1}\otimes\ldots\otimes\|1\rangle_{N}$ and we set $N_{g0}=\frac{1}{2}$. Figure \[Fig4\](a) shows the damped $\langle j_{z}(t)\rangle_{E}$ for several different $\eta$. It is seen that for $\eta\ll1$, i.e., the very weak qubit-qubit interaction, $\langle j_{z}(t)\rangle_{E}$ is similar to that of single qubit, meaning that multiple qubits act almost independently. When $\eta$ is increased, i.e., the interqubit coupling becomes strong, the relaxation time $T_{1}$ of $\langle j_{z}(t)\rangle_{E}$ is dramatically reduced because the qubits get correlated and their dynamics are affected with each other. When one qubit relaxes, the gate-charge bias is disturbed and this influences the dynamics of other qubits, leading to an enhanced collective relaxation. As $\eta$ is further increased, the interqubit interactions become very strong. Surprisingly, $\langle j_{z}(t)\rangle_{E}$ relaxes much more slowly than that of single qubit and the oscillation behavior of $\langle j_{z}(t)\rangle_{E}$ also disappears. It is understandable from the aspect of interaction-induced detunings [@JModOpt:Yu2016]. Diagonalizing the Hamiltonian (\[H2\]) leads to the eigenstates of the system. For the noninteracting system ($C_{c}=0$), the eigenstates with the same total number of the qubits in $\|1\rangle$ are degenerate. The interqubit coupling removes this degeneracy and gives rise to the energy-level shifts of eigenstates. The much strong interactions among qubits enhance the qubit-state shifts to the values comparable or even larger than the Josephson energy $E_{J}$. Thus, the system operating point moves far away from the sweet spot $N_{g}=\frac{1}{2}$, resulting in a large qubit detuning and the suppression of oscillation behavior of $\langle j_{z}(t)\rangle_{E}$. In addition, according to the relaxation-time formula derived from the Fermi’s golden rule [@RMP:Makhlin2001], $T_{1}$ is extended when the system moves away from the optimal working point. The dependence of $T_{1}$ on $\eta$ is displayed in figure \[Fig4\](b). It is shown that $T_{1}$ starts to rise after $\eta\approx1$ where, as illustrated by equation (\[H2\]), the energy-level shift (detuning) $\frac{E^{2}_{C}}{V}$ is equal to the driving strength $E_{J}$. We should point out that although $T_{1}$ is enhanced in the large interqubit-coupling regime, the qubit dephasing time $T_{2}$ is strongly suppressed. At the sweet spot $N_{g}=\frac{1}{2}$, the qubit is minimally sensitive to the $1/f$ fluctuation in the quantum circuit. When the qubit working point departs from $N_{g}=\frac{1}{2}$, the $1/f$ noise significantly reduces $T_{2}$ [@PRL:Astafiev2004; @PRL:Yoshihara2006; @PRA:Koch2007]. Nevertheless, the influence of the $1/f$ noise may be weakened via stabilizing the superconducting qubit to a high-$Q$ resonator [@Nature:Vijay2012] and the feedback control method [@NPJ:Yu2018; @PRA:Yu2018], which potentially extends the dephasing time $T_{2}$. ![(a) Rabi oscillation $\langle j_{z}(t)\rangle_{E}$ of the dissipative homogeneous system where all charge qubits are initialized in $\|n_{k=1,\ldots,N}=1\rangle$. The relaxation time $T_{1}$ is extracted from the decay envelope. (b) $T_{1}$ and steady-state ${\cal{C}}_{zz}(t\rightarrow\infty)$ vs. $\eta$. (c) Time-dependent probability density (PD) distribution of the trajectory ensemble of $j_{z}(t)$ and histogram of steady-state ensemble of $j_{z}(t\rightarrow\infty)$ for several different $\eta$, where the solid curves correspond to the line fittings. For $\eta=0.08$, $0.77$, and $7$, the corresponding standard deviations $\Delta j_{z}(t\rightarrow\infty)$ are derived as $0.45$, $0.18$, and $0.18$, respectively. For all curves, the system parameters are same to figure \[Fig1\] and the ensemble size is chosen to be $10^{3}$.[]{data-label="Fig4"}](Fig4 "fig:"){width="11.0cm"}\ After a long enough time $(t\gg T_{1})$, the multiple-qubit system loses the memory of its initial condition and the ensemble expectation value $\langle j_{z}(t)\rangle_{E}$ approaches a steady-state value [@JOSAB:Molmer1993; @RMP:Plenio1998]. We use the symbol $t\rightarrow\infty$ to denote a large time scale after which the distribution of $j_{z}(t)$ becomes steady, i.e., the ensemble-averaged observables reach steady-state values. As shown in figure \[Fig4\](a), $\langle j_{z}(t\rightarrow\infty)\rangle_{E}$ is equal to zero and independent of the coupling parameter $\eta$. Nevertheless, the steady-state histogram of the ensemble of $j_{z}(t\rightarrow\infty)$ relies on $\eta$. As depicted in figure \[Fig4\](c), the trajectory ensemble of $j_{z}(t\rightarrow\infty)$ is distributed over the entire range from -1 to 1 in the weak-coupling limit ($\eta\ll1$). In contrast, as the qubit-qubit interaction is increased, although the qubits are still equally populated on $\|0\rangle$ and $\|1\rangle$, the spread of the statistical distribution of $j_{z}(t\rightarrow\infty)$ is narrowed, indicating the reduction of measurement uncertainty and the enhanced interqubit correlation. Generally, the expectation value of a product of two operators ($O_{1}$ and $O_{2}$) is different from the product of the expectation values of individual operators, i.e., $\langle\psi\|O_{1}O_{2}\|\psi\rangle\neq\langle\psi\|O_{1}\|\psi\rangle\langle\psi\|O_{2}\|\psi\rangle$, which is attributed to the correlation between two quantities. Accordingly, since the qubits are coupled via the $zz$-interaction \[see equation (\[H1b\])\], we define $${\cal{C}}_{zz}(t)=\textstyle{\left\langle\frac{1}{N(N-1)}\sum_{k_{1}\neq k_{2}}\left[\langle\psi(t)\|\sigma^{(k_{1})}_{z}\sigma^{(k_{2})}_{z}\|\psi(t)\rangle-\langle\psi(t)\|\sigma^{(k_{1})}_{z}\|\psi(t)\rangle\langle\psi(t)\|\sigma^{(k_{2})}_{z}\|\psi(t)\rangle\right]\right\rangle_{E}},$$ to measure the total interqubit correlation in the system [@NewJPhys:Schachenmayer2015]. For independent qubits one has ${\cal{C}}_{zz}(t\rightarrow\infty)=0$. As exhibited in figure \[Fig4\](b), ${\cal{C}}_{zz}(t\rightarrow\infty)$ approximates zero in the limit of $\eta\ll1$, indicating that the qubits behave independently. As $\eta$ is increased, ${\cal{C}}_{zz}(t\rightarrow\infty)$ goes up strongly and is saturated eventually. We should note that ${\cal{C}}_{zz}(t)$ differs from the variance $\Delta j^{2}_{z}(t)$ of the ensemble of trajectories $j_{z}(t)$, the definition of which is $\Delta j^{2}_{z}(t)=\left\langle j^{2}_{z}(t)\right\rangle_{E}-\left\langle j_{z}(t)\right\rangle_{E}^{2}$. $\Delta j_{z}(t)$ weights the ensemble spread of $j_{z}(t)$. Figure \[Fig4\](c) shows the time-dependent ensemble distribution of $j_{z}(t)$ and steady-state histogram of $j_{z}(t\rightarrow\infty)$ for several different $\eta$. The distribution of $j_{z}(t)$ diffuses rapidly when $\eta$ is increased from zero, corresponding to the strong reduction of $T_{1}$. However, the width of steady-state distribution of $j_{z}(t\rightarrow\infty)$ becomes narrow, i.e., $\Delta j_{z}(t\rightarrow\infty)$ is suppressed, meaning that the interqubit correlation is enhanced. For $\eta\gg1$, the ensemble of $j_{z}(t)$ spreads slowly \[see $j_{z}(t)$ with $\eta=0.77$ and $\eta=7$ in figure \[Fig4\](c)\] and the histogram of $j_{z}(t\rightarrow\infty)$ barely changes compared with that of $\eta\approx1$, denoting the extension of $T_{1}$ and the saturation of ${\cal{C}}_{zz}(t\rightarrow\infty)$. Actually, the spreading rate of the histogram of $j_{z}(t)$ corresponds to the relaxation rate of the multi-qubit system. As shown in figure \[Fig4\](c), the system is initially prepared in the ground state $\|1\rangle_{1}\otimes\ldots\otimes\|1\rangle_{N}$. The histogram width of the trajectory ensemble $j_{z}(t)$ at $t=0$ is zero in principle. As the time $t$ is increased, the histogram of $j_{z}(t)$ becomes broader due to the fluctuation in the quantum circuit. For the system with small (large) $T_{1}$, the histogram of $j_{z}(t)$ reaches the steady-state distribution fast (slowly). Inhomogeneous Circuit --------------------- So far, we have only focused on the homogeneous system with identical qubits. However, in the practical circumstance it is extremely difficult to fabricate an array of identical Josephson junctions. Thus, we have to return to the inhomogeneous system described by equations (\[H1a\]) and (\[H1b\]). To take into account this nonideal-fabrication-induced inhomogeneity, we assume a normal distribution with a standard deviation $\lambda$ for different junction areas whose mean value is normalized to be unity [@PRL:Kakuyanagi2016]. We use $C_{j}$ and $E_{J}$ to respectively denote the mean values of self-capacitances $C_{j,k=1,\ldots,N}$ and Josephson energies $E_{J,k=1,\ldots,N}$, i.e., $C_{j}=\frac{1}{N}\sum_{k}C_{j,k}$ and $E_{J}=\frac{1}{N}\sum_{k}E_{J,k}$. The corresponding standard deviations are given by $\lambda C_{j}$ and $\lambda E_{J}$ since both $C_{j,k}$ and $E_{J,k}$ are proportional to the area of the $k$-th junction. For the charging energies $E_{C,k=1,\ldots,N}$, the mean value is $E_{C}=\frac{(2e)^{2}}{2C_{\Sigma}}$ with $C_{\Sigma}=C_{g}+C_{j}+C_{c}$ and the standard deviation is derived as $\lambda\frac{C_{j}}{C_{\Sigma}}E_{C}$ in the limit of $C_{\Sigma}\gg C_{j}$. In addition, the ensemble average $\langle\ldots\rangle_{E}$ needs to involve this extra inhomogeneity, for which we assume that a group of $N$-qubit systems are prepared in the same initial condition and $E_{C,k=1,\ldots,N}$ and $E_{J,k=1,\ldots,N}$ of each system fulfill the corresponding normal distributions. As shown in figure \[Fig5\](a), the inhomogeneity caused by nonidentical Josephson junctions strongly accelerates the relaxation of the collective Rabi oscillation even in the weak-coupling limit ($\eta\ll1$). This corresponds to the inhomogeneous broadening, where the unsynchronized Rabi dynamics of individual qubits destructively interfere with each other. Moreover, the extra inhomogeneity reduces the interqubit correlation and broadens the trajectory distribution \[figure \[Fig5\](b)\]. Therefore, maximally minimizing the inhomogeneity arising from the defective fabrication technology becomes indispensable for the application of multiple-qubit circuits in QIP. Nevertheless, this inhomogeneous system may be still potentially applied to study the quantum many-body localization. We choose two spin states of the $k$-th qubit as $\|\uparrow\rangle_{k}=\frac{1}{\sqrt{2}}(\|0\rangle_{k}+\|1\rangle_{k})$ and $\|\downarrow\rangle_{k}=\frac{1}{\sqrt{2}}(\|0\rangle_{k}-\|1\rangle_{k})$ and define the new $x$- and $y$-components of Pauli matrix as $\tilde{\sigma}_{x,k}\equiv\sigma_{z,k}$ and $\tilde{\sigma}_{z,k}\equiv\sigma_{x,k}$. Then, the inhomogeneous Hamiltonian $H$ can be mapped onto the Ising model [@NatPhys:Smith2016] $$H_{Ising}=\textstyle\sum_{k<k'}J_{k,k'}\tilde{\sigma}_{x,k}\tilde{\sigma}_{x,k'}-\frac{B}{2}\sum_{k}\tilde{\sigma}_{z,k}-\sum_{k}\frac{D_{k}}{2}\tilde{\sigma}_{z,k}+\textstyle\sum_{k}\Delta_{k}\tilde{\sigma}_{x,k},$$ where $J_{k,k'}=\frac{E_{C,k}E_{C,k'}}{4NV}$ corresponds to the long-range spin-spin interaction, $B=E_{J}$ plays the role of the external field, and the inhomogeneity $D_{k}=E_{J,k}-E_{J}$ acts as the site-dependent disordered potential. Unlike the superconducting circuit in [@arXiv:Xu2017], the nearest-neighbor $J_{k,k\pm1}$ do not dominate. The last term in $H_{Ising}$ with $$\Delta_{k}=\textstyle\frac{E_{C,k}}{2}(1-2N_{g})\left(1+\sum_{k'}\frac{E_{C,k'}}{NV}\right),$$ vanishes at the sweet point $N_{g}=\frac{1}{2}$. However, the nonzero $\delta N_{g}(t)$ deviates $H_{Ising}$ from the standard disordered Ising model. In addition, the disorder term $D_{k}$ is sampled from a normal distribution, rather than a uniform random variable [@arXiv:Xu2017]. ![Inhomogeneous multiple-qubit system and its application in many-body localization. (a) Upper: Relaxation time $T_{1}$ vs. standard deviation $\lambda$ with $\eta=0.08$. Lower: Rabi oscillation $\langle j_{z}(t)\rangle_{E}$ for several different $\lambda$. (b) Upper: ${\cal{C}}_{zz}(t\rightarrow\infty)$ as a function of $\lambda$ with $\eta=1.53$. Lower: Histogram of $j_{z}(t\rightarrow\infty)$ with $\lambda=0.4$ and $\Delta j_{z}(t\rightarrow\infty)=0.26$. (c) $\langle{\cal{D}}(t)\rangle_{E}$ of the disordered dissipative system with $N_{g0}=\frac{1}{2}$ and $\lambda=0.5$. (d) $\langle{\cal{D}}(t)\rangle_{E}$ of the disordered nondissipative system with $N_{g}=\frac{1}{2}$ and $\lambda=0.5$. The dashed lines are the results from the curve fitting based on $\langle{\cal{D}}(t\rightarrow\infty)\rangle_{E}(1-e^{-\gamma t})$. The localization rate approximates $\gamma\approx2\pi\times1.6$ GHz for different $\eta$. The values of $\langle{\cal{D}}(t\rightarrow\infty)\rangle_{E}$ are about 0.01, 0.1, 0.3, and 0.38 for $\eta=0.77$, 7, 39, and 71, respectively. Inset: Histogram of the ratio $r$ of adjacent energy-level gaps, where the solid curve corresponds to the Poisson distribution. The ensemble size is chosen to be $10^{3}$ for all curves. All system parameters are same to figure \[Fig1\].[]{data-label="Fig5"}](Fig5 "fig:"){width="12.0cm"}\ According to [@NatPhys:Smith2016], in the weak-coupling limit ($\eta\ll1$), we have $\lambda E_{J}\gg\textrm{max}(J_{k,k'})$ and the system may emerge the multiple-spin localization, which is quantified by the normalized Hamming distance [@PRB:; @Hauke2015] $${\cal{D}}(t)=\textstyle\frac{1}{2}-\frac{1}{2N}\sum_{k}\langle\psi(0)\|e^{i\frac{H_{Ising}}{\hbar}t}\tilde{\sigma}_{z,k}e^{-i\frac{H_{Ising}}{\hbar}t}\tilde{\sigma}_{z,k}\|\psi(0)\rangle,$$ with the initial state $\psi(0)$ being the Néel state. ${\cal{D}}(t\rightarrow\infty)$ arrives at $\frac{1}{2}$ for a thermalizing state and remains 0 at a fully localized state. For the disordered dissipative system with ($\lambda\neq0$, $\delta N_{g} (t)\neq0$), the Hamming distance ${\cal{D}}(t\rightarrow\infty)$ always approaches $\frac{1}{2}$, reaching the thermal equilibrium \[see figure \[Fig5\](c)\]. This is attributed to the effect of environmental fluctuations $\delta N_{g} (t)$. Thus, besides QIP, suppressing the environmental noise is also the key issue for the application of superconducting circuits in the many-body simulation. Figure \[Fig5\](d), where we have artificially set $\delta N_{g}(t)=0$, exhibits that the disordered nondissipative system with ($\lambda\neq0$, $\delta N_{g} (t)=0$) stays nearly fully localized product states in the weak-coupling limit ($\eta\ll1$) while ${\cal{D}}(t\rightarrow\infty)$ rises gradually up to $\frac{1}{2}$ as the qubit-qubit interactions become strong. Using the diagonalization method, we have also checked the statistics of the ratio parameter $r$ of adjacent energy-level gaps for the disordered nondissipative system [@PRB:Oganesyan2007; @PRB:Pal2010]. It is seen that $r$ follows the Poisson distribution when $\eta\ll1$ \[the inset of figure \[Fig5\](d)\], manifesting the many-body localization, and apparently violates the Poisson distribution for a larger $\eta$, suppressing the localization effect. As experimentally demonstrated in [@arXiv:Xu2017], the many-body localized state may be still attainable within a short time scale when the thermalization rate of the system with ($\lambda=0$, $\delta N_{g} (t)\neq0$) coupling to the environment is much slower than the localization rate of the system with ($\lambda\neq0$, $\delta N_{g} (t)=0$) (i.e., the localization rate measures how fast a disordered nondissipative system reaches the localized state from an initially-prepared state). The localization rate $\gamma$ of the system with ($\lambda\neq0$, $\delta N_{g} (t)=0$) may be roughly estimated by assuming that the envelop of ${\cal{D}}(t)$ follows the exponential law, i.e., $\langle{\cal{D}}(t\rightarrow\infty)\rangle_{E}(1-e^{-\gamma t})$ \[see the dashed lines in figure \[Fig5\](d)\]. The time scale that measures the thermalization rate of the system with ($\lambda=0$, $\delta N_{g} (t)\neq0$) coupling to the environment is given by $T_{1}(\lambda=0)$. The many-body localized state is potentially observed in the disordered dissipative system when $\gamma\gg T^{-1}_{1}(\lambda=0)$ which may be fulfilled for a large $\eta$. In comparison between figure \[Fig5\](c) and \[Fig5\](d), we find that for $\eta\gg1$ the Hamming distance ${\cal{D}}(t)$ of the disordered dissipative system rapidly reaches a metastable value smaller than $\frac{1}{2}$ (i.e., the system arrives at the localized state) and then slowly grows up to $\frac{1}{2}$ (i.e., the system approaches the thermalizing state). Thus, the many-body localization phenomenon can be observed within a short time scale for $\eta\gg1$. Increasing the disorder strength $D_{k}$, i.e., increasing $\lambda$, may enhance the localization rate of the inhomogeneous system, leading to an easy access to the many-body localized phase. However, a large $\lambda$ strongly reduces $\frac{E_{C,k}}{E_{J,k}}$, i.e., some qubits may not operate in the charging limit ($\frac{E_{C,k}}{E_{J,k}}\gg1$). Conclusion ========== We have studied a multiple-charge-qubit system, where the Cooper-pair boxes are all capacitively linked. The qubit dynamics are interfered with environmental fluctuations which are mapped onto the gate voltage source that biases all qubits. The collective Rabi oscillation has been numerically simulated for both homogeneous and inhomogeneous systems. We find the interqubit coupling strongly varies the energy-relaxation rate of the quantum circuit consisting of identical Josephson junctions. For the weak coupling system, the qubit relaxation time $T_{1}$ is significantly reduced since the dynamics of one qubit is unavoidably influenced by the fluctuations of other qubits. In contrast, $T_{1}$ of the homogeneous system in the strong coupling regime can be enhanced to a value much larger than that of a free qubit. This result is caused by the strong energy-level shifts of qubit states (i.e., the large interaction-induced detunings) and consistent with the expectation of Fermi’s golden rule. In QIP, transferring quantum information between two qubits in a homogeneous multi-qubit network is an essential process. The resulting fidelity is limited by the system’s decoherence time. Thus, the enhancement of $T_{1}$ is crucially relevant to QIP. The nonideal-fabrication-induced inhomogeneity always expedites the multi-qubit system’s collective decay. In addition, we mapped the inhomogeneous system onto the disordered Ising model to probe the many-body localization effect. This enables us to investigate the role of the environmental noise in the quantum simulation. For the inhomogeneous system with the localization rate faster than the thermalization rate, the many-body-localization regime is still accessible. Acknowledgements ================ This research has been supported by the National Research Foundation Singapore & by the Ministry of Education Singapore Academic Research Fund Tier 2 (Grant No. MOE2015-T2-1-101). 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--- author: - \| Robert Brooks[^1]\ Department of Mathematics\ Technion– Israel Institute of Technology\ Haifa, Israel - \| Orit Davidovich\ Department of Mathematics\ Technion– Israel Institute of Technology\ Haifa, Israel date: ' [January, 2002]{}' title: Isoscattering on Surfaces --- \#1[[[\[\#1\]]{}]{}]{} \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \#1 In this paper, we give a number of examples of pairs of non-compact surfaces $S_1$ and $S_2$ which are isoscattering, to be defined below. Our basic construction is based on a version of Sunada’s Theorem [@Su], which has been refined using the technique of transplantation ([@Be], [@Zel]) so as to be applicable to isoscattering. See [@BGP] and [@BP] for this approach, which is reviewed below. Our aim here is to present a number of examples which are exceptionally simple in one or more senses. Thus, the present paper can be seen as an extension of [@BP], where the aim was to construct isoscattering surfaces with precisely one end. We will show: \[genus\] [(a)]{} There exist surfaces $S_1$ and $S_2$ of genus 0 with eight ends which are isoscattering. [(b)]{} There exist surfaces $S_1$ and $S_2$ of constant curvature $-1$ which are of genus 0 and have fifteen ends. [(c)]{} There exist surfaces $S_1$ and $S_2$ of genus $1$ with five ends, or genus $2$ with three ends, which are isoscattering. [(d)]{} There exist surfaces $S_1$ and $S_2$ of constant curvature $-1$ which are of genus 1 with thirteen ends, or genus 2 with five ends, or genus 3 with three ends, which are isoscattering. [(e)]{} There exist surfaces $S_1$ and $S_2$ of genus 3 with one end, or constant curvature of genus 4 with one end, which are isoscattering. Part (e) is just a statement of the results of [@BP], and is recorded here for the sake of completeness. It will not be discussed further in this paper. The nature of the ends in Theorem \[genus\] is not too important. In the cases where the curvature is variable, they can be taken to be hyperbolic funnels or Euclidean cones, or to be hyperbolic finite-area cusps. In the constant curvature $-1$ cases, they can be taken either to be infinite-area funnels or finite-area cusps. Recall that a surface $S$ is called a [congruence surface]{} if $S= {{{\Bbb{H}}}}^2/\Gamma$, where $\Gamma$ is contained in $PSL(2, {{\Bbb{Z}}})$ and contains a subgroup $$\Gamma_k = \left\{ {\left( \begin{array}{cc}}a&b \\ c & d { \end{array}\right)}\equiv \pm {\left( \begin{array}{cc}}1 &0\\ 0&1 { \end{array}\right)}({{\hbox{mod}}\ }k) \right\}$$ for some $k$. In other words, the group $\Gamma$ is the inverse image of a subgroup of $PSL(2, {{\Bbb{Z}}}/k)$ under the natural map $$PSL(2, {{\Bbb{Z}}}) \to PSL(2, {{\Bbb{Z}}}/k).$$ We then have: \[cong\] There exist two congruence surfaces $S_1$ and $S_2$ which are isoscattering. Theorem \[cong\] answers a question which was raised to us by Victor Guillemin. The point here is that congruence surfaces have a particularly rich structure of eigenvalues embedded in the continuous spectrum. On the other hand, subgroups of $PSL(2, {{\Bbb{Z}}}/k)$ have a very rigid structure [@Di], and it is not [a priori]{} clear that the finite group theory is rich enough to support the Sunada method. A version of Theorem \[genus\] was announced without proof in an appendix to [@BJP]. Theorems \[s4\] and \[D\] are certainly well-known to finite-group theorists. We hope that the explicit treatment given here will be useful to spectral geometers. The first author would like to thank MIT for its warm hospitality and the Technion for its sabbatical support for the period in which this paper was written. He would also like to thank Victor Guillemin for his interest, and Peter Perry for his suggestion to pursue these questions in the context of the paper [@BJP]. Transplantation and Isoscattering ================================= We recall the approach to the Sunada Theorem given in [@BGP]. Recall that a Sunada triple $(G, H_1, H_2)$ consists of a finite group $G$ and two subgroups $H_1$ and $H_2$ of $G$ satisfying $$\label{dagger} {\hbox{for all}}\ g \in G, \#([g] \cap H_1 ) = \#([g] \cap H_2),$$ where $[g]$ denotes the conjugacy class of $g$ in $G$. \[sun\] Suppose that $M$ is a manifold and $\phi: \pi_1(M) \to G$ a surjective homomorphism. Let $M^{H_1}$ and $M^{H_2}$ be the coverings of $M$ with fundamental groups $\phi^{-1}(H_1)$ and $\phi^{-1}(H_2)$ respectively. Then there is a linear isomorphism $${{\cal{T}}}: C^{\infty}(M^{H_1}) \to C^{\infty}(M^{H_2})$$ such that ${{\cal{T}}}$ and ${{\cal{T}}}^{-1}$ commute with the Laplacian. We first remark that condition (\[dagger\]) is equivalent to the following: if we denote by $L^2(G/H_i)$ the $G$-module of functions on the cosets $G/H_i$, then $$L^2(G/H_1)\ {\hbox{is isomorphic to}}\ L^2(G/H_2)\ {\hbox{as}}\ G-{\hbox{modules}}.$$ We may further rewrite this by noting that $L^2(G)$ has two $G$-actions, on the left and on the right, so that we may write $$L^2(G/H_i) \equiv (L^2(G))^{H_i},$$ where the equivariance under $H_i$ is taken with respect to the left $G$-action, and $G$-equivariance is taken with respect to the right $G$-action. Equation (\[dagger\]) is then equivalent to $$\label{dags} (L^2(G))^{H_1}\ {\hbox{is $G$-isomorphic to}}\ (L^2(G))^{H_2}.$$ We may further rewrite this equation as saying that there is a $G$-equivariant map $T: L^2(G) \to L^2(G)$ which induces an isomorphism $(L^2(G))^{H_1} \to (L^2(G))^{H_2}$. Now any $G$-map is determined by its value on the delta function, which in turn can be described by a function $$c: G \to R,$$ so that the $G$-module map is given by $$T(f)(x) = \sum_{g \in G} c(g) f(g\cdot x).$$ The requirement that the image of this map lies in $(L^2(G))^{H_2}$ can be expressed in terms of $c$ by the condition that $$\label{star} c(gh)= c(g)\ \quad {\hbox{for}}\ h \in H_2.$$ We may therefore express the condition (\[dagger\]) as the existence of a function $c$ on $G$ which satisfies (\[star\]), and which furthermore induces an isomorphism as in (\[dags\]). Given such a function $c$, we may then write out the function ${{\cal{T}}}$ as follows: let $M^{id}$ be the covering of $M$ whose fundamental group is $\phi^{-1}(id)$. Then we may identify $C^{\infty}(M^{H_i})$ with $(C^{\infty}(M^{id}))^{H_i}$. The desired expression for ${{\cal{T}}}$ is then given by $${{\cal{T}}}(f)(x) = \sum_g c(g) f(g\cdot x).$$ We emphasize that all of this makes perfectly good sense for any function $c$ satisfying (\[star\]). The condition that it induces an isomorphism is the crucial property we need. Clearly, ${{\cal{T}}}$ and its inverse take smooth functions to smooth functions, and also commute with the Laplacian, since both statements are true of the action by $g$ and taking linear combinations. This establishes the theorem. We now consider the case when the manifold $M$ is complete and non-compact. We will discuss here the case where $M$ is hyperbolic outside of a compact set, the case of Euclidean ends having been discussed in [@BP]. We begin with a complete surface $M_0$, and consider a conformal compactification of $M_0$, consisting of one circle for each funnel and a point for each cusp. We also pick a [ defining function]{} $\rho$ on $M_0$, that is, a function which is positive on $M$ and vanishes to first order on the boundary of $M_0$. If $0 < \lambda <1/4$, then we choose real $s$ so that $$\lambda = (s)(1-s).$$ Then, if $f \in C^{\infty}({\partial}M_0)$, there exist unique functions $u$ on $M_0$ and ${{\cal{S}}}_s(f) \in C^{\infty}({\partial}M_0) $ such that [(i)]{} $ {\Delta}(u) = \lambda u.$ [(ii)]{} $u \sim (\rho)^s {{\cal{S}}}_s(f) + (\rho)^{1-s}f + {{\cal{O}}}(\rho)$ as $\rho \to 0$. The operator ${{\cal{S}}}_s$ is the [scattering operator]{} for $s$, and continues for all $s$ to be a meromorphic operator. Two surfaces $M_0$ and $M_1$ will be [isoscattering]{} if they have poles of the same multiplicity at the same values of $s$. We now pick $M$ as in Theorem \[sun\], and lift the defining function $\rho$ on $M$ to defining functions on $M^{H_1}$, $M^{H_2}$, and $M^{id}$. Note that we may identify ${{\cal{S}}}_s$ on $M^{H_i}$ with the operator ${{\cal{S}}}_s$ on the $H_1$-invariant part of $M^{id}$. We then have: The surfaces $M^{H_1}$ and $M^{H_2}$ are isoscattering. [[[Proof]{}]{}:]{}If we are given $f \in C^{\infty}({\partial}M^{id})$, then clearly $${{\cal{T}}}(u) \sim (\rho)^s {{\cal{T}}}(S_f(f)) + (\rho)^{1-s} {{\cal{T}}}(f) + {{\cal{O}}}(\rho),$$ or, in other words, $${{\cal{T}}}({{\cal{S}}}_s(f)) = {{\cal{S}}}_s({{\cal{T}}}(f)).$$ Thus, ${{\cal{T}}}$ intertwines ${{\cal{S}}}_s$ for all $s$, and hence ${{\cal{S}}}_s$ on $M^{H_1}$ and $M^{H_2}$ have poles (with multiplicities) at the same values. This completes the proof. The Group $PSL(3, {{\Bbb{Z}}}/2)$ ================================= It is a rather remarkable fact that most of the examples of isospectral surfaces ([@BT]) as well as all of the examples of Theorem \[genus\], can be constructed from one Sunada triple. This is the triple $(G, H_1, H_2)$ , where $$G= PSL(3, {{\Bbb{Z}}}/2),$$ and $$H_1 = {\left( \begin{array}{ccc}}{}* & {}* & {}* \\ 0 & {}* & {}* \\ 0 & {}* & {}* {{ \end{array}\right)}}\quad H_2 = {\left( \begin{array}{ccc}}{}* & 0 & 0\\ {}* & {}* & {}* \\ {}* & {}* & {}* {{ \end{array}\right)}}.$$ Note that the outer automorphism $$A \to (A^{-1})^t$$ takes $H_1$ to $H_2$, and also takes elements of $H_1$ to conjugate elements. This is enough to show that $(G, H_1, H_2)$ is a Sunada triple. In this section, we will present the necessary algebraic facts to prove Theorem \[genus\]. Many of these facts are proved easily by noting the isomorphism $$PSL(3, {{\Bbb{Z}}}/2) \cong PSL(2, {{\Bbb{Z}}}/7).$$ It is somewhat difficult to see the subgroups $H_1$ and $H_2$ in $PSL(2, {{\Bbb{Z}}}/7)$. The outer automorphism which takes $H_1$ to $H_2$ is, however, easy to describe. It is the automorphism $${\left( \begin{array}{cc}}a& b\\ c& d { \end{array}\right)}\to {\left( \begin{array}{cc}}a & -b\\ -c & d { \end{array}\right)}= {\left( \begin{array}{cc}}-1 & 0 \\ 0 & 1 { \end{array}\right)}{\left( \begin{array}{cc}}a & b \\ c & d { \end{array}\right)}{\left( \begin{array}{cc}}-1 & 0 \\ 0 & 1 { \end{array}\right)}.$$ The fact that this cannot be made an inner automorphism follows from the fact that $-1$ is not a square $({{\hbox{mod}}\ }7)$. We now describe the conjugacy classes of $PSL(2, {{\Bbb{Z}}}/7)$: Every element of $PSL(2, {{\Bbb{Z}}}/7)$ is of order $1, 2, 3, 4$, or $7$. [(a)]{} The only element of order 1 is the identity. [(b)]{} Every element of order $2$ is conjugate to ${\left( \begin{array}{cc}}0 & 1\\ -1 & 0 { \end{array}\right)}$. [(c)]{} Every element of order $3$ is conjugate to ${\left( \begin{array}{cc}}1 & 1\\ -1 & 0 { \end{array}\right)}$. [(d)]{} Every element of order $4$ is conjugate to ${\left( \begin{array}{cc}}2 & 1 \\ 1 & 1 { \end{array}\right)}$. [(e)]{} Every element of order $7$ is conjugate to either ${\left( \begin{array}{cc}}1 & 1\\ 0 & 1 { \end{array}\right)}$ or ${\left( \begin{array}{cc}}1 & -1 \\ 0 & 1 { \end{array}\right)}$. Translating back into the group $PSL(3, {{\Bbb{Z}}}/2)$ gives The elements of $PSL(3, {{\Bbb{Z}}}/2)$ satisfy: \[con\] [(a)]{} Every element of order $2$ is conjugate to $${\left( \begin{array}{ccc}}1 & 1 & 0\\ 0&1&0 \\ 0&0&1 {{ \end{array}\right)}}.$$ [(b)]{} Every element of order $3$ is conjugate to $${\left( \begin{array}{ccc}}0&1&0 \\ 0&0&1\\ 1&0&0 {{ \end{array}\right)}}.$$ [(c)]{} Every element of order $4$ is conjugate to $${\left( \begin{array}{ccc}}1&1&0 \\ 0&1&1 \\ 0&0&1 {{ \end{array}\right)}}.$$ [(d)]{} Every element of order $7$ is conjugate to either $${\left( \begin{array}{ccc}}1 & 1 & 1 \\ 1& 1&0 \\ 0 & 1 & 1 {{ \end{array}\right)}}$$ or $${\left( \begin{array}{ccc}}1&0&1 \\ 1&1&1\\ 1&1& 0 {{ \end{array}\right)}}.$$ [[[Proof]{}]{}:]{}It suffices to check that each matrix has the order indicated. We remark that a simple criterion for an element to be of order $7$ is that adding 1 to the diagonal entries produces a non-singular matrix. To check that the two matrices in (d) above are not conjugate, we observe that their characteristic polynomials are distinct. Identifying $G/H_1$ as non-zero row vectors, we now may calculate the action of an element of $G$ on $G/H_1$ as a permutation representation. We will be interested in the cycle structure of this representation, which clearly only depends on the conjugacy class of the element. It follows from the above that the same calculation is also valid for the permutation representation on $G/H_2$. Let $g \in PSL(3, {{\Bbb{Z}}}/2)$. Then the cycle structure of the permutation representation of $g$ on $G/H_1$ and $G/H_2$ is given by: [(a)]{} If $g$ is of order $2$, then $g$ acts as the product of two cycles of order 2 and three 1-cycles. [(b)]{} If $g$ is of order $3$, then $g$ acts as two $3$-cycles and a $1$-cycle. [(c)]{} If $g$ is of order $4$, then $g$ acts as a $4$-cycle, a $2$-cycle, and a $1$-cycle. [(d)]{} If $g$ is of order $7$, then $g$ acts as a $7$-cycle. The proof is just an evaluation in each case of the representatives in Lemma \[con\]. Proof of Theorem \[genus\] ========================== In this section, we will prove Theorem \[genus\]. Our method will be to find an orbifold surface $M$ and a surjective homomorphism from $\pi_1(M)$ to a Sunada triple $(G, H_1, H_2)$. We will take $G$ to be $PSL(3, {{\Bbb{Z}}}/2) \cong PSL(2, {{\Bbb{Z}}}/7)$, and $H_1$ and $H_2$ the corresponding subgroups. We then would like to study the corresponding coverings $M^{H_1}$ and $M^{H_2}$. Let $x$ be a singular point of $M$, and let $g_x$ be the element of $\pi_1(M)$ corresponding to going one around $x$, which is well-defined up to conjugacy. Then the points of $M^{H_1}$ (resp. $M^{H_2}$) lying over $x$ are in 1-to1 correspondence with the cycle decomposition of $g_x$ on $G/H_1$ (resp. $G/H_2$). If $g_x$ acts freely on the cosets, then we may choose the orbifold singularity at $x$ so that it smooths out to a regular (i.e. nonsingular) point in $G/H_1$ (resp. $G/H_2)$. By Theorem \[con\], this will happen only if $g_x$ is the identity or of order $7$, in which case there will be precisely one point lying over $x$. We wish to calculate the genus of $M^{H_1}$ and $M^{H_2}$ as topological surfaces (that is, forgetting the orbifold structure). Our strategy will be the following: we remove all the singular points of $M$ and the points lying over them, so that the covering is now a regular (i.e. non-orbifold) covering. We then multiply the Euler characteristic of $M$ with the points removed by $7$, the index $[G:H_1] = [G:H_2]$. We then add back in the points lying over the singular points. By Theorem \[con\], there will be five such points if $g_x$ is of order 2, three such points if $g_x$ is of order 3 or order 4, and one if $g_x$ is of order $7$. We may now compute the genus of $M^{H_1}$ (resp. $M^{H_2}$) from the Euler characteristic. We now have to calculate the number of ends. We must throw out those singular points lying over a singular points of oreder 2 (there are five of these), order 3, or order 4 (there are three of these in both cases). We need not throw out the point lying over a singular point of order $7$. We then have to worry about whether $M^{H_1}$ and $M^{H_2}$ are distinct. As argued, for instance, in [@Su] or [@BGP], that if we choose a variable metric on $M$, then for a generic choice of such metrics $M^{H_1}$ and $M^{H_2}$ will be non-isometric. If we want constant curvature metrics, this will in general fail if $M$ is a sphere with three singular points, as examples in [@BT] show, but if $M$ is a sphere with $n$ singular points, $n \ge 4$, or a surface of higher genus with an arbirary number of singular points, then choosing a generic conformal structure on $M$ and generically placed points will produce $M^{H_1}$ and $M^{H_2}$ distinct. Let us first take the case where $M$ is a sphere with three singular points. Choosing these singular points to be of order $2, 3$, and $7$, we must find matrices $A, B,$ and $C$ in $PSL(2, {{\Bbb{Z}}}/7)$ such that: [(i)]{} $A$ is of order 2, $B$ is of order $3$, and $C$ is of order $7$. [(ii)]{} $ABC= {\hbox{id}}$. [(iii)]{} $A$, $B$, and $C$ generate $PSL(2, {{\Bbb{Z}}}/7)$. A simple choice is $$A = {\left( \begin{array}{cc}}0&1\\ -1 & 0 { \end{array}\right)}\quad B= {\left( \begin{array}{cc}}1 & 1\\ -1 & 0 { \end{array}\right)}$$ $$C= {\left( \begin{array}{cc}}1&0 \\ -1& 1 { \end{array}\right)}.$$ The computation of the genus of $M^{H_1}$ (resp. $M^{H_2}$) proceeds as follows: the thrice-punctured sphere has Euler characteristic $\chi = -1$. Hence $M^{H_i}$ without the singular points has $\chi = -7$. Putting in the five singular points lying over the singular point of order $2$, the three singular points lying over the point of order $3$, and the point lying over the point of order $7$ adds $5 + 3 + 1=9$ to this, yielding an Euler characteristic of $2$. Hence $M^{H_1}$ and $M^{H_2}$ are of genus $0$. We must make ends out of the points lying over the singular points of orders $2$ and $3$, to give a total of eight ends. This establishes Theorem \[genus\] (a). If we had used two singular points of order $7$ and one singular point of order $2$ (resp. 3 or 4), we would obtain for $M^{H_1}$ and $M^{H_2}$ surfaces of genus 1 (resp. 2) with five (resp. 3) ends, provided we can find the corresponding generators. But $$B = {\left( \begin{array}{cc}}1 & k\\ 0 & 1 { \end{array}\right)}\quad {\hbox{and}}\ C= {\left( \begin{array}{cc}}1 & 0\\ l & 1 { \end{array}\right)}$$ generate $PSL(2,{{\Bbb{Z}}}/7)$ for any choice of $k,l$ prime to $7$, and their product has trace $2+ kl$. Hence, for appropriate choice of $k$ and $l$, we may find $A$ of order $2, 3,$ or $4$. This establishes (c). We now investigate what happens when we choose $M$ to be a sphere with four singular points. Choosing the singular points to be of order $2, 2, 2$, and $7$ respectively, we calculate the Euler characteristic of $M^{H_i}$ as $$\chi = 7(-2) + 3\cdot 5 + 1 = 2,$$ so again the $M^{H_i}$ have genus $0$, now with fifteen ends, provided we can find matrices $A, B, C,$ and $D$ of these orders which generate $PSL(2, {{\Bbb{Z}}}/7)$ and whose product is 1. To do this, we first observe that we may find two matrices $B'$ and $C'$ of order $2$ such that their product is of order $3$. One choice is $$B'= {\left( \begin{array}{cc}}0 & 1 \\ -1 & 0 { \end{array}\right)}, \quad C'= {\left( \begin{array}{cc}}0 & 2 \\ 3 & 0 { \end{array}\right)}.$$ We then conjugate $B'$ and $C'$ so that their product is ${\left( \begin{array}{cc}}1 & 1 \\ -1 & 0 { \end{array}\right)}$. We may then choose $A$ and $D$ to be ${\left( \begin{array}{cc}}0 & 1\\ -1 & 0 { \end{array}\right)}$ and ${\left( \begin{array}{cc}}1 & 0 \\ -1 & 1 { \end{array}\right)}$ as above. This establishes (b). To establish the first part of (d), we search for matrices of order $2,2,3$, and $7$ whose product is 1. Choosing $$C = {\left( \begin{array}{cc}}1 & 1 \\ -1 & 0 { \end{array}\right)}, \quad D= {\left( \begin{array}{cc}}1 & 0\\ k & 1 { \end{array}\right)},$$ we have that $C$ and $D$ generate $PSL(2, {{\Bbb{Z}}}/7)$, and for an appropriate choice of $k$ the product is of order $3$. We may then choose $A$ and $B$ as above to be two matrices of order two whose product of order $3$ is the inverse of this matrix. To establish the second and third parts of (d), we proceed differently. The base surface $M$ will be of genus 1 with one singular point. If the singular point is of order $2$, the resulting $M^{H_i}$ will be of genus $2$, with five ends. If the singular point is of order $3$ or $4$, the resulting surface is of genus $3$, with three ends. We therefore seek matrices $A$ and $B$ which generate $PSL(2, {{\Bbb{Z}}}/7)$, such that their commutator is of order $2$ (resp. $3$ or $4$). Choosing $$A= {\left( \begin{array}{cc}}0 & 1\\ -1 & 0 { \end{array}\right)}, \quad B = {\left( \begin{array}{cc}}1 & 0\\ k& 1 { \end{array}\right)},$$ we get $$\left[A,B\right]= {\left( \begin{array}{cc}}1 + k^2 & -k\\ -k & 1{ \end{array}\right)},$$ and $A$ and $B$ generate $PSL(2,{{\Bbb{Z}}}/7)$. Choosing, for instance, $k=2$ gives a commutator of order $3$. Choosing $$A = {\left( \begin{array}{cc}}0 & 1\\ -1 & 0 { \end{array}\right)}\quad B= {\left( \begin{array}{cc}}4& 1\\ 0 & 2 { \end{array}\right)}$$ gives $$\left[A, B\right] = {\left( \begin{array}{cc}}3 & 2\\ 2&4 { \end{array}\right)},$$ which is of order $2$. We see no elegant way of seeing that $A$ and $B$ generate $PSL(2, {{\Bbb{Z}}}/7)$, but $$A(BAB^2)(\left[A,B\right])= {\left( \begin{array}{cc}}1& 0\\ 2 & 1 { \end{array}\right)},$$ and this matrix and $A$ generate. This concludes the proof of the second and third parts of (d), and hence Theorem \[genus\]. Proof of Theorem \[cong\] ========================= In this section, we construct congruence surfaces which are isoscattering. There are several difficulties in this setting which are not present in the general setting. First of all, congruence surfaces are constructed out of subgroups of the finite groups $PSL(2, {{\Bbb{Z}}}/k)$, and such groups are rather special. The subgroups of $PSL(2, {{\Bbb{Z}}}/p)$ have been classified, and are given in Dickson’s List [@Di]. It is not [a priori]{} evident, for instance, that $PSL(2,{{\Bbb{Z}}}/p)$ contains Sunada triples for general $p$. Of course, the case of $PSL(3, {{\Bbb{Z}}}/2) \cong PSL(2, {{\Bbb{Z}}}/7)$ occurs as a very special example, but we will need a richer collection of examples. Secondly, given such a Sunada triple $(G, H_1,H_2)$, we do not have the freedom of choosing a homomorphism $\pi_1(M) \to G$ as we did previously. It is given to us canonically. Finally, we must worry about “extra isometries,” since we do not have the freedom to change parameters to guarantee that $M^{H_1}$ and $M^{H_2}$ will be distinct. We begin our discussion with the group $G= PSL(2, {{\Bbb{Z}}}/7)$, and $H_1$ and $H_2$ the two subgroups as above. Taking $$\Gamma= PSL(2, {{\Bbb{Z}}}),$$ and considering the natural projection $\phi: \Gamma \to G$, we first notice that the $\phi^{-1}(H_i)$ contain torsion elements, so that the ${{{\Bbb{H}}}}^2/\phi^{-1}(H_i)$ are singular surfaces. To remedy this problem, and also to introduce a technique we will use later, we note that for $k=k_1 k_2$, with $k_1$ and $k_2$ relatively prime, we have $$PSL(2, {{\Bbb{Z}}}/k)= P( SL(2, {{\Bbb{Z}}}/k_1) \times SL(2, {{\Bbb{Z}}}/k_2)).$$ Choosing $k=14$, we see that $$PSL(2, {{\Bbb{Z}}}/14)= PSL(2,{{\Bbb{Z}}}/2)\times PSL(2,{{\Bbb{Z}}}/7),$$ noting that the “$P$” in $PSL(2, {{\Bbb{Z}}}/2)$ is trivial. Furthermore, the kernel $\Gamma_2$ of $PSL(2, {{\Bbb{Z}}}/2)$ satisfies that ${{{\Bbb{H}}}}^2/\Gamma_2$ is a (non-singular) thrice-punctured sphere. Hence, we resolve the issue of singularities by restricting to $\Gamma_2$. Now let $S^{i}= {{{\Bbb{H}}}}^2/\Gamma^{H_i},$ where $$\Gamma^{H_i} = \Gamma_2 \cap \phi^{-1}(H_i).$$ Then, as before, $S^{1}$ and $S^{2}$ are isoscattering. They are, however, also isometric. This can be seen by noting that $H_1$ and $H_2$ are conjugate under the automorphism $$\tau: {\left( \begin{array}{cc}}a& b\\ c& d { \end{array}\right)}\to {\left( \begin{array}{cc}}a & -b \\ -c & d { \end{array}\right)}.$$ Furthermore, this $\tau$ induces an orientation-reversing isometry of $H^2/\Gamma$ which is reflection in the line ${\hbox{Re}}(z)=0$ in the usual fundamental domain for ${{{\Bbb{H}}}}^2/\Gamma$, and therefore lifts to an orientation-reversing isometry of $S^1$ to $S^2$. See the genus 3 example of [@BT], where a similar problem occurs. We will handle this problem in the following way: let us assume for some $p$ different from $2$ or $7$, there is a subgroup $K$ of $SL(2, {{\Bbb{Z}}})$ such that $K$ and $\tau(K)$ are not conjugate in $SL(2, {{\Bbb{Z}}}/p)$. We may now choose our subgroups $$\begin{array}{ll} G &= P({\hbox{id}}\times SL(2, {{\Bbb{Z}}}/7) \times SL(2, {{\Bbb{Z}}}/p))\\ H_1 &= P({\hbox{id}}\times H_1 \times K)\\ H_2 &= P({\hbox{id}}\times H_2 \times K), \end{array}$$ and let $\widetilde{S}^1$ and $\widetilde{S}^2$ be the coverings of ${{{\Bbb{H}}}}^2/PSL(2,{{\Bbb{Z}}})$ corresponding to $\phi^{-1}(H_1)$ and $\phi^{-1}(H_2)$. $\widetilde{S}^1$ and $\widetilde{S}^2$ are isoscattering, but we want to show that they are not isometric. Any such isometry between them must be given by conjugation by some matrix $C$, by the involution $\tau$, or by a composition of the two. The first possibility cannot obtain, because restricting to $PSL(2, {{\Bbb{Z}}}/7)$, $C$ will give a conjugacy of $H_1$ to $H_2$. The second and third possibilities also cannot hold, since if such a matrix $C'$ exists, restricting to the $PSL(2, {{\Bbb{Z}}}/p)$ factor, it will give a conjugacy from $K$ to $\tau(K)$. Thus, Theorem \[cong\] will follow once we find such a $p$ and $K$. We now examine Dickson’s list [@Di] for likely subgroups $K$ of $PSL(2, {{\Bbb{Z}}}/p)$ for which $K$ is not conjugate to $\tau(K)$. We remark that $\tau$ is given by the outer automorphism $${\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}\to {\left( \begin{array}{cc}}-1 & 0 \\ 0 &1 { \end{array}\right)}{\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}{\left( \begin{array}{cc}}-1 & 0 \\ 0 & 1 { \end{array}\right)},$$ and for $p \equiv 1\ ({{\hbox{mod}}\ }4),$ we have that $-1$ is a square root $({{\hbox{mod}}\ }p)$, so $\tau$ is actually an inner automorphism for such $p$. To understand our choice of $K$, we observe that the subgroups $H_1$ and $H_2$ are isomorphic to the symmetric group $S(4)$ on four elements (a more detailed discussion will be given below). We will show: \[s4\] For $ p \equiv 7\ ({{\hbox{mod}}\ }8)$, there exist subgroups $K$ of $PSL(2, {{\Bbb{Z}}}/p)$ isomorphic to $S(4)$, such that $K$ is not conjugate to $\tau(K)$ in $PSL(2, {{\Bbb{Z}}}/p)$. [[[Proof]{}]{}:]{}We first discuss the restriction $p \equiv 7\ ({{\hbox{mod}}\ }8)$. Indeed, $S(4)$ contains the cyclic subgroup ${{\Bbb{Z}}}/4$. In order for $PSL(2, {{\Bbb{Z}}}/p)$ to contain a cyclic subgroup of order $4$, we must have $p \equiv \pm 1 ({{\hbox{mod}}\ }8)$. The plus sign is ruled out by the condition $p \not\equiv 1({{\hbox{mod}}\ }4)$. To construct a subgroup $K$ isomorphic to $S(4)$ in $PSL(2, {{\Bbb{Z}}}/p)$, we first note that $S(4)$ contains the dihedral group $$\{ A, D: A^4 =1, \quad D^2=1, \quad DAD= A^{-1} \},$$ given by $A= (1,2,3,4)$, $D=(1,2)(3,4)$. We now seek such a subgroup in $PSL(2, {{\Bbb{Z}}}/p)$. A convenient choice for $A$ is $$A = {\left( \begin{array}{cc}}\alpha & \alpha\\ -\alpha & \alpha { \end{array}\right)},\quad \alpha^2 \equiv 1/2\ ({{\hbox{mod}}\ }p).$$ Note that $2$ is a square $({{\hbox{mod}}\ }p)$ by the condition that $ p \equiv 7 ({{\hbox{mod}}\ }8)$ and quadratic reciprocity. Note also that $$A^2= C_1 = {\left( \begin{array}{cc}}0&1\\- 1&0 { \end{array}\right)}.$$ We now seek a matrix $D$ such that $A$ and $D$ generate a dihedral group of order 8. That means that $$\label{dihed} D^2 = {\left( \begin{array}{cc}}1 &0 \\ 0&1 { \end{array}\right)}, \quad DAD= A^{-1}.$$ The second relation implies that $D$ commutes with $C_1$. We observe that any matrix $Z$ commuting with $C_1$ must either be of the form $$\label{ccomm1} Z = {\left( \begin{array}{cc}}x&y \\-y & x { \end{array}\right)}, \quad x^2 + y^2 =1$$ or $$\label{ccomm2} Z= {\left( \begin{array}{cc}}\beta & \gamma\\ \gamma & -\beta { \end{array}\right)}, \quad \beta^2 + \gamma^2 = -1.$$ Since any matrix of the form (\[ccomm1\]) commutes with $A$, $D$ must be of the form (\[ccomm2\]). Furthermore, any matrix of the form (\[ccomm2\]) will satisfy (\[dihed\]). For later reference, we remark that neither $\beta$ nor $\gamma$ can be zero, since $-1$ is not a square $({{\hbox{mod}}\ }p)$, and we may choose $\beta + \gamma -1 \not \equiv 0 ({{\hbox{mod}}\ }p)$, by changing the sign of $\gamma$ if necessary. We now set $$C_2 = C_1 D = {\left( \begin{array}{cc}}-\gamma & \beta \\ \beta & \gamma { \end{array}\right)}.$$ We may embed this dihedral group in $S(4)$ by setting $$\begin{array}{ll} A &\to (1,2,3,4)\\ C_1 &\to(1,3)(2,4)\\ D&\to (1,2)(3,4)\\ C_2 &\to (1,4)(2,3). \end{array}$$ We now seek an element $E$ of $PSL(2, {{\Bbb{Z}}}/p)$ corresponding to the element $(1,2,3)$. Thus, $E$ must satisfy the conditions $$\begin{array}{ll} E^3 &=1\\ EC_1E^{-1} &= D\\ EDE^{-1} &= C_2\\ EC_2E^{-1}&= C_1\\ AEA^{-1} &= E^{-1}C_1. \end{array}$$ To do this, let us tentatively set $$\widetilde{E} = {\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}.$$ We will want $\widetilde{E}$ to send the fixed points ${{\hbox{Fix}}}(C_1)$ of $C_1$ (viewed as a linear fractional tranformation) to ${{\hbox{Fix}}}(D)$, the fixed points ${{\hbox{Fix}}}(D)$ of $D$ to ${{\hbox{Fix}}}(C_2)$, and ${{\hbox{Fix}}}(C_2)$ to ${{\hbox{Fix}}}(C_1)$. We calculate: $$\begin{array}{ll} {{\hbox{Fix}}}(C_1) &= \pm i,\\ {{\hbox{Fix}}}(D) &= \frac{\beta \pm i}{\gamma}\\ {{\hbox{Fix}}}(C_2) &= \frac{ -\gamma \pm i}{\beta}, \end{array}$$ where we have denoted by $i$ a square root of $-1$ in the field ${{\Bbb{F}}}_{p^2}$. Choosing the plus sign in each case, we seek $\widetilde{E}$ satisfying $$\begin{array}{ll} \widetilde{E}(i) = \frac{\beta + i}{\gamma}\\ \widetilde{E}( \frac{\beta + i}{\gamma}) = \frac{-\gamma + i}{\beta}\\ \widetilde{E}(\frac{-\gamma+ i}{\beta}) = i. \end{array}$$ After some tedious linear algebra, we find $$\widetilde{E} = {\left( \begin{array}{cc}}\frac{\gamma + \beta^2}{\beta + \gamma - 1} & \frac{-\beta + \beta \gamma - \gamma}{\beta + \gamma -1}\\ \frac{ \beta \gamma - 1}{\beta + \gamma -1} & \frac{\beta + \gamma^2}{\beta + \gamma -1} { \end{array}\right)}.$$ Note that with these choices of $a,b,c,$ and $d$, we have $$a+d =1, \quad c-b=1, \quad ad-bc=1.$$ The first relation assures that $\widetilde{E}$ is of order $3$, and it is easily checked that the conjugacies of $C_1$, $C_2$, and $D$ by $\widetilde{E}$ are as desired. It remains to check the condition $$A\widetilde{E}A^{-1} = \widetilde{E}^{-1}C_1.$$ This unappetizing calculation can be carried out by writing $$A\widetilde{E} A^{-1} \widetilde{E} = {\left( \begin{array}{cc}}\alpha^2 & 0\\ 0& \alpha^2 { \end{array}\right)}{\left( \begin{array}{cc}}1& 1\\ -1 &1 { \end{array}\right)}{\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}{\left( \begin{array}{cc}}1&-1\\ 1&1 { \end{array}\right)}{\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)},$$ using $a +d = 1$ and $c-b =1$ to eliminate $c$ and $d$, and $ad-bc =1$ to replace the quadratic terms $a^2 + b^2$ by $a-b-1$. We find that $$A\widetilde{E}A^{-1}\widetilde{E} =C_1$$ as desired. Setting $E= \widetilde{E}$, we now have constructed a subgroup $G$ of $PSL(2, {{\Bbb{Z}}}/p)$ isomorphic to $S(4)$. We now must show that $\tau(G)$ is not conjugate to $G$ in $PSL(2,{{\Bbb{Z}}}/p)$. But if $\psi: G \to G$ is any isomorphism, we may assume, by replacing $\psi$ by a conjugate, that $$\psi(A) = A.$$ $\psi(C_1)$ must then be $C_1$, and after conjugating by $A$ if necessary, we must have $$\psi(D)= D, \quad \psi(C_2)= C_2.$$ It then follows that $$\psi(E) =E.$$ We now show that under these assumptions, we cannot have $$\psi(X) = Z\tau(X) Z^{-1}, \quad X \in G.$$ Noting that $\tau(C_1)= C_1$, this gives us $$C_1 = Z C_1 Z^{-1},$$ so that $Z$ must be of the form (\[ccomm1\]) or(\[ccomm2\]). Noting that $\tau(A)= A^{-1}$, this gives $$A = Z A^{-1}Z^{-1},$$ so that $Z$ must be of the form $$Z= {\left( \begin{array}{cc}}x & y\\ y & -x { \end{array}\right)}, \quad x^2 + y^2 = -1.$$ We now consider the equation $$D= Z \tau(D) Z^{-1},$$ which we write out as $$\pm {\left( \begin{array}{cc}}\beta & \gamma \\ \gamma& -\beta { \end{array}\right)}= {\left( \begin{array}{cc}}x& y\\ y&-x { \end{array}\right)}{\left( \begin{array}{cc}}-\beta & \gamma\\ \gamma & \beta { \end{array}\right)}{\left( \begin{array}{cc}}x&y\\ y&-x { \end{array}\right)}.$$ The term on the right is computed to be $${\left( \begin{array}{cc}}-x^2\beta + 2xy\gamma + y^2 \beta & -2xy\beta + y^2 \gamma - x^2\gamma\\ -2xy\beta -x^2\gamma + y^2 \gamma & -y^2\beta -2xy\gamma + x^2 \beta { \end{array}\right)}.$$ Taking first the plus sign, we get from the upper-left entry the equation $$-x^2 \beta + 2xy\gamma + y^2\beta = \beta = \beta(-x^2 -y^2),$$ or $$2xy\gamma + 2y^2\beta=0.$$ Since $y \neq 0$, this gives $$x\gamma + y\beta =0.$$ Similarly, the lower-left entry gives $$-x\beta + y\gamma=0.$$ Solving these equations gives $$\frac{x}{\gamma}(\gamma^2 + \beta^2)=0.$$ But this implies $x=0$, a contradiction. Now we take the minus sign. We get the equations $$-x^2 \beta + 2xy\gamma + y^2\beta = -\beta = \beta(x^2 + y^2)$$ and $$-x^2 \gamma -2xy\beta + y^2 \gamma = \gamma(x^2 + y^2).$$ These become the two equations $$y\gamma = x\beta$$ and $$-y\beta = x\gamma,$$ which again yields a contradiction. This contradiction proves Theorem \[s4\], and hence also Theorem \[cong\]. Another approach to Theorem \[cong\] may be based on $p \equiv 1 \ ({{\hbox{mod}}\ }4)$. For ${{\cal{D}}}\in {{\Bbb{Z}}}/p$, let $${\tau_{{{\cal{D}}}}}{\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}= {\left( \begin{array}{cc}}{{\cal{D}}}& 0\\ 0& 1 { \end{array}\right)}{\left( \begin{array}{cc}}a&b\\ c&d { \end{array}\right)}{\left( \begin{array}{cc}}{{\cal{D}}}^{-1} & 0\\ 0&1 { \end{array}\right)}= {\left( \begin{array}{cc}}a& {{\cal{D}}}b\\ \frac{1}{{{\cal{D}}}} c & d { \end{array}\right)}.$$ When ${{\cal{D}}}$ is a square $({{\hbox{mod}}\ }p)$, ${\tau_{{{\cal{D}}}}}$ is inner. We will show: \[D\] Let $p \equiv 1\ ({{\hbox{mod}}\ }4)$, and let ${{\cal{D}}}$ be a non-square $({{\hbox{mod}}\ }p)$. Then there exists a subgroup $K$ of $PSL(2,{{\Bbb{Z}}}/p)$ isomorphic to $S(4)$, such that $K$ and ${\tau_{{{\cal{D}}}}}(K)$ are not conjugate in $PSL(2,{{\Bbb{Z}}}/p)$. Given Theorem \[D\], we set $$\Gamma^1 = \Gamma_2 \cap \phi^{-1}(K)$$ and $$\Gamma^2= \Gamma_2 \cap \phi^{-1}({\tau_{{{\cal{D}}}}}(K)).$$ Then $S^1 = {{{\Bbb{H}}}}^2/\Gamma^1$ and $S^2={{{\Bbb{H}}}}^2/\Gamma^2$ are isoscattering. The orientation-reversing isometry $\tau$ of ${{{\Bbb{H}}}}^2/PSL(2,{{\Bbb{Z}}})$ lifts to an isometry of $S^i$ to itself, $i=1, 2$, since $-1$ is a square $({{\hbox{mod}}\ }p)$, so Theorem \[D\] suffices to show that they are not isometric. To prove Theorem \[D\], we set $$A = {\left( \begin{array}{cc}}\alpha(1+i) &0\\ 0 & \alpha(1-i) { \end{array}\right)}\quad C_1 = {\left( \begin{array}{cc}}i&0\\ 0&-i { \end{array}\right)},$$ and $$D= {\left( \begin{array}{cc}}0&1\\ -1 & 0 { \end{array}\right)},$$ where $i$ is a square root of $-1$ $({{\hbox{mod}}\ }p)$, and $\alpha^2 = \frac{1}{2}\ ({{\hbox{mod}}\ }p)$. We may then solve for $E$ as above, to find $$E= {\left( \begin{array}{cc}}\frac{1-i}{2} & \frac{1-i}{2}\\ -\frac{(1+i)}{2} & \frac{1+i}{2} { \end{array}\right)}.$$ Then $${\tau_{{{\cal{D}}}}}(A)= A, \quad {\tau_{{{\cal{D}}}}}(C_1)= C_1,$$ and ${\tau_{{{\cal{D}}}}}(D) = {\left( \begin{array}{cc}}0 & {{\cal{D}}}\\ - \frac{1}{{{\cal{D}}}} & 0 { \end{array}\right)}.$ We seek $Z$ such that $$ZAZ^{-1}=A,\quad ZC_1Z^{-1}= C_1,$$ and $$Z{\left( \begin{array}{cc}}0 & {{\cal{D}}}\\ -\frac{1}{{{\cal{D}}}} & 0 { \end{array}\right)}Z^{-1} = {\left( \begin{array}{cc}}0&1\\ -1 & 0 { \end{array}\right)}.$$ But $Z$ must be of the form $$Z= {\left( \begin{array}{cc}}x&0\\ 0 & \frac{1}{x} { \end{array}\right)},$$ so that $$Z {\left( \begin{array}{cc}}0 & \cal{D}\\ -\frac{1}{{{\cal{D}}}} &0 { \end{array}\right)}Z^{-1} = {\left( \begin{array}{cc}}0 & x^2 {{\cal{D}}}\\ -\frac{1}{x^2{{\cal{D}}}} & 0 { \end{array}\right)},$$ which can’t be made equal to $C_1$. REFERENCES [Be]{} P. Berard, “Transplantation et Isospectralité,” Math.Ann. 292 (1992), pp. 547-560. [BGP]{} R. Brooks, R. Gornet, and P. Perry, “Isoscattering Schottky Manifolds,” GAFA 10 (2000), pp. 307-326. [BJP]{} D. Borthwick, C. Judge, and P. Perry, “Determinants of Laplacians and Isopolar Metrics on Surfaces of Infinite Area,” preprint. [BP]{} R. Brooks and P. Perry, “Isophasal Scattering Manifolds in Two Dimensions,” Comm. Math. Phys. 223 (2001), pp. 465-474. [BT]{} R. Brooks and R. Tse, “Isospectral Surfaces of Small Genus,” Nagoya Math. J. 107 (1987), pp. 13-24. [Di]{} L. E. Dickson, [Linear Groups,]{} 1901. [Su]{} T. Sunada, “Riemannian Coverings and Isospectral Manifolds,” Ann. Math. 121 (1985), pp. 169-186. [Zel]{} S. Zelditch, “Kuznecov Sum Formulae and Szegö Limit Formulas on Manifolds,” Comm. P. D. E. 17 (1992), pp. 221-260. [^1]: Partially supported by the Israel Science Foundation and the Fund for the Promotion of Research at the Technion
--- author: - \| Ashley Joy$^a$, Matthew Wing$^a$$^b$[^1], Steffen Hauf$^c$, Markus Kuster$^c$, and Tonn R[ü]{}ter$^c$\ University College London (UCL), Department of Physics and Astronomy,\ Gower Street, London WC1E 6BT, United Kingdom\ also at Deutsches Elektronensynchrotron (DESY) Hamburg and Universität Hamburg\ European X-ray Free Electron Laser Facility GmbH,\ Albert-Einstein-Ring 19, 22761 Hamburg, Germany\ E-mail: , , , , title: 'X-CSIT: a toolkit for simulating 2D pixel detectors' --- =1 Introduction {#sec:xxx} ============ X-CSIT is a toolkit for creating simulations of 2D semiconductor pixel detectors. It provides physics simulations from incident photons to the readout electronics. The toolkit is modular and adaptable, as its design goal is to provide a common simulation framework for the wide variety of detectors to be used at the European XFEL (XFEL.EU) [@XFEL1; @XFEL2; @detectors1; @detectors], which are summarised in table \[tab:2\]. Detectors to be simulated by X-CSIT at the European XFEL include: the Adaptive Gain Integrating Pixel Detector (AGIPD) [@AGIPD1; @AGIPD2], the Large Pixel Detector (LPD) [@LPD1; @LPD2], the DePFET sensor with Signal Compression (DSSC) [@DSSC1; @DSSC2; @DSSC3] as well as pnCCDs [@pnCCD1; @pnCCD2] and FastCCDs [@fastCCD]. X-CSIT will be integrated into the European XFEL’s software framework Karabo [@Karabo]. As part of the integration, XFEL.EU users will be provided with pre-built simulations of XFEL.EU detectors. Karabo will allow X-CSIT simulations to run on XFEL.EU’s computer network and transparently handle concurrent processing. Furthermore, simulations can be tied to the data processing pipelines implemented in Karabo, allowing simulated data to be processed in the same way as measured data. LPD AGIPD DSSC pnCCD FastCCD ------------------------- --------------------------- --------------------------- ------------------------------- -------------------------------------- ------------------------------------- Pixels 500$\mu$m square 200$\mu$m square 204$\mu$m hexagonal 75$\mu$m square 30$\mu$m square Depth 500$\mu$m 500$\mu$m 450$\mu$m 300$\mu$m 200$\mu$m Dynamic range 1$\times$10$^5$ at 12 keV 1$\times$10$^4$ at 12 keV 6000 at 1 keV 1000 at 2 keV, dE: 130 eV at 5.9 keV 1000 at 500 eV, dE: 400 eV at 5 keV Dynamic range technique Triple gain profile Pre-amplifier chosen gain DePFET non-linear gain Linear Linear Sensor size 32$\times$128 pixels 512$\times$128 pixels 256$\times$128 pixels 200$\times$128 pixels 1920$\times$960 pixels Photon energy range 12 keV optimal, 1–24 keV 3–13 keV 0.5–6 keV optimal, 0.5–24 keV 0.1–15 keV 0.25–6 keV : Selected specifications for some of the detectors to be used at the European XFEL. Note that the pnCCD and FastCCD can also be used for imaging spectroscopy. Accordingly, an energy resolution $\mathrm{dE}$ is given for these detectors. \[tab:2\] Design of X-CSIT ================ Design goals and objectives --------------------------- X-CSIT was conceived from the desire to have a single common, well validated simulation of the detectors to be used at European XFEL, see table \[tab:2\], capable of handling large amounts of data in a streaming fashion. This requirement necessitates a versatile and adaptable simulation and contrasts with existing simulation tools, which usually require processing all data in sequential steps, e.g. HORUS [@HORUS] or MEGALib [@MEGAlib]. X-CSIT achieves this by providing a set of physics simulations based upon user provided detector definitions. Because X-CSIT is based on an object orientated design, code can be customised without affecting the rest of the simulation. The inherent versatility of X-CSIT and its ability to create simulations of a wide variety of detectors makes it useful beyond its original scope of the detectors at XFEL.EU. Once testing has been performed, and the physics simulations used within X-CSIT have been validated, the code will be made available as open source. Other groups can benefit from X-CSIT, which will provide a faster and more convenient means of creating a simulation than starting from scratch, and will already be validated. Simulation details ------------------ ![The layout and components of the X-CSIT sub-simulations. As input a list of primary particles to generate is passed to the particle simulation. After all simulation steps have been completed, images containing events in detector units are output. []{data-label="fig:3"}](layout.png){width="\textwidth"} X-CSIT consists of three simulations (figure \[fig:3\]), each covering a separate physical regime within the detector. Data passes sequentially through these simulations, although all three can be run concurrently on different data packets, thus constituting a pipeline. Between these simulations data is passed using container classes that define the interface that X-CSIT uses. By deriving from these classes, user-defined data storage classes may be defined. This allows for the expansion of the simulation, e.g. to include more detailed reporting of certain simulation aspects, or storage of data in a user specified way. Karabo integration of X-CSIT at XFEL.EU utilises this functionality to make use of Karabo’s native data storage methods for optimisation of in-framework data transport. ### Particle simulation The particle simulation simulates individual energetic particles interacting with the experimental setup. To produce the presented results Geant4 [@geant403; @geant406] version 10.0 was used for this simulation stage. Using the Livermore physics list [@geantphysics], photon and energetic electron interactions including the photo electric effect, Compton and Rayleigh scattering as well as fluorescence and Auger emissions can be simulated. These processes have been validated for version 9.6 down to $\mathrm{1\,keV}$ [@validation1; @validation2]. In-house validation will be conducted for the Geant4 version integrated within X-CSIT. Users provide the particle simulation with a description of the physical geometry to be used by Geant4 and identify the sensitive detector regions. X-CSIT then manages the Geant4 simulation and handles the streaming of output data from Geant4 to the subsequent simulation components. In this way partial data can already be passed on to the charge and electronics simulations, rather than having to wait for the full Geant4 simulation to finish. ### Charge simulation The charge simulation models the behaviour of charge clouds in the semiconductor layer and the sharing of charge between pixels. Because of the large number of electron–hole pairs created, of the order of 275 per keV of initial energy in silicon, the simulation uses a statistical distribution, rather than a Monte Carlo simulation as used for the particle simulation. Additionally, two regimes are considered. In most cases separate charge clouds do not interact with each other noticeably, and even the influence of interactions between electrons and holes is small compared to that of drift and diffusion. If, due to the number of charges, only drift and diffusion need to be considered a statistical distribution of charge can be assumed. This distribution is a Gaussian normal distribution for which the standard deviation, $\sigma$, is proportional to the root of the distance between the interaction point and the collection point, $d$, over the electric field due to the bias voltage, $E$, as given by $$\label{eq:1} \sigma(d)=\sqrt{\frac{2kT}{qE}d}\,.$$ This is a valid approximation for strongly overbiased detectors, where interactions do not happen close to the electrode ($d$ > $\mathrm{100\,\mu m}$) [@Fowler]. [0.27]{} ![Charge sharing calculations in X-CSIT.[]{data-label="fig:4"}](charge_sharing.png "fig:"){width="75.00000%"} [0.31]{} ![Charge sharing calculations in X-CSIT.[]{data-label="fig:4"}](gaussian.png "fig:"){width="75.00000%"} [0.39]{} ![Charge sharing calculations in X-CSIT.[]{data-label="fig:4"}](SpreadWidth2.png "fig:"){width="75.00000%"} The two dimensional spread is calculated as a pair of orthogonal one dimensional Gaussian distributions in the X and Y axes. The proportion of charge crossing a pixel boundary can then be calculated with the Gaussian cumulative distribution (see figure \[fig:4\]). In the case of very high charge carrier densities the charges begin to screen themselves from the electric field inside the sensor, constituting a plasma. This plasma releases charge slower than in the non-plasma case, thus increasing the spread of the charge cloud as well as the charge collection time. Before the advent of FEL’s, charge clouds resulting from X-ray photon interactions in semiconductor detectors rarely reached the densities required for plasmas to occur [@Becker], and could frequently be neglected in simulations. In contrast, a simulation of plasma effects is required for use at XFEL.EU, since intensities reached there are expected to generate electron–hole plasmas. The plasma model to be used will be developed empirically based on data due to be taken at XFEL.EU in 2015. ### Electronics simulation The electronics simulation consists of a set of modular components, each of which simulate an electronic component or effect such as amplification, digitisation or noise. A simulation of the electronics of a detector is created by chaining together these components to create a functional representation of the detector circuitry as seen by the collected charge moving through it. An example is shown in figure \[fig:5\]. The empirical models used in these components require parameters given either as expected detector performance characteristics, or derived from calibration data taken from an existing detector. In the latter case either a specific detector or, by randomising the input data, a detector of a given detector type may be simulated. X-CSIT will come with a set of general components which can be arranged to model common detector front-end electronics designs. Additional components to model specific electronics can be added. At the European XFEL each component of the electronics simulation is integrated into Karabo as a separate device, conceptually similar to a plugin. In a future release of Karabo this will allow users to layout the electronics simulation of their detector graphically within Karabo’s GUI. [0.21]{} ![Electronics simulation diagrams for the pnCCD and LPD detectors.[]{data-label="fig:5"}](pnCCD_elec.png "fig:"){width="\textwidth"} [0.73]{} ![Electronics simulation diagrams for the pnCCD and LPD detectors.[]{data-label="fig:5"}](LPD_elec.png "fig:"){width="\textwidth"} Initial Testing =============== Initial testing of X-CSIT has been conducted with data taken by pnSensor GmbH using a pnCCD [@pnCCD1; @pnCCD2] (for specifications see table \[tab:2\]) and an Fe-55 source. A simulation of the pnCCD was implemented in X-CSIT (see figure \[fig:5.5\]) and the data from the real and simulated detectors were run through the same analysis pipeline previously validated against an existing calibration and characterisation pipeline used for the CCD of the CERN Axion Solar Telescope [@CAST]. [0.5]{} ![Diagram of the setup of pnCCD Fe-55 measurements as simulated in X-CSIT. The aluminum attenuator was used to reduce the photon intensity such that event pile-up on the sensor is avoided.[]{data-label="fig:5.5"}](setup.png "fig:"){width="\textwidth"} ![Top panel: the uncorrected energy spectra for the measured and simulated data given in detector units. Lower panel: the relative deviation between the two, given in terms of $1\,\sigma$ uncertainties. Visible peaks from left to right are: noise peak, Al fluorescence, $\mathrm{K_{\alpha}}$ and $\mathrm{K_{\beta}}$ $\mathrm{^{55}Mn}$ decay lines at $\mathrm{5.9\,keV}$ and $\mathrm{6.5\,keV}$.[]{data-label="fig:7"}](full_spectrum2.png){width="90.00000%"} Figure \[fig:7\] shows the histograms produced from the uncorrected measured and simulated data. The residuals between the two data sets shown are given by $\mathrm{(measured-simulated)/\sqrt{measured}}$. These spectra are not normalised with respect to each other. Instead, using the known source characteristics, the simulations were run to produce a number of primary photons that is equal to the number of photons incident on the detector in the measurements. Good agreement can be seen between the measured and simulated data above the noise peak ($\mathrm{3000\,ADU}$). The data in the regions just below the $\mathrm{^{55}Mn}$ peaks are underestimated by the simulation, while the region below this, $\mathrm{8000\,ADU}$ to $\mathrm{9500\,ADU}$, and the shoulder of the noise peak are overestimated. This could indicate that the simulation creates an excess of charge sharing events with the majority of the charge in the primary pixel of the split event. The low energy cutoff in the pnCCD is not simulated. [0.45]{} ![Left panel: scatter plot giving the distribution of charge shared between the two pixels of a double pattern (simulated data). The data shown has been offset and common mode corrected. Points are colour coded by fractional Y position on the sensor, the position of the event along the readout direction. Right panel: Relative deviation between measured and simulated data given in terms of $\mathrm{1\,\sigma}$ uncertainties on the measured data.[]{data-label="fig:6"}](Simulated_doubles.png "fig:"){width="\textwidth"} [0.45]{} ![Left panel: scatter plot giving the distribution of charge shared between the two pixels of a double pattern (simulated data). The data shown has been offset and common mode corrected. Points are colour coded by fractional Y position on the sensor, the position of the event along the readout direction. Right panel: Relative deviation between measured and simulated data given in terms of $\mathrm{1\,\sigma}$ uncertainties on the measured data.[]{data-label="fig:6"}](banana_relative.png "fig:"){width="125.00000%"} The left panel of figure \[fig:6\] shows a scatter plot of simulated events, where a single photon has shared charge between two pixels. The same features visible in the plot for the simulated data exist for the measured data (not shown), notably the Al fluorescence line, the escape peak, the $\mathrm{^{55}Mn}$ $\mathrm{K_{\alpha}}$ and $\mathrm{K_{\beta}}$ lines and pile-up events. The right panel shows the deviation between simulated and measured data in terms of $\mathrm{1\,\sigma}$ uncertainties. In the central region the simulation slightly but systematically underestimates the number of events by $\approx\mathrm{2\,\sigma}$, while at the edges a significant systematic overestimation occurs. This corresponds to the previous observation, that in the simulation a larger fraction of charge is attributed to the primary pixel of the split event. Additionally, it is apparent that the simulation has fewer background events, i.e. events which cannot be attributed to the aforementioned features. Whilst only double patterns extending perpendicular to the readout direction are shown here, the behaviour is similar for patterns extending along the readout-direction. In figure \[fig:8\] (a–c) spectra are shown for the different event types, characterised by the shape of the spread on the pixel grid. Additionally, first singles, i.e. singles that are also the first event in a readout column per transfer and thus has not been affected by the residual of a charge transfer of another event, are shown. These show qualitatively similar features on both the simulated and measured results in all four event types, however quantitatively the ratios of event types do not match. Events in three regions were compared for each event type: the low energy background region, ranging up to energies of $\mathrm{2000\,ADU}\simeq\mathrm{1.58\,keV}$, the continuum region, in which split partners will largely be found ($\mathrm{2000\,ADU}$–$\mathrm{6000\,ADU}$ $\simeq\mathrm{4.75\,keV}$) and the peak region, containing the $\mathrm{^{55}Mn}$ photo-peaks ($>\mathrm{6000\,ADU}$). For each region mean deviations in terms of sigma uncertainties of the measured data have been calculated. These values are given in table \[tab:stats\]. Overall, i.e. across the complete spectrum, triple and quadruple patterns are both over estimated (by 0.40% and 13.60%) while double patterns are underestimated (20.32%). The inclusion of background noise in the statistics of the measured singles and first singles but not the simulated singles makes comparison of these patterns ($\mathrm{-51.03\%}$ and $\mathrm{-52.73\%}$ in simulation) difficult. When considering only the immediate peak region for singles, i.e. events above 7000 ADU, thus eliminating the low energy background in the measured data, simulation singles and first singles are underestimated ($\mathrm{-40.06\%}$ and $\mathrm{-42.34\%}$). For the individual regions it is found that the shape of the continuum is generally represented well by the simulation, although a consistent underestimation exists, except for quadruple split patterns. The peak region is consistently overestimated by the simulation, both for singles and higher multiplicity patterns. The large deviations in the background region can be attributed to the fact that a simplified geometry was simulated and as such scattered photons from experimental components other than the detector and the source are missing. This is a systematic effect, as is evident from the residuals shown in the plots and the large variance of deviations in this region. Figure \[fig:8\] (d) shows a comparison between fully calibrated measured and simulated data, which has been corrected for offset, common-mode and charge transfer inefficiency (CTI), alongside fits to the $\mathrm{^{55}Mn}$ $\mathrm{K_{\alpha}}$ and $\mathrm{K_{\beta}}$ lines. The the numerical components of these fits and how they match to the expected emission lines are given in table \[tab:5\]. Additionally, the residuals between the two data sets are shown as $\mathrm{(measured - simulated)/\sqrt{measured}}$. The charge sharing excess in the simulation causes a deficit of singles, this causes higher uncertainties in the simulation fit. A match within 1$\sigma$ of fitted energy is seen between the simulated and measured data in both peaks as well as the width of the $\mathrm{^{55}Mn}$ $\mathrm{K_{\beta}}$ line. The width of the fit of the $\mathrm{^{55}Mn}$ $\mathrm{K_{\alpha}}$ line is larger for the simulated data set by a statistically significant amount, but the source of this deviation has not yet been identified. The deviation between the fully calibrated simulated and measured first singles match the deviations observed for non-calibrated first single events. [1]{} ![Plots (a–c): Energy histograms of the four different event patterns observed, defined by the shape of the spread on the pixel grid in any rotation, as shown in the legend at the top. Invalid corresponds to any other event shape, or patterns with a diagonally offset event adjacent to it. The data has been offset and common mode corrected. Each pixel of an event contributes to the histograms. Plot (d): Photo peak region showing a comparison of fully corrected data alongside the fit models used to determine the gain. []{data-label="fig:8"}](hist_pattern_spec_legend3.png "fig:"){width="50.00000%"} [0.5]{} ![Plots (a–c): Energy histograms of the four different event patterns observed, defined by the shape of the spread on the pixel grid in any rotation, as shown in the legend at the top. Invalid corresponds to any other event shape, or patterns with a diagonally offset event adjacent to it. The data has been offset and common mode corrected. Each pixel of an event contributes to the histograms. Plot (d): Photo peak region showing a comparison of fully corrected data alongside the fit models used to determine the gain. []{data-label="fig:8"}](singles.png "fig:"){width="\textwidth"} [0.5]{} ![Plots (a–c): Energy histograms of the four different event patterns observed, defined by the shape of the spread on the pixel grid in any rotation, as shown in the legend at the top. Invalid corresponds to any other event shape, or patterns with a diagonally offset event adjacent to it. The data has been offset and common mode corrected. Each pixel of an event contributes to the histograms. Plot (d): Photo peak region showing a comparison of fully corrected data alongside the fit models used to determine the gain. []{data-label="fig:8"}](doubles.png "fig:"){width="\textwidth"} [0.5]{} ![Plots (a–c): Energy histograms of the four different event patterns observed, defined by the shape of the spread on the pixel grid in any rotation, as shown in the legend at the top. Invalid corresponds to any other event shape, or patterns with a diagonally offset event adjacent to it. The data has been offset and common mode corrected. Each pixel of an event contributes to the histograms. Plot (d): Photo peak region showing a comparison of fully corrected data alongside the fit models used to determine the gain. []{data-label="fig:8"}](quads.png "fig:"){width="\textwidth"} [0.5]{} ![Plots (a–c): Energy histograms of the four different event patterns observed, defined by the shape of the spread on the pixel grid in any rotation, as shown in the legend at the top. Invalid corresponds to any other event shape, or patterns with a diagonally offset event adjacent to it. The data has been offset and common mode corrected. Each pixel of an event contributes to the histograms. Plot (d): Photo peak region showing a comparison of fully corrected data alongside the fit models used to determine the gain. []{data-label="fig:8"}](peaks.png "fig:"){width="\textwidth"} Region Background Continuum --------------- --------------------- ------------------- ------------------ -- Singles $9.3\,(\;\;\;80.1)$ $1.3\,(\;\;1.0)$ $4.9\,(\;23.9)$ First singles $14.1\,(191.4)$ $1.6\,(\;\;2.4)$ $9.1\,(\;61.0)$ Doubles $2.3\,(\;\;\;33.2)$ $3.5\,(\;\;2.0)$ $0.9\,(\;14.4)$ Triples $-2.1\,(190.5)$ $2.7\,(\;\;1.3)$ $-4.2\,(\;28.3)$ Quads $-14.8\,(424.2)$ $-7.7\,(\;\;8.6)$ $-6.1\,(\;59.6)$ Invalids $-11.1\,(958.8)$ $-0.7\,(\;\;2.4)$ $-4.2\,(\;45.2)$ : Mean deviations between simulated and measured data in terms of $\mathrm{1\,\sigma}$ uncertainties of the measured data for different event types and regions. The values in brackets give the variance of the data in a specific region. \[tab:stats\] Simulated Measured ------------------------------------------------- --------------------------------- --------------------------------- $E$ ($\mathrm{^{55}Mn}$ $\mathrm{K_{\alpha}}$) $\mathrm{(5892.78\pm2.61)\,eV}$ $\mathrm{(5891.41\pm2.46)\,eV}$ FWHM ($\mathrm{^{55}Mn}$ $\mathrm{K_{\alpha}}$) $\mathrm{(151.63\pm0.58)\,eV}$ $\mathrm{(141.20\pm0.35)\,eV}$ $E$($\mathrm{^{55}Mn}$ $\mathrm{K_{\beta}}$) $\mathrm{(6490.19\pm3.37)\,eV}$ $\mathrm{(6488.13\pm3.05)\,eV}$ FWHM ($\mathrm{^{55}Mn}$ $\mathrm{K_{\beta}}$) $\mathrm{(158.50\pm1.81)\,eV}$ $\mathrm{(157.67\pm1.17)\,eV}$ : Tabulated data of the fits to the $\mathrm{^{55}Mn}$ peaks in figure \[fig:8\] (d). \[tab:5\] Conclusion and outlook ====================== X-CSIT is a toolkit for creating simulations of 2D semiconductor pixel detectors. These simulations include the simulation of photon interactions in the semi-conductor material, charge sharing between pixels and the response of the electronic readout. X-CSIT has been designed for adaptability to different detector designs. While this is required to take into account the variations in design of the European XFEL detectors it makes the toolkit useful for other groups as well. At the European XFEL the integration of X-CSIT into Karabo will make it available to XFEL.EU users with pre-set up simulations of the available detectors. Currently, an early working version of X-CSIT has been used to simulate a pnCCD which is also being used to do measurements for early testing and comparisons. Results indicate reasonable agreement between measurements and simulation, within 3$\sigma$ on the uncorrected spectrum. Systematic deviations, especially concerning charge sharing, remain and are under investigation. Later, as more sources and detectors become available at European XFEL, additional physics elements of the toolkit will be validated. After the toolkit has been validated it will be made available as open source for other groups to use. [9]{} M. Altarelli, et al., European X-ray Free Electron Laser, European XFEL GmbH, Technical Design Report (2006). Th. Tschentscher, et al., Photon beam properties at the European XFEL, European XFEL GmbH Technical Report TR-2011-006 (2012). A. Koch, et al, Detector Development for the European XFEL: Requirements and Status, J. Phys. 425, 6, 2013. H. Graafsma Requirements for and development of 2 dimensional X-ray detectors for the European X-ray Free Electron Laser in Hamburg, . Th. Tschentscher, et al., Layout of the X-Ray Systems at the European XFEL, TECHNICAL NOTE XFEL.EU TN-2011-001 (2011). G. Potdevin, et al., Performance simulation of a detector for 4th generation photon sources: The AGIPD, Nucl. Instr. and Meth. A 607(1), 51–54, 2009. M. Hart, et al., Development of the LPD, a high dynamic range pixel detector for the European XFEL, Proceedings of the Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), Anaheim, CA, USA, 534–537, 2012. A. Koch, et al., Performance of an LPD prototype detector at MHz frame rates under Synchrotron and FEL radiation, M. Porro, et al., Development of a DEPFET sensor with signal compression: A large format X-ray imager with mega-frame readout capability for the European XFEL, IEEE Trans. Nucl. Sci., 59, No 6 (2012) 3339. M. Porro, et al., Expected performance of the DEPFET sensor with signal compression: A large format X-ray imager with mega-frame readout capability for the European XFEL, Nucl. Instr. and Meth. A, 624, 509–519, 2010. G. Lutz, et al., DEPFET sensor with intrinsic signal compression developed for the use at the XFEL free electron laser radiation source, Nucl. Instr. and Meth. A, 624, 528–532, 2010. N. Meidinger, et al., pnCCD for photon detection from near-infrared to X-rays, Nucl. Instr. and Meth. A, 565, 251–257, 2006. L. Strüder, et al., Large-format, high-speed, X-ray pnCCDs combined with electron and ion imaging spectrometers in a multipurpose chamber for experiments at 4th generation light sources, Nucl. Instr. and Meth. A, 614, 483–496, 2010. D. Doering, et al., Development of a compact fast CCD camera and resonant soft x-ray scattering endstation for time-resolved pump-probe experiments, Rev. Sci. Instrum., 82, 2011. G. Potdevin, U. Trunk, and H. Graafsma, HORUS, an HPAD X-ray detector simulation program, A. Zoglauer, et al., MEGAlib: simulation and data analysis for low-to-medium-energy gamma-ray telescopes, SPIE Astr. Tel. and Instrum., 70113F–70113F, 2008. B. C. Heisen, et al., Karabo: an integrated software framework combining control, data management and scientific computing tasks, ICALEPS Proceedings, San Francisco, USA, 2013. S. Agostinelli, et al., Geant4 - a simulation toolkit, Nucl. Instr. and Meth. A, 506, 250–303, 2003. J. Allison, et al., Geant4 developments and applications, IEEE Trans. Nucl. Sci., 53, 270–278, 2006. CERN, Phyics reference manual version: geant4 10.0, [PhysicsReferenceManual.pdf](http://geant4.web.cern.ch/geant4/UserDocumentation/UsersGuides/ PhysicsReferenceManual/fo/PhysicsReferenceManual.pdf), Dec, 2013. M. Batic, et al., Photon elastic scattering simulation: validation and improvements to Geant4, IEEE Trans. Nucl. Sci., 59, 1636–1664, 2012. G. A. P. Cirrone, et al., Validation of the Geant4 electromagnetic photon cross-sections for elements and compounds, Nucl. Instr. and Meth. A 618, 315-322, 2010. R. F. Fowler, et al., Computational modelling of semiconducting X-ray detectors, Nucl. Instr. and Meth. A, 477, 226–231, 2002 J. Becker, et al., Impact of plasma effects on the performance of silicon sensors at an X-ray FEL, Nucl. Instr. and Meth. A, 615, 230–236, 2010 M. Kuster, et al., The X-ray Telescope of CAST, New J. of Phys., 9.6, 169, 2007 [^1]: M. Wing acknowledges the support of the Alexander von Humboldt Stiftung.
--- abstract: 'We present a systematic study of vertex corrections in the homogeneous electron gas at metallic densities. The vertex diagrams are built using a recently proposed positive-definite diagrammatic expansion for the spectral function. The vertex function not only provides corrections to the well known plasmon and particle-hole scatterings, but also gives rise to new physical processes such as generation of two plasmon excitations or the decay of the one-particle state into a two-particles-one-hole state. By an efficient Monte Carlo momentum integration we are able to show that the additional scattering channels are responsible for the bandwidth reduction observed in photoemission experiments on bulk sodium, appearance of the secondary plasmon satellite below the Fermi level, and a substantial redistribution of spectral weights. The feasibility of the approach for first-principles band-structure calculations is also discussed.' author: - 'Y. Pavlyukh' - 'A.-M. Uimonen' - 'G. Stefanucci' - 'R. van Leeuwen' title: 'Vertex corrections for positive-definite spectral functions of simple metals' --- Starting from the introduction of the notion of quasiparticle ($qp$) [@landau_collected_1967] as an elemental excitation in Fermi liquids [@nozieres_theory_1999] we have almost a complete picture of its on-shell properties [@giuliani_quantum_2005; @stefanucci_nonequilibrium_2013]. Quantum Monte Carlo simulations [@moroni_static_1995; @holzmann_momentum_2011], phase diagrams [@wigner_interaction_1934; @overhauser_new_1959; @ortiz_zero_1999; @trail_unrestricted_2003; @baguet_hartree-fock_2013; @zhang_hartree-fock_2008], structure factors [@gori-giorgi_analytic_2000], and effective interparticle interactions [@gori-giorgi_short-range_2001] contributed to our knowledge of Fermi liquids and to the development of the density functionals [@petersilka_excitation_1996; @onida_electronic_2002]. Despite these surpluses we still have a poor knowledge of the energy- and momentum-resolved spectral function $A(k,\omega)$ away from the on-shell manifold, i.e., when $\omega\not\approx \varepsilon_k$. In angular resolved photoemission this is the regime where electrons with reduced energy (as compared to the prediction based on band-structure and energy balance) are observed. Self-consistent ($sc$) perturbation theory, e.g., $sc$-$GW$ [@hedin_new_1965], accurately predicts total energies [@almbladh_variational_1999; @garcia-gonzalez_self-consistent_2001; @dahlen_self-consistent_2005] and it is fully conserving at the one-particle level, a crucial property in the description of transport phenomena [@stan_levels_2009]. However, for spectral properties $sc$ schemes do not show the expected improvement over simpler one-shot calculations [@lundqvist_single-particle_1967; @lundqvist_single-particle_1967-1; @lundqvist_single-particle_1968]. In fact, they suffer from serious drawbacks: the incoherent background in the spectral function gains weight at the expenses of the $qp$ peak [@holm_fully_1998], the $qp$ energy does not agree with experiments (overestimating the bandwidth of simple metals) [@yasuhara_why_1999; @takada_inclusion_2001], and the screened interaction does not obey the $f$-sum rule [@kwong_real-time_2000; @pal_conserving_2009]. It was then proposed that self-energy (SE) diagrams with vertex corrections may cancel the spurious $sc$ effects [@mahan_electron-electron_1989; @mahan_gw_1994; @bobbert_lowest-order_1994; @holm_fully_1998; @schindlmayr_systematic_1998]. This fueled a number of notable attempts to include the vertex function in a model fashion: using the plasmon model for the screened interaction [@minnhagen_aspects_1975], neglecting the incoherent part of the electron spectral function [@shirley_self-consistent_1996], employing the Ward identities and a model form of the exchange-correlation kernel [@takada_inclusion_2001; @takada_dynamical_2002; @bruneval_many-body_2005; @maebashi_analysis_2011], or the sc cumulant expansion [@holm_self-consistent_1997; @kas_cumulant_2014; @caruso_cumulant_2016]. Although these methods clarified a number of issues, they did not provide an exhaustive picture [^1]. The major obstacle for a full-fledged vertex calculation, besides numerical complexity, is the issue of negative spectral densities, first noted by Minnhagen [@minnhagen_vertex_1974; @minnhagen_aspects_1975] and only recently solved by us using a positive-definite diagrammatic expansion [@PSDtot; @uimonen_diagrammatic_2015]. Our solution merges many-body perturbation theory (MBPT) and scattering theory, thus returning a positive-semidefinite (PSD) spectral function by construction. With the PSD tool at our disposal, in this Letter we investigate the influence of vertex corrections on the spectral function of the homogeneous electron gas (HEG), paving the way towards first-principles correlated calculations of band-structures. We demonstrate that the vertex function leads to a number of novel physical phenomena which cannot be reduced to mere self-consistency cancellations. Stochastic methods, long been used to describe integral properties, are shown to be well suited for the calculation of spectral features too. In fact, our Monte Carlo momentum integration is so efficient that the numerical part of the calculations does not pose any difficulty. ![\[fig:diag\](Color online) (a-c) The half-diagrams emerging from the bisection of the $\Sigma^{(2)}$ partitions (wiggly lines denote the screened interaction). (d-f) Three partitions of the PSD self-energy and (g-i) their momentum space representation.](Fig1){width="\columnwidth"} Let us motivate and discuss the PSD approximation used in this work. In terms of [ dressed]{} electronic propagators and [screened]{} interaction $W$ there is a single second order SE diagram $\Sigma^{(2)}=$. Its straightforward inclusion, however, yields negative spectra in some frequency regions. This prohibits the usual probability interpretation and, even worse, it jeopardizes sc calculations since the resulting Green’s function (GF) has the wrong analytic structure [@KvL.2016]. The key idea of the PSD scheme [@PSDtot; @uimonen_diagrammatic_2015] consists in (1) writing a SE diagram as the sum of its [partitions]{}, i.e., diagrams with particle and hole propagators, (2) bisect each partition into two half-diagrams, (3) add the missing half-diagrams to form a perfect square, and (4) glue the half-diagrams back. For $\Sigma^{(2)}$ the half-diagrams, see Fig. \[fig:diag\](a-c), contain scatterings with up to three particles and two holes in the final state [@PSDtot]. The SE partitions stem from the interference between these scatterings and after the PSD treatment one obtains partitions up to the [fourth]{} order in $W$, see Ref. for the full list. Among them there are three which deserve special attention. $\Sigma_{aa}$ in Fig. \[fig:diag\](d) results from the interference of scattering (a) with itself. As illustrated in Fig. \[fig:diag\](g) $\Sigma_{aa}$ involves a particle-hole ($ph$) pair (orange area) or a plasmon in the final state (this is the first order effect described by the $GW$ SE). The plus and minus vertices in the SE partitions have the purpose of distinguishing the constituent half-diagrams (resulting from the cut of all propagators with $+/-$ and $-/+$ vertices). $\Sigma_{a\bar{a}}$ in Fig. \[fig:diag\](e) is formed by the interference between the scattering (a), leading to two-particle-one-hole ($p_{f_1}$-$p_{f_2}$-$q_f$) final state, and the same scattering with interchanged particle momenta (indicated with $\bar{a}$), see Fig. \[fig:diag\](h). Finally, $\Sigma_{c\bar{c}}$ in Fig. \[fig:diag\](f) is formed by the interference between the scattering (c), in which a particle loses its energy by exciting 2 $ph$ pairs, 2 plasmons or a mixture of them, and the same scattering with intechanged particle and hole momenta, see Fig. \[fig:diag\](i). Plasmon generation is a dominant second order scattering process although it has a severely limited phase-space (dark blue line and light-blue area) due to energy and momentum conservation. Higher order terms in $W$ (up to fourth order) arise from other interferences and are needed to assure the overall positivity [@PSDtot]. In general the PSD procedure leads to a manifestly positive Fermi Golden rule form of the SE, $\Sigma^{<}(k,\omega)\sim\sum_{n,f} \Gamma^{(n)}(k,\omega)\|1+ r_s\gamma^{(n)}_1+r_s^2\gamma^{(n)}_2+\ldots\|^2\delta(\omega+\epsilon_k-E_{f}^{(n)})$, where the sum runs over all final states of energy $E_{f}^{(n)}$ with $(n+1)$-particles and $n$-holes ($r_{s}$ being Wigner-Seitz radius). The role of high order diagrams is two-fold: they bring new scattering mechanisms into play (hence new rates $\Gamma^{(n)}$) and renormalize them through the perturbative corrections $\gamma^{(n)}_i$. We already mentioned that one of the motivations for including diagrams beyond $GW$ is the excessive broadening of the spectral features when the level of self-consistency is increased, e.g., $G^{(0)}W^{(0)}\to GW^{(0)} \to GW$. As a full $sc$ calculation of the PSD SE of Ref. [@PSDtot] is out of reach, we partially account for self-consistency by using a $G^{(0)}W^{(0)}$ GF (finite $qp$-broadening and plasmon satellites) and an RPA screened interaction. Our calculations indicate that higher-order diagrams (aside from bringing in new spectral features) counteract the undesired $sc$ effects, thus suggesting the occurrence of sizable cancellations. ![\[fig:Ek\] (Color online) (a) Rate $\Gamma(k,{\omega})$ for $k=1.25k_{F}$ calculated from the PSD SE of Ref. with $G^{(0)}W^{(0)}$ GF (red dots) and from the SE $\Sigma=\Sigma_{a\bar{a}}+\Sigma_{c\bar{c}}+\Sigma_{aa}$ with $qp$ GF (dashed). (b) $qp$ energy correction $\Delta\epsilon_{k}=\epsilon_{k}-\epsilon_{k}^{(0)}$ and (c) plasmon dispersions for $G^{(0)}W^{(0)}$ (dotted) and our vertex approximation (full). The corrections to $\mu$ (with respect to the mean-field value) are $\Delta\mu=-1.76\epsilon_F$ and $\Delta \mu=-1.81\epsilon_F$ respectively. ](Fig2){width="\columnwidth"} We then explore the possibility of producing the PSD results with [less]{} diagrams and [bare]{} GF’s. In the bare GF the chemical potential $\mu$ is iteratively adjusted by imposing that the energy of states on the Fermi sphere (where the discontinuity in the momentum distribution $n_k$ occurs) is exactly equal to $\mu$ [^2]. In Fig. \[fig:Ek\](a) we compare the rate $$\Gamma(k,\omega)\equiv i[\Sigma^{\rm R}(k,\omega)-\Sigma^{\rm A}(k,\omega)], \label{gamma}$$ as obtained from the PSD diagrams of Ref. [@PSDtot] with $G^{(0)}W^{(0)}$ GF and from the much simpler $\Sigma=\Sigma_{a\bar{a}}+\Sigma_{c\bar{c}}+\Sigma_{aa}$ with $qp$ GF (in both cases we used an RPA $W$). The left flank and the hight of the peak are in perfect agreement. At energies in the region of plasmon satellites the full PSD rate decays faster but the trend is similar and the impact of this discrepancy on the spectral function is only minor. More calculations at different $k$ (not shown) confirm the agreement between the two SEs. We therefore infer that the relevant scattering mechanisms for a positive-conserving, leading-order vertex correction are those of Fig. \[fig:diag\](g-i). This reduction of diagrams represents an important advance in view of correlated band-structure calculations of solids. In the following we use the vertex correction of Fig. \[fig:diag\](d-f) to calculate $qp$ and plasmon energy dispersions, spectral function, scattering rate, renormalization factor and momentum distribution. #### Results ![\[fig:Akw\] (Color online) Energy- and momentum-resolved spectral function without (top) and with (bottom) vertex corrections. Solid lines denote the free electron dispersion. Dots denote the solutions of the real Dyson equation, see main text. For some momentum values multiple solutions (marked with different colors) are obtained.](Fig3){width="0.9\columnwidth"} The electron density and the dimensionality completely determine the properties of the HEG; in the 3d case they fix the Fermi momentum and energy as follows: $k_F=(\alpha r_s)^{-1}$ and $\epsilon_F=1/2(\alpha r_s)^{-2}$ with $\alpha=[4/(9\pi)]^{1/3}$. We consider the case of metallic densities $r_s=4.0\,a_B$ appropriate for, e.g., bulk Na metal. Angle-resolved photoemission experiments have pointed out a substantial narrowing of the occupied band in sodium [@lyo_quasiparticle_1988]. In Ref. [@yasuhara_why_1999] the origin of this narrowing was ascribed to the effects of electron correlations on the unoccupied bands. Such conclusion is by no means obvious as fully $sc$-$GW$ results [@holm_fully_1998] indicate the opposite trend (bandwidth larger by 20% as compared to the noninteracting electron dispersion). As it was pointed out in Ref. [@takada_inclusion_2001], vertex corrections rather than screening are crucial to reproduce the experimentally observed dispersion. Our diagrammatic approximation confirms this fact, providing a bandwidth reduction of 27.5% (as compared to the noninteracting case), see Fig. \[fig:Ek\](b). Fig. \[fig:Ek\](c) also shows the dispersion of plasmon satellites (red and blue curves). In the calculations with vertex corrections (solid) the high-energy plasmon branch smears out and, unlike in the $G^{(0)}W^{(0)}$ approximation (dotted), there is no real solution to the Dyson equation. We further observe an upward renormalization of the $G^{(0)}W^{(0)}$ low-energy plasmon branch, in agreement with experiment [@aryasetiawan_multiple_1996], as well as the emergence a second branch at lower energy (orange). Vertex corrections have a sizable impact on the energy and momentum resolved spectral function $A(k,\omega)=i[G^\mathrm{R}-G^\mathrm{A}](k,\omega)$. In Fig. \[fig:Akw\] we display the color plot of $A(k,\omega)$ without (top) and with (bottom) vertex corrections. The dotted lines denote the solutions of the real part of the Dyson equation (used to produce the curves in Fig. \[fig:Ek\]): $\omega+\Delta\mu-\epsilon_k={\text{Re}\,}[\Sigma^\mathrm{R}(k,\omega)]$. Appearance of the 2nd plasmon satellite below $\mu$, redistribution of the spectral weight between 1st and 2nd satellite, and further broadening of plasmonic spectral features above $\mu$ are the most important findings of this work. They confirm the plasmon-pole model analysis of Ref. [@shirley_self-consistent_1996] that predicted only hole satellites and much broader particle features. Our results call into question the cumulant parameterization of the spectral function in Ref. [@holm_self-consistent_1997] where no distinction between hole and particle features is made. It is interesting to notice that vertex corrections make the $qp$-peak sharper. This can be inferred from the explicit SE expression or from the rate $\Gamma$ of Eq. (\[gamma\]) which we plot in Fig. \[fig:sgm\](a-c) for three different values of the momentum. Plasmons do not contribute to the on-shell properties at energies around $\mu$ because they carry finite energy at zero momentum ($\omega_{pl}(q=0)=1.881\epsilon_F$ for $r_s=4$). Instead, the life-time of $qp$s in the vicinity of the Fermi sphere is mainly determined by $\Sigma_{aa}$ ($GW$ SE) involving $ph$ production or by $\Sigma_{a\bar{a}}$ ($qp\rightarrow 3 qp$). The latter, shown as yellow shaded curve in Fig. \[fig:sgm\](a-c), contributes with negative sign and leads to the observed reduction of $\Gamma$ (hence an enhancement of the $qp$ peak) [^3]. Such a behavior (alternating series in $\alpha r_s$) is typical of many perturbation theories. Notice that $\Sigma_{aa}+\Sigma_{a\bar{a}}$ also dominates the asymptotic (${\omega}\to\infty$) behavior. The scattering with generation of 2 plasmons, contained in $\Sigma_{c\bar{c}}$, plays a crucial role for the off-shell properties as it gives rise to new spectral peaks, see green shaded curve. ![image](Fig4){width="\textwidth"} In the vicinity of a $qp$ or plasmon ($pl$) peak the spectral function acquires the form [@pavlyukh_initial_2013]: $$\begin{aligned} A(k,\omega)&=Z^{(\alpha)}(k)\frac{1/\tau^{(\alpha)}_k} {(\omega-\omega^{(\alpha)}_k)^2+1/(2\tau_k^{(\alpha)})^2},\end{aligned}$$ where $\alpha=qp,pl$ and $\omega^{(qp)}_{k}=\epsilon_{k}$ whereas $\omega^{(pl)}_{k}=\omega_{pl}(k)$ is the dispersion of plasmon satellites. This expression contains two quantities of physical interest that we computed using our vertex function: the renormalization factor $$\begin{aligned} Z^{(\alpha)}(k)=\left(1-\frac{\partial}{\partial\omega} \left.{\text{Re}\,}\Sigma^\mathrm{R}(k,\omega)\right\|_{\omega=\omega^{(\alpha)}_k}\right)^{-1}, \label{eq:Zk}\end{aligned}$$ and the broadening of the $qp$ or $pl$ excitations $1/\tau^{(\alpha)}_k=Z^{(\alpha)}(k)\Gamma(k,\omega^{(\alpha)}_k)$. The renormalization factor is shown in Fig. \[fig:sgm\](d). At the band bottom ($k=0$) $G^{(0)}W^{(0)}$ gives only one plasmon satellite $Z^{(pl)}=0.382$ whereas our vertex approximation gives two satellites with comparable weight $Z^{(pl)}=0.217$ and $Z^{(2pl)}=0.207$. Furthermore, the $qp$ weight is reduced from $Z^{(qp)}=0.578$ (in $G^{(0)}W^{(0)}$) to $Z^{(qp)}=0.550$ indicating that the incoherent part of the spectrum gains weight. These two effects cannot be seen in the cumulant expansion scheme [@holm_self-consistent_1997] which suppresses the $Z$ of higher plasmon satellites according to the Poissonian distribution [@aryasetiawan_multiple_1996] and, due to the neglect of the coupling between particle and hole seas, yields the same $Z^{(qp)}$ as in $G^{(0)}W^{(0)}$. For $k=k_{F}$ vertex corrections reduce only slightly the $G^{(0)}W^{(0)}$ $qp$ renormalization factor. It is known that $sc$-$GW$ overestimates ($Z^{(qp)}=0.793$) the already good $G^{(0)}W^{(0)}$ value $Z^{(qp)}=0.638$ (our calculation) or $Z^{(qp)}=0.646$ (Hedin [@hedin_new_1965]). The proposed approximation to the vertex gives $Z^{(qp)}=0.628$, which thus remains rather close to QMC results $0.64$ to $0.69$ (at $r_s=3.99\,a_B$) [@holzmann_momentum_2011]. At the Fermi momentum $Z^{(qp)}$ can also be deduced from the discontinuity of the momentum distribution function $n_k$ [@mahaux_theoretical_1992; @gori_giorgi_momentum_2002; @olevano_momentum_2012]. In Fig. \[fig:sgm\](e) we show $n_{k}$ as obtained by a straightforward integration of the smooth part of the spectral function, $n_k=\int_{-\infty}^{\mu}\frac{d\omega}{2\pi}A(k,\omega)$, and by adding the singular contributions analytically. The $G^{(0)}W^{(0)}$ and vertex results are almost indistinguishable. We finally analyze in Fig. \[fig:sgm\](f) the quasiparticle life-time, a measure of electronic correlations [@echenique_theory_2000; @qian_lifetime_2005]. In ab initio calculations for realistic systems this quantity is typically estimated using the $G_0W_0$ approximation [@zhukov_lifetimes_2002; @pavlyukh_decay_2008; @pavlyukh_communication:_2011]. However, there have also been attempts to go beyond this level of theory by, e.g., including $T$-matrix diagrams. In Ref. [@nechaev_variational_2005] a reduction of $qp$ life-time (increase of $\Gamma$ by 50% (70%) in relation to $GW$ for $r_s =2.07 (4.86)$) has been predicted and explained by “the multiple scattering”. Our findings show the opposite trend, i.e., an increase of the $qp$ life-time (reduction of $\Gamma$ by $-$50% in relation to $GW$ for $r_s=4$). #### Conclusions Numerous authors emphasized that the inclusion of the vertex function should remedy the drawbacks of self-consistent calculations [@holm_fully_1998; @mahan_electron-electron_1989; @verdozzi_evaluation_1995; @takada_inclusion_2001]. Using our recently proposed diagrammatic analysis we have been able to confirm these expectations and show that this is only a part of the whole picture. Additionally, other second-order processes appear. They can be best described in the language of scattering theory with the link provided by the PSD formalism [@PSDtot; @uimonen_diagrammatic_2015]. We fully characterized the spectral function of 3d HEG in the $k-\omega$ plane. The main original features that we found are: a second plasmon satellite for holes, redistribution of the spectral weight between hole satellites, reduction of the plasmon spectral weight for particles, bandwidth reduction of the main $qp$-band. So far these effects have only been partially captured by other, non-diagrammatic methods. Our proposed approach has a universal character and can be extended to first-principle calculations of metals. In fact, in going from continuous to discrete translational symmetry the functions simply turn into matrix functions (e.g., $\Sigma({\mathbf{k}},\omega)\to \Sigma_{GG'}({\mathbf{k}},\omega)$), something which does not pose any conceptual difficulties for Monte Carlo momentum integration [^4]. Alkali metals for which a vertex function was partially included (typically using a model exchange-correlation kernel [@northrup_theory_1987; @lischner_effect_2014]) is a logical next step for our method. Acknowledgements {#acknowledgements .unnumbered} ---------------- We acknowledge CSC - IT Center for Science, Finland, for computational resources. Y.P. acknowledges support by the DFG through grant No. PA 1698/1-1. G.S. acknowledges funding by MIUR FIRB Grant No. RBFR12SW0J and EC funding through the RISE Co-ExAN (GA644076). R.vL. would like to thank the Academy of Finland for support. [71]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop []{} (, , ) @noop []{}, Advanced book classics (, , ) @noop []{} (, , ) @noop []{} (, , ) @noop [**, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} in @noop []{}, , Vol. (, ) p. @noop [**, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [ ()]{} @noop [, ()]{} @noop @noop [, ()]{} @noop [ ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [ ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [, ()]{} @noop [**, ()]{} [^1]: For instance the cumulant expansion is exact for deep core states interacting with plasmons and leads to the spectrum with equally spaced satellites [@langreth_singularities_1970]. Yet, this assumption is less justified for the valence band excitations overestimating the weight of higher order plasmon satellites (something that can be partially cured by taking multiple plasmon branches and their dispersion into account [@cini_coherent_1986; @guzzo_multiple_2014]). [^2]: We start with zeroth approximation $\Delta\mu^{(0)}={\text{Re}\,}\Sigma(k_F,1/2k_F^2)$ and perform two more calculations for $k=k_F\pm\Delta k$, where $\Delta k$ is a small number, typically a few percents of the Fermi momentum. The refined chemical potential shift is then given by $\Delta\mu=\Delta\mu^{(0)}+\frac12(\Delta\epsilon_{k_F+\Delta k}+\Delta\epsilon_{k_F-\Delta k})$, where $\Delta\epsilon_{k}$ is the correlational shift. [^3]: If $W$ in $\Sigma_{a\bar{a}}$ is replaced by the bare Coulomb interaction the so-called second-order exchange SE is obtained. Its on-shell value is density independent and it is known analytically [@onsager_integrals_1966; @ziesche_self-energy_2007; @glasser_analysis_2007], $\Sigma_{2x}(k_F,1/2k_F^2)=\big[2\pi^2\ln(2)/3-3\zeta(3)\big]/4\pi^2$. This result represents a useful check for our numerical algorithms. [^4]: Due to the long-range character of the Coulomb interaction (bare and to a lesser extent screened one) the integrations need to be extended beyond the boundaries of the first Brillouin zone. For $GW$ calculations the fast Fourier method have been proposed [@rojas_space-time_1995]. For higher dimensional integrals, such as in the present calculations, this approach becomes impractical, but can be remedied by the Monte Carlo integration featuring excellent scalability.
--- abstract: 'The Dark Sky Simulations are an ongoing series of cosmological N-body simulations designed to provide a quantitative and accessible model of the evolution of the large-scale Universe. Such models are essential for many aspects of the study of dark matter and dark energy, since we lack a sufficiently accurate analytic model of non-linear gravitational clustering. In July 2014, we made available to the general community our early data release, consisting of over 55 Terabytes of simulation data products, including our largest simulation to date, which used $1.07 \times 10^{12}~(10240^3)$ particles in a volume $8h^{-1}\mathrm{Gpc}$ across. Our simulations were performed with HOT, a purely tree-based adaptive N-body method, running on 200,000 processors of the Titan supercomputer, with data analysis enabled by . We provide an overview of the derived halo catalogs, mass function, power spectra and light cone data. We show self-consistency in the mass function and mass power spectrum at the 1% level over a range of more than 1000 in particle mass. We also present a novel method to distribute and access very large datasets, based on an abstraction of the World Wide Web (WWW) as a file system, remote memory-mapped file access semantics, and a space-filling curve index. This method has been implemented for our data release, and provides a means to not only query stored results such as halo catalogs, but also to design and deploy new analysis techniques on large distributed datasets.' author: - \| Samuel W. Skillman, Michael S. Warren, Matthew J. Turk,\ Risa H. Wechsler, Daniel E. Holz, P. M. Sutter bibliography: - 'zotero.bib' - 'sws.bib' - 'mjt.bib' title: 'Dark Sky Simulations: Early Data Release' --- Introduction ============ In the past 40 years we have witnessed tremendous growth in the availability and utility of computational techniques and resources. This has led to an equally astounding increase in the quality and accuracy of N-body simulations of cosmological structure formation, starting with $\sim1000$ particles in early works to the current work (Figure \[fig:ds9\_slice\]) with 9 orders of magnitude more particles [@peebles70; @press74; @davis85; @efstathiou85; @efstathiou90; @jenkins98; @klypin99; @springel05; @springel05a; @2011ApJ...740..102K; @angulo12; @alimi12]. N-body simulations form a major pillar of our understanding of the cosmological standard model; they are an essential link in the chain which connects particle physics to cosmology, and similarly between the first few seconds of the Universe to its current state. Predictions from numerical models are now critical to almost every aspect of precision studies of dark matter and dark energy, due to the intrinsic non-linearity of the gravitational evolution of matter in the Universe. Current and upcoming optical surveys that probe baryon acoustic oscillations (BAO), the galaxy power spectrum, weak gravitational lensing, and the abundances of galaxy clusters all require support from numerical simulations to guide observational campaigns and interpret their results [@1996ApJ...471...30H; @2005ApJ...633..560E; @2013arXiv1309.5382K; @2013PhR...530...87W; @2014MNRAS.441...24A; @2008PhR...462...67M; @2013arXiv1309.5385H]. Surveys in the X-ray and sub-millimeter wavelengths provide an alternate view of the high-temperature plasma that sits in the deep gravitational potential wells of the dark matter, and provide an alternate constraint on the abundances of the most massive collapsed objects in the Universe [@2013arXiv1303.5080P; @2014MNRAS.440.2077M]. Existing and future observational data motivate our work to understand theoretically a variety of spatial scales. The next generation of surveys will span very large volumes; for example, the Dark Energy Spectroscopic Instrument (DESI) [@2013arXiv1308.0847L] will survey $\sim$ 30 million galaxies and quasars over 14000 sq. degrees beyond $z \sim 2.3$, spanning a volume of $\sim$ 50 (Gpc $h^{-1}$)$^3$. The Large Synoptic Survey Telescope (LSST) [@ivezic08] will survey half the sky, detecting $L^$ galaxies over a volume of roughly 100 (Gpc $h^{-1}$)$^3$. Planck is already able to identify massive galaxy clusters over the entire sky, beyond $z \sim 1$ [@2013arXiv1303.5089P]. Achieving the science goals of these surveys requires realistic mock catalogs based on numerical simulations that calculate the non-linear evolution of structure and predict the dependence of survey observables on cosmological parameters. They also require that these simulations be of sufficently large volumes to calculate the statistics of rare objects and to calculate covariances between observables—this requires multiple realizations as large as survey volumes. On the largest scales, the universe is populated by clusters of galaxies, connected by filaments, bordering cosmic voids. The statistics of the distribution of these structures can be used in a variety of methods to constrain fundamental cosmological parameters. For example, the number of objects in the Universe of a given mass, the mass function, is sensitive to cosmological parameters such as the matter density, $\Omega_m$, the initial power spectrum of density fluctuations, and the dark energy equation of state. Especially for very massive clusters (above $10^{15}$ solar masses \[$M_\odot/h$\]) the mass function is a sensitive probe of cosmology. For these reasons, the mass function is a major target of current observational programs. Precisely modeling the mass function at these scales is an enormous challenge for numerical simulations, since both statistical and systematic errors conspire to prevent the emergence of an accurate theoretical model (see [@reed13] and references therein). The dynamic range in mass and convergence tests necessary to model systematic errors requires multiple simulations at different resolutions, since even a $10^{12}$ particle simulation does not have sufficient statistical power by itself. While galaxies and clusters of galaxies account for large concentrations of mass, cosmic voids that grow from regions of local divergence are the underdense regions that comprise most of the volume in the Universe [@2014MNRAS.438.3177S]. Today, over two thousand voids have been detected in galaxy redshift surveys <http://www.cosmicvoids.net> and they offer excellent probes of cosmology via their size distributions, shapes, internal dynamics, and correlations with the Cosmic Microwave Background, as well as unique probes of magnetic fields and galaxy evolution. Voids are only observed in the galaxy distributions, and galaxies are sparse, biased tracers of the underlying dark matter. However, voids are typically studied from a theoretical perspective only in dark matter $N$-body simulations. The identification of voids is sensitive to survey density and geometry in a highly non-trivial fashion; to make direct contact with observed voids we must perform large-volume, high-resolution simulations to capture the structure and dynamics of dark matter, map the dark matter to a galaxy population, place the galaxies on a lightcone, apply realistic survey masks, and identify and characterize the voids. It is theorized that small perturbations, referred to as baryon acoustic oscillations (BAO) and possibly excited during an inflationary epoch, launched sound waves in the photon-dominated baryon plasma. As the Universe expanded and the plasma cooled, eventually these perturbations were “frozen-in” at the time of recombination, and are seen as the fluctuations in the Cosmic Microwave Background [@bennett12; @planckcollaboration13; @2013arXiv1308.0847L]. These small fluctuations are thought to lead to an imprint in the spatial distribution of large scale structure, which can be measured directly by a number of galaxy surveys [@2005ApJ...633..560E; @2014MNRAS.441...24A; @2011MNRAS.416.3017B; @2013arXiv1308.0847L] and in upcoming low-frequency radio surveys [@johnston08; @dewdney09]. The BAO signal has been detected at $\sim10\sigma$ in the Sloan Digital Sky Survey (SDSS-III) Data Release 11 (DR11) Baryon Oscillation Spectroscopic Survey (BOSS) galaxy samples [@2014MNRAS.441...24A]. In principle the precise structure of the galaxy distribution can be used to probe cosmological parameters. Our theoretical models need to keep up with the tremendous advances in observational data, and high quality dark matter simulations can be used provide a bridge between observational and theoretical cosmology. The most basic statistical measures of galaxy clustering are the power spectrum and the two-point correlation function. By producing high-quality databases of galaxy tracers (i.e. “mock catalogs”), cosmological simulations are able to probe observed galaxy distributions [@2000MNRAS.318.1144P; @2006ApJ...652...71W; @2012ApJ...745...16T; @2014ApJ...783..118R]. Galaxy velocities can also be used for directly testing cosmology [@johnston12]. The rigorous statistical and systematic demands of upcoming surveys requires the computational cosmology community to design and deliver high quality simulations that can be used to further our understanding of cosmological theory and the large scale structure of our universe. Our measurements of the Universe are sufficiently refined such that we now require both large statistical volumes and accurate, high-performance methods. [@kuhlen12] reviews the prior state-of-the-art in numerical simulations of the Dark Universe, the largest being the “DEUS FUR” 550 billion particle simulation [@alimi12] performed with the `RAMSES` adaptive particle-mesh code [@teyssier02]. Other simulations/codes at the $10^{11}$ particle scale are HR3 [@kim11] using `GOTPM` [@dubinski04], Millennium-XXL [@angulo12] with `GADGET3` [@springel05], Jubilee [@watson13] with `CUBEP3M` [@harnois-deraps12], and Bolshoi/BolshoiP [@2011ApJ...740..102K] with `ART` [@kravtsov97]. Other codes that have advertised capability at the $10^{12}$ particle scale are `HACC` [@habib13] and Gordon Bell prize winner `Greem` [@ishiyama12]. The method most commonly used for accessing these simulations are relational databases which allow SQL queries to halo catalogs or other data from the simulations [@lemson06; @riebe13]. A potentially disturbing observation is that the research cycle associated with cutting-edge simulations is dominated not by the runtime of the simulation itself, but in the time to both extract scientific results and (if at all) make the simulation data publicly available. This indicates that porting existing analysis tools, validating results, and developing new analysis techniques have not received the same attention as our primary simulation codes. This is not a new phenomenon; in our own work for the 1992 Gordon Bell prize [@warren92], it took another two years for scientific results to become available [@zurek94]. Recent examples of a similar timescale are the Millennium-XXL simulation completed in Summer 2010 with results submitted in March, 2012 [@angulo12], and the DEUS FUR simulation completed in March 2012, with results in [@rasera14]. This highlights the fact that software development and data-analysis are less amenable to acceleration from advances in computer hardware, and warrants additional attention regarding the most productive allocation of resources to support sustainable scientific software and simulations. Computational techniques have progressed so rapidly that the time to run a state-of-the-art simulation following $13.8$ billion years of cosmic history takes only days on modern supercomputer architectures. However, this progress has not come without a cost. The time to design, develop, debug, and deploy a simulation can take years. After the completion of a simulation, the time to disseminate the main results takes months or years. Public data releases can take years to happen, if at all. There are many reasons for the significant delays in dissemination, both technical and social. Simulations of these types produce raw datasets that are measured in hundreds of Terabytes or even Petabytes, with even reduced halo catalogs reaching Terabytes in size. Even on high speed networks, data transfer of just a single snapshot can take as long as the original simulation. Socially, there is both the concern that a mistake will be discovered and the worry that others will make important discoveries with the publicly released data, thereby curtailing the scientific accomplishments of the simulators. However, despite these concerns, we believe it is of immense value to the community for everyone to release their simulation data as soon as possible. A mistake found by another can be fixed, and a new iteration of a simulation can be undertaken. A discovery made by outsiders is only possible through the hard work of the data creators, and correct attribution makes that known. Open source software projects have led the open data field, as a rapid increase in the availability of software developed in the open has led to burgeoning communities of developers in many astrophysical projects (`yt`, `astropy`, and `sunpy` to only name a few of the Python-based projects). A primary goal of this project is to adopt some of the fundamental concepts of the open source community, and translate them to open data for state of the art cosmological simulations. We aim to decrease the barrier to entry for accessing and analyzing data, and increase the speed of iteration and pace of scientific inquiry. Through a set of ongoing simulations, we begin this progress by releasing both reduced and raw data products from a set of cosmological simulations, including a simulation utilizing over a trillion particles. We hope to use input and feedback from the broader community to help shape our future data releases, and ultimate the types of simulations to be run. Our hope is that these data products enable discovery pertaining to many areas of research in cosmology. In particular, our trillion particle simulation is included in our current release; this is state-of-the-art in terms of mass resolution for its cosmological volume, and should offer tremendous insight into the largest scales and structures in our Universe. While experts can be trained to interact with large datasets on parallel file systems, we aimed to create an interface to the data that is novel, simple, and extensible, with the aims of allowing people with a wide range of technical abilities and interests to explore the data. In principle, high school students interested in physics and/or computation should be capable of accessing subsets of a trillion particle dataset and studying the structure of the dark matter potential in a sample galaxy cluster. Researchers in large scale data visualization should be able to load halo catalogs into 3D models of our universe, and explore alternate representation methods for high-dimensional datasets. Digital artists or game designers may even be inclined to use our data as input for their personal work. In what follows we describe the simulation setup, data analysis pipeline, and data access methods. We also describe our initial data validation through the analysis of the $z=0$ mass function, power spectra, and a brief comparison to the Planck Sunyaev-Zel’dovich galaxy cluster catalog. We end with a proposed set of community standards for fostering growth in computational cosmology both within and exterior to the confines of the Dark Sky Simulations project. Software & Hardware {#sec:Methods} =================== HOT {#sub:2hot} --- HOT is an adaptive treecode N-body method whose operation count scales as $N \log N$ in the number of particles. It is described in [@warren13], which we summarize here, and offer additional details relevant to the Dark Sky Simulations Early Data Release. Almost 30 years ago, the field of N-body simulations was revolutionized by the introduction of methods which allow N-body simulations to be performed on arbitrary collections of bodies in a time much less than $O(N^2)$, without imposition of a lattice [@appel85; @barnes86a; @greengard87]. They have in common the use of a truncated expansion to approximate the contribution of many bodies with a single interaction. The resulting complexity is usually determined to be $O(N)$ or $O(N \log N)$, which allows computations using orders of magnitude more particles. These methods represent a system of $N$ bodies in a hierarchical manner by the use of a spatial tree data structure. Aggregations of bodies at various levels of detail form the internal nodes of the tree (cells). These methods obtain greatly increased efficiency by approximating the forces on particles. Properly used, these methods do not contribute significantly to the total solution error. This is because the force errors are exceeded by or are comparable to the time integration error and discretization error. Treecodes offer the best computational efficiency when force resolution at small scales is important. HOT is distinguished from other current cosmology simulation approaches at the petascale by not having a particle-mesh component, using a pure treecode avoiding the potentially problematic transition scale between PM and tree forces inherent with TreePM approaches, and offering additional flexibility for high-resolution simulations with a large dynamic range in particle masses. The code has been evolving for over 20 years on many computational platforms. We began with the very earliest generations of distributed-memory message-passing parallel machines (an architecture which now dominates the arena of high-performance computing), the Intel iPSC/860, Ncube machines, and the Caltech/JPL MarkIII [@warren88; @warren92a]. This original version of the code was abandoned after it won a Gordon Bell Performance Prize in 1992 [@warren92], due to various flaws inherent in the code, which was ported from a serial version. A new version of the code was initially described in [@warren93]. Since then, our hashed oct-tree (`HOT`) algorithm has been extended and optimized to be applicable to more general problems such as incompressible fluid flow with the vortex particle method [@ploumhans02] and astrophysical gas dynamics with smoothed particle hydrodynamics (SPH) [@fryer06]. The code also won the Gordon Bell performance prize again in 1997, with absolute performance reaching 430 Gflops on ASCI Red on a 320 million particle simulation, as well as obtaining a Gordon Bell price/performance prize on the Loki Beowulf cluster [@warren97a] and the Avalon Beowulf cluster [@warren98]. The basic algorithm may be divided into several stages. First, particles are domain decomposed into spatial groups. Second, a distributed tree data structure is constructed. In the main stage of the algorithm, this tree is traversed independently in each processor, with requests for non-local data being generated as needed. In our implementation, we assign a `Key` to each particle, which is based on Morton ordering [@samet90]. This maps the points in 3-dimensional space to a 1-dimensional list, while maintaining as much spatial locality as possible. The domain decomposition is obtained by splitting this list into pieces. The Morton ordered key labeling scheme implicitly defines the topology of the tree, and makes it possible to easily compute the key of a parent, daughter, or boundary cell for a given key. A hash table is used in order to translate the key into a pointer to the location where the cell data are stored. This level of indirection through a hash table can also be used to catch accesses to non-local data, and allows us to request and receive data from other processors using the global key name space. We have developed an efficient mechanism for latency hiding in the tree traversal phase of the algorithm, which is critical to high performance. A recent major effort on code development has added many additional features to the code, being designated in the naming transition from `HOT` to HOT. Accuracy and error behavior have been improved significantly for cosmological volumes through the use of a technique to subtract the uniform background density [@warren13], correcting for small-scale discretization error, and using a Dehnen $K1$ compensating smoothing kernel [@dehnen01] for small-scale force softening. We use an adaptive symplectic integrator [@quinn97] and an efficient implementation of periodic boundary conditions using a high-order ($p=8$) multipole local expansion [@challacombe97; @metchnik09] which accounts for the periodic boundary effects to near single-precision floating point accuracy (one part in $10^{-7}$). We adjust the error tolerance parameter to limit absolute errors to 0.1% of the rms peculiar acceleration. Our code and parameters have been extensively tested and refined with thousands of simulations to test accuracy and convergence across multiple dimensions of timestep, smoothing length, smoothing type, error tolerance parameters, and mass resolution. The HOT code is written in the `C` programming language. We utilize a variety of `gcc` extensions, the most important of which is the `vector_size` attribute, which directs the compiler to use `SSE` or `AVX` vector instructions on Intel architectures. Using gcc with `vector_size` has eliminated the need to write the gravitational inner loops in assembly language to obtain good performance on CPU-only architectures. We have implemented the GPU portions of our code in both `CUDA` and `OpenCL`, with the `CUDA` versions performing somewhat better at present. We use a purely message-passing programming model, implemented in `MPI`. In order to hide latency, the tree-traversal phase of our algorithm uses an active message abstraction implemented inside `MPI` with our own “asynchronous batched messages” routines. Our HOT software does not depend on any external libraries. Treecodes place heavy demands on the various subsystems of modern parallel computers. This results in very poor performance for algorithms which have been designed without careful consideration of message latency, memory bandwidth, instruction-level parallelism and the limitations inherent in deep memory hierarchies. The space-filling curve domain decomposition* approach we proposed in [@warren93] has been widely adopted in both application codes (e.g. [@griebel99; @fryxell00; @springel05; @gittings08; @jetley08; @wu12]) and more general libraries [@parashar96; @macneice00]. Our claim that such orderings are also beneficial for improving memory hierarchy performance has also been validated [@mellor-crummey99; @springel05]. Our method converts a $d$-dimensional set of data elements into a 1-dimensional list, while maintaining as much spatial locality in the list as possible. This allows us to neatly domain decompose any set of spatial data. The idea is simply to cut the one-dimensional list of sorted elements into $N_p$ (number of processors) equal pieces, weighted by the amount of work corresponding to each element. The implementation of the domain decomposition is practically identical to a parallel sorting algorithm, with the modification that the amount of data that ends up in each processor is weighted by the work associated with each item. The mapping of spatial co-ordinates to integer keys converts the domain decomposition problem into a generalized parallel sort. The method we use is similar to the sample sort described in [@solomonik10]. Note that after the initial decomposition, the `Alltoall` communication pattern is very sparse, since usually elements will only move to a small number of neighboring domains during a timestep. This also allows significant optimization of the sample sort, since the samples can be well-placed with respect to the splits in the previous decomposition. In [@warren95a] we describe a tree traversal abstraction which enables a variety of interactions to be expressed between “source” and “sink” nodes in tree data structures. This abstraction has since been termed dual-tree traversal [@yokota12]. The dual-tree traversal is a key component of our initial approach to increase the instruction-level parallelism in the code to better enable GPU architectures. It is also relevant to a number of data analysis tasks, such as neighbor-finding and computing correlation functions. In this earlier work we used the fact that particles which are spatially near each other tend to have very similar cell interaction lists. By updating the particles in an order which takes advantage of their spatial proximity, we improved the performance of the memory hierarchy. Going beyond this optimization with dual-tree traversal, we can bundle a set of $m$ source cells which have interactions in common with a set of $n$ sink particles (contained within a sink cell), and perform the full $m \times n$ interactions on this block. This further improves cache behavior on CPU architectures, and enables a simple way for GPU co-processors to provide reasonable speedup, even in the face of limited peripheral bus bandwidth. We can further perform data reorganization on the source cells (such as swizzling from an array-of-structures to a structure-of-arrays for SIMD processors) to improve performance, and have this cost shared among the $n$ sinks. In an $m \times n$ interaction scheme, the interaction vector for a single sink is computed in several stages, which requires writing the intermediate results back to memory multiple times. For current architectures, the write bandwidth available is easily sufficient to support the $m \times n$ blocking. This is the key to our approach on Titan enabling high performance from the GPUs: we bundle multiple particles with a single interaction list to increase the computational intensity. This allows the full GPU performance to be sustained without being severely limited from PCI-Express bandwidth. In more detail, the computational intensity of our inner loop is 2 flops per byte. With an achievable PCI bandwidth of 5 Gbytes/sec on Titan, we need to increase the flops/byte by a factor of 200 to support a 2 Tflop GPU. We achieve this by packaging 200 or more particles with the same interaction list and sending them to the GPU. Finding more than 200 particles which share the same cell interactions is only possible given the framework of the HOT code, which provides the grouping and multipole acceptance tests to arrange the computation suitably. Even then, this technique only works for about 80% of the interactions, with the rest near the leaf nodes of the tree not being able to be sufficiently grouped. A second round of GPU optimizations was required to manage most of the remaining interactions near the leaf nodes. In particular, pre-staging a large block of particle positions and terminating the tree traversal once less than 80 particles remained in a cell (handling the rest with direct interactions) and bundling partial lists for leaf level quadrupole and hexadecapole interactions allows us to perform 97% of the gravitational interactions on the GPU, with all of the tree traversal logic handled by the CPU. As a scaling comparison (see Table \[tab:one\_node\]), we have run the HOT code on the same small cosmology problem using a single node of the Titan supercomputer, as well as Eos (an Intel Xeon E5-2670 processor-based Cray XC30 also at Oak Ridge National Laboratory) a desktop Haswell processor, and an Amazon Elastic Compute Cloud (EC2) current generation [c3.8xlarge]{} and older [c1.large]{} instance. We quote performance in particles updated per second per node (p/s/n), where our 5.9 Petaflop result described in Section \[sub:ds14\_a\] below corresponds to $10240^3/110/12288 = 7.94 \times 10^5$ p/s/n, or 64% efficiency scaling from 1 node to 12288 nodes. The most scientifically relevant metric for evaluating gravitational N-body simulations is not Petaflops, but how many particles are updated per second, at an accuracy sufficient to accurately represent the physics involved. In 1997, we obtained a performance of 3 million particles updated per second at an RMS force accuracy better than $10^{-3}$ [@warren97a]. Our current performance results are about 8 billion particles per second, with an equivalent force accuracy about 10 times better. Whether this accuracy is sufficient, or if accuracy can be sacrificed without adversely affecting the scientific results, is an area of current research. Node Description Cores (MHz) Perf. (p/s/n) ------------------- ------- ------- -------------------- Opteron 6274/K20x 30 2200 $12.25 \cdot 10^5$ Opteron 6274 16 2200 $ 2.54 \cdot 10^5$ Xeon E5-2670 (HT) 32 2600 $ 6.37 \cdot 10^5$ Xeon E5-2670 16 2600 $ 5.78 \cdot 10^5$ EC2 c3.8xlarge 32 2800 $ 3.91 \cdot 10^5$ EC2 c1.xlarge 8 1800 $ 1.00 \cdot 10^5$ Core i5-4570 4 3200 $ 2.50 \cdot 10^5$ : Performance measured in particles updated per second per node (p/s/n) for a variety of computational platforms. The top line is a Titan node (including the NVIDIA K20x GPU counted as 14 cores), which is 4.8x faster than the second line (without the GPU). The third and fourth lines compare a Cray XC30 node with and without hyperthreading. The Amazon EC2 results had a well defined price, which was \$0.263 per hour for the c3.8xlarge instance, and \$0.064 for the c1.xlarge instance. Performing an equivalent number of particle updates as our large ds14\_a run using Amazon EC2 resources would have cost at least \$200,000 at the current Amazon spot price, or \$1.3 million for on-demand (\$1.68/hr) c3.8xlarge resources. \[tab:one\_node\] `SDF` {#sub:SDF} ----- We use the Self-Describing File (SDF) interface, originally designed and implemented for our early parallel simulations [@warren92], with an implementation of the interface recently released under an open-source license [@sdf2014]. The basic aims of the SDF library are to be simple, flexible, extensible and most importantly, scalable to millions of processing elements. By writing analysis software which uses the SDF interface, the differences between data formats can be encapsulated, allowing software to read multiple data layouts and formats, without requiring recompilation each time a field is added, or a new data format needs to be supported. The SDF format consists of a human readable ASCII header followed by raw binary data. The header is intended to support metadata that describes the data, its pedigree, checksums of the data contained within it, and its layout on disk or in memory. The header also provides all the information needed for any processor in a parallel machine to independently read its own portion of a dataset. Note that the interface is capable of describing other existing data formats (such as the GADGET and Tipsy formats commonly used for cosmological datasets). In this regard, it is distinguished from libraries such as HDF5 [@folk99], which can not describe existing data without going through a conversion process. Each line in the header that includes an `=` is interpreted as a key-value dictionary pair. Other lines are interpreted as comments or internal SDF parameters. The structure of the binary data is encoded using a structure descriptor closely analogous to a C language structure declaration. One is allowed to have as many fixed-length arrays as desired followed by an optional arbitrary length array as the last entry (whose length can be determined by the structure size and file length). We note that while the majority of the data presented in this EDR follows the array-of-structures (AoS) layout, structure-of-array (SoA) is also enabled by the SDF format. This data format is portable and we provide both a C and Python library for reading the headers and binary data. We also provide a Python interface for specific spatial queries such as bounding boxes and spheres through `yt`, described in Section \[sub:yt\]. In addition, frontends for the raw particle and halo catalog SDF datasets have been added to `yt`. The majority of our data is stored in spatial Morton order (also known as `Z` or ordering), meaning that the sequential particles or halos on disk lie along a space filling curve. We further expose this embedded spatial structure in the file by creating an auxiliary file, which we refer to as the Morton index `midx` file. This `midx` file is constructed by first choosing a level in the oct-tree hierarchy to bin particles. The full particle dataset is then searched to find the first particle offset and the total number of particles within each leaf node in the tree. These values are mapped to the Morton index `Key`. This file has an extension with the naming convention `.midx%i % level`, where `level` corresponds to the level of the oct-tree that was constructed to bin the particles. For example, “.midx7” corresponds to a Morton index file that bins particles into $(2^7)^3$ cells. The `midx` file itself is also stored in the SDF format. We typically use higher level `midx` files for progressively larger data. Because the size of an individual `SDF` file can be many Terabytes, we have utilized the concept of a memory-mapped file in two separate implementations. The first is exposed through the Python interface using a `Numpy` `memmap`. This creates an “out-of-core” view of the complete data file. We have utilized this technique for the majority of our early science results, and have been satisfied with its performance and ease-of-use. Our second implementation extends this concept to file stored on the World Wide Web (WWW), and utilizes a local page-cache mechanism to address a remote file through a thin wrapper that exposes binary data to `Numpy`. This approach transparently takes advantage of the enormous investment in multiple technologies which have been developed to improve the performance and reliability of generic WWW resources. We released this software at the time of this manuscript’s submission [@Turk:10773]. This is the first instance that we are aware of that directly exposes binary data hosted on the WWW into local memory in a running Python session. We note that this interface may be useful for future large astronomical surveys, as we will later describe how we use this technology to address individual files that are 34 TB[^1] in size, similar to the expected size of individual data products from SKA-1 survey in 2020 [@2014arXiv1403.2801K]. {#sub:yt} From the data perspective, HOT can be thought of as a highly efficient method to create vast amounts of unstructured data. We therefore required an analysis framework that is both capable of ingesting Terabytes of data and allows for rapid design and development phases for analysis. For this reason we have utilized and extended , an open-source analysis and visualization package written primarily in `Python`, which our team has experience in applying to both large unigrid simulations (such as the $3600^3$ radiation-hydrodynamics simulation mentioned in [@2013arXiv1306.0645N]) and deep Adaptive Mesh Refinement simulations of the first stars [@2009Sci...325..601T]. is parallelized using mpi4py [@dalcin08], which we have exposed on $2048$ nodes ($32768$ cores) of the Titan supercomputer in this work. `yt` was originally designed to manage data output by patch-based Adaptive Mesh Refinement (AMR) astrophysical simulations. Recent versions have restructured the underlying engine to shift the focus from AMR simulations to other forms of data such as octree, unstructured mesh, and as used here, particle-based datasets. During this transition, the focus of `yt` has shifted to enable faster and more flexible indexing methods, which are utilized here to great extent, ensuring that even the very largest of datasets can be analyzed with proper care taken to enable multi-level indexing and on-demand data loading. In this work we have used as a base upon which we build access methods to both local and remote (on the WWW via `thingking`) Petascale datasets. We required a system that enabled ease of deployment, ease of access, and minimized the burden on the individual researcher interested in examining the data — and perhaps most importantly, we rejected any solution that obfuscated the data in any way. For this reason, rather than presenting a SQL frontend, or a science gateway for exploring pre-selected data products and results, we have instead taken a hybrid approach that preserves access to the underlying data, while still making accessible the reduced data products. Our extensions include adding an `SDF` frontend that can utilize the `midx` index for spatially querying large datasets (up to 34 TB at the time of this writing). These additions are being actively reviewed through the projects peer review system, and in the meantime can be used through a fork that can be located through our project website (<http://darksky.slac.stanford.edu>). The current version of `yt` also provides support for loading halo catalogs generated by ROCKSTAR as particle datasets, enabling selection, processing and visualization of the halo objects. We envision users loading the halo catalogs, determining the regions of interest to them in the full dataset, and then (transparently) utilizing the multi-level indexing system described above to load only the subselection of particles that are relevant to their research questions. While it would normally be completely intractable for a researcher to analyze a 34 TB file of particles, this approach will make it convenient and straightforward. Halo Finding {#sub:HaloFinding} ------------ To identify dark matter halos and substructure we use the ROCKSTAR algorithm [@behroozi13]. This halo-finding approach is based on adaptive hierarchical refinement of friends-of-friends groups in both position and velocity. It has been tested extensively and compared with other halo finders in [@knebe11], showing excellent, and in many cases superior, performance when compared with other approaches. While we have previously used the basic ROCKSTAR code successfully for simulations with 69 billion particles using 10,000 CPUs, several features of the code were not ideal for the computing environment on the Titan system. In particular, the client-server model of computation in ROCKSTAR requires using a file descriptor for each pair of communicating processes, and the limit on open file descriptors for each CPU on Titan is 32768. In addition, we have found it preferable to use a smaller number of CPUs to process the data in pieces, which enables the use of less capable computing resources for analysis (rather than requiring a machine with over 70 Tbytes of RAM to process the simulation all at once). In our implementation, we take advantage of the modularity of the ROCKSTAR algorithm and use only the function interface `find_subs()` via the `yt-3.0` halo finding interface. We process each spatial domain independently, with `yt` loading the spatially indexed particle data for a domain and finding the initial FOF groups. We add a buffer region to each domain to contain any particles from a halo near the domain edge ($6h^{-1}\mathrm{Mpc}$ is sufficient for halo masses up to $10^{16}h^{-1}M_\odot$). These groups are then passed to ROCKSTAR `find_subs()`. For strict spherical overdensity masses, `yt` performs the same post-processing steps to assign particles which were missed by the FOF group to halos, and to identify parent/sub-halo relationships. We have validated our implementation by comparing it with unmodified ROCKSTAR for smaller ($4096^3$) simulations. For a 1.07 trillion particle dataset, our complete halo finding process takes about 6 hours on 1024 CPUs. Computational Hardware ---------------------- The complete Titan Cray XK7 system [@bland12] at Oak Ridge National Laboratory contains 18,688 compute nodes, each containing a 16-core 2200 MHz AMD Opteron 6274 with 32 GB of 1600 MHz DDR3 memory, paired across a PCI-Express 2.0 bus with an NVIDIA Tesla K20x GPU with 6 GB of memory. The Cray Gemini interconnect [@alverson10] provides roughly 8 Gbytes/sec of bi-directional bandwidth per node at the hardware level, with MPI latencies quoted as 1.5 microseconds or less. Titan host nodes currently run the SUSE Linux Enterprise Server 11 SP1 (x86\_64) with current kernel 2.6.32.59. Processing nodes run Cray’s Compute Node Linux, designed to minimize interference between operating system services and application scalability. Additional Software ------------------- All of our software was compiled with system installed gcc [@stallman89] version 4.8.2 20131016 (Cray Inc.), with the addition of NVIDIA’s LLVM-based [@lattner04] nvcc V5.5.0 for two CUDA-specific files. Significant software dependencies used via the system modules interface include cray-mpich/6.2.0 [@snir98; @gropp03; @bosilca02], fftw/3.3.0.3 [@frigo98] and gsl/1.16 [@galassi07]. We additionally installed roughly 40 additional packages that were not available on the system, or were out-of-date including gdb 7.6.90 (system version 7.5.1), Mercurial [@osullivan09] (not available on system), git 1.8.5 [@torvalds05; @loeliger12] (system version 1.6.0), Python 2.7.6 [@vanrossum95] (system version 2.6.8), MPI for Python [@dalcin08], Cython [@behnel11], Numpy [@oliphant06] and matplotlib [@hunter07]. We have additionally made extensive use of the `bbcp` program [@hanushevsky01] to transfer hundreds of Tbytes of data between Oak Ridge, LANL, SLAC and NERSC, as well as efficiently copy data between local systems. The Dark Sky Simulations {#sec:DarkSkySimulations} ======================== We have performed a series of calculations that vary in particle number from $2048^3$ to $10240^3$ and in comoving cosmological volumes from $100h^{-1}\mathrm{Mpc}$ to $8h^{-1}\mathrm{Gpc}$ on a side, as shown in Table \[tab:simulations\]. The set of simulations released in this Early Data Release (EDR) are part of a larger effort [@Warren:10777] enabled through a 2014 INCITE[^2] computing grant at Oak Ridge National Laboratory. Additionally, one of the small volume boxes was run on the LANL Mustang supercomputer. All of these simulations utilize the exact same cosmology with $(\Omega_m, \Omega_b, \Omega_\Lambda, h_{100}, \sigma_8) = (0.295, 0.0468, 0.705, 0.688, 0.835)$, detailed in Table \[tab:cosmological\_parameters\]. Initial conditions were calculated using a modified version of 2LPTic [@crocce06]. The input power spectrum was calculated from a million step Markov Chain Monte Carlo calculation using MontePython [@audren12a] and CLASS, the Cosmic Linear Anisotropy Solving System [@blas11]. Observational constraints used as inputs included Planck [@planckcollaboration13], BOSS [@delubac14] and BICEP2 [@bicep2collaboration14]. The raw Markov Chain data the input cosmology was based on is publicly available [@warren14a]. Simulation $\sqrt[3]{N}$ L \[$h^{-1}\mathrm{Mpc}$\] $M_p$ \[$h^{-1}M_\odot$\] $\epsilon~[h^{-1}\mathrm{kpc}]$ $z_{\mathrm{init}}$ timesteps --------------------- --------------- ---------------------------- --------------------------- --------------------------------- --------------------- ----------- ds14\_a 10240 8000 $3.9 \cdot 10^{10}$ 36.8 93 543 ds14\_g\_1600\_4096 4096 1600 $4.9 \cdot 10^{9}$ 18.4 135 563 ds14\_g\_800\_4096 4096 800 $6.1 \cdot 10^{8}$ 9.2 183 983 ds14\_g\_200\_2048 2048 200 $7.6 \cdot 10^{7}$ 4.6 240 1835 ds14\_g\_100\_2048 2048 100 $9.5 \cdot 10^{6}$ 2.3 297 3539 : Dark Sky Simulations. For each simulation, the name, particle count, box size, particle mass, Plummer equivalent softening length, starting redshift and number of timesteps are shown above.[]{data-label="tab:simulations"} \ Param best-fit mean$\pm\sigma$ 95% lower 95% upper --------------------- ----------- ------------------------------- ----------- ----------- $100~\omega_{b }$ $2.201$ $2.214_{-0.025}^{+0.023}$ $2.165$ $2.263$ $\omega_{cdm }$ $0.118$ $0.1175_{-0.0015}^{+0.0014}$ $0.1146$ $0.1204$ $H_0$ $68.46$ $68.81_{-0.68}^{+0.64}$ $67.48$ $70.13$ $10^{+9}A_{s }$ $2.18$ $2.187_{-0.058}^{+0.052}$ $2.076$ $2.296$ $n_{s }$ $0.9688$ $0.9676_{-0.0054}^{+0.0054}$ $0.9568$ $0.9785$ $\tau_{reio }$ $0.08755$ $0.09062_{-0.013}^{+0.012}$ $0.06518$ $0.1156$ $r$ $0.1511$ $0.1737_{-0.042}^{+0.035}$ $0.09785$ $0.2526$ $\Omega_{\Lambda }$ $0.7012$ $0.7048_{-0.0081}^{+0.0085}$ $0.688$ $0.7211$ $YHe$ $0.2477$ $0.2477_{-0.00011}^{+0.0001}$ $0.2475$ $0.2479$ $\sigma_8$ $0.8355$ $0.8344_{-0.012}^{+0.011}$ $0.811$ $0.858$ : Cosmological Parameters. See @warren14a for details. Our simulations use the parameters in the mean value column. The values of any parameters not specified are available in the `class.ini` file in the data repository.[]{data-label="tab:cosmological_parameters"} \ The ds14\_a and ds14\_g Simulations {#sub:ds14_a} ----------------------------------- On April 18-19, 2014, we ran the ds14\_a simulation from redshift $z=93$ to $z=0$ with 1,073,741,824,000 ($10240^3$) particles on 12,288 nodes (196,608 CPU cores and 12,288 NVIDIA K20x GPUs) of the Titan system at Oak Ridge National Laboratory, which represents approximately 2/3 of the total machine. The simulation was of a cubical region of space $8,000h^{-1}\mathrm{Mpc}$ (comoving) across; a region large enough to contain the entire visible Universe older than 2.8 billion years in a light cone to a redshift of 2.3 for an observer at the center of the simulation volume. The simulation carried out $3.14 \times 10^{20}$ floating point operations (0.3 zettaflops). We saved 16 particle dumps totaling 540 Tbytes, as well as 69 subsamples of the data totaling 34 Tbytes, and 34 Tbytes of data in two light cones (one from the center, and one from the lower left corner). Had we attempted the same calculation with a simple $O(N^2)$ algorithm, it would have taken about ten million times as many operations and approximately 37 thousand years on the same hardware to obtain the answer. During the initial stages of the simulation, a single timestep required about 110 seconds, for a performance of 5.9 Petaflops. At the end of the simulation, where significant clustering increases the tree traversal overhead we perform a timestep in 135 seconds, for a performance of 4.3 Petaflops[^3]. Our aggregate performance over the entire 33 hour and 50 minute period was 2.58 Petaflops, due to nearly 40% overhead from disk I/O. This overhead was due in large part to the fact that MPI I/O on the Titan system was limited to only 160 of the roughly 1000 Lustre Object Storage Targets (OST) on each of the two Titan Atlas filesystems. We have demonstrated I/O performance using equivalent POSIX I/O benchmarks nearly 8 times higher, which will reduce I/O overhead to less than 5% for our next run (with an as yet undetermined cost in development time to replace what we thought was the proper MPI interface to use to avoid performance surprise). We also saved more checkpoint files than were necessary, given the new input (surprising to us!) that we encountered no failures during a 34 hour run. In addition to the `ds14_a` simulation, we have performed a number of lower particle count simulations with progressively better spatial and mass resolution. These simulations have thus far been used to provide means for convergence tests, but in and of themselves enable a number of additional projects. At this time, only $z=0$ data is available, but we expect to release additional snapshots, halo catalogs, and merger trees in the future. Early Science Results {#sec:DataVerificationValidation} ===================== While this manuscript’s primary purpose is to announce the Dark Sky Simulations campaign and encourage peer review of the data and development of compatible tools and feedback on its direction, we present present several state-of-the-art scientific results that have been enabled by these simulations. In the following two sections we present the $M_{200b}$ halo mass function, followed by an initial power spectra analysis and a comparison to the Planck Sunyaev-Zel’dovich all-sky galaxy cluster catalog. Halo Mass Function {#sub:HaloMassFunction} ------------------ One of our primary scientific goals is to improve upon the spherical overdensity (SO) mass function derived in [@tinker08], which has a quoted accuracy for virial masses of 5% up to $10^{15} h^{-1} M_\odot$. To give some sense of the computational advances enabling this improvement, in the 30 years since the seminal work of [@press74], the number of particles in a simulation has increased from one thousand to one trillion (a factor of $10^9$). We rely on progress in creating initial conditions [@crocce06; @blas11], algorithms [@warren13], computing capability [@bland12] and analysis [@turk11a; @behroozi13] to increase the volume simulated and analyzed at high fidelity. We also use a better-constrained input cosmology [@2011MNRAS.416.3017B; @bennett12; @2014MNRAS.441...24A; @delubac14; @bicep2collaboration14; @audren12a; @warren14a], reducing the effects of non-universality in applying the [@tinker08] mass function to the current standard cosmological model. While the mass function is almost independent of epoch, cosmological parameters and initial power spectrum (as suggested by [@press74] and demonstrated by [@jenkins01] at the 10-30% level), more detailed work has shown this universality is not obtained more precisely [e.g. @tinker08; @courtin11; @bhattacharya11]. [ ]{} For large masses, the [@tinker08] fit obtains most of its statistical power from the 50 realizations of the larger-volume WMAP1 L1280 ($(1280h^{-1}\mathrm{Mpc})^3$) simulations. Since the WMAP1 cosmology lies considerably outside current observational constraints, any non-universality in the translation from WMAP1 to the currently favored cosmology should be added to the 5% error budget. We analyze here roughly 5 times as much volume (512 $h^{-3}\mathrm{Gpc}^3$) as [@tinker08], with particle masses ($3.9 \times 10^{10} h^{-1} M_{\odot}$) about 15x smaller than the L1280 simulations. It remains to be demonstrated if we are within reach of the sub-percent statistical accuracy required for discriminating Dark Energy models from future surveys, as calculated by [@wu10]. [@reed13] present a number of detailed tests of requirements for the recovery of the mass function to percent level accuracy, and present parameter guidelines for doing so. Their requirements for the “very challenging prospect” of a simulation with a light cone to replicate the volume accessible to future cluster surveys with sufficient mass resolution to avoid sensitivity to simulation parameters are essentially met by our current simulation (we obtain 800 particles per halo of mass $M = 10^{13.5}h^{-1}M_\odot$ rather than the suggested 1000, but our mass resolution is better than those of the large-volume tests they present, which reduces the sensitivity somewhat). As a stringent test of both simulation and halo-finding consistency, we use the ROCKSTAR halo finder [@behroozi13] to select halos in a series of five simulations with particle masses varying between each by a factor of 8, for an overall dynamic range in mass of 4096 ($9.53 \times 10^6 h^{-1} M_\odot$ to $3.90 \times 10^{10} h^{-1} M_\odot$). The largest systematic error we have identified primarily affects the mass of the smallest halos (less than 800 particles), and is related to the initial evolution of clustering on the inter-particle scale. Representing a mode with too few particles results in it growing more slowly than it should. This loss of small-scale power is made worse by starting at higher redshift (because the error has longer to accumulate before non-linear clustering takes over) so the use of 2LPT initial conditions is required to allow a simulation to start at an appropriate redshift. However, even 2LPT initial conditions will lose small-scale power if started at a redshift higher than they need. The correction term to the truncation error is exactly the same mathematical form as deconvolving a cloud-in-cell density interpolation, so we enable the `CORRECT_CIC` flag in the initial conditions generator, even when starting with particles on a grid. Note that this correction will not apply to codes which compute short-range forces with a Fourier method, since the force kernel may already apply “sharpening” with the same effect. In order to compare our theoretical and numerical models with the universe, we require a reliable calibration of the correlation between an observable and a quantity measurable from our simulation data. Since observational data do not provide halo masses neatly derived at a fixed overdensity, the relationship between measurable and observable is non-trivial. At the levels of accuracy required for next-generation surveys, the details are important. In particular, the best definition of “halo mass” in a simulation will be influenced by the computational complexity of the cosmological parameter estimation it is used for, as well as the ability to connect it with an observed halo mass. Even algorithmic choices will affect the precise definition of a spherical overdensity halo mass at the 1% level [@knebe11]. A measure which yields mass functions which are self-similar and parameterized simply in terms of the details of the cosmological parameters and initial conditions would be ideal, but none of the currently favored measures of halo mass meets this goal. Part of our motivation for providing raw particle data and the framework to apply different mass measures is to encourage exploration of alternative definitions which may suit either theoretical models or particular observational programs better than the currently used spherical overdensity. We have also identified halos in the light cone dataset, which places an observer at the center of the `ds14_a` simulation volume. At a radius of $4 h^{-1}~\mathrm{Gpc}$, this yields data out to $z\sim2.3$. Within this volume, we identify 1.85 billion halos with 20 particles or more. This data can be used in a variety of contexts, both observational and theoretical. Observationally, this dataset can be used for creating mock galaxy catalogs [@2002ApJ...575..587B], weak lensing predictions, and large scale clustering predictions. Of particular interest is the distribution and clustering of the largest halos in the Universe. Figure \[fig:planck\] shows halo masses as a function of redshift in the `ds14_a_lc000` dataset. The largest halo has an $M_{200b}$ mass of $4.35 \times 10^{15} h^{-1} M_\odot$, and is at a distance of $631h^{-1}Mpc$ (a redshift of $z=0.22$). This can be compared with theoretical predictions such as @holz12 directly, where they determined the most massive object in the universe should be $M_{200b} = 3.8\times10^{15} M_\odot$. We note that they assumed a slightly different cosmology, and notably a slightly lower $\sigma_8=0.801$. A detailed analysis of the statistics of the most massive halos is forthcoming.\ Power Spectra ------------- The matter power spectrum is a convenient statistic for probing the time-evolution of spatial clustering of matter in the Universe. Measuring the matter and galaxy power spectrum (and its inverse Fourier transform, the correlation function), on linear and non-linear scales, both directly and indirectly, is one of the major goals of many ongoing and proposed observational projects; Pan-STARRs [@kaiser02], , BOSS [@2014MNRAS.441...24A], DES [@descollaboration05], SPT [@hou14], WiggleZ [@marin13], Planck [@planckcollaboration13], LSST [@ivezic08], SKA [@dewdney09], DESI [@2013arXiv1308.0847L], Euclid [@amendola13]. Notable theoretical approaches for calculating the non-linear power spectrum from an initial linear spectrum and the cosmological parameters are the ‘scaling Ansatz’ of [@hamilton91] and its extension to the HALOFIT model of [@smith03], with further recent refinement by [@takahashi12]. At mildly non-linear scales perturbation theory approaches have been successful (e.g. [@taruya12]). Other approaches are based on fitting the results of N-body simulations with an “emulator” [@heitmann08; @heitmann14] or neural network [@agarwal14]. \[sub:PowerSpectra\] In addition to being sensitive to cosmological parameters, the power spectrum is influenced by the volume, mass resolution and code parameters used for the simulation. A necessary (but not sufficient) condition for a well-behaved simulation model is that the measurement of relevant physical quantities be independent of these non-physical parameters. It is often difficult to measure these sensitivities, since performing a simulation with a large volume (to reduce statistical errors) and high mass resolution (to probe small spatial scales) soon becomes prohibitively expensive to compute. Within the current suite of Dark Sky Simulations, we have both large volume and good mass resolution, providing an opportunity to check the sensitivity of the power spectrum to non-physical parameters. These tests are complementary to the previous comparison of the power spectrum evolved from the same initial conditions using the HOT and GADGET [@springel05] codes that were presented in [@warren13]. In Figure \[fig:powspec\] we show the power spectrum measured from five simulations differing in mass resolution by factors of 8. It is to be noted on the log-log scale of the top panel that the measured power spectra differ by less than the width of the line over most of the spatial scale. To minimize distortions from the precise form of the mass interpolation at small scales, we perform large ($4096^3$–$8192^3$) FFTs, and perform multiple FFTs with the spectrum folded up to a factor of 8 to probe high $k$ without approaching the Nyquist limit of the FFT. The lower panel of Figure \[fig:powspec\] shows the power spectra on a linear scale, divided by a spline fit to our data. The spline uses the $8000h^{-1}\mathrm{Mpc}$ data up to $k=2$, $800h^{-1}\mathrm{Mpc}$ from $k=2$ to $k=10$, and $200h^{-1}\mathrm{Mpc}$ data beyond $k=10$. The most obvious 10% difference between simulations is due to the “cosmic variance” of the smaller 100 and $200h^{-1}\mathrm{Mpc}$ volumes. The three larger volume simulations show remarkable agreement (well below the 1% level) over the range from where the mode variance is small, down to spatial scales near the mean inter-particle spacing. The vertical ticks represent the Nyquist frequency of the initial conditions, representing the spatial scale below which there is no information when the simulation begins. Our convergence results near the mean inter-particle spacing echo the conclusions of [@splinter98], that increased force resolution must be accompanied by sufficient mass resolution, and that results below the inter-particle scale are subject to larger systematic errors. In Figure \[sub:PowerSpectra\] we show our results compared with FrankenEmu [@heitmann14], HALOFIT [@takahashi12] as generated from the CAMB code [@lewis00], RegPT [@taruya12] and the linear power spectrum. The matching of the wiggles at the BAO scale in HALOFIT differs from our results at the 2% level. The agreement at large scales with the FrankenEmu+h emulator is very good. FrankenEmu also matches up to k=10 within their quoted 5% accuracy. Since FrankenEmu was not calibrated using this specific cosmology within the emulator framework, errors at the 5% level are expected by $k=10$, as was the case for their `M000` test cosmology. Perturbation theory results from RegPT are not a particularly good match at $z=0$ where the system is highly evolved, but at $z=1$ in the right panel of the figure where perturbation theory would be expected to match to higher k, RegPT matches our results extremely well up to $k=0.3$. Comparison to Planck SZ Cluster Catalogs ---------------------------------------- One motivation for the large $8^{-1}\mathrm{Gpc}$ volume is given by full sky galaxy cluster surveys. For example, @reed13 point out that the full-sky volume corresponding to $z=2$ could be achieved through a simulation with a box size of $8h^{-1}\mathrm{Gpc}$ on a side, with enough mass resolution to populate halos with $M \gtrsim 10^{13.5} h^{-1} M_\odot$ with $N\sim1000$ dark matter particles. `ds14_a` nearly achieves this, having $N\sim800$ particles in the desired mass halo. As such, this simulation provides a unique opportunity for comparison with on-going surveys. In order to showcase this opportunity, we demonstrate its use in creating a simple mock SZ cluster catalog. Properly constructing a full SZ mock catalog is beyond the scope of this work, and the current presentation should be viewed as a demonstration. We use the results shown in Figure 3 of @2013arXiv1303.5080P to construct a minimum mass filter (as defined w.r.t $M_{500c}$) as a function of redshift. Here we use both the “Deep” and “Mean” estimates from their study. In Figure \[fig:planck\], we show the positions on the simulated sky, the histogram of clusters as a function of redshift, and the masses of the galaxy clusters as a function of redshift. Figure \[fig:planck\] can be directly compared with Figure $2$ and $24$ from @2013arXiv1303.5089P. We see a quite striking agreement between the Planck “Mean” estimate with their (Figure $24$) Planck-MCXC sample, and between the Planck “Deep” estimate with the combined set of their “Planck clusters with redshift” distribution. Available Data Products {#sec:DataProducts} ======================= In this first data release our goal is to provide enough public data to enable state-of-the-art scientific research, as well as provide a testbed for the public interface. A list of available data products as of July 2014 is listed in Table \[tab:datasets\]. Raw data can be downloaded directly from the web without authentication. While the exact HTTP address may migrate as server usage is evaluated, we will keep an updated python package `darksky_catalog`[^4] that can be used to alias a given simulation data product to its current uniform resource locator (URL). Updated information will also be kept on our project website (<http://darksky.slac.stanford.edu>). As well as directly accessing data through a web browser, we provide advanced access methods through `yt`, detailed at the end of this Section. Simulation Description Size --------------------- ---------------------------- -------- ds14\_a $z=0$ Particle Data 34 TB ds14\_a $z=0$ Halo Catalog 349 GB ds14\_a Lightcone Data $(z < 2.3)$ 16 TB ds14\_a Lightcone Halo Catalog 155 GB ds14\_g\_1600\_4096 $z=0$ Particle Data 2 TB ds14\_g\_800\_4096 $z=0$ Particle Data 2 TB ds14\_g\_200\_2048 $z=0$ Particle Data 256 GB ds14\_g\_100\_2048 $z=0$ Particle Data 256 GB : Available datasets as of July, 2014. A full listing is available at the project website.[]{data-label="tab:datasets"} \ The datasets in this EDR fall roughly into three categories: raw particle data, halo catalogs, and reduced data. Raw particle data, and halo catalogs are stored in single `SDF` files for each snapshot/redshift. This leads to individual files that are up to 34 TB in size. In addition to the primary `SDF` file, we create a “Morton index file” (`midx`), detailed in \[sub:SDF\] that allows for efficient spatial queries (see Figure \[fig:morton\_particles\]). By combining the raw particle dataset and a `midx` file, sub-selection and bounding box queries can be executed on laptop-scale computational resources while addressing an arbitrarily large simulation. Particle datasets store particle position $(x,y,z)$ and particle velocity $(vx, vy, vz)$ each as a 32-bit `float`, and a particle unique identifier, $id$, as a 64-bit `int`. Halo catalogs store a variety of halo quantities calculated using Rockstar [@behroozi13], described in Section \[sub:HaloFinding\]. [ ]{} In addition to raw particle snapshots, during runtime we output two light cone datasets, one from a corner of the simulation volume and one from the center. At this time we are releasing the lightcone from the center of the box, which covers the full sky out to a redshift of $z=2.3$. Halos found from this lightcone are also released. A slice through the halos found in the lightcone dataset is shown in Figure \[fig:lightcone\]. [ ]{} Data can be browsed and directly downloaded using a common web browser. However, given the data volume of the raw particle data, we do not recommend directly downloading an entire file. Instead, we have provided a simple remote interface through the commonly used `yt` analysis and visualization software. In this way, a researcher may directly query a sub-volume of the domain and download only what is needed for their analysis. These fast spatial queries are enabled by storing the majority of the data products (raw particle, halo catalogs) in “Morton order” on the server. Each particle is described by 32 bytes, and therefore loading a million particles around a halo in `ds14_a` requires downloading $32 \rm{MB}$. The average broadband internet speed in the U.S. (July, 2014) is 25.1 Mbps[^5], meaning that an average household could load this data in less than 10 seconds. Well-connected institutions would presumably be faster. Given a bounding box and Morton index file, offsets and lengths can be used with an HTTP range request to return the desired data. These queries are cached locally such that once loaded, particle data can be analyzed in-memory without any special requirements on the server. An example analysis is shown in Figure \[fig:halo\_particle\_load\], that utilizes helper functions from the `darksky_catalog` to remotely load the particles around the most massive halo in the `ds14_a` simulation into `yt`, and project the dark matter density along the line of sight. [ ]{} Community Standards {#sec:CommunityStandards} =================== The full stack of open-source software utilized to create this data—from the operating system, to the file systems, the compilers, numerical libraries, and even the software used to typeset this paper—is the product of the efforts of thousands upon thousands of individuals. In most of these cases, we are afforded the benefit of decades of an enormous and largely intangible investment in shared software development. We are releasing this data in the spirit of open science and open software development, with the intention of making it immediately usable to individuals with a wide range of skills and interests, encouraging the sharing and reuse of derived data products, and perhaps most importantly, removing ourselves as obstacles to its use. By doing so, we are attempting to participate in a material way in the development of future scientific endeavors, as we have benefited from the open release of software and data in the past. The value and quantity of this data vastly exceeds our ability to mine it for insight; we do not wish to see its utility bottlenecked by our own limitations. On one hand, we would like the data to be usable with as little bureaucratic or viral licensing overhead as possible. On the other, we do not wish that it be used for unfair advantage. To support these goals, we hope to foster the continued growth of a community of individuals using the data, contributing to exploration of the data, and developing tools to further analyze and visualize this and other data within the field. We suggest below some guidelines for usage to foster this community, and as a basis for further discussion of this new facet of data-intensive scientific inquiry enabled by the incredible progress in computational hardware and software. We believe that the longevity and success of this and future endeavors in the release of data for open science will be influenced by the spirit of openness and collaboration with which this and other path-finding projects are received. Citation: Key to ensuring both “credit” (see @katz14 for more detailed discussion) and good feelings between individuals is proper attribution of other, related or foundational work upon which discovery is based. We expect that discoveries and data products that are enabled though data from the Dark Sky Early Data Release cite this manuscript. This will enable others to track the provenance of the data products, replicate the discovery themselves, and also ensure that any impact this work has can be measured and evaluated. The canonical references to work related to this release are HOT [@warren13], [@turk11a] and SDF [@sdf2014]. Work using halo catalogs should include [@behroozi13]. In addition, we appreciate being informed if we have overlooked the citation of a significant contributor to our own work. Particular care is required for highly interdisciplinary scientific and computational research, since standards for citation do vary across fields. Collaboration: In some instances, enabling the use of this data to address new questions will require effort that is best formalized through the establishment of a scientific collaboration, with the intent being co-authorship for those who provide significant additional investment. We welcome additional collaborators, and can be contacted at the e-mail addresses provided. Mirroring Data: We encourage other projects and individuals to share their own copies of the data, and request that they both notify us (so that we can provide links to the mirrors) and refer back to the original website (to attribute as well as help ensure that individuals relying on the mirror can identify “upstream” changes or enhancements). All data released here is available for both commercial and non-commercial use, under the terms of the Creative Commons Attribution 4.0 License; <http://creativecommons.org/licenses/by/4.0/>. Derived Catalogs: Derived data, such as catalogs, generated from the datasets released here are not required to be made publicly available, although we would encourage individuals creating those data catalogs to do so. Any individuals generating derived data products from this data may request that their derived data products either be hosted alongside or linked to from the canonical data repository. Feature Enhancements: The source code for each component of the analysis pipeline of DarkSky is hosted in a publicly accessible, version-controlled repository. We welcome and encourage the development of new features and enhancements, which can be contributed using the pull request mechanism. We also encourage feature enhancements be documented and published (even if through a service such as Zenodo or Figshare) so that contributors are able to collect credit for their work. The value of this data can be greatly expanded by enabling the development of platforms on which it can be analyzed [@Turk2014] and through the review and acceptance of feature enhancements we intend to develop this value. Discussion Forum: We have established a discussion forum at [email protected] and a Google plus community at <http://DarkSkySimulations.info>. We encourage interested individuals to subscribe and to post questions, feedback, and general discussion. Conclusions =========== In this work we present the first public data release from the ongoing “Dark Sky Simulations” cosmological simulation campaign. These first five simulations include one of the largest simulations carried out to date, with $10240^3$ particles in an $(8h^{-1}\mathrm{Gpc})^3$ volume. The main findings from our work can be summarized as the following: - A single-method hierarchical tree approach to the N-body gravitational problem is computationally feasible, accurate, and performant on modern HPC architectures. - Using this method, we’ve carried out a suite of state-of-the-art cosmological simulations, hereafter referred to as the Dark Sky Simulations. - We present results comparing the mass function and power spectra to demonstrate the quality of our simulations. We find internal consistency between different box sizes at the 1% level over more than 3 orders of magnitude in particle number. Comparisons with results in the literature agree at the 1-10% level depending on scale. - The `ds14_a` light cone dataset and associated halo catalog provide a unique resource to make predictions for the large volumes probed by current and upcoming sky surveys; we use all-sky Sunyaev-Zel’dovich effect cluster counts as an example application. - Interacting with data from Petascale supercomputing simulations is algorithmically challenging; We have designed and implemented a novel data access approach that is simple, extensible, and demonstrated its capability of interacting with individual files that are 34 TB in size over the Internet. - We have reduced the time to the dissemination phase of our research by providing open access to a significant portion of the raw data from our simulations less than three months after the simulation was run, totalling more than 55 TB of publicly accessible datasets. We encourage the use and further analysis of these data as well as feedback on the accuracy and accessibility of the data. We will be updating and adding to the contents of the data release, including new simulations, and encourage feedback on which data products and simulation suites would be most useful. We will notify the community through the project website and discussion forum. The computational resources required to generate these data were only a small fraction of our currently allocated computing time, so we expect the amount of and variety of scientifically useful data products to grow rapidly. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to Andrey Kravtsov and Matt Becker for useful discussions regarding the mass function, and particularly thank Matt Becker for providing data to compare with our early mass function results. We thank the Institutional Computing Program at LANL for providing the computing resources used for the initial development and simulations for this project. This research used resources of the Oak Ridge Leadership Computing Facility at Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract DE-AC05-00OR22725. We thank Wayne Joubert, our INCITE project computational project liason at ORNL, and the rest of the OLCF support staff for their help in utilizing the Titan system. We thank the scientific computing team at SLAC for their continued support related to hosting data through the `darksky` server, in particular Stuart Marshall, Adeyemi Adesanya, and Lance Nakata. This research also used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This research was performed under the auspices of the National Nuclear Security Administration of the U. S. Department of Energy under Contract DE-AC52-06NA25396. SWS was supported by a Kavli Fellowship at Stanford. MSW gratefully acknowledges the support of the U.S. Department of Energy through the LANL/LDRD Program and the Office of High Energy Physics. MJT was supported by NSF ACI-1339624 and NSF OCI-1048505. RHW received support from the U.S. Department of Energy under contract number DE-AC02-76SF00515, including support from a SLAC LDRD grant, from the National Science Foundation under NSF-AST-1211838, and from HST-AR-12650.01, provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555. DEH acknowledges support from the National Science Foundation CAREER grant PHY-1151836. He was also supported in part by the Kavli Institute for Cosmological Physics at the University of Chicago through NSF grant PHY-1125897 and an endowment from the Kavli Foundation and its founder Fred Kavli. [^1]: $1 \mathrm{TB} \equiv 10^{12} \mathrm{bytes}$ [^2]: <http://www.doeleadershipcomputing.org/incite-awards/> [^3]: We perform somewhat more flops per timestep at late times [^4]: <https://bitbucket.org/darkskysims/darksky_catalog> [^5]: <http://www.netindex.com/download/2,1/United-States/>
--- abstract: \| A tree $T_{uni} $ is $m$-universal for the class of trees if for every tree $T$ of size $m$, $T$ can be obtained from $T_{uni} $ by successive contractions of edges. We prove that a $m$-universal tree for the class of trees has at least $m\ln (m) + (\gamma - 1)m + O(1)$ edges where $\gamma $ is the Euler’s constant and we build such a tree with less than $m^c$ edges for a fixed constant $c = 1.984...$ author: - Olivier Bodini title: 'On the Minimum Size of a Contraction-Universal Tree' --- Introduction ============ What is the minimum size of an object in which every object of size $m$ embeds? Issued from the category theory, questions of this kind appeared in graph theory. For instance, R. Rado [@Ra] proved the existence of an “initial countable graph”. Recently, Z. Füredi and P. Komjàth [@FK] studied a connected question. We use here the following definition : given a sub-class $C$ of graphs (trees, planar graphs, etc.), a graph $G_{uni} $ is $m$-universal for $C$ if for every graph $G$ of size $m$ in $C, G$ is a minor of $G_{uni} ,$ i.e. it can be obtained from $G_{uni}$ by successive contractions or deletions of edges. Inspired by the Robertson and Seymour work [@RS] on graph minors, P. Duchet asked whether a polynomial bound in $m$ could be found for the size of a $m$-universal tree for the class of trees. We give here a positive sub-quadratic answer. From an applied point of view, such an object would possibly allows us to define a tree from the representation of its contraction. The main results of this paper are the following theorems which give bounds for the minimum size of a $m$-universal tree for the class of trees : \[th1\] A $m$-universal tree for the class of trees has at least $m\ln (m) + (\gamma - 1)m + O(1)$ edges where $\gamma$ is the Euler’s constant. \[th2\] There exists a $m$-universal tree $T_{uni} $ for the class of trees with less than $m^c$ edges for a fixed constant $c = 1.984...$ Our proof follows a recursive construction where large trees are obtained by some amalgamation process involving simpler trees. With this method, the constant $c$ could be reduced to 1.88... but it seems difficult to improve this value. We conclude the paper with related open questions. Terminology =========== Our graphs are undirected and simple (with neither loops nor multiple edges). We denote by $G(V,E)$ a graph (its vertex set is $V(G)$ and its edge set is $E(G)$ (a subset of the family of all the $V(G)$-subsets of cardinality 2)). Referring to C. Thomassen [@Th], we recall some basic definitions that are useful for our purpose: We denote by $P_n $ the path of size $n.$ If $x$ is a vertex then $d(x),$ the degree of $x,$ is the number of edges incident to $x.$ Let $e$ be an edge of $E(G)$, the graph denoted by $G - e$ is the graph on the vertex set of $G$, whose edge set is the edge set of $G$ without $e$. We call classically this operation deletion. Let $e = \left\{ {a,b} \right\}$ be an edge of $G(V,E)$, we name contraction of $G$ along $e,$ the graph denoted by $G / e = H(V',E')$, with ${V}' = \left( {V / \left\{ {a,b} \right\}} \right) \cup \left\{ c \right\}$ where $c$ is a new vertex and $E'$ the edge set which contains all the edges of the sub-graph $G_1 $ on $V / e$ and all the edges of the form $\left\{ {c,x} \right\}$ for $\left\{ {a,x} \right\}$ or $\left\{ {b,x} \right\}$ belonging to $E$. We say that $H$ is a minor of $G$ if and only if we can obtain it from $G$ by successively deleting and /or contracting edges, in an other way, we can define the set $M(G)$ of minors of $G$ by the recursive formula : $$M\left( G \right) = G \cup \left( {\bigcup\limits_{e \in E\left( G \right)} {M\left( {G / e} \right)} } \right) \cup \left( {\bigcup\limits_{e \in E\left( G \right)} {M\left( {G - e} \right)} } \right)$$ The notion of minor induces a partial order on graphs. We write $A\preceq B$ to mean “$A$ is a minor of $B$”. For technical reasons, we prefer to use the size of a tree (edge number) rather than its order (vertex number). Finally, let us recall that, a graph $G_{uni} $ is $m$-universal for a sub-class $C$ of graphs if for every element $G$ of $C$ with $m$ edges$, G$ is a minor of $G_{uni} $. A Lower Bound ============= In this section, we prove that a $m$-universal tree $T_{uni} $ for the trees has asymptotically at least $m\ln (m)$ edges. We use the fact that $T_{uni} $ has to contain all spiders of size $m$ as minors. A spider $S$ on a vertex $w$ is a tree such that $\forall v \in V\left( S \right)\backslash \left\{ w \right\}, d(v) \le 2$. We denote the spider constituted by paths of lengths $1 \le m_1 \le ... \le m_k $ by $Sp(m_1 ,...,m_k )$ (Fig.1). ![image](unitrdes1.eps){width="1.75in" height="0.79in"} \[fig1\] Fig.1. $Sp(2,2,2,3,3)$ Let $T$ be a tree, we denote by $\partial T$ the subtree of $T$ with $V(\partial T) = V(T)\backslash A$, where $A$ is the set of the leaves of $T$. Also, we denote by $\partial ^k$ the $k$-th iteration of $\partial $. \[lm1\] $Sp(m_1 ,...,m_k )\preceq T$ involves that $\partial Sp(m_1 ,...,m_k )\preceq \partial T$. Moreover, if for all $i$, $m_i = 1$ then $\partial Sp(m_1 ,...,m_k )$ is a vertex. Otherwise, put $a$ the first value such that $m_a > 1$, we have $\partial Sp(m_1 ,...,m_k )= Sp(m_a - 1,...,m_k - 1)$ excepted for $k = 1$, in this last case we have $\partial Sp(m_1 ) = Sp(m_1 - 2)$. This just follows from an observation. \[lm2\] For every tree $T$, $Sp(m_1 ,...,m_k )\preceq T \Rightarrow T$ has at least $k$ leaves. Trivial. \[th3\] A $m$-universal tree $T_{uni} $ for the class of trees has at least $\sum\limits_{i = 1, i \ne 2}^m {\left\lfloor {\frac{m}{i}} \right\rfloor } $ edges. A $m$-universal tree $T_{uni} $ for the class of trees has to contain as minors all spiders of size $m$. So, for all $p$ it contains as minors the spiders $Sp(p,...,p)$ where we have $\left\lfloor {\frac{m}{p}} \right\rfloor $ times the letter $p$. By the lemma \[lm1\], for all $p \leq \frac{m}{2}$, $Sp(1,...,1)\preceq \partial ^{p - 1}T_{uni} $ and if $m$ is odd, $Sp(1)\preceq \partial ^{\left\lfloor {\frac{m}{2}} \right\rfloor-1}T_{uni}$. Moreover, it is clear that the terminal edges of the $\partial ^pT_{uni} $ constitute a partition of $T_{uni} $. By the lemma \[lm2\], this involves that $T_{uni} $ has at least $\sum\limits_{ i = 1}^{\left\lfloor {\frac{m}{2}} \right\rfloor } {\left\lfloor {\frac{m}{i}} \right\rfloor }$ edges if $m$ is even and $1+\sum\limits_{ i = 1}^{\left\lfloor {\frac{m}{2}} \right\rfloor} {\left\lfloor {\frac{m}{i}} \right\rfloor}$ edges if $m$ is odd. An easy calculation proves that these values are always equal to $\sum\limits_{i = 1, i \ne 2}^m {\left\lfloor {\frac{m}{i}} \right\rfloor}$. (of the theorem \[th1\]) it follows from the usual estimate $\sum\limits_{i = 1}^n {\frac{1}{i}} \sim \ln \left( n \right) + \gamma + O\left( {\frac{1}{n}} \right)$ and the inequality $\sum\limits_{i = 1, i \ne 2}^m {\left\lfloor {\frac{m}{i}} \right\rfloor} \ge 1 + \sum\limits_{ i = 1, i \ne 2}^{m - 1} {\left( {\frac{m}{i} - 1} \right)} $. What the above proof shows, in fact, is the following : A minimum $m$-universal spider for the class of spiders has $\sum\limits_{ i = 1, i \ne 2}^m {\left\lfloor {\frac{m}{i}} \right\rfloor } $ edges. The spider $Sp\left( {\left\lfloor {\frac{m}{m}} \right\rfloor ,\left\lfloor {\frac{m}{m-1}} \right\rfloor ,...,\left\lfloor {\frac{m}{2}} \right\rfloor,\left\lceil {\frac{m}{2}}\right\rceil } \right)$ is clearly a $m$-universal spider of size $\sum\limits_{ i = 1, i \ne 2}^m {\left\lfloor {\frac{m}{i}} \right\rfloor } $ for the class of spiders, and by theorem \[th3\] it is a minimum value. The Main Stem ============= In the sequel, we deal with rooted graph, i.e. graph $G$ where we can distinguish a special vertex denoted by $r(G)$, called the root. Conventionally, any contracted graph ${G}'$ of same rooted graph $G$ will be rooted at the unique vertex which is the image of the root under the contraction mapping, we say in this case that the rooted graph ${G}'$ is a rooted contraction of $G$. Note that, the contraction operator suffices to obtain all minor trees of a tree. So, we can now define the following new notion for sub-classes of rooted trees : a rooted tree $T_{uni} $ is strongly $m$-universal for a sub-classes $C$ of rooted trees if for every rooted tree $T$ in $C$ of size $m, T$ is a rooted contraction of $T_{uni} $. The concept of root is introduced to avoid problems with graph isomorphisms that, otherwise would greatly impede our inductive proof. For every edge $e$ of a tree $T$, the forest $T\backslash e$ has two connected components. We call $e$-branch, denoted by $B_e $, the connected component of ${T}'$ which does not contain $r\left( T \right)$, we define the root of $B_e $ as $e \cap V\left( {B_e } \right).$ A main stem of a rooted tree of size $m$ is defined as a path $P$ which is issued from the root and such that for all $e$-branches $B_e $ with $e \notin E\left( C \right)$, we have $\left\| {E\left( {B_e } \right)} \right\| < \left\lfloor {\frac{m}{2}} \right\rfloor $ (Fig.2). ![image](unitrdes2.eps){width="1.64in" height="1.06in"} \[fig2\] Fig.2. A main stem in bold The following lemma suggests the procedure which will be used to find a sub-quadratic upper bound for universal trees. Roughly speaking, it endows every tree with some recursive structure constructed with the help of main stems. Every rooted tree has a main stem. By induction on the size of the rooted tree. Let $T$ be a rooted tree, if $T$ has one or two edges, it is trivial. Otherwise let us consider the sub-graph $T\backslash r\left( T \right)$, which is a forest. We choose a connected component $T_1 $ with maximum size and we denote by $b_1 $ the unique vertex of $T_1$ which is adjacent to $r(T)$. Tree $T_1 $, rooted in $b_1 $, has, by the induction hypothesis, a main stem $B.$ Then the path $\left( {V\left( B \right) \cup \left\{ {r\left( T \right)} \right\},E\left( B \right) \cup \left\{ {\left\{ {r\left( T \right),b_1 } \right\}} \right\}} \right)$ is a main stem of $T$. A tree may possess in general several main stems. Let us notice also that a main stem is not necessarily one of the longest paths which contain the root. The Upper Bound =============== We need some new definitions. A rooted brush (Fig.3) is a rooted tree such that the vertices of degree greater than 2 are on a same path $P$ issued from the root. ![image](unitrdes3.eps){width="1.35in" height="1.22in"} \[fig3\] Fig.3. A rooted brush A rooted comb $X$ (Fig.4) is a rooted brush with $d\left( {r\left( X \right)} \right) \le 2$ and $\forall v \in V\left( X \right)$, $d\left( v \right) \le 3$. ![image](unitrdes4.eps){width="1.35in" height="1.13in"} \[fig4\] Fig.4. A rooted comb The length of a rooted comb corresponds to the length of the longest path $P$ issued from the root which contains all vertices of degree greater than 2. To obtain an upper bound, we consider two building processes : the first one, a brushing $M_B $, maps rooted trees with a main stem into rooted brushes, the second one, a ramifying $M_T $, consists in obtaining a sequence of rooted trees, assuming that we have an increasing sequence of rooted combs. We note $M_T^k $ the $k$-th element of the sequence. These building processes will possess the following fundamental property: \[pr1\] Let $\left( {T,\sigma } \right)$ a rooted tree with a main stem $\sigma $ and $\left( {X_n } \right)_{n \in \mathbb{N}} $ a sequence of rooted combs : $$\left( {\forall {T}'\preceq T,M_B \left( {{T}',\sigma } \right)\preceq X_{\left\| {E\left( {T}' \right)} \right\|} } \right) \Rightarrow T\preceq M_T^{\left\| {E\left( T \right)} \right\|} \left( {\left( {X_n } \right)_{n \in \mathbb{N}} } \right).$$ If building processes verify the property \[pr1\] and if for all $i$, the rooted comb $X_i $ is strongly $i$-universal for the class of rooted brushes then the rooted tree $M_T^m \left( {\left( {X_n } \right)_{n \in \mathbb{N}} } \right)$ is strongly $m$-universal for the class of rooted trees. It is just an interpretation of the property. We now establish the existence of building processes which satisfy property \[pr1\]. Brushing $M_{B}$ (Fig.5). Let $T$ be a rooted tree with a main stem $\sigma $. We are going to associate a rooted brush $B$ with it, denoted $M_B \left( {T,\sigma } \right)$ of the same size built from the same main stem $\sigma $ with the following process: every $e$-branch $B_e $ connected to the main stem by edge $e$ is replaced by a path of length $\left\| {E\left( {B_e } \right)} \right\|$ connected by the same edge. ![image](unitrdes5.eps){width="2.92in" height="1.48in"} \[fig5\] Fig.5. Ramifying $M_T^k $. For the second building process we work in two steps : First step. Given rooted trees $T_1 ,...,T_k $ with disjoint vertex sets, we build another rooted tree $T$, denoted $\left[ {T_1 ,...,T_k } \right]$, in the following way : $$V(T) = \bigcup\limits_{i = 1}^k {V\left( {T_i } \right)} \cup \left\{ {v_1 ,...,v_{k + 1} } \right\},$$ $$E(T) = \bigcup\limits_{i = 1}^k {E\left( {T_i } \right)} \cup \left\{ {\left\{ {v_1 ,r\left( {T_1 } \right)} \right\},...,\left\{ {v_k ,r\left( {T_k } \right)} \right\}} \right\} \cup \left\{ {\left\{ {v_1 ,v_2 } \right\},...,\left\{ {v_k ,v_{k + 1} } \right\}} \right\},$$ and $r(T) = v_1 $. If $T_i = \emptyset $, conventionally $\left\{ {v_i ,r\left( {T_i } \right)} \right\} = \emptyset $. Prosaically, from a path $P_k = \left[ {v_1 ,...,v_{k + 1} } \right]$ of size $k$ and from $k$ rooted trees $T_1 ,...,T_k $, we build a rooted tree joining a branch $T_i $ to the vertex $v_i $ of $P$ (Fig.6). ![image](unitrdes6.eps){width="1.73in" height="1.19in"} \[fig6\] Fig.6. A rooted comb $\left[ {T_1 ,T_2 ,T_3 } \right]$ Second step. By convention, $P_{ - 1} = \emptyset $. We are going to construct rooted trees $T_k $ in the following way :\ $T_{ - 1} = \emptyset$, $T_0 = X_0$, and $\forall i,$ $1 \le i \le k, T_i = \left[ {T_{\min \left( {u_1 ,i - 1} \right)} ,...,T_{\min \left( {u_{n_i } ,i - 1} \right)} } \right]$ if $X_i = \left[ {P_{u_1 } ,...,P_{u_{n_i } } } \right]$. We can now define $M_T^k$ : $$M_T^k \left( {\left( {X_n } \right)_{n \in \mathbb{N}} } \right) = T_k.$$ The building processes described above verify the property \[pr1\]. First, note that $M_T \left( {\left( {X_n } \right)_{n \in \mathbb{N}} } \right)$ is an increasing sequence. We prove the lemma by recurrence on the size $m$ of $T$. When $m = 0$ or $m = 1$, this is trivial. We suppose the property is verified for $T$ with size $m < m_0 $. Let $T$ be a rooted tree of size $m_0 $ with a stem $\sigma $, we note $e_1 ,...,e_k $ the edges of $T$ issued from $\sigma $ which do not belong to $\sigma $. To each $e$-branch of $T$ with $e \in \left\{ {e_1 ,...,e_k } \right\}$ corresponds by $M_B $ a $e$-branch (it is a path of same size) in $M_B \left( {T,\sigma } \right)$. So there exists $k$ distinct $e$-branches $R_1 ,...,R_k $ in $X_{m_0 } $ that we can respectively contract to obtain each $e$-branch with $e = e_1 ,...,e_k $ in $M_B \left( {T,\sigma } \right)$. By recurrence hypothesis, we have for $1 \le i \le k, B_{e_i } \preceq M_T^{\left\| {E\left( {B_{e_i } } \right)} \right\|} \left( {\left( {X_n } \right)_{n \in \mathbb{N}} } \right)$ and we have also $M_T^{\left\| {E\left( {B_{e_i } } \right)} \right\|} \left( {\left( {X_n } \right)_{n \in \mathbb{N}} } \right)\preceq M_T^{\left\| {E\left( {R_i } \right)} \right\|} \left( {\left( {X_n } \right)_{n \in \mathbb{N}} } \right)$. So each $e$-branch of $T$ is a minor contraction of $M_T^{\left\| {E\left( {R_i } \right)} \right\|} \left( {\left( {X_n } \right)_{n \in \mathbb{N}} } \right)$. By associativity of contraction map, we have $T\preceq M_T^{\left\| {E\left( T \right)} \right\|} \left( {\left( {X_n } \right)_{n \in \mathbb{N}} } \right)$. In this phase, we determine a sequence of rooted combs $\left( {X_i } \right)_{i \in \mathbb{N}} $ such that the rooted combs $X_i $ are strongly $i$-universal for the rooted brushes. In order to achieve this result, we define $F_p $ as the set of functions $f$ : $\left\{ {1,...,p} \right\} \to \left\{ {1,...,\left\lfloor {\frac{p}{2}} \right\rfloor } \right\}$ satisfying the following property : $$\left( {\forall n \in \left\{ {1,...,p} \right\}} \right)\left( {\forall i \le \left\lfloor {\frac{n}{2}} \right\rfloor } \right)\left( {\exists k \in \mathbb{N}} \right)\left( {n - i + 1 \le k \le n\mbox{ and }f(k) \ge i} \right)$$ $F_p $ is not empty, it contains the following function $\varphi _p $, defined for $1 \le i \le p$ by : $$\varphi _p \left( i \right) = \min \left( {2^{\upsilon _2 \left( i \right) + 1} - 1,\left\lfloor {\frac{p}{2}} \right\rfloor ,i - 1} \right)$$ where $\upsilon _2 \left( k \right)$ is the 2-valuation of $k$ (i.e. the greatest power of 2 dividing $k)$. The verification is obvious. \[lm:b\] For every sequence $F = \left( {f_1 ,f_2 ,...} \right)$ of functions such that $f_i \in F_i $ for $i \ge 1$ and $f_i \left( k \right) \le f_{i + 1} \left( k \right)$ for all $i \ge 1$ and $1 \le k \le i$, the rooted comb defined by $Comb_m^F = \left[ {Pf_1^m ,...,Pf_m^m } \right]$ where $Pf_i^m $ designs the path of size $f_m (m + 1 - i) - 1$, for $1 \le i \le m$ is strongly $m$-universal for the rooted brushes. By induction on $m$ : $Comb_1^F $ is strongly 1-universal for the rooted brushes. Suppose that $Comb_i^F $ has all rooted brushes with $i - 1$ edges as rooted contractions. We consider two cases depending on the shape of a rooted brush $B$ of size $i$ : case 1 case 2 ![image](unitrdes7.eps){width="3.80in" height="2.05in"} Brushes of case 1 are clearly rooted contractions of the rooted comb $Comb_i^F $ (${B}'\preceq Comb_{i - 1}^F $, so $B\preceq \left[ {P_0 ,Pf_1^{i - 1} ,...,Pf_{i - 1}^{i - 1} } \right]\preceq Comb_i^F )$. Let us study case 2 : $B'$ is by induction hypothesis a rooted contraction of the rooted comb $Comb_{i - j}^F $, moreover $Comb_{i - j}^F \preceq \left[ {Pf_{j + 1}^i ,...,Pf_i^i } \right].$ Finally, by the property of $f_i $, there exists $1 \le \alpha \le j$, such that $Pf_\alpha ^i $ has more than $j$ edges. Linking these two points, we can conclude that the rooted brush $B$ is always a rooted contraction of the rooted comb $Comb_i^F $. The rooted comb built as in lemma \[lm:b\] will be said to be associated to the sequence $F$ and denoted by $Comb_m^F $. A minimum strongly $m$-universal rooted brush for the rooted brushes has $O(m\ln (m))$ edges. Proceeding as for theorem \[th1\], we obtain, mutatis mutandis, that a $m$-universal brush for the brushes has at least $m\ln (m) + O(m)$ edges. This order of magnitude is precisely the size of the strongly $m$-universal rooted comb $Comb_m^F$ for the class of rooted brushes. We have this immediate corollary : A minimum $m$-universal brush for the brushes has $O(m\ln (m))$ edges. By convention, we put $Comb_0^F = P_0 $ (tree reduced in a vertex) We define $Tree_m^F = M_T^m \left( {\left( {Comb_n^F } \right)_{n \in \mathbb{N}} } \right)$. As before, we will say that the tree built in such a way is recursively associated to the sequence $F$ and denoted by $Tree_m^F $. Thus, we have : The rooted tree $Tree_m^F $ is strongly $m$-universal for the class of rooted trees. We now analyze the size of $Tree_m^F $. Let $F = \left( {f_1 ,f_2 ,...} \right)$ be a sequence of functions such that $f_i \in F_i $ for $i \ge 1$. The size of a $m$-universal tree constructed from the sequence is given by the following recursive formula : $u_{ - 1} = - 1, u_0 = 0$ and $u_k = 2k - 1 + \sum\limits_{i = 1}^k {u_{f_k \left( i \right) - 1} } $ It derives from the following observation :\ $m$ edges constitute the main stem, we have to add $m - 1$ edges to link branches to the main stem and $\sum\limits_{i = 1}^k {u_{f_k \left( i \right) - 1} } $ edges for the branches. There is a sequence of functions $G = \left( {g_1 ,g_2 ,...} \right)$ such that $g_i \in F_i $ and $\left\| {E\left( {Tree_m^G } \right)} \right\| < \left( {2m} \right)^c$ where $c = 1.984...$ is the unique positive solution of the equation $\frac{1}{2^c} + \frac{1}{2^{2c}} + \frac{1}{2^{\left( {c - 1} \right)} - 1} - \frac{1}{2^c - 1} = 1$. We take the following sequence of functions :\ $g_m \left( i \right) = \min \left( {2^{\upsilon _2 \left( i \right) + 1},i} \right)$ if $i < m$ and $i$ even, $g_m \left( i \right) = 1$ if $i$ odd and $g_m \left( m \right) = \left\lfloor {\frac{m}{4}} \right\rfloor $. It is clear that, if $m$ is a power of 2, the comb $Comb_m^G $ is strongly $m$-universal for the brushes. In fact, the function $g_m $ takes the value $2^{\upsilon _2 \left( i \right) + 1}$ when $i$ is not a power of 2, otherwise it is equal to $i$. Thanks to this remark and with $u_m < m + \sum\limits_{i = 1}^m {u_{f_m \left( i \right)} } $, (the sequence of sizes is increasing), we obtain $u_{2^n} < 2^n + 2^{n - 1} + \sum\limits_{i = 2}^{n - 1} {2^{n - i}u_{2^i} } - \sum\limits_{i = 2}^{n - 1} {u_{2^i} } + u_{2^{n - 1}} + u_{2^{n - 2}} $. Thus, in evaluating the sums and reorganizing the terms, we obtain : $$u_{2^n} < \alpha _n + 2^{nc}\beta$$ with $$\alpha _n = 2^{n - 1} + 1 + 2^c + \frac{1}{2^c - 1} - \left( {\frac{2^n}{2^{\left( {c - 1} \right)} - 1} + 2^{n\left( {c - 1} \right)}} \right)$$ $$\beta = \frac{1}{2^c} + \frac{1}{2^{2c}} + \frac{1}{2^{\left( {c - 1} \right)} - 1} - \frac{1}{2^c - 1}$$\ Now $\alpha _n < 0$ when $m > 1$ and $\beta \le 1$ by definition of $c.$\ So $u_{2^n} < 2^{nc}$, hence $u_m < \left( {2m} \right)^c$. We observe that $c = \frac{\ln \left( x \right)}{\ln \left( 2 \right)}$, where $x$ is the positive root of $X^4 - 5X^3 + 4X^2 + X - 2 = 0$. Theorem \[th2\] then follows since any rooted tree which is strongly $m$-universal for the rooted trees is also clearly $m$-universal for the class of trees. Conclusion and Related Questions ================================ When using the sequence $\Phi = \left( {\varphi _1 ,\varphi _2 ,...} \right)$ of lemma \[lm:b\], the induction step leads to involved expressions that do not allow us to find the asymptotic behavior of the corresponding term $u_m $. A computer simulation gives that such a $m$-universal tree for the trees has less than $m^{1.88}$ edges. In any case, the constructive approach we proposed here, seems to be hopeless to reach the asymptotic best size of a $m$-universal tree for the trees. The minimal size of a $m$-universal tree for the trees is $m^{1 + o\left( 1 \right)}$. As a possible way to prove such a conjecture, it would be interesting to obtain an explicit effective coding of a tree of size $m$ using a list of contracted edges taken in a $m$-universal tree for the trees. A variant of our problem consists in determining a minimum tree which contains as a subtree every tree of size $m.$ This is closely related to a well known still open conjecture due to Erdös and Sös (see [@ES]). [9]{} R. Rado, Universal graphs and universal functions, Acta Arith., 9 (1964), 331-340. Z. Füredi and P. Komjàth, Nonexistence of universal graphs without some trees, Combinatorica, 17, (2) (1997), 163-171. N. Robertson and P.D. Seymour, series of papers on Graph minors, Journal of combinarotics, serie B, (1983-...). C. Thomassen, Embeddings and Minors, chapter 5 in Handbook of Combinatorics, ( R. Graham, M. Grötschel and L. Lovàsz, eds.), Elsevier Science B.V., 1995, 301-349. P. Erdös and T. Gallai, On maximal paths and circuits of graphs, Acta Math. Sci. Hungar. 10 (1959), 337-356.
--- abstract: \| A variational principle is applied to 4D Euclidean space provided with a tensor refractive index, defining what can be seen as 4-dimensional optics (4DO). The geometry of such space is analysed, making no physical assumptions of any kind. However, by assigning geometric entities to physical quantities the paper allows physical predictions to be made. A mechanism is proposed for translation between 4DO and GR, which involves the null subspace of 5D space with signature $(-++++)$. A tensor equation relating the refractive index to sources is established geometrically and the sources tensor is shown to have close relationship to the stress tensor of GR. This equation is solved for the special case of zero sources but the solution that is found is only applicable to Newton mechanics and is inadequate for such predictions as light bending and perihelium advance. It is then argued that testing gravity in the physical world involves the use of a test charge which is itself a source. Solving the new equation, with consideration of the test particle’s inertial mass, produces an exponential refractive index where the Newtonian potential appears in exponent and provides accurate predictions. Resorting to hyperspherical coordinates it becomes possible to show that the Universe’s expansion has a purely geometric explanation without appeal to dark matter. author: - \| J. B. Almeida\ \ \ [E-mail:[`[email protected]`](mailto:[email protected])]{} bibliography: - 'Abrev.bib' - 'aberrations.bib' - 'assistentes.bib' title: Euclidean formulation of general relativity --- Introduction ============ According to general consensus any physics theory is based on a set of principles upon which predictions are made using established mathematical derivations; the validity of such theory depends on agreement between predictions and observed physical reality. In that sense this paper does not formulate a physical theory because it does not presume any physical principles; for instance it does not assume speed of light constancy or equivalence between frame acceleration and gravity. This is a paper about geometry. All along the paper, in several occasions, a parallel is made with the physical world by assigning a physical meaning to geometric entities and this allows predictions to be made. However the validity of derivations and overall consistency of the exposition is independent of prediction correctness. The only postulates in this paper are of a geometrical nature and can be condensed in the definition of the space we are going to work with: 4-dimensional space with Euclidean signature $(++++)$. For the sole purpose of making transitions to spacetime we will also consider the null subspace of the 5-dimensional space with signature $(-++++)$. This choice of space does not imply any assumption about its physical meaning up to the point where geometric entities like coordinates and geodesics start being assigned to physical quantities like distances and trajectories. Some of those assignments will be made very early in the exposition and will be kept consistently until the end in order to allow the reader some assessment of the proposed geometric model as a tool for the prediction of physical phenomena. Mapping between geometry and physics is facilitated if one chooses to work always with non-dimensional quantities; this is easily done with a suitable choice for standards of the fundamental units. In this work all problems of dimensional homogeneity are avoided through the use of normalising factors for all units, listed in Table \[t:factors\], defined with recourse to the fundamental constants: Planck constant, gravitational constant, speed of light and proton charge. ----------------------------------------------------------------------------------------------------------------------------------- Length Time Mass Charge --------------------------------------------- ------------------------------ --------------------------------------------- -------- $\displaystyle \sqrt{\frac{G \hbar}{c^3}} $ $\displaystyle \sqrt{\frac{G $\displaystyle \sqrt{\frac{ \hbar c }{G}} $ $e$ \hbar}{c^5}} $ ----------------------------------------------------------------------------------------------------------------------------------- : \[t:factors\]Normalising factors for non-dimensional units used in the text; $\hbar \rightarrow$ Planck constant divided by $2 \pi$, $G \rightarrow$ gravitational constant, $c \rightarrow$ speed of light and $e \rightarrow$ proton charge. This normalisation defines a system of non-dimensional units with important consequences, namely: 1) all the fundamental constants, $\hbar$, $G$, $c$, $e$, become unity; 2) a particle’s Compton frequency, defined by $\nu = mc^2/\hbar$, becomes equal to the particle’s mass; 3) the frequent term ${GM}/({c^2 r})$ is simplified to ${M}/{r}$. The particular space we chose to work with can have amazing structure, providing countless parallels to the physical world; this paper is just a limited introductory look at such structure and parallels. The exposition makes full use of an extraordinary and little known mathematical tool called geometric algebra (GA), a.k.a. Clifford algebra, which received an important thrust with the introduction of geometric calculus by David Hestenes [@Hestenes84]. A good introduction to GA can be found in @Gull93 and the following paragraphs use basically the notation and conventions therein. A complete course on physical applications of GA can be downloaded from the internet [@Lasenby99] with a more comprehensive version published recently in book form [@Doran03] while an accessible presentation of mechanics in GA formalism is provided by @Hestenes03. Introduction to geometric algebra ================================= We will use Greek characters for the indices that span 1 to 4 and Latin characters for those that exclude the 4 value; in rare cases we will have to use indices spanning 0 to 3 and these will be denoted with Greek characters with an over bar. The geometric algebra of the hyperbolic 5-dimensional space we want to consider $\mathcal{G}_{4,1}$ is generated by the frame of orthonormal vectors $\{\mathrm{i},\sigma_\mu \}$, $\mu = 1 \ldots 4$, verifying the relations $$\mathrm{i}^2 = -1,~~~~ \mathrm{i} \sigma_\mu + \sigma_\mu \mathrm{i} =0,~~~~ \sigma_\mu \sigma_\nu + \sigma_\nu \sigma_\mu = 2 \delta_{\mu \nu}.$$ We will simplify the notation for basis vector products using multiple indices, i.e. $\sigma_\mu \sigma_\nu \equiv \sigma_{\mu\nu}$. The algebra is 32-dimensional and is spanned by the basis $$\begin{array}{cccccc} 1, & \{\mathrm{i},\sigma_\mu \}, & \{\mathrm{i} \sigma_\mu,\sigma_{\mu\nu} \}, & \{\mathrm{i} \sigma_{\mu\nu},\sigma_{\mu\nu\lambda} \}, & \{\mathrm{i}I, \sigma_\mu I \}, & I; \\ \mathrm{1~ scalar} & \mathrm{5~ vectors} & \mathrm{10~ bivectors} & \mathrm{10~ trivectors} & \mathrm{5~ tetravectors} & \mathrm{1~ pentavector} \end{array}$$ where $I \equiv \mathrm{i}\sigma_1 \sigma_2 \sigma_3\sigma_4$ is also called the pseudoscalar unit. Several elements of this basis square to unity: $$(\sigma_\mu)^2 = 1,~~~~ (i \sigma_\mu)^2=1,~~~~ (\mathrm{i}\sigma_{\mu\nu})^2 =1, ~~~~ (\mathrm{i}I)^2 =1;$$ and the remaining square to $-1$: $$\mathrm{i}^2 = -1,~~~~(\sigma_{\mu\nu})^2 = -1,~~~~ (\sigma_{\mu\nu\lambda})^2 = -1,~~~~(\sigma_\mu I)^2,~~~~I^2=-1.$$ Note that the symbol $\mathrm{i}$ is used here to represent a vector with norm $-1$ and must not be confused with the scalar imaginary, which we don’t usually need. The geometric product of any two vectors $a = a^0 \mathrm{i} + a^\mu \sigma_\mu$ and $b = b^0 \mathrm{i} + b^\nu \sigma_\nu$ can be decomposed into a symmetric part, a scalar called the inner product, and an anti-symmetric part, a bivector called the exterior product. $$ab = a {\! \cdot \!}b + a {\! \wedge \!}b,~~~~ ba = a {\! \cdot \!}b - a {\! \wedge \!}b.$$ Reversing the definition one can write internal and exterior products as $$a {\! \cdot \!}b = \frac{1}{2}\, (ab + ba),~~~~ a {\! \wedge \!}b = \frac{1}{2}\, (ab - ba).$$ When a vector is operated with a multivector the inner product reduces the grade of each element by one unit and the outer product increases the grade by one. There are two exceptions; when operated with a scalar the inner product does not produce grade $-1$ but grade $1$ instead, and the outer product with a pseudoscalar is disallowed. Displacement and velocity ========================= Any displacement in the 5-dimensional hyperbolic space can be defined by the displacement vector $$\label{eq:displacement} \mathrm{d}s =\mathrm{i} \mathrm{d}x^0 + \sigma_\mu \mathrm{d}x^\mu;$$ and the null space condition implies that $\mathrm{d}s$ has zero length $$\mathrm{d}s^2 = \mathrm{d}s {\! \cdot \!}\mathrm{d}s = 0;$$ which is easily seen equivalent to either of the relations $$\label{eq:twospaces} (\mathrm{d}x^0)^2 = \sum (\mathrm{d}x^\mu)^2;~~~~ (\mathrm{d}x^4)^2 = (\mathrm{d}x^0)^2 - \sum (\mathrm{d}x^j)^2.$$ These equations define the metrics of two alternative spaces, one Euclidean the other one Minkowskian, both equivalent to the null 5-dimensional subspace. A path on null space does not have any affine parameter but we can use Eqs.(\[eq:twospaces\]) to express 4 coordinates in terms of the fifth one. We will frequently use the letter $t$ to refer to coordinate $x^0$ and the letter $\tau$ for coordinate $x^4$; total derivatives with respect to $t$ will be denoted by an over dot while total derivatives with respect to $\tau$ will be denoted by a “check”, as in $\check{F}$. Dividing both members of Eq.(\[eq:displacement\]) by $\mathrm{d}t$ we get $$\label{eq:euclvelocity} \dot{s} = \mathrm{i} + \sigma_\mu \dot{x}^\mu = \mathrm{i} + v.$$ This is the definition for the velocity vector $v$; it is important to stress again that the velocity vector defined here is a geometrical entity which bears for the moment no relation to physical velocity, be it relativistic or not. The velocity has unit norm because $\dot{s}^2 =0$; evaluation of $v{\! \cdot \!}v$ yields the relation $$\label{eq:vsquare} v {\! \cdot \!}v = \sum (\dot{x}^\mu)^2 = 1.$$ The velocity vector can be obtained by a suitable rotation of any of the $\sigma_\mu$ frame vectors, in particular it can always be expressed as a rotation of the $\sigma_4$ vector. At this point we are going to make a small detour for the first parallel with physics. In the previous equation we replace $x^0$ by the greek letter $\tau$ and rewrite with $\dot{\tau}^2$ in the first member $$\label{eq:dtau2} \dot{\tau}^2 = 1 - \sum (\dot{x}^j)^2.$$ The relation above is well known in special relativity, see for instance @Martin88; see also @Almeida02:2 [@Montanus01] for parallels between special relativity and its Euclidean space counterpart.[^1] We note that the operation performed between Eqs.(\[eq:vsquare\]) and (\[eq:dtau2\]) is a perfectly legitimate algebraic operation since all the elements involved are pure numbers. Obviously we could also divide both members of Eq. (\[eq:displacement\]) by $\mathrm{d}\tau$, which is then associated with relativistic proper time; $$\check{s} = \mathrm{i}\check{x}^0 + \sigma_j \check{x}^j + \sigma_4.$$ Squaring the second member and noting that it must be null we obtain $(\check{x}^0)^2 - \sum (\check{x}^j)^2 = 1$. This means that we can relate the vector $\mathrm{i}\check{x}^0 + \sigma_j \check{x}^j$ to relativistic 4-velocity, although the norm of this vector is symmetric to what is usual in SR. The relativistic 4-velocity is more conveniently assigned to the 5D bivector $\mathrm{i}\sigma_4\check{x}^0 + \sigma_{j4} \check{x}^j$, which has the necessary properties. The method we have used to make the transition between 4D Euclidean space and hyperbolic spacetime involved the transformation of a 5D vector into scalar plus bivector through product with $\sigma_4$; this method will later be extended to curved spaces. Equation (\[eq:euclvelocity\]) applies to flat space but can be generalised for curved space; we do this in two steps. First of all we can include a scale factor $(v = n \sigma_\mu \dot{x}^\mu)$, which can change from point to point $$\label{eq:refindex} \dot{s} = \mathrm{i} + n \sigma_\mu \dot{x}^\mu.$$ In this way we are introducing the 4-dimensional analogue of a refractive index, that can be seen as a generalisation of the 3-dimensional definition of refractive index for an optical medium: the quotient between the speed of light in vacuum and the speed of light in that medium. The scale factor $n$ used here relates the norm of vector $\sigma_\mu \dot{x}^\mu$ to unity and so it deserves the designation of 4-dimensional refractive index; we will drop the “4-dimensional” qualification because the confusion with the 3-dimensional case can always be resolved easily. The material presented in this paper is, in many respects, a logical generalisation of optics to 4-dimensional space; so, even if the paper is only about geometry, it becomes natural to designate this study as 4-dimensional optics (4DO). Full generalisation of Eq. (\[eq:euclvelocity\]) implies the consideration of a tensor refractive index, similar to the non-isotropic refractive index of optical media $$\label{eq:vgeneral} \dot{s} = \mathrm{i} + {n^\mu}_\nu \dot{x}^\nu \sigma_\mu;$$ the velocity is then generally defined by $v = {n^\mu}_\nu \dot{x}^\nu \sigma_\mu$. The same expression can be used with any orthonormal frame, including for instance spherical coordinates, but for the moment we will restrict our attention to those cases where the frame does not rotate in a displacement; this poses no restriction on the problems to be addressed but is obviously inconvenient when symmetries are involved. Equation (\[eq:vgeneral\]) can be written with the velocity in the form $v = g_\nu \dot{x}^\nu$ if we define the refractive index vectors $$\label{eq:gmu} g_\nu = {n^\mu}_\nu \sigma_\mu.$$ The set of four $g_\mu$ vectors will be designated the refractive index frame. Obviously the velocity is still a unitary vector and we can express this fact evaluating the internal product with itself and noting that the second member in Eq. (\[eq:vgeneral\]) has zero norm. $$\label{eq:vsquaregen} v {\! \cdot \!}v = {n^\alpha}_\mu \dot{x}^\mu {n^\beta}_\nu \dot{x}^\nu \delta_{\alpha \beta}=1.$$ Using Eq. (\[eq:gmu\]) we can rewrite the equation above as $g_\mu {\! \cdot \!}g_\nu \dot{x}^\mu \dot{x}^\nu = 1$ and denoting by $g_{\mu \nu}$ the scalar $g_\mu {\! \cdot \!}g_\nu$ the equation becomes $$g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu =1.$$ The generalised form of the displacement vector arises from multiplying Eq.(\[eq:vgeneral\]) by $\mathrm{d}t$, using the definition (\[eq:gmu\]) $$\label{eq:dsgeneral} \mathrm{d}s = \mathrm{i}\mathrm{d}t + g_\mu \mathrm{d}x^\mu.$$ This can be put in the form of a space metric by dotting with itself and noting that the first member vanishes $$\label{eq:dt2general} (\mathrm{d}t)^2 = g_{\mu \nu} \mathrm{d}x^\mu \mathrm{d}x^\nu.$$ Notice that the coordinates are still referred to the fixed frame vectors $\sigma_\mu$ and not to the refractive index vectors $g_\mu$. In GR there is no such distinction between two frames but @Montanus01 clearly separates the frame from tensor $g_{\mu\nu}$. We are going to need the reciprocal frame [@Doran03] $\{-\mathrm{i},g^\mu\}$ such that $$\label{eq:recframe} g^\mu {\! \cdot \!}g_\nu = {\delta^\mu}_\nu.$$ From the definition it becomes obvious that $g_\mu g^\nu = g_\mu {\! \cdot \!}g^\nu + g_\mu {\! \wedge \!}g^\nu$ is a pure bivector and so $g_\mu g^\nu = -g^\nu g_\mu$. We now multiply Eq. (\[eq:dsgeneral\]) on the right and on the left by $g^4$, simultaneously replacing $x^4$ by $\tau$ to obtain $$\mathrm{d}s g^4 = \mathrm{i}g^4 \mathrm{d}t + g_jg^4 \mathrm{d}x^j + \mathrm{d}\tau;~~~~g^4 \mathrm{d}s = g^4 \mathrm{i} \mathrm{d}t + g^4g_j \mathrm{d}x^j + \mathrm{d}\tau.$$ When the internal product is performed between the two equations member to member the first member vanishes and the second member produces the result $$\label{eq:transition1} (\mathrm{d}\tau)^2 = g^{44} \left[(\mathrm{d}t)^2 -g_{jk}\mathrm{d}x^j \mathrm{d}x^k \right].$$ If the various $g_\mu$ are functions only of $x^j$ the equation is equivalent to a metric definition in general relativity. We will examine the special case when $g_\mu = n_\mu \sigma_\mu$; replacing in Eq. (\[eq:transition1\]) $$\label{eq:dtau2gen} (\mathrm{d}\tau)^2 = \frac{1}{(n_4)^2}\, (\mathrm{d}t)^2 -\sum\left( \frac{n_j}{n_4}\, \mathrm{d}x^j \right)^2.$$ This equation covers a large number of situations in general relativity, including the very important Schwarzschild’s metric, as was shown in @Almeida04:1 and will be discussed below. Notice that Eq.(\[eq:dt2general\]) has more information than Eq. (\[eq:transition1\]) because the structure of $g_4$ is kept in the former, through the coefficients $g_{\mu 4}$, but is mostly lost in the $g^{44}$ coefficient of the latter. The sources of space curvature ============================== Equations (\[eq:dt2general\]) and (\[eq:transition1\]) define two alternative 4-dimensional spaces; in the former, 4DO, $t$ is an affine parameter while in the latter, GR, it is $\tau$ that takes such role. The geodesics of one space can be mapped one to one with those of the other and we can choose to work on the space that best suits us. The geodesics of 4DO space can be found by consideration of the Lagrangian $$L = \frac{g_{\mu \nu} \dot{x}^\mu \dot{x}^\nu}{2} = \frac{1}{2}\ .$$ The justification for this choice of Lagrangian can be found in several reference books but see for instance @Martin88. From the Lagrangian one defines immediately the conjugate momenta $$v_\mu = \frac{\partial L}{\partial \dot{x}^\mu} = g_{\mu \nu} \dot{x}^\nu.$$ The conjugate momenta are the components of the conjugate momentum vector $v = g^\mu v_\mu$ and from Eq. (\[eq:recframe\]) $$v = g^\mu v_\mu = g^\mu g_{\mu \nu} \dot{x}^\nu = g_\nu \dot{x}^\nu.$$ The conjugate momentum and velocity are the same but their components are referred to the reverse and refractive index frames, respectively. The geodesic equations can now be written in the form of Euler-Lagrange equations $$\dot{v}_\mu = \partial_\mu L;$$ these equations define those paths that minimise $t$ when displacements are made with velocity given by Eq. (\[eq:vgeneral\]). Considering the parallel already made with general relativity we can safely say that geodesics of 4DO spaces have a one to one correspondence to those of GR in the majority of situations. We are going to need geometric calculus which was introduced by @Hestenes84 as said earlier; another good reference is provided by @Doran03. The existence of such references allows us to introduce the vector derivative without further explanation; the reader should search the cited books for full justification of the definition we give below $$\Box = g^\mu \partial_\mu.$$ The vector derivative is a vector and can be operated with any multivector using the established rules; in particular the geometric product of $\Box$ with a multivector can be decomposed into inner and outer products. When applied to vector $a$ $(\Box a = \Box {\! \cdot \!}a + \Box {\! \wedge \!}a)$ the inner product is the divergence of vector $a$ and the outer product is the exterior derivative, related to the curl although usable in spaces of arbitrary dimension and expressed as a bivector. We also define the Laplacian as the scalar operator $\Box^2 = \Box {\! \cdot \!}\Box$. In this work we do not use the conventions of Riemanian geometry for the affine connection, as was already noted in relation to Eq. (\[eq:recframe\]). For this reason we will also need to distinguish between the curved space derivative defined above and the ordinary flat space derivative $$\nabla = \sigma^\mu \partial_\mu = \sum \sigma_\mu \partial_\mu.$$ When using spherical coordinates, for instance, the connection will be involved only in the flat space component of the derivative and we will deal with it by explicitly expressing the frame vector derivatives. Velocity is a vector with very special significance in 4DO space because it is the unitary vector tangent to a geodesic. We therefore attribute high significance to velocity derivatives, since they express the characteristics of the particular space we are considering. When the Laplacian is applied to the velocity vector we obtain a vector $$\label{eq:current} \Box^2 v = T.$$ Vector $T$ is called the sources vector and can be expanded into sixteen terms as $$T = {T^\mu}_\nu \sigma_\mu \dot{x}^\nu = (\Box^2 {n^\mu}_\nu) \sigma_\mu \dot{x}^\nu.$$ The tensor ${T^\mu}_\nu$ contains the coefficients of the sources vector and we call it the sources tensor; it is very similar to the stress tensor of GR, although its relation to geometry is different. The sources tensor influences the shape of geodesics but we shall not examine here how such influence arises, except for very special cases. Before we begin searching solutions for Eq. (\[eq:current\]) we will show that this equation can be decomposed into a set of equations similar to Maxwell’s. Consider first the velocity derivative $\Box v = \Box {\! \cdot \!}v + \Box {\! \wedge \!}v$; the result is a multivector with scalar and bivector part $G = \Box v$. Now derive again $\Box G = \Box {\! \cdot \!}G + \Box {\! \wedge \!}G$; we know that the exterior derivative of $G$ vanishes and the divergence equals the sources vector. Maxwell’s equations can be written in a similar form, as was shown in @Almeida04:2, with velocity replaced by the vector potential and multivector $G$ replaced by the Faraday bivector $F$; @Doran03 offer similar formulation for spacetime. An isotropic space must be characterised by orthogonal refractive index vectors $g_\mu$ whose norm can change with coordinates but is the same for all vectors. We usually relax this condition by accepting that the three $g_j$ must have equal norm but $g_4$ can be different. The reason for this relaxed isotropy is found in the parallel usually made with physics by assigning dimensions $1$ to $3$ to physical space. Isotropy in a physical sense need only be concerned with these dimensions and ignores what happens with dimension 4. We will therefore characterise an isotropic space by the refractive index frame $g_j = n_r \sigma_j$, $g_4 = n_4 \sigma_4$. Indeed we could also accept a non-orthogonal $g_4$ within the relaxed isotropy concept but we will not do so in this work. We will only investigate spherically symmetric solutions independent of $x^4$; this means that the refractive index can be expressed as functions of $r$ in spherical coordinates. The vector derivative in spherical coordinates is of course $$\Box = \frac{1}{n_r}\, \left(\sigma_r \partial_r + \frac{1}{r}\, \sigma_\theta \partial_\theta + \frac{1}{r \sin \theta}\, \sigma_\varphi \partial_\varphi \right) + \frac{1}{n_4}\, \sigma_4 \partial_4.$$ The Laplacian is the inner product of $\Box$ with itself but the frame derivatives must be considered $$\begin{aligned} \partial_r \sigma_r = 0, ~~~~ &\partial_\theta \sigma_r = \sigma_\theta, ~~~~ &\partial_\varphi \sigma_r = \sin \theta \sigma_\varphi, \nonumber \\ \partial_r \sigma_\theta = 0, ~~~~ &\partial_\theta \sigma_\theta = -\sigma_r, ~~~~ &\partial_\varphi \sigma_\theta = \cos \theta \sigma_\varphi, \\ \partial_r \sigma_\varphi = 0, ~~~~ &\partial_\theta \sigma_\varphi = 0, ~~~~ &\partial_\varphi \sigma_\varphi = -\sin \theta\, \sigma_r - \cos \theta\, \sigma_\theta. \nonumber\end{aligned}$$ After evaluation the Laplacian becomes $$\label{eq:laplacradial} \Box^2 = \frac{1}{(n_r)^2}\, \left(\partial_{rr} + \frac{2}{r}\, \partial_r - \frac{n'_r}{n_r}\, \partial_r + \frac{1}{r^2}\, \partial_{\theta \theta} +\frac{\cot \theta}{r^2}\, \partial_\theta + \frac{\csc^2 \theta}{r^2}\, \partial_{\varphi \varphi} \right) + \frac{1}{(n_4)^2}\, \partial_{\tau \tau}.$$ In the absence of sources we want the sources tensor to vanish, implying that the Laplacian of both $n_r$ and $n_4$ must be zero; considering that they are functions of $r$ we get the following equation for $n_r$ $$n^{''}_r + \frac{2 n'_r}{r} - \frac{(n'_r)^2}{n_r} = 0,$$ with general solution $n_r = b \exp(a/r)$. We can make $b =1$ because we want the refractive index to be unity at infinity. Using this solution in Eq.(\[eq:laplacradial\]) the Laplacian becomes $$\Box^2 = \mathrm{e}^{-a/r}\left(\mathrm{d}^2 + \frac{2}{r}\, \mathrm{d} + \frac{a }{r^2}\, \mathrm{d}\right).$$ When applied to $n_4$ and equated to zero we obtain solutions which impose $n_4 = n_r $ and so the space must be truly isotropic and not relaxed isotropic as we had allowed. The solution we have found for the refractive index components in isotropic space can correctly model Newton dynamics, which led the author to adhere to it for some time [@Almeida01:4]. However if inserted into Eq.(\[eq:dtau2gen\]) this solution produces a GR metric which is verifiably in disagreement with observations; consequently it has purely geometric significance. The inadequacy of the isotropic solution found above for relativistic predictions deserves some thought, so that we can search for solutions guided by the results that are expected to have physical significance. In the physical world we are never in a situation of zero sources because the shape of space or the existence of a refractive index must always be tested with a test particle. A test particle is an abstraction corresponding to a point mass considered so small as to have no influence on the shape of space. But in reality a test particle is always a source of refractive index and its influence on the shape of space may not be negligible in any circumstances. If this is the case the solutions for vanishing sources vector may have only geometric meaning, with no connection to physical reality. The question is then how do we include the test particle in Eq.(\[eq:current\]) in order to find physically meaningful solutions. Here we will make one add hoc proposal without further justification because the author has not yet completed the work that will provide such justification in geometric terms. The second member of Eq. (\[eq:current\]) will not be zero and we will impose the sources vector $$\label{eq:statpart} J = -\nabla^2 n_4 \sigma_4.$$ Equation (\[eq:current\]) becomes $$\label{eq:gravitation} \Box^2 v = -\nabla^2 n_4 \sigma_4;$$ as a result the equation for $n_r$ remains unchanged but the equation for $n_4$ becomes $$n^{''}_4 + \frac{2 n'_4}{r} - \frac{n'_r n'_4}{n_r} = - n^{''}_4 + \frac{2 n'_4}{r}.$$ When $n_r$ is given the exponential form found above the solution is $n_4 = \sqrt{n_r}$. This can now be entered into Eq. (\[eq:dtau2gen\]) and the coefficients can be expanded in series and compared to Schwarzschild’s for the determination of parameter $a$. The final solution, for a stationary mass $M$ is $$\label{eq:refind} n_r = \mathrm{e}^{2M/r},~~~~n_4 = \mathrm{e}^{M/r}.$$ Equation (\[eq:gravitation\]) can be interpreted in physical terms as containing the essence of gravitation. When solved for spherically symmetric solutions, as we have done, the first member provides the definition of a stationary gravitational mass as the factor $M$ appearing in the exponent and the second member defines inertial mass as $\nabla^2 n_4$. Gravitational mass is defined with recourse to some particle which undergoes its influence and is animated with velocity $v$ and inertial mass cannot be defined without some field $n_4$ acting upon it. Complete investigation of the sources tensor elements and their relation to physical quantities is not yet done. It is believed that the 16 terms of this tensor have strong links with homologous elements of stress tensor in GR but this will have to be verified. Finally we turn our attention to hyperspherical coordinates. The position vector is quite simply $x = \tau \sigma_\tau$, where the coordinate is the distance to the hypersphere centre. Differentiating the position vector we obtain the displacement vector, which is a natural generalisation of 3D spherical coordinates case $$\mathrm{d}x = \sigma_\tau \mathrm{d} \tau + \tau \sigma_\rho \mathrm{d} \rho + \tau \sin \rho \sigma_\theta \mathrm{d}\theta + \tau \sin \rho \sin \theta \sigma_\varphi \mathrm{d} \varphi;$$ $\rho$, $\theta$ and $\varphi$ are angles. The velocity in an isotropic medium should now be written as $$\label{eq:hypervelocity} v = n_4 \sigma_\tau \dot{\tau} + n_r \tau (\sigma_\rho \dot{\rho} + \sin \rho \sigma_\theta \dot{\theta} + \sin \rho \sin \theta \sigma_\varphi \dot{\varphi}).$$ In order to replace the angular coordinate $\rho$ with a distance coordinate $r$ we can make $r = \tau \rho$ and derive with respect to time $$\dot{r} =\rho \dot{\tau} + \tau \dot{\rho} = \frac{r}{\tau}\, \dot{\tau} + \tau \dot{\rho}.$$ Taking $\tau \dot{\rho}$ from this equation and inserting into Eq.(\[eq:hypervelocity\]), assuming that $\sin \rho$ is sufficiently small to be replaced by $\rho$ $$\label{eq:veluniverse} v = n_4 \left(\sigma_\tau - \frac{r}{\tau}\, \sigma_r \right) \dot{\tau} + n_r (\sigma_r \dot{r} + r \sigma_\theta \dot{\theta} + r \sin \theta \sigma_\varphi \dot{\varphi}).$$ we have also replaced $\sigma_\rho$ by $\sigma_r$ for consistency with the new coordinates. We have just defined a particularly important set of coordinates, which appears to be especially well adapted to describe the physical Universe, with $\tau$ being interpreted as the Universe’s age or its radius; note that time and distance cannot be distinguished in non-dimensional units. When $r \dot{\tau}/\tau$ is small in Eq. (\[eq:veluniverse\]), the refractive index vectors become orthogonal and we use $n_4$ and $n_r$ in conjunction with Eq.(\[eq:dtau2gen\]) to obtain a GR metric whose coefficients are equivalent so Schwarzschild’s on the first terms of their series expansions. When $r \dot{\tau}/\tau$ cannot be neglected, however, the equation can explain the Universe’s expansion and flat rotation curves in galaxies without dark matter intervention. A more complete discussion of this subject can be found in Ref.[@Almeida04:1]. Conclusions =========== Euclidean and Minkowskian 4-spaces can be formally linked through the null subspace of 5-dimensional space with signature $(-++++)$. The extension of such formalism to non-flat spaces allows the transition between spaces with both signatures and the paper discusses some conditions for metric and geodesic translation. For its similarities with optics, the geometry of 4-spaces with Euclidean signature is called 4-dimensional optics (4DO). Using only geometric arguments it is possible to define such concepts as velocity and trajectory in 4DO which become physical concepts when proper and natural assignments are made. One important point which is addressed for the first time in the author’s work is the link between the shape of space and the sources of curvature. This is done on geometrical grounds but it is also placed in the context of physics. The equation pertaining to the test of gravity by a test particle is proposed and solved for the spherically symmetric case providing a solution equivalent to Schwarzschild’s as first approximation. Some mention is made of hyperspherical coordinates and the reader is referred to previous work linking this geometry to the Universe’s expansion in the absence of dark matter. [^1]: Montanus first proposed the Euclidean alternative to relativity in 1991, nine years before the author started independent work along the same lines.
--- abstract: 'We propose a new type of molecular transistor, the Quantum Interference Effect Transistor (QuIET), based on tunable current suppression due to quantum interference. We show that any aromatic hydrocarbon ring has two-lead configurations for which current at small voltages is suppressed by destructive interference. A transistor can be created by providing phase relaxation or decoherence at a site on the ring. We propose several molecules which could tunably introduce the necessary dephasing or decoherence, as well as a proof of principle using a scanning tunneling microscope tip. Within the self-consistent Hartree-Fock approximation, the QuIET is shown to have characteristics strikingly similar to those of conventional field effect and bipolar junction transistors.' author: - 'D. M. Cardamone, C. A. Stafford, S. Mazumdar' title: The Quantum Interference Effect Transistor --- Introduction ============ Although there has been considerable scientific and commercial interest, a small molecular transistor has yet to be discovered. Needless to say, such a device is crucial to transferring existing technology to smaller length scales. Current industrial fabrication techniques have more or less exhausted the possibilities for purely classical phenomena to solve this problem. We must therefore turn to quantum mechanical effects in the search for smaller transistors. The solution, therefore, is also of fundamental interest to mesoscopic and molecular physicists: we find that the “small transistor” problem is an engaging way of posing the question, “What happens to multi-lead, many-body electronic systems when their size is such that quantum effects are important?” We propose a new type of device, the Quantum Interference Effect Transistor (QuIET), capable of filling the role of the Field Effect Transistor (FET) and Bipolar Junction Transistor (BJT) at length scales $\lesssim$1nm. The QuIET consists simply of a hydrocarbon ring and a mechanism to tunably introduce phase relaxation at a particular site. Unlike the Single Electron Transistor (SET), the transistor behavior can occur over a large range of base voltages. Its $I-\mathcal{V}$ characteristic is a single, broad resonance, strikingly similar to those of macroscopic transistors over a domain of several volts. This is to be taken in contrast to the SET’s $I-\mathcal{V}$, a series of many sharp peaks. The operating principle of the QuIET is that bias applied to a third lead can modulate an otherwise complete conductance suppression across a hydrocarbon ring. This conductance suppression is a simple, single-particle effect of quantum mechanics. The modulation from a third lead can be achieved either through direct coupling to the molecule (Fig. \[transistor\]a), as in the case of a scanning tunneling microscope (STM) tip, or via the introduction of an appropriate intermediary molecular complex, as shown in Fig. \[transistor\]b. In either case, the resultant device is an excellent molecular-scale transistor. ![Schematic diagrams of two types of QuIET. In each, base voltage modulates the coherent suppression of current between emitter (E) and collector (C) leads. In (a), base voltage controls the distance $x$ between the benzene ring and base lead (B), for example an STM tip. This in turn controls the coupling of the ring to the base lead. In (b), a base complex is introduced between the ring and base lead. The electrostatic effect of the base lead’s bias on this molecule alters its coupling to the benzene ring.[]{data-label="transistor"}](transistor.eps){width="\columnwidth"} Section \[model\] gives an outline of the extended Hubbard model Hamiltonian and Green function formalism we use to treat the problem. The multi-terminal current formula allows extraction of current. Section \[ring\] explains the tunable coherent conductance suppression by which the QuIET works. Section \[dba\] focuses on the use of a acceptor-donor molecule as a scalable method of tuning interference. Section \[numerics\] presents numerical results which indicate the viability of this idea. Finally, Section \[conclusion\] contains our conclusions. Model ===== The Hamiltonian of the system can be written as the sum of three terms: $H=H_m+H_l+H_{tun}$. The first is the extended Hubbard model molecular Hamiltonian $$\label{Hm} H=\sum_{i\sigma}\varepsilon_i d_{i\sigma}^\dagger d_{i\sigma}+\sum_{\langle ij\rangle\sigma}t_{ij}\left(d_{i\sigma}^\dagger d_{j\sigma}+\mathrm{H.c.}\right)+\sum_{ij}\frac{U_{ij}}{2}Q_iQ_j,$$ where $d_{i\sigma}$ annihilates an electron on atomic site $i$ with spin $\sigma$, $\varepsilon_i$ are the site energies, and $t_{ij}$ are the tunneling matrix elements. The final term of Eq. (\[Hm\]) contains intersite and same-site Coulomb interactions, as well as the electrostatic effects of the leads. The interaction energies are modeled according to the Ohno parameterization [@ohno]: $$\label{interactions} U_{ij}=\frac{11.13\mathrm{eV}}{\sqrt{1+.6117\left(R_{ij}/\mathrm{\AA}\right)^2}},$$ where $R_{ij}$ is the distance between sites $i$ and $j$. $Q_i$ is an effective charge operator for atomic site $i$: $$Q_i=\sum_\sigma d_{i\sigma}^\dagger d_{i\sigma}-\sum_\alpha\frac{C_{i\alpha}\mathcal{V}_\alpha}{e}-1.$$ The second term represents the polarization charge on site $i$ due to capacitive coupling with lead $\alpha$. Here $C_{i\alpha}$ is the capacitance between site $i$ and lead $\alpha$, chosen to correspond with the interaction energies of Eq.(\[interactions\]), and $\mathcal{V}_\alpha$ is the voltage on lead $\alpha$. $e$ is the magnitude of the electron charge. The QuIET is intended for use at room temperature and above, a temperature range far beyond the regime in which lead-molecule or lead-lead correlations play an important role. As such, we have followed the method of Ref. [@capacitance] and treated electrostatic interactions between molecule and leads at the level of capacitance parameters. The electronic situation of the leads is thus completely determined by the externally controlled voltages $\mathcal{V}_\alpha$, along with the leads’ temperatures and Fermi energies. Each lead possesses a continuum of states, and their total Hamiltonian is $$\label{Hl} H_l=\sum_{\alpha}\sum_{\substack{k\in\alpha\\ \sigma}}\varepsilon'_kc_{k\sigma}^\dagger c_{k\sigma},$$ where $\varepsilon'_k$ are the energies of the single-particle levels $k$ in lead $\alpha$, and $c_{k\sigma}$ are the annihilation operators for the states in the leads. Tunneling between molecule and leads is provided by the final term of the Hamiltonian: $$\label{Htun} H_{tun}=\sum_{\langle i\alpha\rangle}\sum_{\substack{k\in\alpha\\ \sigma}}\left(V_{ik}d_{i\sigma}^\dagger c_{k\sigma}+\mathrm{H.c.}\right).$$ $V_{ik}$ are the tunneling matrix elements for moving from a level $k$ within lead $\alpha$ to the nearby site $i$. Coupling of the leads to the ring via inert molecular chains, as may be desirable for fabrication purposes, can be included in the effective $V_{ik}$, as can the effect of the substituents used to bond the leads to the molecule. A system whose Hamiltonian includes such terms as Eqs. (\[Hl\]) and (\[Htun\]) requires an infinite-dimensional Fock space, but we are concerned mainly with the behavior of the discrete molecule suspended between the leads. We therefore adopt a Green function approach, in which Dyson’s Equation gives the Green function of the full system $$\label{Dyson} G(E)=\left[G_m^{-1}(E)-\Sigma(E)\right]^{-1},$$ where $G_m$ is the Green function of the isolated molecular system. With the use of an appropriate self-energy $\Sigma$, Equation (\[Dyson\]) is true both for the retarded Green function $G^r$ as well as for its $2\times 2$ Keldysh counterpart. The retarded self-energy due to the leads is $$\Sigma_{ij}^r(E)=-\frac{i}{2}\Gamma_i(E)\delta_{ij},$$ where the energy widths are given by Fermi’s Golden Rule $$\label{FGR} \Gamma_i(E)=2\pi\sum_\alpha\sum_{k\in\alpha}\|V_{ik}\|^2\delta\left(E-\varepsilon'_{ k}\right).$$ We take the broad-band limit of Eq. (\[FGR\]) and treat each of the $\Gamma_i$ as a constant parameter characterizing the lead-site coupling. The only effect of $H_l$ and $H_{tun}$ in this limit is to shift the poles of the Green function into the complex plane. This causes the density of states $\rho(E)=-\frac{1}{\pi}\mathrm{Im}\mathrm{Tr}G^r(E)$ to change from a discrete spectrum of delta functions to a continuous, width-broadened function. Due to the open nature of the system, electrons can occupy all energies. The retarded Green function gained via Eq. (\[Dyson\]) contains all information regarding the dynamics of the system. In particular, the current in lead $\beta$ is given by the familiar multi-terminal current formula [@buettiker]: $$\label{l-b} I_\beta=\frac{2e}{h}\sum_\alpha\int_{-\infty}^\infty dE\;T_{\alpha\beta}(E)\left[f_\alpha(E)-f_\beta(E)\right],$$ where $f_\alpha$ is the Fermi function for lead $\alpha$. The transmission probability is $$T_{\alpha\beta}(E)=\Gamma_a\Gamma_b\|G_{ab}^r(E)\|^2.$$ Here $a$($b$) is the site with hopping to lead $\alpha$($\beta$). We note that Eq. (\[l-b\]) is an exact result of the Keldysh formalism in cases, like ours, consisting only of elastic processes. In order to arrive at $G^r(E)$, we must consider electron-electron interactions. Here we do so via the well known self-consistent Hartree-Fock method. The Hamiltonian is replaced by its mean-field approximation $$\label{hfham} H_m^\mathrm{HF}=\sum_{i\sigma}\left(\varepsilon_i-\sum_{j\alpha}U_{ij}\frac{C_{j\alpha}\mathcal{V}_\alpha}{e}\right)d_{i\sigma}^\dagger d_{i\sigma}+\sum_{\langle ij\rangle\sigma}t_{ij}\left(d_{i\sigma}^\dagger d_{j\sigma}+\mathrm{H.c.}\right) +\sum_{ij\sigma\rho}U_{ij}\left(\langle d_{j\rho}^\dagger d_{j\rho}\rangle d_{i\sigma}^\dagger d_{i\sigma}-\langle d_{j\rho}^\dagger d_{i\sigma}\rangle d_{i\sigma}^\dagger d_{j\rho}\delta_{\sigma\rho}\right).$$ In this approximation, the retarded Green function is $G^r_m(E)=\left(E-H^\mathrm{HF}_m+i0^+\right)^{-1}$. Equation (\[hfham\]) gives the mean-field Hamiltonian as a function of the diagonal and off-diagonal equal-time correlation functions $\langle d_{i\sigma}^\dagger d_{j\sigma}\rangle$. To complete a self-consistent loop, we require an expression for these quantities in terms of Green functions. They are given in the Keldysh formalism by the equal-time limit of the “$<$” Green function $$G_{i\sigma,j\sigma}^<(t,t')=i\langle d_{i\sigma}^\dagger (t)d_{j\sigma}(t')\rangle=\int_{-\infty}^\infty\frac{d\omega}{2\pi}G_{i\sigma,j\sigma}^<(\omega)\mathrm{e}^{-i\omega(t-t')}.$$ From Dyson’s Equation (\[Dyson\]) it follows [@keldysh] that $$G^<=G^r\Sigma^<G^{r\dagger}+(1+G^r\Sigma^r)G_m^<(1+\Sigma^{r\dagger}G^{r\dagger}).$$ The second term is a purely equilibrium property, and can be related to the total charge on the molecule when the three lead biases are equal. The “$<$” self-energy is given by $$\Sigma^<_{ab}(\omega)=i\Gamma_af_\alpha(\omega)\delta_{ab}.$$ The desired relation between the equal-time correlation functions and $G^r$, $$\langle d_{i\sigma}^\dagger d_{j\sigma}\rangle=\sum_a\Gamma_a\int_{-\infty}^\infty\frac{d\omega}{2\pi}G^r_{i\sigma,a\sigma}(\omega)G^{r}_{a\sigma,j\sigma}(\omega)f_\alpha(\omega),$$ is now readily computed, and the self-consistent loop is complete. Tunable conductance suppression {#ring} =============================== ![(a) Two-lead experiment to measure the conductance of benzene when the leads are in the meta configuration. Shown are the two most direct paths a carrier can take from the emitter lead to the collector. These two cancel exactly, as do all other paths with the same endpoints in a similar pairwise fashion. (b) Example of a new path allowed when a base complex or lead is included. Such paths are not canceled, and so contribute to the total current between emitter and collector.[]{data-label="2paths"}](paths.eps){width="\columnwidth"} In the two-lead device shown in Fig. \[2paths\]a, a single carrier (electron or hole) is injected into the ring by the emitter and exits via the collector some time later. In the path integral formulation of quantum mechanics, it traverses all paths around the ring allowed by the connectivity of the system during this process. We operate the QuIET in the regime where there is little charge transfer between it and the leads. In the linear response, the carrier has momentum equal to the Fermi momentum of the ring $k_F=\frac{\pi}{2a}$, where $a=1.397\mathrm{\AA}$ is the intersite spacing of benzene. Clearly, then, the phase difference between paths and is $\pi$, and they cancel exactly. Similarly, all of the paths through the ring from emitter to collector exactly cancel in a pairwise fashion. Therefore, transport of carriers is forbidden in linear response. It is a consequence of Luttinger’s Theorem [@luttinger] that this coherent suppression of current persists into the interacting regime, as demonstrated in Fig. \[suppression\]a. The transmission is calculated by the self-consistent Hartree-Fock model outlined in Section \[model\]. In this figure the base coupling $\Gamma_B=0$, and so transmission at the Fermi energy is wholly suppressed by coherence. Figure \[suppression\]b shows how this result changes as $\Gamma_B$ is increased. Current is allowed to flow due to two new phenomena. The first is that new paths, such as the one shown in Fig.\[2paths\]b, are added. These paths have no particular phase relationship to other paths, and so support transmission. Additionally, nonzero $\Gamma_B$ denotes decoherence, and therefore a departure from the perfect coherent current suppression of $\Gamma_B=0$. This is the basic operating principle of the QuIET: coherent current suppression can be tunably broken by the introduction of decoherence and dephasing from a base complex or third lead. In fact, Figure \[suppression\]c shows that the transmission varies nearly linearly for $\Gamma_B\le t$. (3,1) (0,0)[![image](transmission.eps){width=".65\columnwidth"}]{} (1,0)[![image](blowup.eps){width=".65\columnwidth"}]{} (2,0)[![image](varygb.eps){width=".65\columnwidth"}]{} (0,.85)[(a)]{} (.95,.85)[(b)]{} (1.95,.85)[(c)]{} Clearly, the transistor behavior based on the mechanism outlined above is requisite upon an assumption that the device be operated well within the gap of benzene. The numerical simulations discussed in Section \[numerics\] indicate that the best transistor results are found for collector-emitter bias $\lesssim 1-2\mathrm{V}$. Another, related, consideration is that in equilibrium, charge transfer between the molecule and leads not play an important role. For this to be true, the work function of the metallic leads must be comparable to the chemical potential of benzene. Fortunately, this is the case with many bulk metals, among them palladium, iridium, platinum, and gold [@crc]. While in this work we have chosen to focus on benzene, the QuIET mechanism applies to any aromatic annulene with leads positioned so the two most direct paths have a phase difference of $\pi$. Furthermore, larger molecules have other possible lead configurations, based on phase differences of $3\pi$, $5\pi$, etc. Figure \[18\] shows the lead configurations for a QuIET based on \[18\]-annulene. Of course, benzene provides the smallest of all possible QuIETs. ![Emitter-collector lead configurations possible in a QuIET based on \[18\]-annulene. The bold lines represent the positioning of the two leads. Each of the four arrangements has a different phase difference associated with it: (a) $\pi$, (b) $3\pi$, (c) $5\pi$, and (d) $7\pi$.[]{data-label="18"}](18.eps){width="\columnwidth"} The position of the third lead affects the degree to which destructive interference is suppressed. For benzene, the most effective location for a third lead is shown in Fig. \[3rdlead\]a. The base may also be placed at the site immediately between the emitter and collector leads, as shown in Fig. \[3rdlead\]b. The QuIET operates in this configuration as well, although since base coupling to the current carriers is less, the transistor effect is somewhat suppressed. The third, three-fold symmetric configuration of leads (Fig.\[3rdlead\]c) completely decouples base from current carriers within benzene. Because of this, the base cannot be used to provide the decoherence or dephasing necessary to QuIET operation in this configuration. For each aromatic hydrocarbon, exactly one three-fold symmetric lead configuration exists and yields no transistor behavior. ![The three different arrangements for a third lead on benzene when the emitter and collector leads (bold) are in the meta configuration. (a) is the choice which couples most strongly to the conducting orbitals of benzene. (b) is a second case which allows for QuIET operation. The third possibility, (c), decouples entirely from the conducting molecular orbitals by symmetry. A third lead in this configuration cannot break the coherent current suppression at all, and so the molecule does not function as a QuIET.[]{data-label="3rdlead"}](leads.eps){width="\columnwidth"} Base Complex {#dba} ============ With the tunable current suppression outlined in the previous section, the working principle of a transistor has become apparent. The most straightforward method of varying $\Gamma_B$ is to change the distance of the third lead from the hydrocarbon molecule, as via an STM tip. It is also possible, however, to interpose an additional molecular complex between annulene and the base lead. If the effect of biasing the base lead is to increase the transparency of the molecular complex, dephasing and decoherence result. Such a QuIET is depicted schematically in Fig. \[transistor\]b. ![Specific example of a QuIET: a dithiol derivative of phenylene–TCNQ. E and C are the emitter and collector leads. The base lead can be either weakly coupled via TCNQ, as for a BJT analogue, or capacitively coupled if FET-like behavior is desired.[]{data-label="QuIET"}](QuIET.eps){width="\columnwidth"} A simple way to engineer the base complex is to include two spatially separated orbitals, with a detuning from each other $\Delta$ and hopping to each other $t_B$. If $\Delta\gg t_B$ for zero bias on the base lead, paths through the base complex are highly suppressed. Since the orbitals are spatially separate, however, one electrostatically couples more strongly to the base lead. Thus, the voltage on that lead can be used to control $\Delta$, and hence change the transparency of the base complex. The result is the broad conductance peak characteristic of two discrete levels coming into resonance. One way to achieve this structure is simply to link the annulene via an inert bridge to a donor or acceptor molecule, e. g.* the dithiol derivative of phenylene–TNCQ shown in Fig. \[QuIET\]. In this case, the effect of a base bias of appropriate sign is to bring the lowest unoccupied molecular orbital (LUMO) of TCNQ into resonance with the neighboring $\pi$ orbital of phenylene. The extra paths allowed within and through the TCNQ break the coherent suppression of current within the benzene ring. Any donor or acceptor molecule with which phenylene bonds in this way will yield similar results, although a donor used in this way will allow emitter-collector current to flow for negative, rather than positive, base voltages. The class of donor-acceptor rectifiers proposed by Aviram and Ratner [@aviram74] have this two-level structure as well. These molecules possess an orbital of increased electron affinity (the acceptor) and one of increased ionization potential (the donor), as well as perhaps an inert bridge orbital. By the influence of a base lead, the donor highest occupied molecular orbital (HOMO) and acceptor LUMO can be brought into resonance, and the rectifier be made to serve as an effective QuIET base complex. In addition to this effect, several many-body mechanisms have been proposed whereby such molecules could exhibit asymmetric current flow [@waldeck93]. For our purposes, however, it is sufficient to note that rectification in general is indicative of a tunable transparency to charge carriers. The variety within this well studied family of molecules (e. g., Refs. [@aviram74; @waldeck93; @geddes90; @metzger97; @metzger03; @heath03]) lends a great deal of versatility to the QuIET. Different choices of donor, acceptor, and bridge molecules allow QuIETs to be fabricated for specific applications. Table \[specs\] gives approximate ranges of parameters available for the use of molecular diodes as base complexes. Parameter Range ------------------------------------- ------------- Donor HOMO energy \[-8,-5\]eV Acceptor LUMO energy \[-3,-1\]eV $t_B$ \[.01,1\]eV $t'$ \[.01,1\]eV Distance between donor and acceptor \[6,10\]Å : Approximate ranges of values available for molecular diodes which could be used as QuIET base complexes.[]{data-label="specs"} Numerical Results and Discussion {#numerics} ================================ We turn now to discussion of numerical results, based on the self-consistent Hartree-Fock model presented in Section \[model\]. For the purposes of illustration, the numerical results are based on the dithiol derivative of phenylene–TCNQ shown in Fig. \[QuIET\] and three bulk gold leads. Similar results can be obtained for many such QuIETs. For the purposes of electrostatic interactions, each lead is considered to be immediately adjacent to its neighboring site. The electrostatic interactions between TCNQ and phenylene are modeled as those between two conducting spheres of appropriate size, while within phenylene the Ohno parameterization (\[interactions\]) is used. We neglect reduction of the $U_{ij}$ due to screening from a third site, an effect which we find to be of the order of 1%. In keeping with our assumption that the leads do not significantly perturb the molecular system, modifications to the interaction parameters due to the presence of the leads are also neglected. (1,2) (0,1)[![Typical $I-\mathcal{V}$ characteristic of two QuIETs, showing emitter current out of the molecule vs. voltage on the base lead for three different biases applied to the collector, with the emitter ground. The calculation is done for the molecule in Fig. \[QuIET\] at room temperature. $\overline{\Gamma}\equiv\frac{\Gamma_E\Gamma_C}{\Gamma_E+\Gamma_C}$ gives the sequential tunneling rate through the device. Here, $\Gamma_E=\Gamma_C=2.4\mathrm{eV}$. In (a), $\Gamma_B=.0024\mathrm{eV}$, and the QuIET amplifies current, similar to a BJT. The dotted curve is the base current into the molecule for the case of $\mathcal{V}_{CE}=.24\mathrm{V}$, multiplied by 10 for clarity. Base currents for other collector voltages are similar. (b) shows the FET limit of the QuIET, with $\Gamma_B=0$.[]{data-label="IV"}](BJT.eps "fig:"){width="\columnwidth"}]{} (0,0)[![Typical $I-\mathcal{V}$ characteristic of two QuIETs, showing emitter current out of the molecule vs. voltage on the base lead for three different biases applied to the collector, with the emitter ground. The calculation is done for the molecule in Fig. \[QuIET\] at room temperature. $\overline{\Gamma}\equiv\frac{\Gamma_E\Gamma_C}{\Gamma_E+\Gamma_C}$ gives the sequential tunneling rate through the device. Here, $\Gamma_E=\Gamma_C=2.4\mathrm{eV}$. In (a), $\Gamma_B=.0024\mathrm{eV}$, and the QuIET amplifies current, similar to a BJT. The dotted curve is the base current into the molecule for the case of $\mathcal{V}_{CE}=.24\mathrm{V}$, multiplied by 10 for clarity. Base currents for other collector voltages are similar. (b) shows the FET limit of the QuIET, with $\Gamma_B=0$.[]{data-label="IV"}](FET.eps "fig:"){width="\columnwidth"}]{} (-.01,1.9)[(a)]{} (-.01,.9)[(b)]{} A typical $I-\mathcal{V}$ diagram for this QuIET is shown in Fig. \[IV\], demonstrating that the QuIET is quite reminiscent in operation to classical transistors. The currents in the emitter and collector leads exhibit a broad resonance as the base voltage is increased. Furthermore, for nonzero $\Gamma_B$, the device amplifies the current in the base lead, providing emulation of the classical BJT. Since a donor, TCNQ, was used in the base complex, the device performs in a manner analogous to an NPN transistor, with base current flowing into the molecule. Use of an acceptor instead, for example TTF, yields PNP-like behavior. For capacitive coupling of the third lead, on the other hand, a FET-like $I-\mathcal{V}$ characteristic is obtained. We interpret this transistor behavior as due to the coherence mechanism discussed in Section \[ring\]. If hopping between the benzene ring and the base complex is set to zero, we find that full coherent current suppression is restored and almost no current flows between the emitter and collector leads. Furthermore, the current step effect persists for arbitrarily small $\Gamma_B$, which is consistent with our interpretation that transport through the non-canceling paths is enhanced by the electrostatic effect of the third lead. Estimation of the $\Gamma_i$’s remains an open question in the field of molecular electronics. Similar quantities are often estimated to be $\lesssim .5\mathrm{eV}$ [@nitzan01; @paulsson03] by the method of Ref.[@mujica94], whereas values as high as 1eV have been suggested [@tian98]. For the emitter and collector broadenings, we have taken values somewhat higher than the norm for the sake of numerical convergence. Fortunately, nothing in the arguments of Sections \[ring\] or \[dba\] depends strongly on the choice of these quantities. So long as they are positive, their magnitude merely determines the scale of the current, and has negligible effect on the overall operating principle and transistor behavior of the QuIET. Figure \[gammabar\] demonstrates the effect of varying the coupling of the benzene ring to emitter and collector leads. As is to be expected, the internal structure of the resonance, due to electrons moving to screen the changing field from the base, sharpens. Furthermore, at low base voltages, less than about .75V, the base complex is strongly off resonance, and the scale of the leakage current through the annulene is set mainly by $\overline{\Gamma}\equiv\frac{\Gamma_E\Gamma_C}{\Gamma_E+\Gamma_C}$, an expected result for systems like the QuIET, which are not near a charge fluctuation resonance [@stafford96]. ![$I_E$ vs. $\mathcal{V}_{BC}$ for different values of $\Gamma_E=\Gamma_C=\Gamma$. Each curve has been scaled so that the leakage current at zero bias is 1. $\Gamma_B=0$, $\mathcal{V}_{CE}=.24\mathrm{V}$, and positive current is out of the molecule. Reducing $\Gamma$ suppresses the leakage current, but does not strongly affect the QuIET’s behavior on-resonance, thus greatly enhancing its overall function.[]{data-label="gammabar"}](scaling.eps){width=".9\columnwidth"} For higher base voltages, the base complex begins to play an important role. Thus, as described by Ref. [@paulsson03], $\overline{\Gamma}$ alone no longer determines the scale of the current. Instead, we find that the rate-limiting process is travel through the base complex, and varying $\Gamma_E$ and $\Gamma_C$ has little effect. Thus, smaller emitter and collector energy widths enhance the QuIET’s current contrast dramatically. Conclusions {#conclusion} =========== We have presented a novel idea for a small single-molecule transistor. The device is based on the coherent suppression of current through an aromatic hydrocarbon ring, and the control of that effect by increasing the contribution of paths outside the ring. Two simple methods of creating that control, either through an STM tip or through a small base complex, were discussed, and numerical results presented. In contrast to SET-style devices, the QuIET is predicted to be an extremely versatile, scalable device. Since it is chemically constructed, it should be possible to fabricate as many identical QuIETs as desired. Furthermore, QuIETs can be designed which mimic the functionality of all major classes of macroscale transistors: FET, NPN, and PNP. Acknowledgments {#acknowledgments .unnumbered} =============== The authors acknowledge support from National Science Foundation Grant Nos. PHY0210750, DMR0312028, and DMR 0406604. [9]{} K. Ohno, Theor. Chim. Acta [2]{}, 219 (1964); M. Chandross et al., PRB [55]{}, 1486 (1997). C. A. Stafford, R. Kotlyar, and S. Das Sarma, PRB [ 58]{}, 7091 (1998). M. Büttiker, Phys. Rev. Lett. [57]{}, 1761 (1986). A.-P. Jauho, N. S. Wingreen, and Y. Meir, PRB [50]{}, 5528 (1994). J. M. Luttinger, Phys. Rev. [119]{}, 1153 (1960). S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge (1995). D. R. Lide, et al., eds., CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, Fla. (2005). A. Aviram and M. Ratner, Chem. Phys. Lett. [29]{}, 277 (1974). D. H. Waldeck and D. N. Beratan, Science [261]{}, 576 (1993). N. J. Geddes et al., App. Phys. Lett. [56]{}, 1916 (1990). R. M. Metzger et al., J. Am. Chem. Soc. [ 119]{}, 10455 (1997). R. M. Metzger et al., J. Phys. Chem. [B107]{}, 1021 (2003). J. R. Heath and M. A. Ratner, Physics Today [56]{}(5), 43 (2003). A. Nitzan, Annu. Rev. Phys. Chem. [52]{},681 (2001). M. Paulsson, F. Zahid, and S. Datta, in Handbook of Nanoscience, Engineering, and Technology, W. A. Goddard III et al., eds., CRC Press, Boca Raton, Fla. (2003). V. Mujica, M. Kemp, and M. A. Ratner, J. Chem. Phys. [ 101]{}, 6849 (1994). W. Tian et al., J. Chem. Phys. [109]{}, 2874 (1998). C. A. Stafford, Phys. Rev. Lett. [77]{}, 2770 (1996).
--- abstract: 'We study the superlinear oscillator equation $\ddot{x}+ \lvert x \rvert^{\alpha-1}x = p(t)$ for $\alpha\geq 3$, where $p$ is a quasi-periodic forcing with no Diophantine condition on the frequencies and show that typically the set of initial values leading to solutions $x$ such that $\lim_{t\to\infty} (\lvert x(t) \rvert + \lvert \dot{x}(t) \rvert) = \infty$ has Lebesgue measure zero, provided the starting energy $\lvert x(t_0) \rvert + \lvert \dot{x}(t_0) \rvert$ is sufficiently large.' author: - 'Henrik Schließauf [^1]' bibliography: - 'bibliography.bib' date: 'November 29, 2018' title: 'Escaping orbits are rare in the quasi-periodic Littlewood boundedness problem' --- Introduction ============ The dynamics of the Duffing-type equation $$\label{duffing eq} \ddot x + G'(x) = p(t),$$ have been studied extensively due to its relevance as a model for the motion of a classical particle in a one-dimensional potential field $G(x)$ affected by an external time-dependent force $p(t)$. In the 1960’s, Littlewood [@Littlewood_unbounded_solutions] asked whether solutions of (\[duffing eq\]) stay bounded in the $(x,\dot{x})$-phase space if either $$\begin{aligned} &(i) \; G'(x)/x \to +\infty \;\; \text{as} \;\; x \to \pm \infty \\ \text{or} \;\; &(ii) \; \operatorname{sign}(x) \cdot G'(x) \to +\infty \;\; \text{and} \;\; G'(x)/x \to 0 \;\; \text{as} \;\; x \to \pm \infty.\end{aligned}$$ Despite it’s harmless appearance, this question turned out to be a quite delicate matter. Whether some resonance phenomena occur, obviously does not only depend on the growth of $G$, but also on the properties of $p$ with respect to regularity and (quasi)-periodicity. The most investigated case is that of a time-periodic forcing $p$. The first affirmative contribution in that regard is due to Morris [@Morris_a_case_of_boundedness], who showed the boundedness of all solutions to $$\ddot x + 2x^3 = p(t),$$ where $p$ is continuous and periodic. Later, Dieckerhoff and Zehnder [@dieckerhoff_zehnder_boundedness_via_twisttheorem] were able to show the same for $$\ddot x + x^{2n+1} + \sum_{j=0}^{2n} p_j(t) x^{j} = 0,$$ where $n\in{\mathbb N}$ and $p_j \in \mathcal{C}^{\infty}$ are $1$-periodic. In the following years, this result was improved by several authors (see [@Bin_1989], [@Laederich_1991], [@levi_quasiperiodic_motions],[@Norris_1992],[@levi_zehnder_quasiperiodic] and the references therein). If however the periodicity condition is dropped, Littlewood [@Littlewood_unbounded_solutions] himself showed that for any odd potential $G$ satisfying the super-/sublinearity condition there exists a bounded forcing $p$ leading to at least one unbounded trajectory. Later, Ortega [@Ortega_2005] was able to prove in a more general context that for any given $\mathcal{C}^2$-potential one can find an arbitrarily small $p\in \mathcal{C}^\infty$ such that most initial conditions (in the sense of a residual set) correspond to unbounded solutions of (\[duffing eq\]). Even in the time-periodic case Littlewood [@Littlewood_unbounded_solutions_periodic_p] constructed $G \in \mathcal{C}^\infty$ and a periodic $p$ such that there is at least one unbounded solution. (Actually both [@Littlewood_unbounded_solutions] and [@Littlewood_unbounded_solutions_periodic_p] contain a computational mistake; see [@Levi_counterexample; @Long] for corrections.) Let us also mention [@Zharnitsky_1997], where Zharnitsky improved the latter result for the superlinear case such that the periodic $p$ can be chosen continuously. These counterexamples show that besides periodicity and regularity assumptions on $p$ an additional hypothesis on $G$ is needed if one hopes for boundedness of all solutions. Indeed, all positive results mentioned above suppose the monotone growth of $G'(x)/x$. This condition guarantees the monotonicity of the corresponding Poincaré map and thus enables the authors to use KAM theory.\ We also want to point out the related problem of the so called Fermi-Ulam “ping pong” [@Fermi_on_the_origin; @ulam1961]. The latter is a model for a particle bouncing elastically between periodically moving walls. In [@Laederich_1991], it was first shown that for sufficiently regular motions the velocity of the particle stays bounded for all time and many results followed ever after.\ In the last twenty years a wealth of works on the Littlewood boundedness problem has been published, including the sublinear, semilinear and other cases (see [@Kuepper_1999; @Li_2001; @Wang_2009; @Liu_2009] and the references therein for some examples). Since those are far too many to be presented here, let us focus on the superlinear oscillator equation $$\label{DGL 0} \ddot x + \lvert x \rvert^{\alpha-1}x = p(t),$$ where $\alpha \geq 3$. In [@levi_zehnder_quasiperiodic], Levi and Zehnder were able to show that for a quasi-periodic forcing $p$ all solutions are bounded, if the frequencies of $p$ satisfy a diophantine condition. In the present paper we shall omit this restriction on the frequencies and investigate the resulting long term behavior of solutions. But since in that case the tools related to invariant curve theorems are not available, one needs a different approach. In [@kunze_ortega_ping_pong], Kunze and Ortega presented a technique applicable in the above situation. Using a refined version of Poincaré’s recurrence theorem due to Dolgopyat [@dolgopyat], they proved that under appropriate conditions almost all orbits of a certain successor map $f$ are recurrent. In particular, they used this theorem to show that quasi-periodic forcing functions $p$ lead to recurrent orbits in the Fermi-Ulam “ping pong”. Here, we want to do the same for (\[DGL 0\]). If $x(t)$ denotes a solution to this equation, we consider the map $$\psi:(v_0,t_0)\mapsto(v_1,t_1),$$ which sends the time $t_0$ of a zero with negative derivative $v_0=\dot{x}(t_0)$ to the subsequent zero $t_1$ of this form and its corresponding velocity $v_1=\dot{x}(t_1)<0$. This map will be well defined for $\lvert v_0 \rvert$ sufficiently large, since in this case the corresponding solution oscillates quickly. (0 , -5.0000) (-0.0314 , -5.0000)– (-0.0628 , -4.9998)– (-0.0942 , -4.9996)– (-0.1257 , -4.9992)– (-0.1571 , -4.9988)– (-0.1885 , -4.9982)– (-0.2199 , -4.9975)– (-0.2513 , -4.9968)– (-0.2827 , -4.9958)– (-0.3141 , -4.9948)– (-0.3454, -4.9935)– (-0.3768, -4.9921)– (-0.4082 , -4.9906)– (-0.4395 , -4.9888)– (-0.4709 , -4.9868)– (-0.5022 , -4.9846)– (-0.5335 , -4.9822)– (-0.5648, -4.9795)– (-0.5961 , -4.9765)– (-0.6273 , -4.9732)– (-0.6586 ,-4.9696)– (-0.6898, -4.9656)– (-0.7210 , -4.9613)– (-0.7521 , -4.9565)– (-0.7832 ,-4.9514)– (-0.8143, -4.9458)– (-0.8454 , -4.9399)– (-0.8764 , -4.9335)– (-0.9074 , -4.9266)– (-0.9384, -4.9191)– (-0.9693 , -4.9111)– (-1.0002 , -4.9024)– (-1.0310 , -4.8931)– (-1.0617 , -4.8830)– (-1.0924, -4.8722)– (-1.1230, -4.8606)– (-1.1535, -4.8481)– (-1.1840, -4.8347)– (-1.2143, -4.8204)– (-1.2446, -4.8051)– (-1.2747, -4.7887)– (-1.3048, -4.7714)– (-1.3347, -4.7529)– (-1.3645, -4.7332)– (-1.3942, -4.7124)– (-1.4237, -4.6904)– (-1.4531, -4.6672)– (-1.4824, -4.6426)– (-1.5114, -4.6167)– (-1.5403, -4.5895)– (-1.5691, -4.5609)– (-1.5976, -4.5308)– (-1.6260, -4.4993)– (-1.6541, -4.4663)– (-1.6821, -4.4318)– (-1.7098, -4.3957)– (-1.7373, -4.3580)– (-1.7645, -4.3188)– (-1.7915, -4.2779)– (-1.8182, -4.2353)– (-1.8447, -4.1910)– (-1.8708, -4.1450)– (-1.8967, -4.0972)– (-1.9223, -4.0476)– (-1.9475, -3.9963)– (-1.9725, -3.9431)– (-1.9971, -3.8881)– (-2.0213, -3.8311)– (-2.0452, -3.7723)– (-2.0687, -3.7116)– (-2.0918, -3.6489)– (-2.1145, -3.5842)– (-2.1368, -3.5176)– (-2.1587, -3.4489)– (-2.1802, -3.3783)– (-2.2012, -3.3055)– (-2.2217, -3.2307)– (-2.2418, -3.1539)– (-2.2614, -3.0750)– (-2.2805, -2.9941)– (-2.2991, -2.9112)– (-2.3172, -2.8264)– (-2.3347, -2.7398)– (-2.3517, -2.6513)– (-2.3682, -2.5610)– (-2.3840, -2.4690)– (-2.3993, -2.3754)– (-2.4140, -2.2801)– (-2.4281, -2.1832)– (-2.4415, -2.0849)– (-2.4544, -1.9850)– (-2.4665, -1.8838)– (-2.4781, -1.7812)– (-2.4890, -1.6773)– (-2.4992, -1.5723)– (-2.5087, -1.4660)– (-2.5176, -1.3587)– (-2.5258, -1.2503)– (-2.5333, -1.1410)– (-2.5400, -1.0308)– (-2.5461, -0.9197)– (-2.5515, -0.8079)– (-2.5562, -0.6954)– (-2.5602, -0.5823)– (-2.5634, -0.4687)– (-2.5659, -0.3546)– (-2.5678, -0.2402)– (-2.5688, -0.1254)– (-2.5692, -0.0105)– (-2.5689, 0.1047)– (-2.5678, 0.2198)– (-2.5660, 0.3350)– (-2.5660, 0.3350) – (-2.5635,0.45)– (-2.5603,0.5648)– (-2.5564,0.6794)– (-2.5517,0.7935)– (-2.5464,0.9072)– (-2.5403,1.0202)– (-2.5336,1.1326)– (-2.5261,1.2441)– (-2.5179,1.3547)– (-2.5091,1.4644)– (-2.4996,1.573)– (-2.4894,1.6805)– (-2.4785,1.7868)– (-2.4669,1.8919)– (-2.4548,1.9957)– (-2.4419,2.0981)– (-2.4285,2.1991)– (-2.4144,2.2986)– (-2.3996,2.3966)– (-2.3843,2.493)– (-2.3684,2.5878)– (-2.3519,2.681)– (-2.3348,2.7724)– (-2.3171,2.8621)– (-2.2989,2.9501)– (-2.2801,3.0362)– (-2.2607,3.1206)– (-2.2409,3.2031)– (-2.2205,3.2838)– (-2.1996,3.3626)– (-2.1782,3.4395)– (-2.1564,3.5145)– (-2.134,3.5876)– (-2.1112,3.6589)– (-2.088,3.7282)– (-2.0643,3.7956)– (-2.0402,3.8612)– (-2.0157,3.9248)– (-1.9908,3.9866)– (-1.9655,4.0466)– (-1.9399,4.1047)– (-1.9139,4.161)– (-1.8875,4.2155)– (-1.8609,4.2682)– (-1.8339,4.3192)– (-1.8066,4.3686)– (-1.779,4.4162)– (-1.7511,4.4622)– (-1.7229,4.5066)– (-1.6945,4.5495)– (-1.6658,4.5908)– (-1.6368,4.6307)– (-1.6076,4.6692)– (-1.5781,4.7062)– (-1.5485,4.7419)– (-1.5186,4.7763)– (-1.4884,4.8094)– (-1.4581,4.8413)– (-1.4276,4.872)– (-1.3969,4.9016)– (-1.366,4.93)– (-1.335,4.9574)– (-1.3037,4.9837)– (-1.2723,5.009)– (-1.2408,5.0334)– (-1.2091,5.0569)– (-1.1772,5.0795)– (-1.1452,5.1012)– (-1.1131,5.1222)– (-1.0809,5.1425)– (-1.0485,5.162)– (-1.016,5.1808)– (-0.9834,5.199)– (-0.9507,5.2167)– (-0.9178,5.2337)– (-0.8849,5.2503)– (-0.8518,5.2664)– (-0.8187,5.2821)– (-0.7855,5.2973)– (-0.7521,5.3123)– (-0.7187,5.3269)– (-0.6852,5.3411)– (-0.6516,5.3551)– (-0.6179,5.3687)– (-0.5842,5.3822)– (-0.5503,5.3954)– (-0.5164,5.4084)– (-0.4824,5.4212)– (-0.4483,5.4339)– (-0.4141,5.4465)– (-0.3798,5.459)– (-0.3455,5.4714)– (-0.3111,5.4837)– (-0.2766,5.496)– (-0.242,5.5082)– (-0.2073,5.5204)– (-0.1726,5.5325)– (-0.1378,5.5447)– (-0.1029,5.5568)– (-0.0679,5.5689)– (-0.0329,5.581)– (0.0023,5.5932)– (0.0375,5.6053)– (0.0727,5.6174)– (0.1081,5.6295)– (0.1435,5.6416)– (0.179,5.6536)– (0.2146,5.6656)– (0.2502,5.6776)– (0.2859,5.6895)– (0.3217,5.7013)– (0.3576,5.7131)– (0.3935,5.7247)– (0.4295,5.7362)– (0.4656,5.7476)– (0.5017,5.7588)– (0.5379,5.7697)– (0.5742,5.7805)– (0.6105,5.791)– (0.6469,5.8013)– (0.6834,5.8112)– (0.7199,5.8208)– (0.7565,5.83)– (0.7931,5.8387)– (0.8299,5.8472)– (0.8666,5.8552)– (0.9034,5.8628)– (0.9403,5.8698)– (0.9772,5.8763)– (1.0142,5.882)– (1.0512,5.887)– (1.0882,5.8912)– (1.1252,5.8945)– (1.1623,5.8968)– (1.1993,5.8981)– (1.2364,5.8983)– (1.2735,5.8973)– (1.3105,5.8951)– (1.3476,5.8916)– (1.3846,5.8868)– (1.4216,5.8805)– (1.4585,5.8726)– (1.4954,5.8632)– (1.5322,5.8522)– (1.5689,5.8394)– (1.6056,5.8248)– (1.6421,5.8084)– (1.6786,5.7901)– (1.7149,5.7698)– (1.751,5.7475)– (1.7871,5.723)– (1.8229,5.6963)– (1.8586,5.6674)– (1.8941,5.6361)– (1.9294,5.6024)– (1.9645,5.5663)– (1.9994,5.5277)– (2.034,5.4865)– (2.0683,5.4426)– (2.1023,5.396)– (2.1361,5.3466)– (2.1695,5.2943)– (2.2026,5.2392)– (2.2353,5.181)– (2.2677,5.1198)– (2.2996,5.0554)– (2.3312,4.9879)– (2.3623,4.9171)– (2.3929,4.843)– (2.4231,4.7655)– (2.4528,4.6846)– (2.482,4.6001)– (2.5106,4.5121)– (2.5387,4.4207)– (2.5662,4.3257)– (2.5931,4.2274)– (2.6194,4.1256)– (2.6451,4.0205)– (2.67,3.9119)– (2.6943,3.8)– (2.7179,3.6849)– (2.7407,3.5665)– (2.7627,3.4449)– (2.784,3.3201)– (2.8045,3.1923)– (2.8241,3.0616)– (2.8429,2.9279)– (2.8609,2.7913)– (2.8779,2.6521)– (2.8941,2.5102)– (2.9094,2.3658)– (2.9238,2.2189)– (2.9372,2.0697)– (2.9497,1.9184)– (2.9612,1.765)– (2.9718,1.6097)– (2.9814,1.4525)– (2.99,1.2938)– (2.9976,1.1335)– (3.0042,0.972)– (3.0098,0.8093)– (3.0144,0.6455)– (3.0179,0.481); (3.0179,0.481)– (3.0204,0.3158)– (3.0219,0.1501)– (3.0223,-0.0158)– (3.0217,-0.1818)– (3.0201,-0.3476)– (3.0174,-0.5131)– (3.0137,-0.6781)– (3.009,-0.8423)– (3.0032,-1.0057)– 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{\mathbb N}$, we study the escaping set $$\label{def escaping set} E = \{ (v_0,t_0) : \lim_{n\to\infty} v_n = -\infty \}.$$ Especially, for any solution $x$ such that $\lim_{t\to\infty} (\lvert x(t) \rvert + \lvert \dot{x}(t) \rvert) = \infty$ there is a time $t_0$ with $(\dot{x}(t_0),t_0) \in E$. Now, let us state the main result. Note however, that due to the vague definition of $\psi$ this formulation of the theorem will still be somewhat imprecise. A rigorous definition of the successor map $\psi$ and its domain $(-\infty,v_)\times {\mathbb R}$ will be given only in the last section. We refer the reader to this part and Theorem \[main theorem\] below for the complete statement. \[main thm short\] Let $\bm{p}\in \mathcal{C}^4({\mathbb T}^N)$ generate the family of forcing functions $$\label{def quasiperiodic p} p_{\bar{\Theta}}(t)=\bm{p}(\overline{\theta_1+t\omega_1}, \ldots, \overline{\theta_N+t\omega_N}), \;\; \bar{\Theta}=(\bar\theta_1,\ldots,\bar\theta_N)\in{\mathbb T}^N,$$ for fixed rationally independent frequencies $\omega_1,\ldots,\omega_N>0$. Let $(v_n,t_n)_{n \in {\mathbb N}_0} = (\psi^n(v_0,t_0))_{n \in {\mathbb N}_0}$ denote a generic complete forward orbit associated to system (\[DGL 0\]) with $p$ replaced by $p_{\bar{\Theta}}$ and let $\mathcal{E}_{\bar{\Theta}}$ denote the corresponding escaping set (\[def escaping set\]). Then, for almost all $\bar{\Theta} \in {\mathbb T}^N$, the set $\mathcal{E}_{\bar{\Theta}}$ has Lebesgue measure zero. Here ${\mathbb T}$ stands for the torus ${\mathbb R}/ {\mathbb Z}$ and $\mathcal{C}^k({\mathbb T}^N)$ denotes the space of functions $\bm{p}\in \mathcal{C}^k({\mathbb R}^N)$ that are $1$-periodic in each argument. For $\alpha>3$ the same holds, if only $\bm{p} \in \mathcal{C}^2({\mathbb T}^N)$ is imposed. Just Theorem \[thm ham transformation\] of the second section requires higher regularity and the latter is needed only for $\alpha=3$. Let us give a short outline of the paper. As mentioned above, the theorem about escaping sets by Kunze and Ortega is the main tool used therein. It deals with certain quasi-periodic successor maps $(t_0,r_0) \mapsto (t_1,r_1)$ on ${\mathbb R}\times (0,\infty)$ and associated maps $f(\bar{\Theta}_0,r_0) = (\bar{\Theta}_1,r_1)$ on ${\mathbb T}^N \times (0,\infty)$. If one can find an adiabatic invariant-type function $W$, such that $$W(f(\bar{\Theta},r)) \leq W(\bar{\Theta},r) + k(r),$$ where $k:(0,\infty) \to {\mathbb R}$ is a decreasing and bounded function such that $\lim_{r\to\infty} k(r)=0$, then most orbits are recurrent. Section \[section escaping sets\] is dedicated to presenting the terminology and general setup needed in order to state this theorem. For a more detailed description as well as the proof, we refer the reader to [@kunze_ortega_ping_pong].\ In the subsequent section we basically follow [@kunze_ortega_stability_estimates] by developing three diffeomorphisms that transform the underlying equation (\[DGL 0\]) into a more convenient form suitable for the setup. First we restate the problem in terms of action-angle coordinates $(\bar{\vartheta},r)$ associated to the unperturbed system, as was already done by Morris [@Morris_a_case_of_boundedness]. Afterwards, we consider a transformation $\mathcal{S}$ by which the time $t=\phi$ and the old Hamiltonian become the new conjugate variables, whereas the old symplectic angle $\vartheta=\tau$ is chosen as the new independent variable. This trick was refined by Levi [@levi_quasiperiodic_motions] in the context of Littlewood’s boundedness problem. In the third part of this section, we present a canonical change of variables $\mathcal{T}$ developed by Kunze and Ortega [@kunze_ortega_stability_estimates] that preserves the structure of the Hamiltonian, while making the new momentum coordinate $\mathcal{I}$ an adiabatic invariant in the sense of the estimate above. Those transformations can be shortly illustrated as follows: $$(x,\dot x;t) \overset{\mathcal{R}}{\rightarrow} (\bar{\vartheta},r;t) \hookrightarrow (\vartheta,r;t) \overset{\mathcal{S}}{\rightarrow} (\phi,I;\tau) \overset{\mathcal{T}}{\rightarrow} (\varphi,\mathcal{I};\tau)$$ The estimate is confirmed in the next section, where we show that the time-$2\pi$ map $\Phi(\varphi_0,\mathcal{I}_0)$ of the resulting system defines an area-preserving successor map. Subsequently we prove that all three transformations retain the quasi-periodic structure if $p_{{\Theta}}(t)$ from (\[def quasiperiodic p\]) is taken as the forcing function. This links the developed set of coordinates $(\varphi,\mathcal{I})$ to the setup of the second section. Finally, the map $\psi$ is defined properly, leading to a restatement of the main result. We end the paper with a proof of this theorem, where it is shown that initial conditions in $\mathcal{E}_{\bar{\Theta}}$ correspond to non-recurrent orbits $(\varphi_n,\mathcal{I}_n)_{n\in{\mathbb N}}$. A theorem about escaping sets {#section escaping sets} ============================= In this section we want to state the aforementioned theorem about escaping sets of Kunze and Ortega. But first we need to introduce some basic notation and terminology. Measure-preserving embeddings ----------------------------- We will write ${\mathbb T}={\mathbb R}/ {\mathbb Z}$ for the standard torus and denote the class in ${\mathbb T}^N$ corresponding to a vector $\Theta=(\theta_1,\ldots,\theta_N) \in {\mathbb R}^N$ by $\bar{\Theta}=(\bar\theta_1,\ldots,\bar\theta_N)$, where $\bar\theta_j = \theta_j + {\mathbb Z}$. Similarly, $\bar{\vartheta}$ will indicate the class in $\mathbb{S}^1 = {\mathbb R}/ 2\pi{\mathbb Z}$ associated to some $\vartheta \in {\mathbb R}$. Denote by $\mu_{{\mathbb T}^N}$ the unique Haar measure such that $\mu_{{\mathbb T}^N}({\mathbb T}^N)=1$. From now on we will consider functions $$f:\mathcal{D}\subset {\mathbb T}^N\times(0,\infty) \to {\mathbb T}^N\times(0,\infty),$$ where $\mathcal{D}$ is an open set. We will call such a function measure-preserving embedding, if $f$ is continuous, injective and furthermore $$(\mu_{{\mathbb T}^N} \otimes \lambda) (f(\mathcal{B}))= (\mu_{{\mathbb T}^N} \otimes \lambda) (\mathcal{B})$$ holds for all Borel sets $\mathcal{B}\subset \mathcal{D}$, where $\lambda$ denotes the Lebesgue measure on ${\mathbb R}$. It is easy to show that under these conditions, $f:\mathcal{D} \to \tilde{\mathcal{D}}$ is a homeomorphism, where $\tilde{\mathcal{D}}=f(\mathcal{D})$.\ Since we want to use the iterations of $f$, we have to carefully construct a suitable domain on which these forward iterations are well-defined. We initialize $\mathcal{D}_1 = \mathcal{D}, \;\; f^1=f$ and set $$\mathcal{D}_{n+1}=f^{-1}(\mathcal{D}_{n}), \;\; f^{n+1}= f^n \circ f \; \text{ for } \; n \in {\mathbb N}.$$ This way $f^n$ is well-defined on $\mathcal{D}_n$. Clearly, $f^n$ is a measure-preserving embedding as well. Also, it can be shown inductively that $\mathcal{D}_{n+1}=\{(\bar{\Theta},r)\in\mathcal{D}:f(\bar{\Theta},r),\ldots,f^n(\bar{\Theta},r)\in \mathcal{D} \}$ and therefore $\mathcal{D}_{n+1}\subset \mathcal{D}_n \subset \mathcal{D}$ for all $n \in {\mathbb N}$. Initial conditions in the set $$\mathcal{D}_\infty = \bigcap\limits_{n=1}^\infty \mathcal{D}_n \subset {\mathbb T}^N\times(0,\infty)$$ correspond to complete forward orbits, i.e. if $(\bar{\Theta}_0,r_0)\in \mathcal{D}_\infty$, then $$(\bar{\Theta}_n,r_n)=f^n(\bar{\Theta}_0,r_0)$$ is defined for all $n \in {\mathbb N}$. It could however happen that $\mathcal{D}_\infty = \emptyset$ or $\mathcal{D}_{n} = \emptyset$ for some $n\geq 2$. Quasi-periodic functions {#subsection quasi-periodic functions} ------------------------ Let $\omega_1,\ldots,\omega_N>0$ be rationally independent and consider the map $$\iota:{\mathbb R}\to {\mathbb T}^N, \;\; \iota(t)=(\overline{t\omega_1},\ldots,\overline{t\omega_N}).$$ For $N>1$ this homomorphism is injective and the image $\iota({\mathbb R})\subset {\mathbb T}^N$ is dense. If $N=1$, then $\iota$ is surjective. Moreover, for a fixed $\bar{\Theta} \in {\mathbb T}^N$ we define the map $$\iota_{\bar{\Theta}}:{\mathbb R}\to {\mathbb T}^N, \;\; \iota_{\bar{\Theta}}(t)= \bar{\Theta} + \iota(t).$$ Let $\mathcal{C}^k({\mathbb T}^N)$ be the space of functions $\bm{u}:{\mathbb R}^N\to{\mathbb R}$, that are $1$-periodic in each argument and have continuous derivatives up to the $k$-th order. We will call a function $u:{\mathbb R}\to {\mathbb R}$ quasi-periodic (with frequency $\omega$), if there is a function $\bm{u} \in \mathcal{C}^0({\mathbb T}^N)$ such that $$u(t)=\bm{u}(\iota(t)) \;\; \text{for all} \;\; t\in {\mathbb R}.$$ Now, consider a measure-preserving embedding $f:\mathcal{D}\subset {\mathbb T}^N\times(0,\infty) \to {\mathbb T}^N\times(0,\infty)$, which has the special structure $$\label{embedding form} f(\bar{\Theta},r)=(\bar{\Theta}+\iota(F(\bar{\Theta},r)),r + G(\bar{\Theta},r)),$$ where $F,G:\mathcal{D}\to {\mathbb R}$ are continuous. For $\bar{\Theta} \in {\mathbb T}^N$ let $${D}_{\bar{\Theta}} = (\iota_{\bar{\Theta}} \times {\textrm{id}})^{-1}(\mathcal{D}) \subset {\mathbb R}\times (0,\infty).$$ On these open sets we define the maps $f_{\bar{\Theta}}: {D}_{\bar{\Theta}} \subset {\mathbb R}\times (0,\infty) \to {\mathbb R}\times (0,\infty)$ given by $$\label{planar maps} f_{\bar{\Theta}}(t,r)=(t+F(\bar{\Theta}+\iota(t),r),r+G(\bar{\Theta}+\iota(t),r)).$$ Then $f_{\bar{\Theta}}$ is continuous and meets the identity $$f \circ (\iota_{\bar{\Theta}} \times {\textrm{id}}) = (\iota_{\bar{\Theta}} \times {\textrm{id}}) \circ f_{\bar{\Theta}} \;\; \text{on} \;\; D_{\bar{\Theta}},$$ i.e. the following diagram is commutative: $$\begin{tikzcd} \mathcal{D} \arrow[r, "f"] & {\mathbb T}^N\times(0,\infty) \\ {D}_{\bar{\Theta}} \arrow[r, "f_{\bar{\Theta}}"] \arrow[u, "\iota_{\bar{\Theta}} \times {\textrm{id}}"] & {\mathbb R}\times (0,\infty) \arrow[u, "\iota_{\bar{\Theta}} \times {\textrm{id}}"] \end{tikzcd}$$ Therefore $f_{\bar{\Theta}}$ is injective as well. Again we define $D_{\bar{\Theta},1} = D_{\bar{\Theta}}$ and $D_{\bar{\Theta},n+1} = f_{\bar{\Theta}}^{-1}(D_{\bar{\Theta},n})$ to construct the set $$D_{\bar{\Theta}, \infty} = \bigcap\limits_{n=1}^{\infty} D_{\bar{\Theta},n} \subset {\mathbb R}\times (0,\infty),$$ where the forward iterates $(t_n,r_n)=f_{\bar{\Theta}}^n(t_0,t_0)$ are defined for all $n\in{\mathbb N}$. This set is equivalently defined through the relation $$\begin{aligned} D_{\bar{\Theta}, \infty} = (\iota_{\bar{\Theta}} \times {\textrm{id}})^{-1}(\mathcal{D}_\infty).\end{aligned}$$ Now we can define the escaping set* $${E_{\bar{\Theta}}}= \{(t_0,r_0)\in{D}_{{\bar{\Theta}},\infty}: \lim_{n\to\infty}r_n=\infty \}.$$ Finally we are in position to state the theorem [@kunze_ortega_ping_pong Theorem 3.1]: \[thm escaping set nullmenge\] Let $f:\mathcal{D}\subset {\mathbb T}^N \times (0,\infty) \to {\mathbb T}^N \times (0,\infty)$ be a measure-preserving embedding of the form (\[embedding form\]) and suppose that there is a function $W=W(\bar{\Theta},r)$ satisfying $W\in \mathcal{C}^1({\mathbb T}^N \times (0,\infty))$, $$0<\beta \leq {\partial}_r W(\bar{\Theta},r) \leq \delta \;\; \text{for} \;\; \bar{\Theta} \in {\mathbb T}^N, \;\; r \in (0,\infty),$$ with some constants $\beta,\delta>0$, and furthermore $$\label{ineq adiabatic inv} W(f(\bar{\Theta},r)) \leq W(\bar{\Theta},r) + k(r) \;\; \text{for} \;\; (\bar{\Theta},r)\in \mathcal{D},$$ where $k:(0,\infty) \to {\mathbb R}$ is a decreasing and bounded function such that $\lim_{r\to\infty} k(r)=0$. Then, for allmost all $\bar\Theta \in {\mathbb T}^N$, the set $E_{\bar{\Theta}}\subset {\mathbb R}\times (0,\infty)$ has Lebesgue measure zero. The function $W$ can be seen as a generalized adiabatic invariant, since any growth will be slow for large energies. Transformation of the problem {#section transformations} ============================= For $\alpha \geq 3$, consider the second order differential equation $$\label{DGL 1} \ddot{x} + \lvert x \rvert^{\alpha-1}x = p(t),$$ where $p\in \mathcal{C}_b^4({\mathbb R})$ is a (in general) non periodic forcing function. Here, $\mathcal{C}_b^k({\mathbb R})$ denotes the space of bounded functions with continuous and bounded derivatives up to order $k$. Solutions $x(t)$ to (\[DGL 1\]) are unique and exist for all time. To see this, we define $$E(t)= \frac{1}{2}\dot{x}(t)^2 + \frac{1}{\alpha +1}\lvert x(t) \rvert^{\alpha+1}.$$ Then $\lvert \dot{E} \rvert = \lvert p(t) \dot{x} \rvert \leq \lvert p(t) \rvert \sqrt{2E}$, and therefore $$\sqrt{E(t)} \leq \sqrt{E(t_0)} + \frac{1}{\sqrt{2}} \left\lvert \int_{t_0}^{t} \lvert p(s)\rvert ds \right\rvert.$$ So $E$ is bounded on finite intervals and thus $x$ can be continued on ${\mathbb R}$. Action-angle coordinates ------------------------ First we want to reformulate (\[DGL 1\]) in terms of the action-angle coordinates of the unperturbed system, that is $$\label{DGL unperturbed} \ddot{x} + \lvert x \rvert^{\alpha-1}x = 0.$$ The orbits of (\[DGL unperturbed\]) are closed curves, defined by $\frac{1}{2}y^2 + \frac{1}{\alpha +1}\lvert x \rvert^{\alpha+1}=\text{const.}$ and correspond to periodic solutions. For $\lambda>0$ let $x_\lambda$ denote the solution of (\[DGL unperturbed\]) having the initial values $$x_\lambda(0)=\lambda, \;\; \dot x_\lambda(0)=0.$$ Using the homogeneity of the problem, we get $x_\lambda(t)=\lambda x_1(\lambda^\frac{\alpha-1}{2} t)$. In particular $x_\lambda$ has a decreasing minimal period $T(\lambda) = \lambda^\frac{1-\alpha}{2} T(1)$. Thus we can find the unique number $\Lambda>0$, such that $T(\Lambda)=2\pi$. We will use the notation $$c(t) = x_{\Lambda}(t), \;\; s(t)=\dot c (t),$$ since in a lot of ways these functions behave like the trigonometric functions $\cos$ and $\sin$: $c$ is even, $s$ is odd, and both are anti-periodic with period $\pi$. Hence they have zero mean value, i.e. $$\int_{0}^{2\pi} c(t) \, dt = \int_{0}^{2\pi} s(t) \, dt = 0.$$ In this case however, $(c(t),s(t))$ spins clockwise around the origin of the $(x,\dot{x})$-plane. Furthermore $c$ and $s$ meet the identity $$\label{Identi} \frac{1}{2}s(t)^2 +\frac{1}{\alpha +1} \lvert c(t) \rvert^{\alpha+1} = \frac{1}{\alpha + 1} \Lambda^{\alpha + 1} \;\; \forall t \in {\mathbb R}.$$ Now we can define a change of variables $\eta: \mathbb{S}^1\times(0,\infty) \to {\mathbb R}^2\setminus\{0\}, \; (\bar{\vartheta},r)\mapsto (x,v)$ by $$x = \gamma r^\frac{2}{\alpha+3} c(\bar{\vartheta}), \;\; v = \gamma^\frac{\alpha+1}{2} r^\frac{\alpha+1}{\alpha +3} s(\bar{\vartheta}),$$ where $\gamma>0$ is determined by $$\gamma^\frac{\alpha+3}{2}\frac{2}{\alpha +3} \Lambda^{\alpha +1} = 1.$$ This choice of $\gamma$ makes $\eta$ a symplectic diffeomorphism, as can be shown by an easy calculation. Moreover, from (\[Identi\]) follows the identity $$\label{eq energy action} \frac{1}{2}v^2 +\frac{1}{\alpha +1} \lvert x \rvert^{\alpha+1} = \kappa_1 r^\frac{2(\alpha+1)}{\alpha+3},$$ where $\kappa_1= \frac{1}{\alpha+1}(\gamma\Lambda)^{\alpha+1}$. Adding a new component for the time, we define the transformation map $$\mathcal{R}:{\mathbb R}^2\setminus\{0\}\times {\mathbb R}\to \mathbb{S}^1\times(0,\infty) \times {\mathbb R}, \;\; \mathcal{R}(x,v;t)=(\eta^{-1}(x,v);t).$$ Going back to the perturbed system, the old Hamiltonian $$h (x,\dot x;t) = \frac{1}{2}\dot{x}^2 +\frac{1}{\alpha +1} \lvert x \rvert^{\alpha+1} - p(t)x$$ expressed in the new coordinates is $$\mathcal{H}(\bar{\vartheta},r ;t) = \kappa_1 r^\frac{2(\alpha+1)}{\alpha+3} - \gamma r^\frac{2}{\alpha+3} p(t) c(\bar{\vartheta}).$$ For simplicity’s sake let us denote the lift of $\mathcal{H}$ onto ${\mathbb R}\times (0,\infty) \times {\mathbb R}$ by the same letter $\mathcal{H}$. The associated differential equations then become $$\label{DGL 3} \begin{cases} \dot{{\vartheta}} &= {\partial}_r \mathcal{H} = \frac{2(\alpha+1)}{\alpha+3}\kappa_1 r^\frac{\alpha-1}{\alpha+3} - \frac{2}{\alpha+3}\gamma r^{-\frac{\alpha+1}{\alpha+3}} p(t) c({\vartheta}) \\ \dot r &= - {\partial}_{{\vartheta}} \mathcal{H} = \gamma r^\frac{2}{\alpha+3} p(t) s({\vartheta}) \end{cases}.$$ It should be noted that solutions to (\[DGL 3\]) only exist on intervals $J \subset {\mathbb R}$, where $r(t)>0$. Therefore, we can only make assertions about solutions of the original problem (\[DGL 1\]) defined on intervals, where $(x,\dot{x}) \neq 0$, when working with these action-angle coordinates. Time-energy coordinates ----------------------- In order to construct a measure preserving embedding one could take the Poincaré map of Hamiltonian system (\[DGL 3\]). However, to fit the setting of subsection \[subsection quasi-periodic functions\], this map would need to have the time (and thus the quasi-periodic dependence of the system) as the first variable. Therefore, we will follow [@levi_quasiperiodic_motions] and take the time $t$ as the new “position”-coordinate, the energy $\mathcal{H}$ as the new “momentum” and the angle $\vartheta$ as the new independent variable. \ Since the first term in (\[DGL 3\]) is dominant for $r\to \infty$ one can find $r_$ such that $$\label{Mono abs} {\partial}_r \mathcal{H}(\vartheta,r;t) \geq 1 \;\; \text{for all} \;\; r\geq r_.$$ \[remark constants\] The value of $r_$ depends upon $\alpha,\gamma, \kappa_1,\lVert c \rVert_{\mathcal{C}_b}$ and $\lVert p\rVert_{\mathcal{C}^4_b}$, where again $\gamma, \kappa_1,\lVert c \rVert_{\mathcal{C}_b}$ are uniquely determined by the choice of $\alpha$. We will call quantities depending only upon $\alpha$ and $\lVert p\rVert_{\mathcal{C}^4_b}$ constants. Let us also point out, that $r_$ can be chosen “increasingly in $\lVert p\rVert_{\mathcal{C}^4_b}$”. By this we mean, that if $r_=r_(\alpha,\lVert p\rVert_{\mathcal{C}^4_b})$ is the threshold corresponding to some $p \in \mathcal{C}^4_b({\mathbb R})$, then (\[Mono abs\]) also holds for any forcing $\tilde{p}\in \mathcal{C}^4_b$ with $\lVert \tilde{p} \rVert_{\mathcal{C}^4_b} \leq \lVert p\rVert_{\mathcal{C}^4_b}$. Indeed, all thresholds we will construct have this property. Now consider a solution $(\vartheta,r)$ of (\[DGL 3\]) defined on an interval $J$, where $r(t)>r_$ for all $t \in J$. Than the function $t\mapsto \vartheta(t)$ is invertible, since $$\dot\vartheta (t) = {\partial}_r \mathcal{H}(\vartheta(t),r(t);t) \geq 1.$$ Adopting the notation of [@kunze_ortega_stability_estimates], we will write $\tau=\vartheta(t)$ and denote the inverse by $\phi$, i.e. $\phi(\tau)=t$. Since $\vartheta(t)$ is at least of class $\mathcal{C}^2$, the same holds for the inverse function $\phi$ defined on $\vartheta(J)$. Let us now define $$I(\tau)=\mathcal{H}(\tau,r(\phi(\tau));\phi(\tau)) \;\; \text{for} \; \; \tau \in \vartheta(J).$$ This function will be the new momentum. It is a well known fact that the resulting system is again Hamiltonian. To find the corresponding Hamiltonian, we can solve the equation $$\mathcal{H}(\vartheta,H;t)=I$$ implicitly for $H(t,I;\vartheta)$. Because of (\[Mono abs\]) this equation admits a solution, which is well-defined on the open set $$\Omega=\{(t,I;\vartheta)\in {\mathbb R}^3 : I > \mathcal{H}(\vartheta, r_ ;t)\}.$$ Indeed, by implicit differentiation it can be verified that $$\phi'={\partial}_I H, \;\; I'=-{\partial}_\phi H,$$ where the prime $'$ indicates differentiation with respect to $\tau$. Using the new coordinates, we have to solve $$\label{Impl Ham 1} \kappa_1 H^\frac{2(\alpha+1)}{\alpha+3} - \gamma H^\frac{2}{\alpha+3} p(\phi) c(\tau) = I$$ or equivalently $$\label{Impl Ham 2} H = I^\frac{\alpha+3}{2(\alpha+1)}\kappa_1^{-\frac{\alpha+3}{2(\alpha+1)}} (1- \kappa_1^{-1} \gamma H^\frac{-2\alpha}{\alpha+3} p(\phi) c(\tau))^{-\frac{\alpha+3}{2(\alpha+1)}}.$$ Since $p\in \mathcal{C}^6$ and $c\in \mathcal{C}^3$, also $H$ will be of class $\mathcal{C}^3$. Moreover, we can find $I_>0$ (depending upon $\alpha,\gamma, \kappa_1,\lVert c \rVert_{\mathcal{C}_b},\lVert p\rVert_{\mathcal{C}_b}$ and $r_$) such that $$\{(\phi,I;\tau)\in {\mathbb R}^3 : I\geq I_* \} \subset \Omega.$$ Furthermore, we can choose $I_$ so large that the solution $H$ of (\[Impl Ham 1\]) satisfies $$\alpha_0 I^\frac{\alpha+3}{2(\alpha+1)} \leq H \leq \beta_0 I^\frac{\alpha+3}{2(\alpha+1)} \;\; \text{for} \;\; I\geq I_$$ for some constants $\alpha_0,\beta_0>0$. Let $$\kappa_0=\kappa_1^{-\frac{\alpha+3}{2(\alpha+1)}}=\left(\frac{2(\alpha+1)}{\alpha+3}\gamma^\frac{1-\alpha}{2}\right)^\frac{\alpha+3}{2(\alpha+1)}.$$ To approximate the solution $H(\phi,I;\tau)$ of (\[Impl Ham 2\]), one can use the Taylor polynomial of degree one for $(1-z)^{-\frac{\alpha+3}{2(\alpha+1)}}$ and then plug in the highest order approximation $\kappa_0 I^\frac{\alpha+3}{2(\alpha+1)}$ for the remaining $H$ on the right-hand side. Therefore we define the remainder function $R \in \mathcal{C}^3(G)$ through the relation $$\label{Ham 3} H(\phi,I;\tau)=\kappa_0 I^\frac{\alpha+3}{2(\alpha+1)} + \frac{(\alpha+3)}{2(\alpha+1)}\gamma \kappa_0^\frac{\alpha+5}{\alpha + 3} p(\phi) c(\tau) I^\frac{3-\alpha}{2(\alpha+1)} + R(\phi,I;\tau).$$ The corresponding system is described by $$\begin{cases} \label{Ham 3 equations} \phi' &= {\partial}_I H = \kappa_0 \frac{\alpha+3}{2(\alpha+1)} I^\frac{1-\alpha}{2(\alpha+1)} + \frac{9-\alpha^2}{4(\alpha+1)^2}\gamma \kappa_0^\frac{\alpha+5}{\alpha+3} p(\phi)c(\tau) I^\frac{1-3\alpha}{2(\alpha+1)} + {\partial}_I R, \\ I' &= -{\partial}_\phi H = -\frac{\alpha+3}{2(\alpha+1)} \gamma \kappa_0^\frac{\alpha+5}{\alpha+3} \dot p(\phi)c(\tau) I^\frac{3-\alpha}{2(\alpha+1)} - {\partial}_\phi R. \end{cases}$$ The change of variables $(\vartheta,r;t)\mapsto(\phi,I;\tau)$ can be realized via the transformation map $\mathcal{S}:{\mathbb R}\times [r_,\infty) \times {\mathbb R}\to {\mathbb R}\times (0,\infty) \times {\mathbb R}$ defined by $$\mathcal{S}(\vartheta,r;t) = (t,\mathcal{H}(\vartheta,r;t);\vartheta).$$ So $\mathcal{S}$ maps a solution $(\vartheta(t),r(t))$ of (\[DGL 3\]) with the initial condition $(\vartheta(t_0),r(t_0))=(\vartheta_0,r_0)$ onto a solution $(\phi(\tau),I(\tau))$ of (\[Ham 3 equations\]) with initial condition $(\phi(\vartheta_0),I(\vartheta_0))=(t_0,\mathcal{H}(\vartheta_0,r_0;t_0))$.\ The following lemma by Kunze and Ortega [@kunze_ortega_stability_estimates Lemma 7.1] shows that $R$ is small in a suitable sense: \[lem remainder\] There are constants $C_0>0$ and $I_{C_0} \geq I_>0$ (depending upon $\lVert p \rVert_{\mathcal{C}_b^2({\mathbb R})}$) such that $$\label{Remainder Abs} \lvert R \rvert + \lvert {\partial}_\phi R \rvert + I\lvert {\partial}_I R \rvert + \lvert {\partial}^2_{\phi\phi} R \rvert + I\lvert {\partial}^2_{\phi I} R \rvert + I^2\lvert {\partial}^2_{II} R \rvert \leq C_0 I^\frac{3(1-\alpha)}{2(\alpha+1)}$$ holds for all $\phi,\tau \in {\mathbb R}$ and $I\geq I_{C_0}$. Now we could use these coordinates and a corresponding Poincaré map for Theorem \[thm escaping set nullmenge\]. But since it can be tricky to find a suitable function $W$, we would like the energy variable $I(\tau)$ itself to be an adiabatic invariant in the sense of (\[ineq adiabatic inv\]) . However, for $\alpha = 3$ we do not have $I'\to0$ as $I\to \infty$. Therefore we have to do one further transformation. For $\alpha > 3$ this last step would not be necessary. A last transformation --------------------- In [@kunze_ortega_stability_estimates Theorem 6.7] Ortega and Kunze constructed a change of coordinates, which reduces the power of the momentum variable in the second term of (\[Ham 3\]) while preserving the special structure of the Hamiltonian. Since in their paper they had to use this transformation several times consecutively, the associated theorem is somewhat general and too complicated for our purpose here. Thus we will cite it only in the here needed form.\ For $\mu>0$ we set $$\Sigma_{\mu}={\mathbb R}\times [\mu,\infty) \times {\mathbb R}.$$ \[thm ham transformation\] Consider the Hamiltonian $H$ from (\[Ham 3\]), i.e. $$H(\phi,I;\tau)=\kappa_0 I^\frac{\alpha+3}{2(\alpha+1)} + f(\phi) c(\tau) I^\frac{3-\alpha}{2(\alpha+1)} + R(\phi,I;\tau),$$ where $f(\phi)= \frac{(\alpha+3)}{2(\alpha+1)}\gamma \kappa_0^\frac{\alpha+5}{\alpha + 3} p(\phi)$, and $I_{C_0}$ from Lemma \[lem remainder\]. Then there exists $I_{*}>I_{C_0}$, $\mathcal{I}_>0$ and a $\mathcal{C}^1$-diffeomorphism $$\mathcal{T}:\Sigma_{I_{}} \to \mathcal{T}(\Sigma_{I_{}} ) \subset \Sigma_{\mathcal{I}_} , \;\; (\phi,I;\tau)\mapsto (\varphi,\mathcal{I};\tau),$$ which transforms the system (\[Ham 3 equations\]) into $\varphi'={\partial}_\mathcal{I}H_1, \mathcal{I}'=-{\partial}_\varphi H_1$, where $$H_1(\varphi,\mathcal{I};\tau) = \kappa_0 \mathcal{I}^\frac{\alpha+3}{2(\alpha+1)} + f_1(\varphi)c_1(\tau)\mathcal{I}^{b_\alpha} + R_1(\varphi,\mathcal{I};\tau).$$ The new functions appearing in $H_1$ satisfy 1. $f_1(\varphi)= -\frac{\alpha+3}{2(\alpha+1)}\kappa_0 \dot f(\varphi) = -\left(\frac{\alpha+3}{2(\alpha+1)}\right)^2 \gamma \kappa_0^\frac{2\alpha+8}{\alpha+3} \dot p (\varphi)$, 2. $c_1 \in \mathcal{C}^4({\mathbb R}), c_1'(\tau)=c(\tau), \int_{0}^{2\pi} c_1(\tau) \, d\tau=0,$ 3. $b_\alpha= -\frac{3\alpha^2 - 2 \alpha - 9}{2(\alpha +3)(\alpha+1)} < \frac{3-\alpha}{2(\alpha+1)} \leq 0$, and 4. $R_1 \in \mathcal{C}^{3}(\Sigma_{\mathcal{I}_{}})$ satisfies (\[Remainder Abs\]) for all $\mathcal{I}\geq \mathcal{I}_$ and with some constant $\tilde{C}_0>0$. The quantities $I_{},\mathcal{I_{}}$ and $\tilde{C}_0$ can be estimated in terms of $\alpha,\kappa_0, \lVert f \rVert_{\mathcal{C}_b^4({\mathbb R})} , \lVert c \rVert_{\mathcal{C}_b({\mathbb R})}$, and $C_0$ from Lemma \[lem remainder\]. Furthermore, the change of variables $\mathcal{T}$ has the following properties: 1. $\mathcal{T}(\cdot,\cdot;\tau)$ is symplectic for all $\tau \in {\mathbb R}$, i.e. $d\varphi \wedge d\mathcal{I} = d\phi \wedge dI$, 2. $\mathcal{T}(\phi,I;\tau+2\pi) = \mathcal{T}(\phi,I;\tau) + (0,0;2\pi)$, and 3. $I/2 \leq \mathcal{I}(\phi,I;\tau) \leq 2I$ for all $(\phi,I;\tau)$. Even if we omit the proof here, let us note that the change of variables can be realized via the generating function $$\Psi(\phi,\mathcal{I}; \tau) = -\mathcal{I}^\frac{3-\alpha}{2(\alpha+1)} f(\phi) c_1(\tau),$$ where $c_1$ is uniquely determined by the conditions in $(b)$. Therefore $\mathcal{T}$ is implicitly defined by the equations $$I = \mathcal{I} + {\partial}_\phi \Psi, \;\; \varphi = \phi + {\partial}_\mathcal{I} \Psi$$ and one can determine $H_1$ through the relation $$H_1(\varphi,\mathcal{I};\tau) = H(\phi,I;\tau) + {\partial}_{\tau} \Psi(\phi,\mathcal{I};\tau).$$ The successor map {#section successor map} ================= Consider the new Hamiltonian from Theorem \[thm ham transformation\], that is $$H_1(\varphi,\mathcal{I};\tau) = \kappa_0 \mathcal{I}^\frac{\alpha+3}{2(\alpha+1)} + f_1(\varphi)c_1(\tau)\mathcal{I}^{b_\alpha} + R_1(\varphi,\mathcal{I};\tau),$$ which is well-defined on the set $\mathcal{T}(\Sigma_{I_{}})$ and $2\pi$-periodic in the time variable $\tau$. The corresponding equations of motion are $$\begin{aligned} \label{Ham 4 equations} \begin{cases} \varphi' &= {\partial}_{\mathcal{I}} H_1 = \kappa_0\frac{\alpha+3}{2(\alpha+1)}\mathcal{I}^{\frac{1-\alpha}{2(\alpha+1)}} +b_\alpha f_1(\varphi)c_1(\tau)\mathcal{I}^{b_\alpha -1} + {\partial}_\mathcal{I} R_1, \\ \mathcal{I}'&= -{\partial}_\varphi H_1 = -\dot f_1 (\varphi) c_1(\tau)\mathcal{I}^{b_\alpha} - {\partial}_\varphi R_1, \end{cases}\end{aligned}$$ where $\dot f_1 (\varphi) = - \left(\frac{\alpha+3}{2(\alpha+1)}\right)^2 \kappa_0^\frac{2\alpha+8}{\alpha+3} \gamma \ddot{p} (\varphi)$.\ Now suppose $(\varphi_0,\mathcal{I}_0;\tau_0)\in \mathcal{T}(\Sigma_{I_{}})$ and denote by $(\varphi(\tau;\varphi_0,\mathcal{I}_0,\tau_0),\mathcal{I}(\tau;\varphi_0,\mathcal{I}_0,\tau_0))$ the solution of (\[Ham 4 equations\]) with the initial data $$\varphi(\tau_0)= \varphi_0, \;\; \mathcal{I}(\tau_0)= \mathcal{I}_0.$$ We want to construct a subset $\Sigma_{\mathcal{I}_{}} = {\mathbb R}\times [\mathcal{I}_{},\infty) \times {\mathbb R}\subset \mathcal{T}(\Sigma_{I_{}})$ such that $(\varphi,\mathcal{I})$ is defined on the whole interval $[\tau_0,\tau_0+2\pi]$ whenever $(\varphi_0,\mathcal{I}_0,\tau_0)\in \Sigma_{\mathcal{I}_{}}$. Similar to [@kunze_ortega_stability_estimates Lemma 4.1], we state: \[lem I abs\] There exists a constant $\mathcal{I_{*}}> \mathcal{I}_$ (depending only upon $\alpha, \lVert f \rVert_{\mathcal{C}_b^4({\mathbb R})} , \lVert c \rVert_{\mathcal{C}_b({\mathbb R})}$ and $\tilde{C}_0$ from Theorem \[thm ham transformation\]) such that $\Sigma_{\mathcal{I}_{}} \subset \mathcal{T}(\Sigma_{I_{}})$ and for any $(\varphi_0,\mathcal{I}_0,\tau_0) \in \Sigma_{\mathcal{I}_{}} $ the solution $(\varphi,\mathcal{I})$ of (\[Ham 4 equations\]) with initial data $$\varphi(\tau_0)= \varphi_0, \;\; \mathcal{I}(\tau_0)= \mathcal{I}_0$$ exists on $[\tau_0,\tau_0+2\pi]$, where it satisfies $$\label{abs I aus lemma existenzint} \frac{\mathcal{I}_0}{4} \leq \mathcal{I}(\tau) \leq 4 \mathcal{I}_0 \;\; \text{for} \;\; \tau \in [\tau_0,\tau_0+2\pi].$$ Suppose $\mathcal{I}_0 \geq \mathcal{I}_{}\geq 4 \mathcal{I}_{}$, then $(iii)$ from Theorem \[thm ham transformation\] yields $\Sigma_{\mathcal{I}_{}} \subset \mathcal{T}(\Sigma_{I_{}})$. Now, let $T>0$ be maximal such that $\mathcal{I}_0/4 \leq \mathcal{I}(\tau) \leq 4 \mathcal{I}_0$ holds for all $\tau \in [\tau_0,\tau_0 + T)$. On this interval we have $$(\mathcal{I}^{1-b_\alpha})'=(1-b_\alpha)\mathcal{I}^{-b_\alpha}\mathcal{I}' = (1-b_\alpha)\mathcal{I}^{-b_\alpha}(-\dot f_1 (\varphi) c_1(\tau)\mathcal{I}^{b_\alpha} - {\partial}_\varphi R_1)$$ and thus $$\lvert (\mathcal{I}^{1-b_\alpha})' \rvert \leq \lvert (1-b_\alpha) \rvert \left( \lVert \dot f_1 \rVert_{\mathcal{C}_b} \; \lVert c_1 \rVert_{\mathcal{C}_b} +\lvert {\partial}_\varphi R_1 \rvert \mathcal{I}^{-b_\alpha} \right) \leq \lvert (1-b_\alpha) \rvert \left( \lVert \dot f_1 \rVert_{\mathcal{C}_b} \; \lVert c_1 \rVert_{\mathcal{C}_b} + \tilde{C}_0 \right) = \hat{C},$$ with $\tilde{C}_0>0$ from $(d)$ of Theorem \[thm ham transformation\], since $b_\alpha = -\frac{3\alpha^2 - 2 \alpha - 9}{2(\alpha +3)(\alpha+1)} > \frac{3(1-\alpha)}{2(\alpha+1)}$. Now assume $T\leq 2\pi$, then for $\mathcal{I}_{}$ sufficiently large we conclude $$\left(\frac{\mathcal{I}_0}{2}\right)^{1-b_\alpha} \leq \mathcal{I}_0^{1-b_\alpha} - 2\pi \hat{C} \leq \mathcal{I}(\tau)^{1-b_\alpha} \leq \mathcal{I}_0^{1-b_\alpha} + 2\pi \hat{C} \leq (2\mathcal{I}_0)^{1-b_\alpha}$$ on the whole interval $[\tau_0,\tau_0 + T)$. This contradicts the definition of $T$ and thus completes the proof. We can therefore consider the Poincaré map $\Phi:{\mathbb R}\times [\mathcal{I}_{},\infty) \to {\mathbb R}^2$ corresponding to the periodic system (\[Ham 4 equations\]), defined by $$\label{Def successor} \Phi(\varphi_0,\mathcal{I}_0) = (\varphi(5\pi/2;\varphi_0,\mathcal{I}_0,\pi/2),\mathcal{I}(5\pi/2;\varphi_0,\mathcal{I}_0,\pi/2)).$$ The choice $\tau_0=\frac{\pi}{2}$ is basically due to computational advantages, since $c(\vartheta)=0$ if and only if $\vartheta = \pi/2 + m\pi$ with $m \in {\mathbb Z}$. Moreover, values of $\tau = \vartheta$ in $\pi/2 + 2\pi {\mathbb Z}$ correspond exactly to those zeros of the solution $x(t)$, where $\dot{x}<0$. $$\Phi(\varphi_0,\mathcal{I}_0) = (\varphi_1,\mathcal{I}_1) .$$ Now that we have defined a suitable successor map, we can prove that $\mathcal{I}$ is an adiabatic invariant in the sense of equation (\[ineq adiabatic inv\]): \[lem adiabatic invariant\] There is a constant $C>0$ (depending only upon $\alpha, \lVert f \rVert_{\mathcal{C}_b^4({\mathbb R})} , \lVert c \rVert_{\mathcal{C}_b({\mathbb R})}$ and $\tilde{C}_0$) such that $$\lvert \mathcal{I}_1 - \mathcal{I}_0 \rvert \leq C \mathcal{I}_0^{b_\alpha}$$ holds for all $(\varphi_0,\mathcal{I}_0) \in {\mathbb R}\times [\mathcal{I}_{},\infty)$. With a similar reasoning like in the proof of Lemma \[lem I abs\] we get $$\begin{aligned} \lvert \mathcal{I}' (\tau) \rvert &= \lvert -\dot f_1 (\varphi) c_1(\tau)\mathcal{I}^{b_\alpha}(\tau) - {\partial}_\varphi R_1 \rvert \leq \lVert \dot f_1 \rVert_{\mathcal{C}_b} \; \lVert c_1 \rVert_{\mathcal{C}_b} \mathcal{I}(\tau)^{b_\alpha} + \tilde{C}_0 \mathcal{I}(\tau)^{ \frac{3(1-\alpha)}{2(\alpha+1)}} \\ &\leq \left( \lVert \dot f_1 \rVert_{\mathcal{C}_b} \; \lVert c_1 \rVert_{\mathcal{C}_b} + \tilde{C}_0 \right)\mathcal{I}(\tau)^{b_\alpha} \leq \left( \lVert \dot f_1 \rVert_{\mathcal{C}_b} \; \lVert c_1 \rVert_{\mathcal{C}_b} + \tilde{C}_0 \right) 4^{-b_\alpha} \mathcal{I}_0^{b_\alpha}. \end{aligned}$$ Now integrating over $[\pi/2,5\pi/2]$ gives us $$\lvert \mathcal{I}_1 - \mathcal{I}_0 \rvert \leq 2\pi \left( \lVert \dot f_1 \rVert_{\mathcal{C}_b} \; \lVert c_1 \rVert_{\mathcal{C}_b} + \tilde{C}_0 \right) 4^{-b_\alpha} \mathcal{I}_0^{b_\alpha}.$$ Quasi-periodicity {#section quasi periodicity} ================= So far all our considerations have dealt with the case of a general forcing function $p \in \mathcal{C}^{4}_b({\mathbb R})$. Now we will replace $p(t)$ by $p_{\bar{\Theta}}(t)$ from (\[def quasiperiodic p\]) and show that the quasi-periodicity is inherited by the Hamiltonian system (\[Ham 4 equations\]). But first let us clarify some notation.\ In this section, we will mark continuous functions with an argument in ${\mathbb T}^N$ with bold letters. Each such function $\bm{u}$ gives rise to a family of quasi-periodic maps $\{u_{\bar{\Theta}}\}_{\bar{\Theta}\in {\mathbb T}^N}$ via the relation $$u_{\bar{\Theta}}(t)=\bm{u}(\iota_{\bar{\Theta}}(t)).$$ Note that since $\iota_{\bar{\Theta}}({\mathbb R})$ lies dense in ${\mathbb T}^N$, the function $\bm{u}\in \mathcal{C}({\mathbb T}^N)$ is also uniquely determined by this property. Moreover, for $\bm{u}\in \mathcal{C}^1({\mathbb T}^N)$ we introduce the notation ${\partial}_\omega = \sum_{i=1}^{N} \omega_i \frac{{\partial}}{{\partial}\theta_i}$, so that $$\frac{d}{d t} u_{\bar{\Theta}}(t)= {\partial}_\omega \bm{u}(\iota_{\bar{\Theta}}(t)).$$ So let us plug the forcing $p_{\bar{\Theta}}(t)$ associated to $\bm{p}$ from the main theorem into (\[DGL 1\]). Then $p_{\bar{\Theta}} \in \mathcal{C}^4_b({\mathbb R})$, because $$\lVert p_{\bar{\Theta}} \rVert_{\mathcal{C}_b^4({\mathbb R})} \leq \max(1, \lVert \omega \rVert^4_\infty) \lVert \bm{p} \rVert_{\mathcal{C}^4({\mathbb T}^N)}$$ holds. Therefore all results of section \[section transformations\] are applicable. Considering Remark \[remark constants\], this also implies that we can find new constants $r_, I_$ etc. depending only upon $\alpha, \omega$ and $\lVert \bm{p} \rVert_{\mathcal{C}^4({\mathbb T}^N)}$ such that corresponding estimates hold uniformly in $\bar{\Theta} \in {\mathbb T}^N$.\ Since $\mathcal{R}$ and $\mathcal{S}$ basically leave the time variable $t$ unchanged, it is straightforward to prove that the transformation to action-angle coordinates as well as the change to the time-energy coordinates $(\phi,I)$, preserves the quasi-periodic structure (cf. [@levi_zehnder_quasiperiodic p. 1242]). Thus we find functions $$\bm{H},\bm{R}:{\mathbb T}^N \times [I_,\infty) \times {\mathbb R}\to {\mathbb R}\times [I_,\infty) \times {\mathbb R}$$ of class $\mathcal{C}^{3}$ depending on $\bm{p}$ such that $$H(\phi,I;\tau) = \bm{H}(\iota_{\bar{\Theta}}(\phi),I;\tau), \;\; R(\phi,I;\tau) = \bm{R}(\iota_{\bar{\Theta}}(\phi),I;\tau).$$ holds for every $\bar{\Theta}\in {\mathbb T}^N$. Let us note, that the functions $H,R$ etc. now depend on the choice of ${\bar{\Theta}}$. Thus it would be more precise to write $H_{\bar{\Theta}},R_{\bar{\Theta}}$ and so on, but for reasons of clarity we will omit the index throughout this section. The functions $\bm{H},\bm{R}$ etc. on the other hand are uniquely determined by $\bm{p}$. However it requires a bit more work, to see that also the transformation $\mathcal{T}$ defined in Theorem \[thm ham transformation\] retains the quasi-periodic properties. We recall that this change of variables is defined by $$I = \mathcal{I} + {\partial}_\phi \Psi, \;\; \varphi = \phi + {\partial}_\mathcal{I} \Psi,$$ where $\Psi(\phi,\mathcal{I};\tau) = - \mathcal{I}^\frac{3-\alpha}{2(\alpha+1)}f(\phi)c_1(\tau)$. Let $\bm{f} \in \mathcal{C}^4({\mathbb T}^N)$ be the function one obtains by replacing $p$ in the definition of $f$ by $\bm{p}$. Inspired by [@Moser_siegel_celmechanics p. 261], we can prove the following lemma: \[lem q quasip\] Suppose $\mathcal{I}_{} \geq 2\pi\max (1, \lVert \omega \rVert^4_\infty)\lVert \bm{f} \rVert_{\mathcal{C}^4({\mathbb T}^N)} \lVert c \rVert_{\mathcal{C}_b({\mathbb R})}$ and consider $$\varphi(\phi,\mathcal{I};\tau) = \phi - \frac{3-\alpha}{2(\alpha+1)} \mathcal{I}^{\frac{1-3\alpha}{2(\alpha+1)}} c_1(\tau) f(\phi),$$ then for $\mathcal{I}\geq \mathcal{I}_{},\tau \in {\mathbb R}$ the inverse can be written in the form $$\phi(\varphi,\mathcal{I};\tau) = \varphi + q(\varphi,\mathcal{I};\tau),$$ where $q(\varphi,\mathcal{I};\tau) = \bm{q}(\iota_{\bar{\Theta}}(\varphi),\mathcal{I};\tau)$ with $\bm{q}\in \mathcal{C}^{3}({\mathbb T}^N\times [\mathcal{I}_{},\infty)\times {\mathbb R})$. First consider the function $w(\phi,\varphi,\mathcal{I},\tau)=\phi - \frac{3-\alpha}{2(\alpha+1)} \mathcal{I}^{\frac{1-3\alpha}{2(\alpha+1)}} c_1(\tau) f(\phi) - \varphi$, where $\mathcal{I}\geq \mathcal{I}_{}$. Since $\mathcal{I}_{} \geq \lVert f \rVert_{\mathcal{C}_b^4({\mathbb R})} \lVert c_1 \rVert_{\mathcal{C}_b({\mathbb R})}$ we have ${\partial}_\phi w \geq \frac{1}{(\alpha+1)}$. Since also $w \in \mathcal{C}^{4}({\mathbb R}^2\times [\mathcal{I}_{},\infty)\times {\mathbb R})$ the equation $w=0$ defines a unique solution $\phi(\varphi,\mathcal{I};\tau)$ of class $\mathcal{C}^{4}$.\ If there was such a function $q$ we should have $$\label{gleichung def q} q(\varphi,\mathcal{I};\tau) = \phi -\varphi = \frac{3-\alpha}{2(\alpha+1)} \mathcal{I}^{\frac{1-3\alpha}{2(\alpha+1)}} c_1(\tau) f(\varphi + q(\varphi,\mathcal{I};\tau)).$$ Let us now fix some $\mathcal{I}\geq \mathcal{I}_{}$ and $\tau \in {\mathbb R}$. For simplicity’s sake we will drop the dependence of this variables and thus write $q(\varphi)$ for $q(\varphi,\mathcal{I};\tau)$ etc. Since $f(\phi)=\bm{f}(\iota_{\bar{\Theta}}(\phi))$, equation (\[gleichung def q\]) translates into $$\bm{q}(\bar{\Omega}) - \frac{3-\alpha}{2(\alpha+1)} \mathcal{I}^{\frac{1-3\alpha}{2(\alpha+1)}} c_1(\tau) \bm{f}(\bar{\Omega} + \iota(\bm{q}(\bar{\Omega}))) = 0 \;\; \text{for all} \;\; \bar{\Omega}\in {\mathbb T}^N.$$ We instead consider the equation $$\bm{q} - \sigma \left( \frac{3-\alpha}{2(\alpha+1)} \mathcal{I}^{\frac{1-3\alpha}{2(\alpha+1)}} c_1(\tau) \bm{f}(\bar{\Omega} + \iota(\bm{q})) \right)= 0, \;\; \sigma\in[0,1]$$ and search for a solution $\bm{q}(\bar{\Omega}; \sigma)$. Differentiation with respect to $\sigma$ yields the ordinary differential equation $$\frac{d \bm{q}}{d \sigma} = \chi (\bar{\Omega}+\iota(\bm{q}); \sigma), \;\; \bm{q}(\bar{\Omega};0)=0,$$ where $$\chi (\bar{\Omega}; \sigma) = \frac{\frac{3-\alpha}{2(\alpha+1)} \mathcal{I}^{\frac{1-3\alpha}{2(\alpha+1)}} c_1(\tau) \bm{f}(\bar{\Omega})}{1 - \sigma \frac{3-\alpha}{2(\alpha+1)} \mathcal{I}^{\frac{1-3\alpha}{2(\alpha+1)}} c_1(\tau) {\partial}_\omega\bm{f}(\bar{\Omega}) }.$$ As above, the denominator is greater than $1-\frac{3-\alpha}{2(\alpha+1)}>0$ by assumption. Thus, the right-hand side $\chi$ is $\mathcal{C}^{3}({\mathbb T}^N\times [0,1])$ for all $\bar{\Omega} \in {\mathbb T}^N$. Therefore the differential equation has a unique solution $\bm{q}(\bar{\Omega},\sigma)$ for all $\sigma \in [0,1]$. Now, the function $q(\varphi)=\bm{q}(\iota_{\bar{\Theta}}(\varphi),1)$ has the desired properties. Without loss of generality we can impose the assumption of Lemma \[lem q quasip\], since $\lVert \bm{f} \rVert_{\mathcal{C}^4({\mathbb T}^N)} \leq \gamma \kappa_0^2 \lVert \bm{p} \rVert_{\mathcal{C}^4({\mathbb T}^N)}$ and thus $\mathcal{I}_{}$ still depends only upon $\alpha, \omega$ and $\lVert \bm{p} \rVert_{\mathcal{C}^4({\mathbb T}^N)}$. Now, since $$H_1(\varphi,\mathcal{I};\tau) = H(\phi,I;\tau) + {\partial}_{\tau} \Psi(\phi,\mathcal{I};\tau),$$ where $ {\partial}_{\tau} \Psi(\phi,\mathcal{I};\tau) = - \mathcal{I}^\frac{3-\alpha}{2(\alpha+1)}f(\phi)c(\tau)$, and because of Lemma \[lem q quasip\] we have $$H_1(\varphi,\mathcal{I};\tau) = H(\varphi + q(\varphi,\mathcal{I};\tau),I;\tau) - \mathcal{I}^\frac{3-\alpha}{2(\alpha+1)}f(\varphi + q(\varphi,\mathcal{I};\tau))c(\tau).$$ Moreover, $I$ can be expressed as $$I =\mathcal{I} - \mathcal{I}^\frac{3-\alpha}{2(\alpha+1)}\dot f(\phi)c_1(\tau) = \mathcal{I} - \mathcal{I}^\frac{3-\alpha}{2(\alpha+1)}\dot f(\varphi + q(\varphi,\mathcal{I};\tau))c_1(\tau).$$ Hereby motivated, we define the $\mathcal{C}^3$-maps $\bm{I},\bm{H_1}:{\mathbb T}^N \times [\mathcal{I}_{},\infty)\times {\mathbb R}\to {\mathbb R}$ by $$\bm{I}(\bar{\Omega},\mathcal{I};\tau) = \mathcal{I} - \mathcal{I}^\frac{3-\alpha}{2(\alpha+1)}{\partial}_\omega \bm{f}(\bar{\Omega} + \iota(\bm{q}(\bar{\Omega},\mathcal{I};\tau)))c_1(\tau)$$ and further $$\begin{aligned} \bm{H_1}(\bar{\Omega},\mathcal{I};\tau) = \bm{H}(\bar{\Omega} + \iota(\bm{q}(\bar{\Omega},\mathcal{I};\tau)),\bm{I}(\bar{\Omega},\mathcal{I};\tau);\tau) - \mathcal{I}^\frac{3-\alpha}{2(\alpha+1)}\bm{f}(\bar{\Omega} + \iota(\bm{q}(\bar{\Omega},\mathcal{I};\tau)))c(\tau).\end{aligned}$$ Then the relation $H_1(\varphi,\mathcal{I};\tau) = \bm{H_1}(\iota_{\bar{\Theta}}(\varphi),\mathcal{I};\tau)$ holds for all $(\varphi,\mathcal{I};\tau) \in {\mathbb R}\times [\mathcal{I}_{},\infty)\times {\mathbb R}$. Proof of the main result {#section main result} ======================== Let us recall equation (\[DGL 0\]), that is $$\ddot x + \lvert x \rvert^{\alpha-1}x = p(t), \;\; \alpha \geq 3,$$ where $p \in \mathcal{C}^4_b({\mathbb R})$. Denote by $x(t;v_0,t_0)$ the solution of this equation to the initial condition $$x(t_0)=0, \;\; \dot{x}(t_0)=v_0.$$ Now set $\mathcal{I}^ = \max\{4 \mathcal{I}_{*}, (2\kappa_1)^{\frac{\alpha+3}{2}} \}$ and define $v_ = -2 \sqrt{\mathcal{I}^}$. Thus $v_$ is a constant depending only upon $\alpha, \omega$ and $\lVert \bm{p} \lVert_{\mathcal{C}^k({\mathbb T}^N)}$. Then, consider the function $\psi$ that maps the initial values $(v_0,t_0)\in (-\infty,v_) \times {\mathbb R}$ to $(v_1,t_1)$, where $v_1 = \dot{x}(t_1;v_0,t_0)$ and $$t_1 = \inf \{s \in (t_0,\infty): x(s;v_0,t_0)=0, \; \dot{x}(s;v_0,t_0)<0 \}.$$ We want to show, that this map is well defined on $(-\infty,v_) \times {\mathbb R}$. To this end, let us recall the transformations of section \[section transformations\]: $$(x,v;t) \overset{\mathcal{R}}{\rightarrow} (\bar{\vartheta},r;t) \hookrightarrow (\vartheta,r;t) \overset{\mathcal{S}}{\rightarrow} (\phi,I;\tau) \overset{\mathcal{T}}{\rightarrow} (\varphi,\mathcal{I};\tau)$$ Since $x=0$ and $v<0$ corresponds to $\bar{\vartheta} = \pi/2$ and therefore $\vartheta = \tau \in \{\pi/2 + 2\pi {\mathbb Z}\}$, one variable becomes redundant if we stay on the lower $y$-axis. Therefore we consider restrictions of the transformation maps onto some $2$-dimensional subspaces, namely: $$\begin{aligned} &\mathcal{R}_0:(-\infty,0) \times {\mathbb R}\to (0,\infty) \times {\mathbb R}, \;\; \mathcal{R}_0(v,t) = (\pi_2(\mathcal{R}(0,v;t)),\pi_3(\mathcal{R}(0,v;t))), \\ &\mathcal{S}_0:[r_,\infty) \times {\mathbb R}\to {\mathbb R}\times (0,\infty), \;\; \mathcal{S}_0(r,t) = (\pi_1(\mathcal{S}(\pi/2,r;t)),\pi_2(\mathcal{S}(\pi/2,r;t))), \;\; \text{and} \\ &\mathcal{T}_0: {\mathbb R}\times [I_{},\infty) \to {\mathbb R}\times (0,\infty), \;\; \mathcal{T}_0(\phi,I) = (\pi_1(\mathcal{T}(\phi,I;\pi/2)),\pi_2(\mathcal{T}(\phi,I;\pi/2))),\end{aligned}$$ where $\pi_j:{\mathbb R}^3 \to {\mathbb R}$ denotes the projection on to the $j$-th component. For $\lvert v_0 \rvert$ sufficiently large, let $\mathcal{R}_0(v_0,t_0) = (r_0,t_0)$, $\mathcal{S}_0(r_0,t_0) = (\phi_0,I_0)$ and $\mathcal{T}_0(\phi_0,I_0)=(\varphi_0,\mathcal{I}_0)$. Plugging $x_0=0$ into (\[eq energy action\]) and the definition of $I_0$ give us $$\label{eq v r I} \frac{1}{2}v_0^2 = \kappa_1 r_0^{\frac{2(\alpha+1)}{\alpha+3}} = \mathcal{H}(\pi/2,r_0;t_0) = I_0,$$ due to $c(\pi/2)=0$. Moreover, Theorem \[thm ham transformation\] yields $2I_0 \geq \mathcal{I}_0 \geq \frac{I_0}{2}$. So in total we have $$\mathcal{I}_0 \geq \frac{I_0}{2} = \frac{v_0^2}{4}.$$ This implies $(\mathcal{T}_0 \circ \mathcal{S}_0 \circ \mathcal{R}_0) ((-\infty,v_)\times {\mathbb R}) \subset {\mathbb R}\times (\mathcal{I}^,\infty)$. For $\mathcal{I}_0>\mathcal{I}^$ on the other hand, Lemma \[lem I abs\] yields the existence of the corresponding solution $(\varphi,\mathcal{I})(\tau;\varphi_0,\mathcal{I}_0,\pi/2)$ to system (\[Ham 4 equations\]) on $[\pi/2,5\pi/2]$ and guarantees $\mathcal{I}(\tau) >\mathcal{I}_{}$. But since $$\label{inklusion transform back} \mathcal{T}_0^{-1}({\mathbb R}\times[\mathcal{I_{}},\infty)) \subset {\mathbb R}\times[{I_},\infty) \text{ and } \mathcal{S}_0^{-1}({\mathbb R}\times[{I_{}},\infty)) \subset {\mathbb R}\times[{r_},\infty),$$ $(\varphi,\mathcal{I})$ can be transformed back to a solution $x(t;v_0,t_0)$ of the original problem (\[DGL 0\]) and $$(v_1,t_1) = (\mathcal{R}_0^{-1} \circ \mathcal{S}_0^{-1} \circ \mathcal{T}_0^{-1})(\varphi(5\pi/2),\mathcal{I}(5\pi/2))$$ has all the desired properties. Thus, $\psi$ can be equivalently defined in the following way: $$\label{def psi} \psi:(-\infty,v_) \times {\mathbb R}\to (-\infty,0)\times {\mathbb R}, \;\; \psi = (\mathcal{R}_0^{-1} \circ \mathcal{S}_0^{-1} \circ \mathcal{T}_0^{-1}) \circ \Phi \circ (\mathcal{T}_0 \circ \mathcal{S}_0 \circ \mathcal{R}_0)$$ Finally we are ready to state and prove the main theorem. \[main theorem\] Let $\bm{p}\in \mathcal{C}^4({\mathbb T}^N)$ generate the family of forcing functions $$p_{\bar{\Theta}}(t)=\bm{p}(\overline{\theta_1+t\omega_1}, \ldots, \overline{\theta_N+t\omega_N}), \;\; \bar{\Theta}=(\bar\theta_1,\ldots,\bar\theta_N)\in{\mathbb T}^N,$$ for fixed rationally independent frequencies $\omega_1,\ldots,\omega_N>0$. Let $(v_n,t_n)_{n \in J} = (\psi^n(v_0,t_0))_{n \in J}$ denote a generic orbit associated to system (\[DGL 0\]) with $p$ replaced by $p_{\bar{\Theta}}$. The escaping set $\mathcal{E}_{\bar{\Theta}}$ consists of those initial values $(v_0,t_0) \in {\mathbb R}^2$ such that 1. $v_0<v_$, 2. the corresponding orbit satisfies ${\mathbb N}\subset J$, and 3. $\lim_{n\to\infty} v_n = -\infty$. Then, for almost all $\bar{\Theta} \in {\mathbb T}^N$, the set $\mathcal{E}_{\bar{\Theta}}$ has Lebesgue measure zero. The proof will be divided in two parts: First we are going to construct a measure-preserving embedding suitable for Theorem \[thm escaping set nullmenge\], which basically translates into the successor map $\Phi$ of system (\[Ham 4 equations\]). Therefore it can be shown that for almost all $\bar{\Theta} \in {\mathbb T}^N$ the corresponding escaping set $E_{\bar{\Theta}}$ has Lebesgue measure zero. In the second part we will prove that initial values in $\mathcal{E}_{\bar{\Theta}}$ correspond to points in $E_{\bar{\Theta}}$ and conclude $\lambda^2(\mathcal{E}_{\bar{\Theta}})=0$ for almost all $\bar{\Theta} \in {\mathbb T}^N$. Escaping orbits of the transformed system ----------------------------------------- For $\bar{\Theta}\in {\mathbb T}^N$, denote by $(\varphi_{\bar{\Theta}}(\tau;\varphi_0,\mathcal{I}_0,\pi/2),\mathcal{I}_{\bar{\Theta}}(\tau;\varphi_0,\mathcal{I}_0,\pi/2))$ the solution to system (\[Ham 4 equations\]) with initial data $\varphi(\pi/2)= \varphi_0, \; \mathcal{I}(\pi/2)= \mathcal{I}_0$ and forcing function $p = p_{\bar{\Theta}}$. Furthermore, we will write $(\varphi(\tau;\bar{\Theta}_0,\mathcal{I}_0,\pi/2),\mathcal{I}(\tau;\bar{\Theta}_0,\mathcal{I}_0,\pi/2))$ for the solution of $$\begin{aligned} \label{DGL on torus} \varphi' = {\partial}_{\mathcal{I}} \bm{H_1}(\bar{\Theta}_0 +\iota(\varphi),\mathcal{I};\tau), \;\; \mathcal{I}'= -{\partial}_\omega \bm{H_1}(\bar{\Theta}_0 +\iota(\varphi),\mathcal{I};\tau), \end{aligned}$$ with $\bm{H_1}$ defined as in the last section and the initial values $\varphi(\pi/2)= 0, \; \mathcal{I}(\pi/2)= \mathcal{I}_0$.\ If we have $\bar{\Theta}_0 = \iota_{\bar{\Theta}}(\varphi_0)$ these solutions meet the identity $$\label{Eq gleiche Losungen} (\varphi_{\bar{\Theta}}(\tau;\varphi_0,\mathcal{I}_0,\pi/2),\mathcal{I}_{\bar{\Theta}}(\tau;\varphi_0,\mathcal{I}_0,\pi/2)) = (\varphi_0 +\varphi(\tau;\bar{\Theta}_0,\mathcal{I}_0,\pi/2),\mathcal{I}(\tau;\bar{\Theta}_0,\mathcal{I}_0,\pi/2)).$$ Let $F,G:\mathcal{D} \to {\mathbb R}$, where $\mathcal{D}={\mathbb T}^N \times (\mathcal{I}^,\infty)$, be defined by $$F(\bar{\Theta}_0,\mathcal{I}_0) = \int_{\frac{\pi}{2}}^{\frac{5\pi}{2}} \varphi'(\tau;\bar{\Theta}_0,\mathcal{I}_0,\pi/2) \, d\tau$$ and $$G(\bar{\Theta}_0,\mathcal{I}_0) = \int_{\frac{\pi}{2}}^{\frac{5\pi}{2}} \mathcal{I}'(\tau;\bar{\Theta}_0,\mathcal{I}_0,\pi/2) \, d\tau,$$ respectively. And consider $g:\mathcal{D}\to {\mathbb T}^N\times(0,\infty)$ given by $$g(\bar{\Theta}_0,\mathcal{I}_0) = (\bar{\Theta}_0 + \iota(F(\bar{\Theta}_0,\mathcal{I}_0) ), \mathcal{I}_0 + G(\bar{\Theta}_0,\mathcal{I}_0) ).$$ Then $F$ and $G$ are continuous, since the solution of (\[DGL on torus\]) depends continuously upon the initial condition and the parameter $\bar{\Theta}_0$. Therefore $g$ has special form (\[embedding form\]). The corresponding family of maps of the plane $\{g_{\bar{\Theta}}\}_{\bar{\Theta}\in {\mathbb T}^N}$ as in (\[planar maps\]) is $$\begin{aligned} &g_{\bar{\Theta}}:D_{\bar{\Theta}}\subset {\mathbb R}\times (0,\infty)\to {\mathbb R}\times (0,\infty),\\ &g_{\bar{\Theta}}(\varphi_0,\mathcal{I}_0) = (\varphi_0 + F(\bar{\Theta}+\iota(\varphi_0), \mathcal{I}_0), \mathcal{I}_0 + G(\bar{\Theta}+\iota(\varphi_0), \mathcal{I}_0)),\end{aligned}$$ where $D_{\bar{\Theta}} = (\iota_{\bar{\Theta}}\times {\textrm{id}})^{-1}(\mathcal{D}) = {\mathbb R}\times (\mathcal{I_{*}},\infty)$. Because of (\[Eq gleiche Losungen\]) these maps coincide with the successor map $\Phi$ of (\[Def successor\]) for the forcing function $p_{\bar{\Theta}}$.\ The injectivity of $g$ is a consequence of the unique resolvability of the initial value problem $$\begin{aligned} \varphi' = {\partial}_{\mathcal{I}} \bm{H}_1(\bar{\Theta}_1 +\iota(\varphi),\mathcal{I};\tau), \;\; \mathcal{I}'= -{\partial}_\omega \bm{H}_1(\bar{\Theta}_1 +\iota(\varphi),\mathcal{I};\tau), \end{aligned}$$ with $\varphi(5\pi/2)= 0$ and $\mathcal{I}(5\pi/2)= \mathcal{I}_1$, where $(\bar{\Theta}_1,\mathcal{I}_1)=g(\bar{\Theta}_0,\mathcal{I}_0)$.\ It remains to proof that $g$ is measure-preserving. Since the maps $g_{\bar{\Theta}}$ correspond to a Hamiltonian flow, Liouville’s theorem yields $\det J_{g_{\bar{\Theta}}}(\varphi_0,\mathcal{I}_0) = 1$, i.e. $$1= \left(1+{\partial}_\omega F(\bar{\Theta}+\iota(\varphi_0), \mathcal{I}_0)\right)\left(1+{\partial}_{\mathcal{I}_0}G(\bar{\Theta}+\iota(\varphi_0), \mathcal{I}_0)\right) - \left({\partial}_{\mathcal{I}_0}F(\bar{\Theta}+\iota(\varphi_0), \mathcal{I}_0)\right)\left({\partial}_\omega G(\bar{\Theta}+\iota(\varphi_0), \mathcal{I}_0)\right).$$ Since this is valid for all $\bar{\Theta} \in {\mathbb T}^N$ and all of $D_{\bar{\Theta}}$, it follows $$1= \left(1+{\partial}_\omega F \right) \left(1+{\partial}_{\mathcal{I}_0}G\right) - \left({\partial}_{\mathcal{I}_0}F\right) \left({\partial}_\omega G\right).$$ However, as shown in [@kunze_ortega_ping_pong Lemma 3.3] this implies that the map $g$ is orientation- and measure-preserving.\ Hence we have shown that $g$ is a measure-preserving embedding. Now, we have to find functions $W,k$ as described in Theorem \[thm escaping set nullmenge\]. Since $C$ from Lemma \[lem adiabatic invariant\] depends only upon $\lVert f \rVert_{\mathcal{C}_b^4({\mathbb R})}$, this constant is uniform in $\bar{\Theta} \in {\mathbb T}^N$. Therefore, if we take $W(\bar{\Theta}_0,\mathcal{I}_0)=\mathcal{I}_0$, Lemma \[lem adiabatic invariant\] implies $$W(g(\bar{\Theta}_0,\mathcal{I}_0)) - W(\bar{\Theta}_0,\mathcal{I}_0) = \mathcal{I}_1 - \mathcal{I}_0 \leq k(\mathcal{I}_0),$$ where $k(\mathcal{I}_0) = C \mathcal{I}_0^{b_\alpha}$ with $C$ as mentioned above and $b_\alpha<0$ from Theorem \[thm ham transformation\]. That way $W$ and $k$ meet all demanded criteria. Thus the measure-preserving embedding $g$ satisfies all conditions of Theorem \[thm escaping set nullmenge\] and we are finally ready to apply it. This gives us $\lambda^2(E_{\bar{\Theta}})=0$ for almost all $\bar{\Theta} \in {\mathbb T}^N$ for the escaping set $${E_{\bar{\Theta}}}= \{(\varphi_0,\mathcal{I}_0)\in{D}_{{\bar{\Theta}},\infty}: \lim_{n\to\infty}\mathcal{I}_n=\infty \},$$ where ${D}_{{\bar{\Theta}},\infty}$ is the set of initial conditions leading to complete forward orbits of $g_{\bar{\Theta}}$ as described in subsection \[subsection quasi-periodic functions\].\ Undoing the transformations --------------------------- Now let $\bar{\Theta}\in {\mathbb T}^N$ be fixed and consider the set $$\tilde{E}_{\bar{\Theta}} = (\mathcal{R}_0^{-1} \circ \mathcal{S}_0^{-1} \circ \mathcal{T}_0^{-1})( {E_{\bar{\Theta}}} ).$$ Our strategy will be to show $\lambda^2(\tilde{E}_{\bar{\Theta}})=0$ and $\mathcal{E}_{\bar{\Theta}} \subset \tilde{E}_{\bar{\Theta}}$.\ Since $\mathcal{T}^{-1}(\cdot,\cdot;\tau)$ is symplectic for all $\tau \in {\mathbb R}$, it follows $\lambda^2(\mathcal{T}_0^{-1}(E_{\bar{\Theta}})) = \lambda^2({E_{\bar{\Theta}}}) = 0$. Also, due to (\[inklusion transform back\]) it is $\mathcal{S}_0^{-1}(\mathcal{T}_0^{-1}( {E_{\bar{\Theta}}} )) \subset [{r_},\infty)\times {\mathbb R}$. But then, since $\mathcal{S}_0(r_0,t_0) = (t_0,\mathcal{H}(0,r_0;t_0))$, the definition of $r_$ (\[Mono abs\]) yields $\left\lvert \det J_{\mathcal{S}_0^{-1}}(\phi,I) \right\rvert = \left\lvert {\partial}_r \mathcal{H} (0,\mathcal{S}_0^{-1}(\phi,I)) \right\rvert^{-1} \leq 1$ on $\mathcal{T}_0^{-1}( {E_{\bar{\Theta}}} )$. Hence $$\lambda^2(\mathcal{S}_0^{-1}(\mathcal{T}_0^{-1}( {E_{\bar{\Theta}}} ))) \leq \lambda^2(\mathcal{T}_0^{-1}({E_{\bar{\Theta}}})) = 0.$$ Finally, due to (\[eq v r I\]),(\[abs I aus lemma existenzint\]) and the definition of $\mathcal{I}^$ we have $$\kappa_1 r_0^{\frac{2(\alpha+1)}{\alpha+3}} > \frac{\mathcal{I}^*}{2} \geq \frac{1}{2}(2\kappa_1)^{\frac{\alpha+3}{2}} \;\; \text{for} \;\; (r_0,t_0) \in \mathcal{S}_0^{-1}(\mathcal{T}_0^{-1}( {E_{\bar{\Theta}}} ))$$ and consequently $\left\lvert \det J_{\mathcal{R}_0^{-1}}(r_0,t_0) \right\rvert =\left\lvert - \sqrt{2 \kappa_1} \frac{\alpha +1}{\alpha + 3} r_0^{-\frac{2}{\alpha+3}}\right\rvert <1$. Therefore we can conclude $$\lambda^2( \tilde{E}_{\bar{\Theta}} ) = 0.$$ It remains to show that $\mathcal{E}_{\bar{\Theta}} \subset \tilde{E}_{\bar{\Theta}}$. If $\mathcal{E}_{\bar{\Theta}}$ is empty there is nothing to prove. Otherwise let $(v_0,t_0) \in \mathcal{E}_{\bar{\Theta}}$ and consider the corresponding orbit $$(v_n,t_n) = \psi^n(v_0,t_0), \;\; n\in{\mathbb N}.$$ Above we have demonstrated $\psi = (\mathcal{R}_0^{-1} \circ \mathcal{S}_0^{-1} \circ \mathcal{T}_0^{-1}) \circ \Phi \circ (\mathcal{T}_0 \circ \mathcal{S}_0 \circ \mathcal{R}_0)$, which means $$(v_n,t_n) = (\mathcal{R}_0^{-1} \circ \mathcal{S}_0^{-1} \circ \mathcal{T}_0^{-1}) (\varphi_n,\mathcal{I}_n)$$ with the notation as usual. Since also $$\mathcal{I}_n \geq \frac{v_n^2}{2}$$ holds for all $n \in {\mathbb N}$, we can conclude $\mathcal{I}_n\to\infty$ as $n \to\infty$. But this implies $(\varphi_0,\mathcal{I}_0) \in E_{\bar{\Theta}}$ and therefore completes the proof. [^1]: Electronic address: `[email protected]`
--- abstract: 'We examine the steady state and dynamic behaviour of an optical resonator comprised of two interlinked fibre loops sharing a common pump. A coupled Ikeda map models with great accuracy the field evolution within and exchange between both fibres over a single roundtrip. We find this supports a range of rich multi-dimensional bistability in the continuous wave regime, as well as previously unseen cavity soliton states. Floquet analysis reveals that modulation and parametric instabilities occur over wider domains than in single-fibre resonators, which can be tailored by controlling the relative dispersion and resonance frequencies of the two fibre loops. Parametric instability gives birth to train of pulses with a peculiar period-doubling behavior.' author: - Calum Maitland - Matteo Conforti - Arnaud Mussot - Fabio Biancalana bibliography: - 'Moebius.bib' title: Stationary States and Instabilities of a Möbius Fibre Resonator --- Introduction ============ Optical resonators are complex physical platforms, exhibiting an even richer range of phenomena than single-pass nonlinear systems due to their driven-dissipative nature. They are of increasing importance in metrology as sources of highly tunable, broadband frequency combs [@DelHaye2007; @Kippenberg2011; @Schliesser2012]. The Kerr cavity soliton is one of the fundamental states responsible for generating these combs [@Leo2013; @Coen2013a; @Herr2014; @Hansson2015; @Bitha2019]. Further, they have been proposed as cryptographic tools due to their chaotic output [@Ramos2000; @Imai2009; @Tunsiri2012]. Nonlinear optical resonators can be precisely modelled by the so-called Ikeda map [@Ikeda1979; @Steinmeyer1995], which describes separately the evolution of the electric field as it propagates through the cavity, and the boundary conditions which account for the injection of pump light and transmission of the cavity field between each round-trip. In this work, we propose a resonator composed of two fiber loops sharing a common pump. This geometry is reminiscent of the Möbius strip. Indeed, the two fiber coils are not closed, but they form an unique loop, in the same way as a Möbius surface has an unique side. The structure of the Möbius fibre resonator we consider is shown in figure \[fig: ResonatorSketch\]. We model the light propagation inside this resonator by means of a coupled Ikeda map. Several works have considered extended/multi-dimensional Ikeda maps and mean-field approximations by Lugiato-Lefever equations with nonlinear coupling between the fields of a single resonator [@Daguanno2016; @DAguanno2017; @Yi2017; @Guo2017; @Woodley2018; @Haelterman1992; @Haelterman1992a; @Haelterman1993], but to our knowledge no study has addressed the scenario of a pair of resonators in which the output of one is fed back into the other and vice versa. The dimensionless Ikeda map connecting the fibre fields between round-trips (labelled with an integer $m$) is $$\label{eq: normalisedIkeda} \begin{split} A_1^{m+1}(z=0, t) = \sqrt{\theta} A_{in} + \sqrt{1-\theta} e^{-i \delta_2} A_2^m(z=1, t)\\ A_2^{m+1}(z=0, t) = \sqrt{\theta} A_{in} + \sqrt{1-\theta} e^{-i \delta_1} A_1^m(z=1, t)\\ \end{split}$$ while intra-fibre propagation is described by a lossy nonlinear Schrödinger equation (NLSE) $$\label{eq: normalisedNLSE} i \partial_z A_n^m = \eta_n {\partial_t}^2 A_n^m + i \beta_{3,n} {\partial_t}^3 A_n^m - {\|A_n^m\|}^2 A_n^m -i \frac{\alpha_{i}}{2} A_n^m.$$ for fibres indexed as $n=1,2$. We work in dimensionless units in which the intra-fibre propagation coordinate $z = \tilde{z}/L$ is scaled to the fibre length $L$ (denoting here and subsequently all physical counterparts of dimensionless quantities with a tilde $\sim$). Time $t = \tilde{t} \sqrt{2/\|\beta_{2,2}\| L}$ is scaled by the second-order dispersion coefficient of the second fibre $\beta_{2,2} \equiv [\partial_\omega^2 \beta_2]_{\omega_{0}}$, given propagation constants $\beta_2(\omega)$ in the second fibre and pump frequency $\omega_0$. Hence $\eta_2 \equiv \text{sgn}(\beta_{2, 2})$ and $\eta_1 = \text{sgn}(\beta_{2, 1})\|\beta_{2, 1} / \beta_{2, 2}\|$ and the dimensionless third order dispersion parameters are related to its physical counterpart $\beta_{3, n} = \sqrt{2/{\|\beta_{2,2}\|}^3 L} [\partial_\omega^3 \beta_n]_{\omega_{0}}/3$. The fibre fields $A_n^m$ are related to physical electric fields as $A_n^m=\sqrt{\gamma L} \tilde{A}_n^m$ where $\gamma$ is the Kerr nonlinear coefficient, and similarly the pump field $A_{in} = \sqrt{\gamma L} \tilde{A}_{in}$. Since we work with a pump which is implicitly a time-independent constant, our frequency variable $\Omega$ expresses a scaled detuning of the physical frequency $\omega$ from $\omega_0$, that is $\Omega = (\omega-\omega_0)\sqrt{\|\beta_{2,2}\| L/2}$. $\theta$ parametrises the transmission of pump light into the fibres and fibre output fields into the bus waveguide. The absorption coefficient is also scaled by the fibre length $\alpha_i = \tilde{\alpha}_i L$, while the detunings $\delta_n=2k\pi-\beta_n(\omega_0)L$ measure the phase difference accumulated per round-trip with respect to the nearest single-loop resonance indexed by the integer $k$. ![[]{data-label="fig: ResonatorSketch"}](MoebiusOffAxisAnnotated.png){width="0.95\linewidth"} In this article, we investigate the power output of the Möbius resonator in different dynamical situations. First we examine steady states in the continuous-wave (time independent) limit, where the output from either fibre is independent of both the round trip number $n$ and the intracavity time coordinate $t$. This reveals an extended set of solutions exhibiting bistability, whose symmetry depends on the relative detuning of the two fibre loops. In the following section we consider the dynamical case in which the output varies over time, with a period corresponding to an integer number of round-trips. Here, allowing the fibre loops to have different dispersive characteristics gives rise to a new class of Kerr cavity solitons. The structure of these solitons is sensitive to the relative second and third order dispersion coefficients in each fibre. Finally we derive the modulation instability (MI) spectrum by applying Floquet analysis, which demonstrates the existence of additional Arnold instability tongues beyond those found in single fibre resonators, arising from dispersion variations between the two loops. Homogeneous Steady States ========================= We first seek the time-independent steady states of the Ikeda map in both fibre loops. A pair of equations for the steady state fields is obtained by first integrating the NLSE Eq. over one round-trip, giving $A_n^{m}(z=1)= \exp(i {\|A_n^m(z=0)\|}^2 L_{\text{eff}} - \alpha_i/2)A_n^{m}(z=0)$. The dimensionless effective length $L_{\text{eff}} = (1-\exp(-\alpha_i))/\alpha_i$ accounts for the power decrease due to propagation losses, which reduces the accumulated nonlinear phase over a round-trip [@Hansson2015]. Using this, we substitute for $A_n^{m}(1)$ in Eq. , imposing the constraint $A_n^{m+1}(0)=A_n^m(0)$. A single equation for the two output powers $Y \equiv {\|A_1\|}^2$, $Z \equiv {\|A_2\|}^2$ can be obtained by eliminating the pump term $\sqrt{\theta} A_{in}$ and multiplying by the complex conjugate of both sides: $$\begin{gathered} \label{eq: exactpumpindSS} \left(1+ 2 e^{-\frac{1}{2}\alpha_i} \sqrt{1-\theta} \cos(Y L_{\text{eff}}-\delta_1) + e^{-\alpha_i}(1-\theta) \right) Y\\ = \left(1+ 2 e^{-\frac{1}{2}\alpha_i} \sqrt{1-\theta} \cos(Z L_{\text{eff}}-\delta_2) + e^{-\alpha_i}(1-\theta) \right) Z.\end{gathered}$$ To isolate a single resonance and obtain an approximated analytical solution of Eq. (\[eq: exactpumpindSS\]), we expand the round-trip phases about the cavity anti-resonance: $ \exp(i (Y L_{\text{eff}}-\delta_1)) \rightarrow -1 - i(Y L_{\text{eff}}-\delta_1-\pi)$ and $\exp(i (Z L_{\text{eff}}-\delta_2)) \rightarrow -1 - i(Z L_{\text{eff}}-\delta_2-\pi)$. Expansion about the anti-resonance in fact corresponds to a resonance in both fibre loops separately, in the same manner as period-doubling instabilities in single-loop resonators [@Haelterman1992; @Haelterman1992a; @Haelterman1993]. We get: $$\begin{gathered} \label{eq: pumpindSS} Y \left( {(1- \sqrt{1-\theta}e^{-\frac{1}{2}\alpha_i})}^2 + (1-\theta)e^{-\alpha_i}{(Y L_{\text{eff}}-\delta_1-\pi)}^2 \right) \\ = Z \left( {(1- \sqrt{1-\theta}e^{-\frac{1}{2}\alpha_i})}^2 + (1-\theta)e^{-\alpha_i}{(Z L_{\text{eff}}-\delta_2-\pi)}^2 \right)\end{gathered}$$ Solving Eq. (\[eq: pumpindSS\]) for one of the powers in terms of the other gives three possible solutions for $Y(Z)$. Some example plots of these solutions with a fixed $\delta_2$ and various $\delta_1$ are shown in figure \[fig: TempSS\]. In the case of a resonator with equivalent fibres, $\delta_1=\delta_2$, the powers follow a symmetric bistability curve identical to that in Fig. 1a) in [@Hill2020]. If the fibres are unequally detuned however, the curve opens asymmetrically. If the detuning imbalance between the two fibres is relatively small, a separate closed orbit exists, close to which the solutions approximate an elliptic curve [@Washington2003]. These approximate local elliptic curves are indicated by red dashed boxes in figure \[fig: TempSS\]. Although in the present paper we do not explore the implications of the existence of such curves in the stationary state diagram (a unique feature of our Möbius resonators), it is interesting to notice that elliptic curves are widely used in practical cryptography due to their amazingly rich group-theoretical structure that allows factoring integers efficiently [@Stein2009; @Lenstra1987]. One could easily speculate that optical systems could be well-suited to study empirically open number-theoretical problems such as the famous Birch and Swinnerton-Dyer conjecture [@Silverman1992]. ![[]{data-label="fig: TempSS"}](Fig2.png){width="0.95\linewidth"} Alternatively, one can resort to numerical methods to solve the exact equation for the steady state powers. Numerically solving this equation (\[eq: exactpumpindSS\]) using Newton’s method shows a greatly extended (yet finite) range of solutions in the $Y, Z$ plane (figures \[fig: exactSS\] and \[fig: exactSS2\]). A large number of quasi-elliptic curve structures appear when asymmetric $Y \neq Z$ solutions are supported. However, solutions lying on repeated sub-structures at higher $Y$ and $Z$ are typically unstable and decay to the lowest branches . ![[]{data-label="fig: exactSS"}](Fig3.png){width="0.8\linewidth"} ![[]{data-label="fig: exactSS2"}](ExactCWSS_LongYAxisLeffFix.png){width="0.8\linewidth"} Alternatively solutions can be plotted against the fibre loop detuning assuming a fixed pump power. A distorted cavity resonance for in the intrafibre powers $(Y, Z)$ appears in the $(\delta_1, \delta_2)$ plane; as with the resonance in a standard Kerr cavity, this becomes broader as the round-trip losses are increased and more tilted for higher pump power or nonlinearity. The power in either fibre becomes asymmetric about $\delta_1=\delta_2$ however with otherwise indentical fibres the opposite asymmetry applies to power in the other fibre, that is $Y(\delta_1, \delta_2)=Z(\delta_2, \delta_1)$. An example is shown in figure \[fig: numericalSS2\]. Scanning both detunings over the $2\pi$ domain while maintaining a constant offset between them allows the resonance to be crossed more than once, with different power dependencies in the two fibres depending on the choice of offset. ![[]{data-label="fig: numericalSS2"}](Fig5.png){width="0.95\linewidth"} Dynamic Steady States & Möbius Cavity Solitons ============================================== If the two fibres have identical parameters and detuning, we recover the typical regimes of ring resonator steady state behaviour, which are well described in the mean-field limit by a Lugiato-Lefever equation [@Coen2013a]. These are illustrated in figure \[fig: DynamSS\], which shows a sequence of homogeneous, modulation instability, chaotic, oscillating and stabilised cavity solitons followed finally by homogeneous steady states as the detuning of both fibres is gradually increased over the resonant interval. ![[]{data-label="fig: DynamSS"}](EqualFibresTDepSS.png){width="0.75\linewidth"} The two fibres within the loop may have completely different dispersive properties. In a first example we examine the steady states within a fibre resonator, where one fibre has normal dispersion $\eta_2=+1$ while the other has stronger anomalous dispersion $\eta_1=-1.5$. Surprisingly similar behaviour regimes to those seen with identical anomalous fibres emerge, with clear transitions from stationary MI patterns to chaos followed by cavity soliton formation (figure \[fig: DynamSS\_UnbalacedB2\]). When multiple solitons form collisions may occur; at lower detuning this results in the solitons merging, whereas they tend to collapse for higher detuning. Reducing the strength of the anomalous dispersion in the second fibre decreases the detuning ranges for which MI and cavity solitons are supported. ![[]{data-label="fig: DynamSS_UnbalacedB2"}](DynamicSSVsDelta_UnbalancedGVD_noBeta3.png){width="0.95\linewidth"} Considering a pair of fibres which have equal and opposite dispersion, $\eta_1=-\eta_2=-1$, neither modulation instability nor chaotic steady states emerge. However, instead of the usual Kerr cavity soliton, a new kind of compact state which we term Möbius cavity soliton (MCS) appears with a complicated yet stable waveform at the output of both fibres. In all that follows, the output states given varying values of detuning $\delta_1=\delta_2$ are recorded after propagating the initial condition over many round-trips of the dual Ikeda map Eq. . Specifically, at each step we increment $\delta_1=\delta_2$ by a small amount and propagate over $500$ round-trips, using the output from the previous $\delta_1=\delta_2$ increment as the initial condition. This is typically sufficient to obtain convergence to the steady state in each case. At the first integration step we choose the initial condition $A_n(t) = 1.2 \sech(1.2 t)$ for $n=1,2$. We also choose $A_{in} = 0.166$, $\theta=2/15$ and $\alpha_i=1/100$. Figure \[fig: DynamSS\_OppGVD\] plots the output power from both fibres as a function of detuning. Under this constraint, the MCS appears only within a small detuning range $\delta \in (0.42, 0.455)$, and its structure changes somewhat within this range. Outside of this range only homgeneous steady states appear, with the modulation instability and chaotic regimes apparently suppresed. On closer examination, plotting the field’s intra-fibre evolution over two round-trips reveals that the stationary fibre outputs are snapshots of a stable periodic state which oscillates smoothly between the two (figure \[fig: MobiusCS\]). The periodic behaviour arises as the field experiences cyclic second-order dispersion as it travels through both loops of the resonator; a similar state was previously found in a dispersion-modulated fibre ring [@Gavrielides2004]. The spectrum of this MCS is considerably wider than that of the standard cavity soliton given the same pump power and detuning, and therefore is a promising seed for a broadband frequency comb when emitted in a pulse train. ![[]{data-label="fig: DynamSS_OppGVD"}](DynamicSSvsDelta_OppGVD_noreinit.png){width="0.95\linewidth"} ![ []{data-label="fig: MobiusCS"}](DynamicSS_OppGVD.png){width="0.95\linewidth"} Including third order dispersion $\beta_{3, n} = 2.6$ in both fibres, the MCS output becomes asymmetric in time and acquires a finite group velocity. It is stable for an extended detuning range $\delta \in (0.34, 0.45)$, however unlike the previous MCS which results from $\beta_{3, n} = 0$ the two fields within the resonator are distinct and do not periodically transform into each other. As a consequence the output from either fibre swaps each round-trip (but appears stationary when examined every second round-trip). From this it is clear that the initial condition affects steady state stability, as the field which first travelled through the anomalously-dispersive fibre supports a MCS to a slightly larger detuning limit $\delta = 0.49$ than that which started in the fibre with normal dispersion, $\delta = 0.46$. See figure \[fig: DynamSS\_OppGVDandsameb3\]. This MCS has a highly oscillatory structure in time and a narrower, peaked spectrum with resonant radiation resulting from the third-order dispersion clearly visible (figure \[fig: Mobius3rdorderCS\]). If the sign of $\beta_{3, n}$ swaps sign between fibres as well $\eta_n$, a similar MCS forms in both fibre loops. The oscillating $\beta_{3, n}$ leads to a reduced MCS group velocity compared to the state formed with a common $\beta_{3, n}$ as well as a reversed $t$-asymmetry between the power outputs (figure \[fig: Mobiusflipping3rdorderCS\]). Further, the resonant radiation spectral peaks do not appear since consistent phase-matching cannot be achieved under these conditions. ![[]{data-label="fig: DynamSS_OppGVDandsameb3"}](DynamicSS_OppGVD_commonBeta3_noreinit.png){width="0.95\linewidth"} ![ []{data-label="fig: Mobius3rdorderCS"}](DynamicSS_OppGVD_commonBeta3.png){width="0.95\linewidth"} ![ []{data-label="fig: Mobiusflipping3rdorderCS"}](DynamicSS_OppGVD_oppBeta3v2.png){width="0.95\linewidth"} Allowing for unequal detuning from resonance in the two fibres $\delta_1 \neq \delta_2$ adds an additional dimension to the resonator’s parameter space and considerably extends the possible existence of dynamical steady states. Performing a similar survey of time dependent steady states over $(\delta_1, \delta_2) \in [-\pi, \pi] \times [-\pi, \pi]$ as was done in figure \[fig: DynamSS\] for a resonator with $\eta_1=-1.5$, $\eta_2=1$ yields map as shown in figure \[fig: DynSS\_2D\]. Time dependent states are concentrated around the joint resonance close to the line $\delta_1+\delta_2=0$. Proceeding from the the negative $(-\pi,-\pi)$ to positive $(\pi,\pi)$ we find the familiar sequence of homogeneous, modulationally unstable, chaotic, solitonic and homogeneous states. These state domains show some curvature in the $(\delta_1, \delta_2)$ plane, which we expect to increase with pump power and round-trip losses as the resonance assuming homogeneous states in figure \[fig: numericalSS2\] does. ![[]{data-label="fig: DynSS_2D"}](StateID2D_v2.png){width="0.8\linewidth"} We note that the dynamical steady-states are dependent on the sequence of parameters tested and their intial values. For example, if the scan of increasing detuning shown in figure \[fig: DynamSS\_OppGVD\] had started with $\delta_1=\delta_2=0.4$, the corresponding steady state would be homogeneous. On increasing $\delta_1=\delta_2$ from this point, we would not observe the formation of Möbius solitons in the same detuning range as in \[fig: DynamSS\_OppGVD\] unless an external perturbation was added. It appears that the chaotic MI phase from which cavity solitons typically emerge is supressed in resonators with $\eta_1=-\eta_2=-1$, meaning they will not appear spontaneously on increasing detuning from the homogeneous state. Likewise, had we examined the same detuning ranges in reverse order a different sequence of states would result. The analysis of bifurcation, hysteresis and multi-stability in single fibre resonators is a complex area of study in itself [@ParraRivas2016; @ParraRivas2017; @ParraRivas2018], so we propose a more thorough examination of these as applied to the Möbius resonator in future works. Modulation Instability ====================== The presence of two fibres in the resonator with distinct dispersive properties raises the possibility of strongly modified modulation instability compared with that observed in homogeneous fibre resonators [@Hansson2015; @Haelterman1992b; @Coen1997]. The step-wise dispersion modulation seen by light over a complete round-trip of the resonator might be expected to result in similar instabilities as a resonator composed of a single fibre with oscillating dispersion, as presented in several previous works [@Conforti2014; @Conforti2016; @Copie2016; @Bessin2019]. To investigate this, we adapt the Floquet analysis of [@Conforti2016] to the Möbius resonator. The only significant extension required is that perturbations in both fibres must be monitored simultaneously to describe a complete round-trip, meaning we have a system of four simultaneous equations to solve rather than two. To facilitate analysis, we incorporate small round-trip losses into the boundary condition of the Ikeda map Eq. (\[eq: normalisedIkeda\]), which now reads: $$\label{eq: IkedaMI} \begin{split} A_1^{m+1}(z=0, t) = \sqrt{\theta} A_{in} + \sqrt{\rho} e^{-i \delta_2} A_2^m(z=1,t)\\ A_2^{m+1}(z=0, t) = \sqrt{\theta} A_{in} + \sqrt{\rho} e^{-i\delta_1} A_1^m(z=1, t)\\ \end{split}$$ $$\label{eq: cavityNLSEMI} \partial_z A_n^m = -i\eta_n {\partial_t}^2 A_n^m + i {\|A_n^m\|}^2 A_n^m$$ with $\rho+\theta<1$. We write the total field in the two resonators as $A_1^m(z,t) = \overline{A}_1 + \check{a}^m(z, t) +i \check{b}^m(z, t)$, $A_2^m(z,t) = \overline{A}_2 + \check{c}^m(z, t) +i \check{d}^m(z, t)$. $\check a, \check b, \check c, \check d$ are real perturbations and the time-independent stationary state in resonator $n$ is $$\label{eq: MItimeindSS} \overline{A}_n(z) = \sqrt{P_n} \exp\left(i P_n z + \psi_n \right).$$ As discussed earlier, there is no analytic solution available for the steady state powers $P_n$ and phases $\psi_n$, though it is straightforward to find them numerically by integrating a time-independent version of the Ikeda map. Considering solely evolution through the first fibre loop, the linearised NLSE for the perturbation coefficients reduces to a $2\times2$ coupled pair of ordinary differential equations, $$\label{eq: linearODE1} \frac{d}{dz} \begin{pmatrix} a^m(z, \omega) \\ b^m(z, \omega) \end{pmatrix} = \begin{pmatrix} 0 & -\eta_1\omega^2 \\ \eta_1 \omega^2+ 2 P_1 & 0 \end{pmatrix} \begin{pmatrix} a^m(z, \omega) \\ b^m(z, \omega) \end{pmatrix} .$$ Note the perturbations have been Fourier transformed into the frequency ($\omega$) domain. This problem has solutions with eigenvalues $$\label{eq: eigval} \pm k_1= \pm \sqrt{\eta_1 \omega^2\left( \eta_1 \omega^2+ 2 P_1 \right) }$$ and corresponding eigenvectors $$\label{eq: eigvec} \mathbf{v}_{+, -}= \begin{pmatrix} \cos(k_1 z)\\ \frac{k_1}{\eta_1\omega^2} \sin(k_1 z) \end{pmatrix}, \begin{pmatrix} -\frac{\eta_1\omega^2}{k_1} \sin(k_1 z)\\ \cos(k_1 z) \end{pmatrix} .$$ The eigenvectors define a fundamental solution matrix for the ODE (\[eq: linearODE1\]) $X_1(z) = ( \mathbf{v}_+ , \mathbf{v}_- )$. A completely analogous solution matrix will exist for the perturbations in the second fibre; $$\label{eq: FSS} X_2(z) = \begin{pmatrix} \cos(k_2 z) & -\frac{\eta_2 \omega^2}{k_2} \sin(k_2 z)\\ \frac{k_2}{\eta_2 \omega^2} \sin(k_2 z) & \cos(k_2 z) \end{pmatrix},$$ These can be combined into a single 4x4 block diagonal matrix which describes the evolution of all perturbations in the $\mathbf{a}^m(z)=(a^m, b^m, c^m, d^m)$ basis as $$\label{eq: masterFSS} X(z_1, z_2) = \left[ \begin{array}{c\|c} X_1 & O_2 \\ \hline O_2 & X_2 \\ \end{array} \right] \\$$ given $O_2$ is the 2x2 null matrix. Meanwhile the boundary conditions can be implemented by a combined rotation $$\label{eq: masterBC} \Gamma = \sqrt{\rho}\left[ \begin{array}{c\|c} O_2 & \Gamma_2 \\ \hline \Gamma_1 & O_2 \\ \end{array} \right] \\$$ where $$\begin{aligned} \label{eq: BC} \Gamma_n &= \begin{pmatrix} \cos(\phi_n) & -\sin(\phi_n) \\ \sin(\phi_n) & \cos(\phi_n) \end{pmatrix} \end{aligned}$$ and $\phi_1= P_1-\delta_1 + \psi_1-\psi_2$, $\phi_2=P_2 -\delta_2 + \psi_2-\psi_1$. The combined matrix describing a complete round-trip evolution is the product of these, $W = \Gamma X(1, 1)$. The eigenvalues and eigenvectors of $W$ can be solved for analytically, however the expressions are not particularly tractable. The four eigenvalues consist of two pairs with equal magnitudes but opposite sign, $\pm \lambda_1, \pm \lambda_2$; since the eigenvalue modulus is what determines instability gain, we need only consider one of each pair. The gain for a particular frequency $\omega$ is then [@Conforti2016] $$\label{eq: MobiusMIgain} g(\omega)=\log\left(\max({\|\lambda_1(\omega)\|, \|\lambda_2(\omega)\|})\right)$$ We plot the gain assuming different values of dispersion in both fibres in figure \[fig: MobiusMIgain\], assuming the detuning in both fibres is the same. At least two instability branches are always present at low frequencies. Increasing the GVD value in the first fibre $\eta_1$ relative to the fixed GVD in the second fibre $\eta_2=1$ alters the curvature of the instability branches; it also gives rise two additional branches at higher frequencies when the difference in dispersion magnitude between the two fibres is sufficiently big. This is consistent with the findings of [@Conforti2016]. ![ []{data-label="fig: MobiusMIgain"}](MobiusMIGain_eta.png){width="0.95\linewidth"} Here we have an additional degree of freedom in the relative detuning between the two fibres, which can modify the both the extent and positions of the branches. For example, choosing $\phi_2=\phi_1+\pi/2$ leads to the modified gain in figure \[fig: MobiusMIgain\_PiOver2Shift\], which shows the branches enabled by dispersion oscillation in figure \[fig: MobiusMIgain\] become significant at low frequencies for any relative dispersion $\eta_{1} / \eta_{2}$. It is interesting to note that when the two loops have dispersion with equal magnitude but opposite signs, the instability branches become flat. This means that for each steady state parametrized by the phase $\phi$, either we have no gain, either the gain is peaked around $\omega=0$. Hence the steady state is either stable or unstable with respect to zero frequency perturbations, that is the unstable state of a multi-stable response. This fact explains why no modulation instability is observed for opposite dispersions, as we found in the previous section (see Fig. \[fig: DynamSS\_OppGVD\]). ![ []{data-label="fig: MobiusMIgain_PiOver2Shift"}](MobiusMIGain_PiOver2ShiftedResonators_eta.png){width="0.95\linewidth"} To check the accuracy of the Floquet analysis’ predictions we numerically simulate MI in a Möbius resonator with $\delta_1=\delta_2=0.6\pi$, $P_1=P_2=0.2$, $\eta_{2, 1} = 10\eta_{2, 2} =1$. The Floquet theory indicates that to first order there should be four pairs of instability bands under these conditions, and indeed the numerically integrated spectrum shows three sidebands developing after $100$ round-trips (see figure \[fig: MobiusMI\_numerics\]). ![ []{data-label="fig: MobiusMI_numerics"}](MobiusMIGain_numerics.png){width="0.95\linewidth"} Instability branches may vary in their period behaviour; MI patterns on certain branches repeat exactly after every two round-trips, whereas those on others repeat every four round-trips. The period-doubling behaviour associated with Faraday-type instability branches is well documented in other works [@Hansson2015; @Conforti2016; @Bessin2019]; an example is presented in figure \[fig: MobiusMI\_PeriodDoubling\], in which the intracavity power in either fibre shows the same MI pattern after two round-trips, but exactly out-of-phase with respect to the original pattern. The power time series in both fibres repeats exactly every four round-trips, corresponding to a full period of the Faraday instability. This contrasts with homogeneous or dispersion oscillating cavities, where the Faraday instability gives rise to a period two pattern. This periodicity is explained by the fact that the unstable eigenvalues of the Floquet matrix $W$ are purely imaginary over these branches, i.e. they have a phase of $\pm\pi/2$. The perturbations are again in-phase after four round-trips, which eventually generates the observed sequence. For comparison an instability pattern that develops with real Floquet eigenvalues, which repeats exactly every two round-trips, is presented in figure \[fig: MobiusMI\_StandardPeriod\]. the A detailed analysis of parametric instabilities of the Möbius resonator and they period doubling is outside the scope of this paper and it will be reported elsewhere. ![ []{data-label="fig: MobiusMI_PeriodDoubling"}](MobiusMI_PeriodDoubling.png){width="0.95\linewidth"} ![ []{data-label="fig: MobiusMI_StandardPeriod"}](1RT_BothFibres.png){width="0.95\linewidth"} Conclusion ========== We have found new dissipative structures in a Möbius optical fibre resonator, which to our knowledge has not been studied previously. When continuous-wave solutions are modulationally stable, their powers define unusual bistability curves which may be approximated by elliptic curves in certain limits. A variety of time-dependent localised and periodic states which cannot be realised by standard fibre resonators are supported, including exotic cavity solitons and extended modulation instability. These are enabled by the ability to tune the resonances and dispersive properties of both fibres in the resonator independently. We anticipate that Möbius cavity solitons will be of interest to researchers working on frequency comb generation, owing to their broadened spectrum compared to the typical Kerr cavity soliton. Acknowledgments {#acknowledgments .unnumbered} =============== C.M. acknowledges studentship funding from EPSRC under CM-CDT Grant No. EP/L015110/1. F.B. acknowledges support from the German Max Planck Society for the Advancement of Science (MPG), in particular the IMPP partnership between Scottish Universities and MPG.
--- abstract: 'KMgBi single crystals are grown by using the Bi flux successfully. KMgBi shows semiconducting behavior with a metal-semiconductor transition at high temperature region and a resistivity plateau at low temperature region, suggesting KMgBi could be a topological insulator with a very small band gap. Moreover, KMgBi exhibits multiband feature with strong temperature dependence of carrier concentrations and mobilities.' author: - 'Xiao Zhang$^{1,\dag}$, Shanshan Sun$^{2,\dag}$, and Hechang Lei$^{2,}$' title: Physical properties of KMgBi single crystals --- Introduction ============ Topological materials have attracted tremendous attentions in the last decade, not only because of their fundamental importance but also because of their potential applications in future technology. After a large number of theoretical and experimental studies, various topological materials have been predicted and confirmed, such as topological insulators (TIs) and superconductors (TSCs),[@Hasan1; @QiXL1] Dirac semimetals (DSMs),[@WangZ1; @WangZ2; @LiuZK1; @LiuZK2] and Weyl semimetals (WSMs).[@Nielsen; @WanX; @XuG; @WengH; @HuangSM; @XuSY; @LvBQ; @YangLX; @HuangX] Moreover, Weyl fermions in the condensed matters can be further classified into two types.[@Soluyanov] For type-I WSMs, there is a topologically protected linear crossing of two bands (Weyl point) at the Fermi energy level ($E_{F}$), resulting in a point-like Fermi surface. In contrast, for type-II WSMs, when the Lorentz invariance is violated, the conical spectrum can be strongly tilted and the Fermi surface is no longer point-like. Instead, it consists of electron and hole pockets with finite density of states at $E_{F}$. In recent years, the type-II WSMs have been theoretically predicted and experimentally verified in several materials, like (W, Mo)Te$_{2}$.[@Soluyanov; @SunY; @Kourtis; @ChangTR; @WangZJ; @HuangL; @Deng; @XuN; @LiangA; @JiangJ; @WangC; @WuY; @Bruno] Although the type-I DSMs, such as Na$_{3}$Bi and Cd$_{3}$As$_{2}$, have been discovered,[@WangZ1; @WangZ2; @LiuZK1; @LiuZK2] the type-II DSMs which are the spin-degenerate counterparts of type-II WSMs are still rare. Very recently, theoretical study predicts that PtSe$_{2}$-type materials are type-II DSMs and it is confirmed by angle-resolved photoemission spectroscopy (ARPES) measurements.[@HuangH; @YanM] On the other hand, the PbClF-structure type is an important structural prototype, which has been found in hundreds of materials. They exhibit various novel physical properties, such as superconductivity in Li/NaFeAs and ferromagnetic semiconducting behavior in Li(Zn,Mn)As etc.[@Tapp; @WangXC; @Parker; @DengZ] Importantly, recent experimental and theoretical studies further indicate that materials with PbClF-structure could also host various topological states. For example, the monolayer of ZrSiO is predicted to be a two-dimensional (2D) TI and many of isostructural compounds with formula of WHM (W = Zr, Hf, or La, H = Si, Ge, Sn, or Sb, and M = O, S, Se, and Te) possess a similar electronic structure.[@XuDN] The ARPES measurements reveal that the topmost unit cell on the (001) surface of ZrSnTe single crystals could be a 2D TI with a curved $E_{F}$,[@LouR] thus confirming the theoretical prediction. Moreover, if the spin-orbit coupling (SOC) can be neglected, they will become three-dimensional (3D) node-line DSMs, which has been observed in ZrSiS.[@Schoop] Very recently, KMgBi with PbClF-structure is theoretically predicted to be a 3D DSM and the Dirac fermions are protected by the $C_{4v}$ point group symmetry.[@LeC] Furthermore, when compared to Na$_{3}$Bi and Cd$_{3}$As$_{2}$, the dispersion of the Dirac fermions in KMgBi is highly anisotropic because of weak dispersion of the $p_{x}$ and $p_{y}$ orbitals of Bi along the $z$-direction. It suggests that KMgBi is located at the edge of type-I and type-II DSM phases. When doping Rb or Cs into K site, K$_{1-x}$R$_{x}$MgBi(R = Rb, Cs) can be tuned between these two DSM phases.[@LeC] Motivated by the theoretical study, in this work, we study the physical properties of KMgBi single crystals in detail. Experimental results indicate that KMgBi exhibits a narrow-band semiconducting behavior with multiband feature, different from the theoretically predicted 3D DSM behavior. Moreover, there is a resistivity plateau appearing at low temperature region, implying that there may be a nontrivial topological surface state in KMgBi. Interestingly, a non-bulk superconducting transition emerges at $T_{c}\sim$ 2.8 K. It could originate either from extrinsic superconducting second phase or from intrinsic filamentary superconductivity of KMgBi. Experimental ============ KMgBi single crystals were grown by using the Bi flux with the molar ratio of K : Mg : Bi = 1 : 1 : 18. K chunk (99.5 $\%$), Mg chip (99.9 $\%$) and Bi shot (99.99 $\%$) were mixed and put into an alumina crucible, covered with quartz wool and then sealed into the quartz tube with partial pressure of argon. The sealed quartz ampoule was heated to and soaked at 600 K for 4 h, then cooled down to 300 K with 3 K/h. At this temperature, the ampoule was taken out from the furnace and decanted with a centrifuge to separate KMgBi crystals from Bi flux. KMgBi single crystals with typical size 3$\times$4$\times$0.3 mm$^{3}$ can be obtained. Because the raw materials and KMgBi are highly air-sensitive, all manipulations were carried out in an argon-filled glovebox with an O$_{2}$ and H$_{2}$O content below 0.1 ppm. X-ray diffraction (XRD) patterns of powdered small crystals and a single crystal were collected using a Bruker D8 X-ray Diffractometer with Cu $K_{\alpha}$ radiation ($\lambda=$ 0.15418 nm) at room temperature. Rietveld refinements of the XRD patterns were performed using the code TOPAS4.[@TOPAS] Electrical transport and heat capacity measurements were carried out in a Quantum Design PPMS. The longitudinal and Hall electrical resistivity were measured using a four-probe method on single crystals cutting into rectangular shape. The current flows in the $ab$ plane of samples. The Hall resistivity was obtained from the difference of the transverse resistivity measured at the positive and negative fields in order to remove the longitudinal resistivity contribution due to voltage probe misalignment, i.e., $\rho_{xy}(H)=[\rho(+H)-\rho(-H)]/2$. Magnetization measurements were carried out by using a Quantum Design MPMS3. Results and discussion ====================== ![(a) Powder XRD pattern of KMgBi. Inset: Crystal structure of KMgBi. The small green, medium blue and big red balls represent Mg, Bi and K atoms, respectively. (b) XRD pattern of a KMgBi single crystal. Inset: photo of a typical KMgBi single crystal. The length of one grid in the photo is 1 mm.](Fig1.eps) The crystal structure of KMgBi is composed of the Mg-Bi and K layers stacking along the $c$-axis direction alternatively (inset of Fig. 1(a)).[@Vogel] In Mg-Bi layer, Each Mg atom is coordinated with four Bi atoms and the tetrahedra of MgBi$_{4}$ are connected each other by edge-sharing. The crystal structure of KMgBi with the space group $P4/nmm$ is similar to the 111 family of iron-based superconductors,[@Tapp; @WangXC] where K, Mg and Bi atoms are replaced by Li, Fe and As atoms, respectively. The main panel of Fig. 1(a) shows the powder XRD pattern and refinement of crushed KMgBi crystals. All reflections can be well indexed using the $P4/nmm$ space group. The determined lattice parameters are $a=$ 4.8808(4) and $c=$ 8.3765(3) , consistent with previous results.[@Vogel] The XRD pattern of a single crystal (Fig. 1(b)) reveals that the crystal surface is normal to the $c$-axis with the plate-shaped surface parallel to the $ab$-plane. The plate-like crystals with square shape (inset of Fig. 1(b)) is consistent with the layered structure and the tetrahedron symmetry of KMgBi. Moreover, the crystals are rather soft and very easy to cleave along the $ab$-plane. ![Temperature dependence of the in-plane resistivity $\rho_{xx}(T)$ of KMgBi single crystal at zero field. Inset: enlarged part of $\rho_{xx}(T)$ (a) between 70 and 140 K and (b) below 5 K.](Fig2.eps) Temperature dependence of zero-field electrical resistivity $\rho_{xx}(T)$ in the $ab$-plane for KMgBi single crystal is shown in Fig. 2. The $\rho_{xx}(T)$ decreases with lowering temperature gradually and then exhibits semiconducting behavior when $T<T_{m}$ ($\sim$ 110 K) (inset (a) of Fig. 2). Moreover, the absolute value of $\rho_{xx}(T)$ is relative small. Thus, it suggests that KMgBi should be a narrow-band semiconductor rather than a semimetal. Assuming the semiconducting behavior is intrinsic and using the thermal activation model $\rho_{xx}(T)=\rho_{0}$exp($E_{g}/2k_{B}T$), the fit of $\rho_{xx}(T)$ between 40 and 100 K gives the band gap $E_{g}\sim$ 11.2(3) meV (130(4) K), which is much smaller than the calculated $E_{g}$ without SOC (362.8 meV).[@LeC] On the other hand, the metal-semiconductor transition at high temperature region can also be explained by this small band gap: once the temperature is comparable to the $E_{g}$, it will become degenerate semiconductor behaving like a bad metal. This phenomenon has been observed in some of doped semiconductors, such as Ga doped ZnO films and indium tin oxide films etc.[@Bhosle; @LiY] Surprisedly, when further decreasing temperature below about 20 K, the slope of $\rho_{xx}(T)$ curve becomes smaller and the resistivity plateau appears. It implies that another conducting channel becomes dominant when the bulk resistance becomes relatively high. This phenomenon is very similar to those observed in TIs, like Bi$_{2}$Te$_{2}$Se and SmB$_{6}$,[@RenZ; @KimDJ] and could be explained by the existence of metallic surface state which is topologically protected. This surface state has been predicted in theory.[@LeC] Moreover, theoretical calculation indicates that when compressing the lattice along the $a$ axis with 2% change, the Dirac nodes in KMgBi will be gapped out and KMgBi will change from a DSM to a three-dimensional (3D) strong TI.[@LeC] It seems more consistent with the present $\rho_{xx}(T)$ result. This surface state needs to be confirmed further by other techniques, such as ARPES measurement. More interestingly, there is a drop in the $\rho_{xx}(T)$ curve when $T<$ 2.8 K (inset (b) of Fig. 2) and this transition shifts to lower temperature under magnetic field (not shown here). It indicates that this drop results from a superconducting transition. Due to the broad transition width and the non-zero $\rho_{xx}(T)$ at 2 K, this superconductivity should be non-bulk, which is confirmed by heat capacity and magnetization measurements shown below. ![(a) Magnetoresistance (MR) of $\rho_{xx}(T,H)$ and (b) MR vs. $H/\rho_{xx}(T,0)$ at various temperatures for $H\parallel c$.](Fig3.eps) The magnetoresistance (MR) (MR $=(\rho_{xx}(T,H)-\rho_{xx}(T,0))/\rho_{xx}(T,0)=\Delta\rho_{xx}/\rho_{xx}(T,0)$) at various temperatures are shown in Fig. 3(a). The MR at 5 K is about 33 % under $H=$ 90 kOe, which is much smaller than topological or compensated SMs, such as Cd$_{3}$As$_{2}$, TaAs, WTe$_{2}$, LaBi etc.[@LiangT; @HuangX; @Ali; @SunSS2] Moreover, the MR at $H=$ 90 kOe does not change with temperature monotonically. With increasing temperature, the MR increases initially and then starts to decrease. The maximum of MR appears at $T\sim$ 60 K. The small MR at low temperatures can be partially related to the rather large $\rho_{xx}(T,0)$. On the other hand, if there is a single type of carrier and the scattering time $\tau$ is same at all points on the Fermi surface, the MR will follow the Kohler’s rule: the MR measured at various temperatures can be scaled into a single curve, MR $=F(H/\rho_{xx}(T,0))$.[@Pippard; @Ziman] Clearly, the scaled MR curves do not fall into one curve for KMgBi (Fig. 3(b)), indicating that the Kohler’s rule is violated. It strongly suggests that there are more than one type of carrier and the carrier concentrations and/or the mobilities of electron and hole are strongly temperature dependent.[@McKenzie; @ZhanY] ![(a) Field dependence of Hall resistivity $\rho_{xy}(H)$ up to $H$ = 90 kOe for $H\Vert c$ at various temperatures. (b) Temperature dependence of Hall coefficient $R_{H}(T)$ at $H$ = 90 kOe. (c) Temperature dependence of fitted electron- and hole-type carrier concentrations $n_{e,h}(T)$. (d) Fitted mobilities of electron and hole $\mu_{e,h}(T)$ as a function of temperature.](Fig4.eps) Fig. 4(a) shows the field dependence of Hall resistivity $\rho_{xy}(H)$ at various temperatures. The $\rho_{xy}(H)$ curves are almost linear with positive slope at high temperatures, and start to bend strongly when $T<$ 60 K. With further decreasing temperature ($T\leq$ 30 K), the $\rho_{xy}(H)$ curves become nearly linear again and the slopes are small. The corresponding Hall coefficient $R_{H}(T)$ ($=\rho_{xy}(T,H)/H$) at $H$ = 90 kOe increases with decreasing temperature at first and then decreases quickly when $T<$ 60 K. Finally, at $T\leq$ 30 K, the $R_{H}(T)$ becomes almost temperature independent. Interestingly, the temperature region where the $\rho_{xy}(H)$ exhibits remarkably nonlinear behavior accompanying with the sharp drop of $R_{H}(T)$ is nearly same as that where the $\rho_{xx}(T)$ shows remarkably semiconducting behavior. According to the two-band model, the $\rho_{xy}(H)$ can be expressed as,[@Pippard; @Ziman] $$\rho_{xy}(H) = \frac{\mu_{0}H}{\|e\|}\frac{(n_{h}\mu_{h}^{2}-n_{e}\mu_{e}^{2})+(n_{h}-n_{e})(\mu_{e}\mu_{h})^{2}(\mu_{0}H)^{2}}{(n_{h}\mu_{h}+n_{e}\mu_{e})^{2}+(n_{h}-n_{e})^{2}(\mu _{e}\mu _{h})^{2}(\mu_{0}H)^{2}}$$ where $n_{e,h}$ and $\mu_{e,h}$ are carrier concentrations and mobilities of electron and hole, respectively. Using this model, the fitted $n_{e,h}$ and $\mu_{e,h}$ as a function of temperature are shown in Fig. 4(c) and (d). The $n_{e,h}(T)$ decrease gradually with lowering temperature but the slope is larger when $T\leq$ 100 K (Fig. 4(c)). Then, the $n_{e,h}(T)$ become almost temperature independent at $T<$ 60 K. The ratio of $n_{h}$ to $n_{e}$ at $T>$ 60 K is larger than 1, indicating that the dominant carrier in KMgBi is hole and consistent with the positive slope of $\rho_{xy}(H)$ at high temperature. In contrast, the $n_{h}/n_{e}$ is slightly smaller than 1 between 20 and 60 K, i.e., there are more electrons than holes, leading to the strong downward bending of $\rho_{xy}(H)$ at high field region. The estimated $n_{e,h}$ at 2 K and 300 K are about 1.3$\times$10$^{15}$ and 3.3$\times$10$^{16}$ cm$^{-3}$, respectively. Such low $n_{e,h}$ are comparable with those in typical TIs, such as Bi$_{2}$Te$_{2}$Se and Sb doped Bi$_{2}$Se$_{3}$,[@RenZ; @Analytis] but much smaller than those in SMs, such as Cd$_{3}$As$_{2}$ and WTe$_{2}$.[@LiangT; @LuoY] Moreover, the strong enhancement of $n_{e,h}(T)$ with temperature at $T>$ 50 K also confirms the semiconducting feature of KMgBi. On the other hand, the $\mu_{e,h}(T)$ exhibit similar temperature dependence to the $R_{H}(T)$. They increase monotonically with decreasing temperature from 300 K to 60 K and this is often observed when the lattice scattering is dominant. In contrast, the $\mu_{e,h}(T)$ decrease quickly when $T<$ 60 K and then become nearly temperature independent below 30 K. The decreases of $\mu_{e,h}(T)$ should be the main cause of the remarkable increase of $\rho_{xx}(T)$ and it could be due to the impurity scattering, which enhances with decreasing temperature and dominates over the lattice scattering at low temperature region.[@Pierret] Moreover, strong temperature dependence of $n_{e,h}(T)$ and $\mu_{e,h}(T)$ should be the reasons that cause the violation of Kohler’s rule in KMgBi (Fig. 3(a)). ![The relationship between $C_{p}/T$ and $T^{2}$ for KMgBi single crystal at low temperature region. The red solid curve represents the fitting result using the formula $C_{p}/T=\gamma+\beta T^{2}+\delta T^{4}$. Inset: temperature dependence of magnetic susceptibility $\chi(T)$ at $H=$ 10 Oe with ZFC and FC modes for $H\parallel ab$.](Fig5.eps) Fig. 5 shows the temperature dependence of heat capacity for KMgBi single crystal. The relation between $C_{p}(T)/T$ and $T^{2}$ is not linear even at very low temperature region (2.2 K $<T<$ 6.8 K). The fit using the formula $C_{p}/T=\gamma+\beta T^{2}+\delta T^{4}$ is rather good and it gives $\gamma$ = 0.2(3) mJ mol$^{-1}$ K$^{-2}$, $\beta$ = 0.68(3) mJ mol$^{-1}$ K$^{-4}$, and $\delta$ = 0.0079(6) mJ mol$^{-1}$ K$^{-6}$. The electronic specific heat coefficient $\gamma$ is almost zero, consistent with the extremely low carrier concentrations of KMgBi from Hall measurement. The Debye temperature $\Theta _{D}$ is estimated to be 212(8) K using the formula $\Theta _{D}$ = $(12\pi^{4}NR/5\beta )^{1/3}$, where $N$ is the atomic number in the chemical formula ($N$ = 3) and $R$ is the gas constant ($R$ = 8.314 J mol$^{-1} $ K$^{-1}$). The absence of the heat capacity jump at $T\sim$ 2.8 K indicates that the superconductivity appearing in KMgBi is not bulk. This non-bulk superconductivity is confirmed by the magnetization measurement (inset of Fig. 5). Although there is a diamagnetic transition in the magnetic susceptibility $\chi(T)$ curves at $T_{c,onset}\sim$ 2.8 K when $H=$ 10 Oe with zero-field-cooling (ZFC) and field-cooling (FC) modes and the transition temperature is close to that corresponding to the drop in the $\rho_{xx}(T)$ curve, the superconducting volume fractions are very small ($\sim$ 1 and 2 % at 1.8 K for ZFC and FC $\chi(T)$ curves). Because the $T_{c}$ is close to that of KBi$_{2}$ ($T_{c}=$ 3.57 K),[@SunSS] this superconducting phase could originate from the tiny of KBi$_{2}$. But it has to be mentioned that we measured at least ten samples that are carefully cleaved, cut and scratched, and all of samples exhibit superconducting transition with similar $T_{c}$. It implies that there is either a tiny KBi$_{2}$ embedded or an intrinsic filamentary superconductivity existing in KMgBi single crystals. If it is intrinsic, the measurements using the surface-sensitive spectroscopy techniques, such as ARPES or scanning tunneling microscope measurements, would further help to distinguish whether the superconductivity is related to the possible non-trivial surface state. Conclusion ========== In summary, we grow the KMgBi single crystals by using the Bi flux successfully. It has a layered structure with space group $P4/nmm$ (PbClF-structure). KMgBi exhibits the metal-semiconductor transition at $T_{m}\sim$ 110 K with a resistivity plateau below 30 K. It strongly suggests that KMgBi should be a TI with small band gap, which is different from the predicted DSM state in theory. The MR and Hall measurements indicate that there are two kinds of carriers in KMgBi with extremely low concentrations. Moreover, the MR does not follow the Kohler’s rule because the carrier concentrations and mobilities of electron and hole significantly depend on temperature. Further analysis implies that the remarkable upturn behavior of $\rho_{xx}(T)$ can be ascribed to the decrease of mobilities. This decrease possibly results from the crossover from lattice scattering to impurity scattering mechanism at low temperature region. Acknowledgments =============== This work was supported by the Ministry of Science and Technology of China (2016YFA0300504), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (RUC) (15XNLF06, 15XNLQ07), and the National Natural Science Foundation of China (Grant No. 11574394). $\dag$ These authors contributed equally to this work. $\ast$ [email protected] [99]{} M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). X. L. Qi and S. C. Zhang, Rev. Mod. Phys. 83, 1057 (2011). Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. Weng, X. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012). Z. Wang, H. Weng, Q. Wu, X. Dai, and Z. Fang, Phys. Rev. B 88, 125427 (2013). 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--- abstract: 'We present a numerical study of the evolution of a non-linearly disturbed black hole described by the Bondi–Sachs metric, for which the outgoing gravitational waves can readily be found using the news function. We compare the gravitational wave output obtained with the use of the news function in the Bondi–Sachs framework, with that obtained from the Weyl scalars, where the latter are evaluated in a quasi-Kinnersley tetrad. The latter method has the advantage of being applicable to any formulation of Einstein’s equations—including the ADM formulation and its various descendants—in addition to being robust. Using the non-linearly disturbed Bondi–Sachs black hole as a test-bed, we show that the two approaches give wave-extraction results which are in very good agreement. When wave extraction through the Weyl scalars is done in a non quasi-Kinnersley tetrad, the results are markedly different from those obtained using the news function.' author: - Andrea Nerozzi - Marco Bruni - Virginia Re - 'Lior M. Burko' bibliography: - 'references.bib' title: \| Towards a wave-extraction method for numerical relativity.\ IV. Testing the quasi-Kinnersley method in the Bondi–Sachs framework --- Introduction {#sec:introduction} ============ Gravitational–wave detection has been gaining much interest over the last decades. Much effort has been correspondingly made in modelling possible sources. Interesting examples of sources of gravitational waves are, e.g., binary systems of merging black holes, spiralling systems of two neutron stars or coalescing black hole–neutron star binaries. Analytical tools able to investigate the dynamics of such sources when the merging takes place do not exist, due to the strong non-linearity of the problem. Numerical simulations are therefore invaluable in order to extract information about the gravitational-wave signal emitted. Typically, numerical relativity studies are done in three stages: First, one specifies initial data that correspond to the physical system of interest, and that satisfy certain constraint equations. Next, one evolves these initial data numerically, using the evolution equations (with or without enforcing the constraints), and finally, one needs to interpret the results of the simulation and extract the relevant physics thereof. This paper—like its prequels [@Beetle04; @Nerozzi04; @Burko04]—is concerned with this last stage, namely, with the problem of wave extraction. In order to interpret the results of numerical relativity simulations—at least within the context of a specific formulation of Einstein’s equations—a useful approach is that based on the characteristic initial value problem, originally introduced by Bondi and Sachs [@Bondi62; @Sachs62]. The characteristic initial value problem has been extensively used in numerical relativity, for spherically symmetric systems [@Corkill83; @Stewart84; @Gomez92a; @Gomez94b; @Clarke94; @Clarke94b; @Gomez96; @Burko97], axisymmetric systems [@Isaacson83; @Gomez94a; @Papadopoulos02] and 3D systems [@Bishop97a; @Bishop99; @Gomez97a; @Gomez97b]. This article is based on the results found in [@Papadopoulos02]—where the characteristic initial value problem, using Bondi coordinates, is used in axisymmetry to study a non-linearly disturbed non-rotating black hole metric. The non-linear response of the Schwarzschild black hole to the gravitational perturbation is embodied in a superposition of angular harmonics propagating outside the source. All the information concerning the angular harmonics and the energy radiated are easily derived by the evolved quantities. In fact, in this particular case one can identify a [news]{} function [@Bondi62], i.e., a function which embodies the information about the gravitational–radiation energy emitted [@Bishop97a; @dInverno96; @dInverno97; @Bartnik:1999sg; @Zlochower:2003yh; @Bishop03; @Babiuc:2005pg]. It is not currently possible to have at hand a quantity such as the news function when using other numerical approaches like, specifically, the widely used 3+1 decomposition of Einstein’s equations [@Arnowitt62]. Indeed, one of the outstanding problems of numerical relativity is that of [wave extraction]{}, i.e., the problem of how to extract the outgoing gravitational waves from the results of numerical simulations. In currently available methods, various approximations are applied to determine the gravitational-wave emission of isolated sources. One of the simplest approaches applies the quadrupole formula (that strictly speaking is valid for weak gravitational fields and slow motions) [@Thorne80; @Schutz86a]. This approach has been used effectively, e.g., in models of stellar collapse [@Dimmelmeier02b]. More sophisticated approaches use the Moncrief formalism [@Moncrief74; @Moncrief74b] to extract first-order gauge invariant variables from a spacetime which is assumed to be a perturbation of a Schwarzschild background at large distances [@Abrahams90; @Abrahams92a; @Alcubierre00b; @Alcubierre01a]. The strength of this approach is that it is gauge invariant, i.e., the information extracted is related to the physics of the system, and not to the coordinates used. Specifically, it avoids the pitfall of misidentifying gauge degrees of freedom as gravitational waves. \[E.g., in the Lorenz gauge all degrees of freedom, including the (residual-) gauge ones, travel at the same speed—the speed of light—such that they might be confused with physical waves.\] This procedure is usually performed under the assumption that the underlying gauge, i.e., the particular choice of the spacetime coordinate system, leads to a metric which is asymptotically Minkowski in its standard form, which is indeed the case for the most commonly used gauges in simulations of isolated systems. The gravitational waveform is determined by integrating metric components over a coordinate sphere at some appropriately large distance from the central source, and then subtracting the spherical part of the field (which is non-radiative). However, as we have already mentioned, such techniques are well defined when the background metric is assumed to be Schwarzschild, while their application to the more generic Kerr background metric can [at best]{} be intended as a very crude approximation. In addition, in the typical numerical relativity simulation, one does not usually have information about the mass and spin angular momentum of the eventual quiescent black hole, and no prescription is currently known to uniquely separate a background from a perturbation. Moreover, once the elimination of the spherical background is performed, what is left does not necessarily satisfy the perturbative field equations, and may, in fact, be quite large [@seidel01]. Another important approach is that of Cauchy–charactestic matching (CCM) schemes [@Clarke94]. The peculiarity of such schemes is that they use a characteristic Bondi-Sachs approach to study the numerical space-time far from the source, where the fields are weak and the probability to form caustics, which would make the code crash, is limited. In this way, using the notion of the Bondi news function, it is possible to extract easily the gravitational wave information. In the strong field part of the computational domain, instead, a usual Cauchy foliation is used, so that the problem of caustic formation is irrelevant. The CCM schemes have been used successfully to simulate cylindrically symmetric vacuum space-times [@Clarke94b] or to study the Einstein–Klein–Gordon system with spherical symmetry [@Gomez96]. A fully 3D application of the CCM scheme is, however, still unavailable. A final approach worth mentioning here aimed at wave extraction is the one involving the Bel–Robinson vector [@Smarr77], which can be considered a generalization to general relativity of the Poynting vector defined in electromagnetism. However its connection with the radiative degrees of freedom is still not entirely clear. A novel approach has been suggested recently [@Beetle02; @Beetle04; @Nerozzi04]: one extracts information about the gravitational radiation through quantities that are gauge and background-independent. One such quantity is the Beetle–Burko scalar, which is also tetrad independent. Specifically, no matter how one chooses to separate the perturbation from the background, the Beetle–Burko scalar remains unchanged. However, as pointed out in various contexts [@Burko04; @Cherubini04; @Berti05], the physical meaning of the Beetle–Burko scalar is non trivial. For example, in the stationary spacetime of a rotating neutron star its non-zero value is due to the deviation of the quadrupole from that of Kerr [@Berti05], while clearly no radiation is present. Thus the Beetle–Burko scalar awaits further study. At any case, the Beetle–Burko scalar—while including information only on the radiative degrees of freedom (when the notion of radiation is defined unambiguously)—describes the latter only partially. To obtain a full description of the radiative degrees of freedom, it is therefore desirable to consider an approach in which one calculates quantities, whose physical interpretation is more straightforward; however, the price to pay for these advantages is that it is harder to obtain such quantities uniquely: in fact, one still needs to break the spin/boost symmetry in a useful way. This approach, which is the basis of this paper, is that of using the Weyl scalars—which, under certain assumptions, isolate the radiative degrees of freedom from the background and gauge ones [@Newman62a; @Sachs62; @Szekeres65]—for wave extraction. Teukolsky [@Teukolsky73] showed that, choosing a particular tetrad, namely the [Kinnersley]{} tetrad [@Kinnersley69], to calculate the Weyl scalars, it is possible to associate the Weyl scalar $\Psi_4$ with the outgoing gravitational radiation for a perturbed Kerr space-time. Other authors have suggested to use the Weyl scalars for wave extraction, most recently in [@Fiske05], or to explore their relation with metric perturbations [@Lousto05]. However, the Weyl scalars depend on the choice of tetrad. Specifically, performing null rotations on the basis vectors of the null tetrad one can change the values of the Weyl scalars. (Recall, that one of the advantages of the Beetle–Burko scalar is that it is tetrad [independent]{}.) In fact, extracting the Weyl scalar $\Psi_4$ is meaningless, unless one also describes how to construct the tetrad to which it corresponds. In most tetrads, the Weyl scalar $\Psi_4$ mixes the information of the outgoing radiation with other information, including gauge degrees of freedom. In our proposal the use of the Weyl scalars is intimately related to the construction of the tetrad in which they are to be calculated—the quasi-Kinnersley tetrad—and therein lies its strength. With the aim of using $\Psi_4$ to extract information from simulations about the gravitational radiation output, starting with a transverse condition $\Psi_1=\Psi_3=0$, recent work [@Beetle04; @Nerozzi04; @Burko04] has addressed the problem of computing Weyl scalars in a tetrad which will eventually converge to the Kinnersley tetrad. This tetrad has been dubbed the [quasi-Kinnersley tetrad]{}, and work is still in progress in order to uniquely identify it for a general metric. Up to now, it is possible, for a general spacetime, to identify a class of tetrads, namely the quasi-Kinnersley [frame]{} [@Beetle04; @Nerozzi04], with the property that in this frame the radiative degrees of freedom (when and where the notion of radiation is unambiguous) are completely separated from the background ones. However, work is still in progress to identify the quasi-Kinnersley tetrad out of this frame. The difficuly in doing so is related to the following property of the tetrad members of the quasi-Kinnersley frame: they are all connected through type III (“spin/boost") rotations, and the spin/boost symmetry needs to be broken before the Weyl scalar $\Psi_4$ can be extracted in the quasi-Kinnersley tetrad. The quasi-Kinnersley frame is therefore a two-parameter family of tetrads, and the value of $\Psi_4$ (and that of $\Psi_0$) depends on the choice of the tetrad member of the frame. Notably, the Beetle–Burko scalar is invariant under type III rotations. That is, all the tetrad members of the quasi-Kinnersley frame share the same Beetle–Burko scalar. We avoid the spin/boost symmetry breaking difficulty in the present paper by applying an [ad hoc]{} technique to find the quasi-Kinnersley tetrad. This ad hoc technique allows us to obtain the Weyl scalar $\Psi_4$ in the quasi-Kinnersley tetrad, where its interpretation according to the gravitational compass is readily available. Specifically, in this paper we use the Bondi–Sachs formalism [@Bondi62], as it turns out that in this special case we can identify a (non-transverse) quasi-Kinnersley tetrad in a simple way, and to compute the Weyl scalars directly. The aim of this work is thus to demonstrate—in the context of this practical numerical example—the applicability, and necessity, of the quasi-Kinnersley tetrad method in using the Weyl scalars as wave extraction tools. The article is organized as follows: Section \[sec:bondi\] introduces our physical scenario, and Section \[sec:weyl\] describes our Weyl scalars computation. In Section \[sec:psinews\] we present the expected result which links the Bondi [news function]{} to $\Psi_4$. Finally results and conclusions are presented in Section \[sec:results\]. The Bondi problem {#sec:bondi} ================= ![image](diagram){width="13cm" height="8cm"} The numerical scenario we are studying is that of a non-linearly perturbed[^1] Schwarzschild black hole using an ingoing null-cone foliation of the space-time. We set up our system of coordinates as follows: a timelike geodesic is the origin of our coordinate system, photons are travelling from the origin in all directions, their trajectories forming null hypersurfaces. The hypersurface foliation is labelled by the coordinate $v$. As $r$ coordinate we choose a luminosity distance, such that the two-surfaces of constant $r$ and $u$ have area $4\pi r^2$. Finally, each null geodesic in the hypersurface is labelled by the two angular variables $\theta$ and $\phi$. We will restrict our attention to an axisymmetric space-time such that $\frac{\partial}{\partial\phi}$ is a Killing vector. Having chosen these variables, the Bondi metric in ingoing coordinates reads $$\begin{aligned} \label{metric} ds^2&=&-\left[\left(1-2\frac{M}{r}\right)e^{2\beta}-U^2r^2e^{2\gamma}\right] \,dv^2\nonumber \\&+&2e^{2\beta}dvdr - 2Ur^2e^{2\gamma}dvd\theta \label{eqn:bondimetric}\\&+& r^2(e^{2\gamma}d\theta^2+e^{-2\gamma}\sin\theta^2d\phi^2), \nonumber\end{aligned}$$ where $M,U,\beta,\gamma$ are unknown functions of the coordinates $\left(v,r, \theta\right)$. Within this framework, the Einstein equations decompose into three hypersurface equations and one evolution equation, as given below in symbolic notation \[eqn:fieldeqn\] $$\begin{aligned} \Box^{\left(2\right)}\psi &=& \mathcal{H}_{\gamma}\left(M,\beta,U,\gamma\right), \label{eqn:gammaeqn} \\ \beta_{,r} &=& \mathcal{H}_{\beta} \left(\gamma\right), \label{eqn:betaeqn} \\ U_{,rr} &=& \mathcal{H}_U \left(\beta, \gamma\right), \label{eqn:ueqn} \\ M_{,r} &=& \mathcal{H}_M \left(U, \beta, \gamma\right), \label{eqn:meqn}\end{aligned}$$ where $\Box^{\left(2\right)}$ is a 2-dimensional wave operator, $\psi=r\gamma$, and the various $H$ symbols are functions of the Bondi variables. We will write here only the expression for $\mathcal{H}_{\beta}$, the simplest of all the functions, which is given by $$\mathcal{H}_{\beta}=\frac{1}{2}r\gamma^2. \label{eqn:hbeta}$$ We will use this expression to describe the numerical algorithm. For exhaustive description of the system (\[eqn:fieldeqn\]) we refer to [@Gomez94a] and [@Isaacson83]. The structure of Eq. (\[eqn:fieldeqn\]) establishes a natural hierarchy in integrating them. By setting the initial value for the function $\gamma$ on the initial hypersurface (in addition to four free parameters), it is possible to integrate Eq. (\[eqn:betaeqn\]) to obtain $\beta$, then, having both $\beta$ and $\gamma$, Eq. (\[eqn:ueqn\]) can be integrated to obtain $U$ and finally $M$ can be derived by integrating Eq. (\[eqn:meqn\]). At this point we have all the metric functions on the initial hypersurface and we can integrate Eq. (\[eqn:gammaeqn\]) to obtain $\gamma$ on the next hypersurface, and the procedure is iterated. The integration of Eq. (\[eqn:fieldeqn\]) introduces constants of integration, which we set to zero (Bondi frame [@Tamburino66; @Isaacson83]), which corresponds to having an asymptotically inertial frame. The metric introduced in Eq. (\[eqn:bondimetric\]) describes a static Schwarzschild black hole if we set, in the Bondi frame, all the functions except $M$ to zero everywhere in the domain. $M$ is chosen to be the Schwarzschild mass $M_0$ of the black hole. Besides, outgoing gravitational radiation as a perturbation is introduced by the function $\gamma$; it turns out that $\gamma$ is a spin-2 field and is actually related to the radiative degree of freedom. Choosing an initial shape for $\gamma$ means in practice choosing the initial profile of outgoing gravitational waves. More specifically, the initial data are chosen in the following way: - We recover the background Schwarzschild geometry by setting $\gamma=\beta= U=0$ over the whole computational domain, while we set $M=M_0$ - We set up the initial data for a gravitational wave outgoing pulse by choosing a gaussian shape (with parameters $r_c$ and $\sigma$) for the function $\gamma$ (the choice of the gaussian shape is dictated by requiring the function to vanish at the outer boundary of the grid): $$\gamma\left(r,\theta\right) = \frac{\lambda}{\sqrt{2\pi}\sigma}e^{-\frac{\left(r -r_c\right)^2}{\sigma^2}} Y_{2lm}\left(\theta\right), \label{eqn:gammapert}$$ where $\lambda$ is the amplitude of the perturbation, and $Y_{lm}$ is the spherical harmonic of spin 2. In our numerical simulations we will set two types of initial data: the first one with $l=2$ and $m=0$ to get a pure quadrupole outgoing perturbation, while in the second one we will set $l=3$ and $m=0$. The integration of the hypersurface equations leads to the problem of the gauge freedom in choosing the integration constants. This apparent freedom is fixed by choosing outer boundary conditions on our numerical grid. As depicted in Fig. (\[fig:diagram\]) we in fact fix the metric at the outer boundary, i.e. on the worldtube $\mathcal{W}$, to be that of a Schwarzschild black hole. This automatically fixes the integration constants to be 0 for $\gamma$, $U$ and $\beta$, and $M_0$ for $M$ (Bondi frame). We point out here that such a boundary condition is well posed only for simulations which are limited in time, so that no relevant outgoing gravitational flow has crossed the worldtube. More information about the evolution routine can be found in [@Papadopoulos94; @Papadopoulos02; @NerozziTh04]. Weyl scalars {#sec:weyl} ============ Once we have the numerically computed metric for the evolved space-time we can derive the Newman-Penrose quantities we need. The Weyl scalars are \[eqn:weylscalars\] $$\begin{aligned} \Psi_0 &=& -C_{abcd}l^am^bl^cm^d, \label{eqn:psi0} \\ \Psi_1 &=& -C_{abcd}l^an^bl^cm^d, \label{eqn:psi1} \\ \Psi_2 &=& -C_{abcd}l^am^b\bar{m}^cn^d, \label{eqn:psi2}\\ \Psi_3 &=& -C_{abcd}l^an^b\bar{m}^cn^d, \label{eqn:psi3} \\ \Psi_4 &=& -C_{abcd}n^a\bar{m}^bn^c\bar{m}^d, \label{eqn:psi4} \end{aligned}$$ where $l^a$, $n^a$, $m^a$ and $\bar{m}^a$ are the Newman-Penrose null vectors. The five scalars defined in Eq. (\[eqn:weylscalars\]) are of course coordinate independent, but they do depend on the particular tetrad choice. We calculate the scalars in a quasi-Kinnersley tetrad, i.e., in a tetrad that converges to the Kinnersley tetrad when space-time settles down to that of an unperturbed black hole. In [@Beetle04; @Nerozzi04] a procedure to find the quasi-Kinnersley frame in a background independent way is given by looking at transverse frames, i.e., those frames where $\Psi_1$ and $\Psi_3$ vanish. In the following we use the word [frame]{} to indicate an equivalence class of tetrads which are connected by a type III spin/boost tetrad transformation. Fixing the right quasi-Kinnersley tetrad means choosing the tetrad in the quasi-Kinnersley frame which shows the right radial behavior for $\Psi_0$ and $\Psi_4$ according to the peeling-off theorem. That is, we seek the tetrad in which these two Weyl scalars peel off correctly. A background independent procedure to single out the tetrad out of the frame currently needs further investigation. In the Bondi–Sachs framework the identification of a quasi-Kinnersley tetrad is simple, and does not need to use the notion of transverse frames. The main reason for this proprety is due to the asymptotic knowledge of the Bondi functions when spacetime approaches Schwarzschild: in fact $\gamma$, $U$ and $\beta$ tend to zero, while $M$ tends to the Schwarzschild mass $M_0$ of the black hole (for further details see the next section). This situation is much different from the typical situation in numerical relativity simulations, for which the wave-extraction methods needs to be background-independent. If we assume to be in the Schwarzschild limit, the background Kinnersley tetrad chosen by Teukolsky [@Teukolsky73] in the perturbative scenario would look, using our coordinates $\left(v,r,\theta,\phi\right)$, as \[eqn:schwarztetradnull\] $$\begin{aligned} \ell^{\mu}&=&\left[\frac{2r}{r-2M},1,0,0\right], \label{eqn:schwarztetradnulll} \\ n^{\mu}&=&\left[0,-\frac{r-2M}{2r},0,0\right], \label{eqn:schwarztetradnulln} \\ m^{\mu}&=&\left[0,0,\frac{1}{\sqrt{2}r},\frac{i} {\sqrt{2}r\sin\theta}\right]. \label{eqn:schwarztetradnullm}\end{aligned}$$ This tetrad has been chosen by letting the $\ell^{\mu}$ and $n^{\mu}$ vector coincide with the repeated principal null directions of the Schwarzschild space-time. Such a condition fixes a frame, i.e. a set of tetrads connected by a type III rotation. The type III rotation parameter is then fixed by setting the spin coefficient $\epsilon$ to be vanishing. Eq. (\[eqn:schwarztetradnull\]) can be used to find the general expression for the tetrad in the full Bondi formalism. Using the asymptotic values of the Bondi functions, we can write down the expression for a general tetrad for the Bondi metric, whose vectors converge to the null vectors written in Eq. (\[eqn:schwarztetradnull\]) in the Schwarzschild limit. The result is given by \[eqn:bonditetradnull\] $$\begin{aligned} \ell^{\mu}&=&\left[\frac{2}{\left[\left(1-2M/r\right)e^{4\beta}- U^2r^2e^{2(\gamma+\beta)}\right]},e^{-4\beta},0,0\right], \nonumber \\ \label{eqn:bonditetradnulll} \\ n^{\mu}&=&\left[0,-\frac{\left[\left(1-2M/r\right)e^{2\beta}- U^2r^2e^{2\gamma}\right]}{2},0,0\right], \label{eqn:bonditetradnulln} \\ m^{\mu}&=&\left[0,\frac{rUe^{(\gamma-2\beta)}} {\sqrt{2}},\frac{1}{\sqrt{2}re^{\gamma}},\frac{i} {\sqrt{2}r\sin\theta e^{-\gamma}}\right]. \label{eqn:bonditetradnullm}\end{aligned}$$ Henceforth we will denote this first tetrad choice, which is supposed to be the successful one, as the tetrad $\mathcal{T}_1$. It is worth pointing out that the tetrad $\mathcal{T}_1$ is not transverse, i.e. $\Psi_1$ and $\Psi_3$ are not vanishing; Nevertheless it satisfies the requirements needed of a quasi-Kinnersley tetrad. (The quasi-Kinnersley tetrad does not have to be transverse, although it does need to be [asymptotically]{} transverse.) As explained above, the choice of $\mathcal{T}_1$ has been driven by the form of the Kinnersley tetrad expressed in Eq. (\[eqn:schwarztetradnullm\]) for the unperturbed black hole. However, this is not always possible in general where tetrad choices are dictated by different criteria. A straightforward example in this case would be that of basing the tetrad on null vectors directly derived from the metric. For instance, in the specific example of the metric (\[metric\]), an apparently natural choice of the tetrad, obtained with the algebraic manipulation packages Maple and GRTensor, is the following: \[eqn:bonditetradnull2\] $$\begin{aligned} \ell^{\mu}&=&\left[ 0,-e^{-4\beta},0,0\right], \nonumber \\ \label{eqn:bonditetradnulll2} \\ n^{\mu}&=&\left[e^{2\beta},\frac{\left[\left(1-2M/r\right)e^{2\beta}- U^2r^2e^{2\gamma}\right]}{2},0,0\right], \nonumber \\ \label{eqn:bonditetradnulln2} \\ m^{\mu}&=&\left[0, \frac{rUe^{(\gamma-2\beta)}}{\sqrt{2}}, \frac{1}{\sqrt{2}re^{\gamma}}, \frac{i}{\sqrt{2}r\sin\theta e^{-\gamma}}\right]. \label{eqn:bonditetradnullm2}\end{aligned}$$ The reason why this tetrad choice [looks]{} more natural than the first one is related to the fact that packages like GRTensor construct this tetrad starting from the $\ell^{\mu}$ vector, which is assumed to be lying on the null foliation, leading to the expression $\ell_{\mu}=\delta_{\mu 0}$. The contravariant components are then given by Eq. (\[eqn:bonditetradnulll2\]). Once $\ell^{\mu}$ is fixed, the other tetrad vector expressions are found by imposing the normalization conditions between the vectors in the Newman-Penrose formalism. The tetrad in Eq. (\[eqn:bonditetradnull2\]) will be hereafter referred to as tetrad $\mathcal{T}_2$. We will show that this different tetrad choice leads to results which, although equivalent from a qualitative point of view, are different than those obtained in the quasi-Kinnersley tetrad $\mathcal{T}_1$. In the numerical results that we are going to present in the next sections we have calculated the Weyl scalars in the two tetrads presented above, to show the reliability of quasi-Kinnersly tetrad method and an example, through a complete comparison, of the different results that we might obtain in a numerical simulation in doing wave extraction using Weyl scalars in different tetrads. The linear regime {#sec:psinews} ================= In the linear regime both the Bondi and the Newman–Penrose formalisms define quantities which provide information about gravitational waves. We will discuss here briefly such definitions in order to get a correspondence between the two approaches, which will be tested numerically in the following section. In the Bondi formalism the initial assumption is that the space-time is asymtotically flat, which leads to the following expansion for the function $\gamma$ at null infinity: $$\gamma=K+\frac{c}{r}+O\left(r^{-2}\right). \label{eqn:gammaradial}$$ Here we assume to be in the Bondi frame [@Bondi62], i.e. we set the integration constant $K$ to be zero. By integrating the hypersurface equations for the other Bondi functions we can get their radial expansion at null infinity. Such integrations in general introduce other integration constants but, again, we assume that in our frame those constants are vanishing, ending up with the following expressions for the remaining Bondi functions [@Isaacson83]: \[eqn:bondiradialexp\] $$\begin{aligned} \beta&=&-\frac{c^2}{4r^2}+O\left(r^{-4}\right), \label{eqn:betaexp} \\ U&=& -\frac{\left[c\sin^2\theta\right]_{,\theta}}{r^2\sin^2\theta}+O\left(r^{-3}\right), \label{eqn:Ueqrad} \\ M&=&M_0+O\left(r^{-1}\right). \label{eqn:Veqrad}\end{aligned}$$ It is trivial to verify that, with this choice of integration constants, the space-time is asymptotically flat. It is possible to define at null infinity a notion of energy, which leads to the result found by Bondi $$E=\frac{1}{4\pi}\oint M\sin\theta d\theta d\phi, \label{eqn:bondimassfunc}$$ and in addition the energy flux per unit solid angle, which is given by $$\frac{d^2E}{dvd\Omega}=-\frac{\left(c_{,v}\right)^2}{4\pi}. \label{eqn:bondienfunc}$$ It is clear that the information about the energy carried by gravitational waves is contained in the Bondi news function $c_{,v}\approx r\gamma_{,v}$, where the approximation is assumed to hold at large distances in the linear regime. Expressing Eq. (\[eqn:bondienfunc\]) in terms of $\gamma$ gives $$\frac{d^2E}{dvd\Omega}\approx-\frac{r^2\left(\gamma_{,v}\right)^2}{4\pi}. \label{eqn:bondienfuncgamma}$$ An analogous derivation can be achieved within the Newman-Penrose formalism. The key point is that $\Psi_4$ can be expressed, when computed in the [quasi-Kinnersley]{} tetrad, directly as a function of Riemann tensor components, i.e. [@Sachs62; @Teukolsky73] $$\left(\Psi_4\right)_{qKT}=-\left(R_{\hat{v}\hat{\theta}\hat{v}\hat{\theta}}- iR_{\hat{v}\hat{\theta}\hat{v}\hat{\phi}}\right). \label{eqn:psi4quasi}$$ The hatted symbols in Eq. (\[eqn:psi4quasi\]) are indicating that the Riemann tensor components are contracted over a tetrad of vectors oriented along the coordinates. In the linear regime however, those vectors can be assumed to be the basis coordinate vectors, as the perturbation is already contained in the Riemann tensor. For this reason we will omit the hatted symbols from now on, and always talk about coordinate components. The components of the Riemann tensor in Eq. (\[eqn:psi4quasi\]) can then be related to the transverse-traceless gauge terms, using $R_{v\alpha v\beta}=-\frac{1}{2}\frac{\partial^2 h_{\alpha\beta}}{\partial v^2}$, which leads to the result $$\left(\Psi_4\right)_{qKT}=-\frac{1}{2}\left(\frac{\partial^2 h_{\theta\theta}} {\partial v^2}-i\frac{\partial^2 h_{\theta\phi}} {\partial v^2}\right). \label{eqn:psi4hmunu}$$ This relation between $\Psi_4$ in the [quasi-Kinnersley]{} tetrad and the transverse–traceless (TT) components of the perturbed metric leads us to a definition of the energy emitted by simply calculating the expression of the energy tensor for the gravitational wave defined by $$T^{GW}_{\mu\nu}=\frac{1}{32\pi}\left[\partial_{\mu}\left(h^{TT}\right)^{\sigma\rho} \partial_{\nu}\left(h^{TT}\right)_{\sigma\rho}\right]. \label{eqn:stressenergygrav}$$ The total energy flux is then given by the formula, which is assumed to hold at null infinity: $$\frac{d^2E}{dvd\Omega}=- r^2{\left(T^{GW}\right)^r}_v=\frac{r^2}{16\pi}\left[\left(\frac{\partial h^{TT}_{\theta\theta}}{\partial v}\right)^2 +\left(\frac{\partial h^{TT}_{\theta\phi}}{\partial v}\right)^2\right], \label{eqn:energyfluxv}$$ and, by substituting our expression in terms of $\Psi_4$ we get the result $$\frac{d^2E}{dvd\Omega}=- \frac{r^2}{4\pi}\left\|\int^v_0\left(\Psi_4\right)_{qKT}dv\right\|^2. \label{eqn:energypsi4}$$ In our specific case of axisymmetry we expect only one polarization state to be present, which is made evident by the presence of a single news function. Correspondingly, we expect $\left(\Psi_4\right)_{qKT}$ to have only its real part non vanishing. Combining Eq. (\[eqn:energypsi4\]) with Eq. (\[eqn:bondienfuncgamma\]) and considering the presence of only one polarization state, we finally obtain that in the linearized regime the relation $$\left(\Psi_4\right)_{qKT}=-\frac{\partial^2\gamma}{\partial v^2} \label{eqn:psi4gamma}$$ must hold. This is the relation we want to verify numerically. The minus sign comes from the negative sign given in Eq. (\[eqn:psi4hmunu\]). We want to stress again the attention to the fact that such relation is strictly true at null infinity, however, we expect it to be well satisfied provided we are at sufficiently large distances from the black hole. As it will be clear in the next section, this assumption turns out to be very well motivated. ![Convergence test for the $R_{22}$ component of the Ricci tensor. The top panel shows this component for two different resolutions for a radial slice on the equatorial plane. As expected the value is converging to zero. In the bottom panel we have tested the second order power law of convergence by multiplying the $1200\text{x}120$ output by a factor of four. The two curves now overlap perfectly, thus proving second order convergence.[]{data-label="fig:ricciconv"}](ricciconv){width="8cm" height="8cm"} ![(a) Top panel: the value for $\Psi_4$ for two different resolutions at time $v=80$. (b) Bottom panel: the value for $\Psi_0$ for two different resolutions at $v=80$. Both values are calculated on the equatorial plane.[]{data-label="fig:convpsi04"}](psiconv){width="8cm" height="8cm"} Numerical Results {#sec:results} ================= In this section we present numerical results for a standard simulation. We have written a code that solves the Bondi equations and calculates the Weyl scalars in the two tetrads $\mathcal{T}_1$ and $\mathcal{T}_2$. Our code makes use of the Cactus infrastructure [@cactusweb]. We set up an initial Schwarzschild black hole, and construct an initial quadrupole perturbation on $\gamma$ using the expression indicated in Eq. (\[eqn:gammapert\]). The values chosen in this case are $\lambda=0.1$, $r_0=3$ and $\sigma=1$, although various tries have been performed varying these parameters, all leading to the same physical results. We emphasize that $\lambda$ represents the amplitude of the initial perturbation: the chosen value is such that the perturbation is somehow realistically small, yet large enough for the full non-linearity of the problem to appear clearly through the harmonic coupling, as we are going to show (see Figg. \[fig:energyl2T1\] and \[fig:energyl3T1\], cf. also [@Papadopoulos02]). All the results presented here are obtained using two different resolutions, the coarser one having 600 points in the radial dimension and 60 points in the angular direction, the finer one having those values doubled. The results which are not convergence test results are all obtained using the finer resolution of 1200 points in the radial direction and 120 points in the angular direction. We will first present some tests in order to verify the robustness of our algorithm, and then we will proceed to a full comparison of our results in the two approaches presented here. The first two following subsections will deal with the calculation of the Weyl scalars in the tetrad $\mathcal{T}_1$ defined in Eq. (\[eqn:bonditetradnull\]); we don’t expect the second tetrad to give different results for what concerns radial fall-offs and convergence. Section \[sec:psi4news\] will instead deal with the relation of $\Psi_4$ with the news function and, within this context, it is very important to show a comparison of results in different tetrads, to have an evident demonstration of how important the choice of the right tetrad is, i.e. the quasi-Kinnersley tetrad, in the process of evaluating the outgoing gravitational wave contribution. Convergence {#sec:convergence} ----------- The first thing we want to test in our code is of course convergence. In order to do so, once the numerical variables are computed, we have calculated independently the values of the Ricci tensor components, which should vanish in vacuum. Such components are suitable for doing convergence tests. In Fig. (\[fig:ricciconv\]) we show the value of the Ricci component $R_{22}$ for two resolutions, the picture shows a radial slice of our space-time on the equatorial plane, for the time value $v=80$. The first figure simply superimposes the two values obtained for the two different resolutions, while in the second picture we have first multiplied the values for the finer resolution by a factor of four, as expected in a second order convergence code. We have found similar results for the other components, which ensured us of the convergence of our algorithm. Radial fall-offs of the scalars {#sec:radialfalloffs} ------------------------------- Figg. (\[fig:convpsi04\]a) and (\[fig:convpsi04\]b) show the numerical output for $\Psi_0$ and $\Psi_4$ for two different numerical resolutions. The outputs show satisfactory convergence for $\Psi_4$, but not for $\Psi_0$. This is because the asymptotic radial behavior for $\Psi_0$ should be $r^{-5}$, as expected from the peeling-off conjecture (see e.g. [@Wald84]) and the linear perturbation analysis [@Teukolsky73], and this gets completely embedded in the numerical error. We believe that this could constitute a serious numerical problem in situations where the initial tetrad chosen for the scalars computation is not the right one, and a tetrad rotation is needed. The numerical error found in $\Psi_0$ would then propagate when other quantities, like the curvature invariants $I$ and $J$, are computed, thus leading to meaningless results. Recall, however, that the curvature invariants $I,J$, in addition to the Coulomb scalar $\chi$ and the Beetle–Burko scalar $\xi$, can be found invariantly and in a background-independent way which is also tetrad-independent, i.e., it does not require finding first the Weyl scalars to find $I,J$ [@Beetle04; @Burko04]. Fig. (\[fig:radpsi04\]a) and (\[fig:radpsi04\]b) emphasize the radial dependence of $\Psi_2$ and $\Psi_4$, which is highlighted very well in our simulations. The two figures show that at late times $\Psi_2$ gets the background contribution with the superposition of a wave whose radial behaviour is $r^{-3}$. We have tested the convergence of such a wave to prove its physical meaning; this is itself a quite interesting result as we don’t have a perturbation equation for $\Psi_2$, and it is entirely due to the full non-linear treatment of the problem. Of course, given its rapid fall-off, the wave contribution from $\Psi_2$ is negligible. $\Psi_4$ shows instead the well expected $r^{-1}$ behaviour. ![ (a) Top panel: the value for $\Omega_2=r^3\Psi_2$ at two different times $v_1=40$ and $v_2=80$. (b) Bottom panel: the value for $\Omega_4=r\Psi_4$ for the same couple of times $v_1=40$ and $v_2=80$.[]{data-label="fig:radpsi04"}](reducedpsi){width="8cm" height="8cm"} Relation of $\Psi_4$ with the Bondi news {#sec:psi4news} ---------------------------------------- In this section we want to show the comparison of $\Psi_4$ with the second time derivative of $\gamma$, where $\Psi_4$ will be calculated in the two tetrads shown in Eq. (\[eqn:bonditetradnull\]) and (\[eqn:bonditetradnull2\]). We first start with the tetrad $\mathcal{T}_1$: Figg. (\[fig:newsbondi\]) and (\[fig:newsdiff\]) verify numerically the equivalence expressed by Eq. (\[eqn:psi4gamma\]): it is clear that the two functions $\Psi_4$ and $-\gamma_{,vv}$ are different in the non-linear regime but converge in the linear regime. In particular Fig. (\[fig:newsdiff\]) shows in logarithmic scale the absolute value of their difference at time $v=80$, well in the linear regime. This numerical result proves the generic assumption that $\Psi_4$ is related to the outgoing gravitational radiation contribution. ![From top to bottom: the comparison of $\Psi_4$ (dashed line) calculated in tetrad $\mathcal{T}_1$ and $-\gamma_{,vv}$ (solid line) for the values of $v_0 = 10$, $v_1=40$, $v_2=60$ and $v_3=80$.[]{data-label="fig:newsbondi"}](news){width="8cm" height="9cm"} ![The function $\Delta=\left\|\Psi_4+\gamma_{,vv}\right\|$ at $v_3=80$.[]{data-label="fig:newsdiff"}](newsdiff){width="80mm" height="40mm"} As a counterexample, we show the same result when $\Psi_4$ is computed in tetrad $\mathcal{T}_2$, which would actually have been our simplest choice hadn’t we applied the concept of a [quasi-Kinnesley tetrad]{}. The results for this calculation are shown in Fig. (\[fig:newsbondi2\]). It is evident that $\Psi_4$ does not get any contribution from the background, meaning that the tetrad we have chosen is part of the [quasi-Kinnersley frame]{}, however, it is evident that the result is rather different from that coming from the Bondi function $\gamma$. ![(a) Top panel: the comparison of $\Psi_4$ (dashed line) calculated in tetrad $\mathcal{T}_2$ and $-\gamma_{,vv}$ (solid line) for $v=80$. (b) Bottom panel: the value for $\Psi_4$ alone.[]{data-label="fig:newsbondi2"}](newst2psi4){width="8cm" height="7cm"} In order to understand what is happening, we need to analyze further the tetrad $\mathcal{T}_2$, and in particular its limit when the space-time approaches a type D one. Using our well known asymptotic limits for the Bondi functions, it is easy to show that tetrad $\mathcal{T}_2$ converges, in the type D limit, to the tetrad \[eqn:bonditetradnull2D\] $$\begin{aligned} \ell^{\mu}&=&\left[ 0,-1,0,0\right], \label{eqn:bonditetradnulll2D} \\ n^{\mu}&=&\left[1,\frac{r-2M}{2r}, 0,0\right], \label{eqn:bonditetradnulln2D} \\ m^{\mu}&=&\left[0,0,\frac{1}{\sqrt{2}r} ,\frac{i} {\sqrt{2}r\sin\theta}\right], \label{eqn:bonditetradnullm2D}\end{aligned}$$ which is different from the Kinnersly tetrad used by Teukolsky, Eq. (\[eqn:schwarztetradnull\]). A simple analysis of the differences let us conclude that the original tetrad Eq. (\[eqn:schwarztetradnull\]) can be obtained by first of all exchanging the two real null vectors $\ell$ and $n$, and then using a boost transformation of the type \[eqn:boost\] $$\begin{aligned} \ell&\rightarrow& A\ell, \label{eqn:boostl} \\ n&\rightarrow& A^{-1}n, \label{eqn:boostn}\end{aligned}$$ where $A$ is a real parameter. It is easy to show that choosing $A=\frac{2r}{r-2M}$ we get that the new real null vectors coincide with the Kinnersley tetrad defined in Eq. (\[eqn:schwarztetradnull\]). It is now straightforward to understand how these differences affect the values of the Weyl scalars. First of all, exchanging $\ell$ and $n$ corresponds to exchanging $\Psi_0$ and $\Psi_4$; this means that if we use the tetrad $\mathcal{T}_2$ we will find the outgoing radiative contribution in $\Psi_0$. This completely clarifies the result found in Fig. (\[fig:newsbondi2\]): it turns out that in this particular tetrad $\Psi_4$ is supposed to have a $r^{-5}$ radial fall-off and, in practice, just like the result shown in Fig. (\[fig:convpsi04\]a) for $\Psi_0$ in tetrad $\mathcal{T}_1$, we are not able to obtain this radial behaviour numerically, and we end up getting just numerical error. In Fig. (\[fig:newsbondi3\]) we show the comparison of $\Psi_0$ with the news function; here the results are in better agreement but we still have no correspondence, the reason for this is to be found in the boost transformation, in fact a transformation like the one written in Eq. (\[eqn:boost\]) changes the value of $\Psi_0$ according to $$\Psi_0\rightarrow A^{-2}\Psi_0. \label{eqn:psi0boost}$$ This leads us to the final conclusion that, in the linearized regime, the following relation must hold: $$\left(\Psi_4\right)_{\mathcal{T}_1} = \left(\frac{r-2M}{2r}\right)^2\left(\Psi_0\right)_{\mathcal{T}_2} = -\frac{\partial^2\gamma}{\partial v^2} \label{eqn:finalrel}$$ ![From top to bottom: the comparison of $\Psi_0$ computed in the tetrad $\mathcal{T}_2$ and $-\gamma_{,vv}$ for the time values of $v_0 = 10$, $v_1=40$, $v_2=60$ and $v_3=80$. The dashed line is $\Psi_0$ while the solid line is $-\gamma_{,vv}$.[]{data-label="fig:newsbondi3"}](newst2psi0){width="8cm" height="9cm"} We test this conclusion in Fig. (\[fig:newsbondi4\]) where we have plotted the value of $\left(\frac{r-2M}{2r}\right)^2\left(\Psi_0\right)_{\mathcal{T}_2}$. We emphasize that all the results that we have obtained in all the tetrads have wave-like profiles, although only one is the correct wave contribution. In practical numerical simulations one should really make sure that the tetrad in which the scalars are computed is a [quasi-Kinnersley]{} tetrad, otherwise the results, even if wave-like shaped, could be wrong. ![From top to bottom: the comparison of $A^{-2}\Psi_0$ (dashed line) calculated in tetrad $\mathcal{T}_2$ and $-\gamma_{,vv}$ (solid line) for the values of $v_0 = 10$, $v_1=40$, $v_2=60$ and $v_3=80$.[]{data-label="fig:newsbondi4"}](newst2psi0red){width="8cm" height="9cm"} Energy calculation {#sec:energy} ------------------ Having made sure that $\Psi_4$ calculated in tetrad $\mathcal{T}_1$ is related, in the linear regime, to the Bondi news function, we can use its expression to calculate the energy radiated from the black hole. In section \[sec:psinews\] we have shown that the expression of the energy flux per unit solid angle is given by $$\frac{d^2E}{dvd\Omega} = -\frac{r^2\Phi^2}{4\pi}, \label{eqn:energyfluxgen}$$ where we denote with $\Phi$ the generic expression for the news function, being it $\gamma_{,v}$ or $\int\left(\Psi_4\right)_{qKT}dv$. We can integrate the expression in Eq. (\[eqn:energyfluxgen\]) on a 2-sphere in order to obtain the energy flux. For the sake of simplicity we take a sphere of radius $r_0$, getting the result $$\frac{dE}{dv} = -\frac{r_0^2}{4\pi}\oint\Phi^2\sin\theta d\theta d\phi, \label{eqn:energyfluxgen2}$$ The computation of the energy flux was the goal of [@Papadopoulos02], where the news function was used to calculate the amount of energy which is carried away by each spin-weighted spherical harmonics of the outgoing radiation. We can perform a similar calculation using $\Psi_4$ and compare our results, in order to have a further demonstration of the validity of our approach. Given the results described in Section \[sec:psi4news\] it is evident that also these results will be in good agreement, however we want to highlight their validity and to show their dependence on the position of the observer. ![From top to bottom: energy contribution of the $l=2$ harmonic initial data. The three graphs represent the outgoing energy contribution for the values of $l=2,4,6$. In each graph the two curves represent the value for the energy calculated using the Bondi news function (upper curve), while the lower curve uses the value of $\Psi_4$ in tetrad $\mathcal{T}_1$, as a function of the position of the observer. It is evident that at late times there is a convergence of the two values. This convergence seems to be less evident for the $l=6$ case (lowest graph), but we expect this phenomenon to be purely numerical, because the numerical error on this multipole component is very high.[]{data-label="fig:energyl2T1"}](energy2){width="8cm" height="8cm"} ![From top to bottom: energy contribution of the $l=3$ harmonic initial data. We show the two graphs corresponding to the dominant terms $l=3,5$ in the emitted gravitational signal. Again here we compare the result coming from the Bondi news function with the one using $\Psi_4$ in tetrad $\mathcal{T}_1$. The results are similar to the ones shown in Fig. (\[fig:energyl2T1\]).[]{data-label="fig:energyl3T1"}](energy3){width="8cm" height="6cm"} Since we are interested in the energy contribution of each spin weighted spherical harmonic, we first have to perform the decomposition of the signal into spin weighted spherical harmonics contributions. This is done by introducing the quantity $\Phi_l$ defined as $$\Phi_l\left(v,r\right)=2\pi\int^1_{-1} \Phi\left(v,r,y\right)Y_{2l0}\left(y\right)dy, \label{eqn:spinangularmode}$$ where $y=-\cos\theta$ and $Y_{2l0}$ is the spin weighted spherical harmonic of spin 2. Using Eqq. (\[eqn:energyfluxgen2\]) and (\[eqn:spinangularmode\]) we can get an expression for the total energy emitted in each angular mode after the evolution to a final time $T$, given by $$E_l\left(T\right)=\frac{r_0}{4\pi}\int^T_0 \left[\Phi_l\left(v,r=r_0\right)\right]^2dv. \label{eqn:enangularmode}$$ We have performed some numerical simulations where the energy, using both the Bondi news function and $\Psi_4$, has been calculated. The results are shown in Figg. (\[fig:energyl2T1\]) and (\[fig:energyl3T1\]). Fig. (\[fig:energyl2T1\]) shows the result for a numerical simulation where the initial profile of the $\gamma$ has been chosen to be quadrupolar, i.e. using the spin-weighted spherical harmonic with $l=2$, $m=0$. The non-linearity of the problem is translated into the fact that the evolution excites higher order multipolar terms. However, simmetry considerations allow only even multipolar terms to be excited. In the picture we show the energy at time $v=80$ for the $l=2,4,6$ terms. Such energy is calculated varying the position of the observer and it is evident that, as soon as we push the observer further from the source, the two energy calculations coincide. On the other hand, numerical errors become stronger when going higher order multipole terms, which explains the not-perfect convergence for the $l=6$ terms. Fig. (\[fig:energyl3T1\]) shows a similar simulation for an initial data with $l=3$, $m=0$. Here again we expect the non-linearity to excite the other harmonics. Differently from the $l=2$ case, we don’t expect to have forbidden modes, however our numerical results show that the highest amplitude modes are the odd ones, so we show only those modes. Anyway the results in this case are qualitatively equivalent to those obtained in the case of quadrupolar initial data. Conclusions {#sec:concl} =========== The problem of correctly extracting the gravitational signal in numerical simulations is of primary importance. We believe that the use of the Weyl scalars of the Newman-Penrose formalism offers a promising method for wave extraction, as it applies to any formulation of Einstein’s equations and, even more important, to any kind of background we end up with, being it Schwarzschild or Kerr. However the problem of identifying the tetrad in which one is to compute the Weyl scalars still awaits a full solution. Recent work [@Beetle04; @Nerozzi04] shows how to make the important first step of identifying an equivalence class of tetrads, the quasi-Kinnersley frame, of which the desired quasi-Kinnersley tetrad is a member. However, the problem of isolating the right tetrad out of this set is still under investigation. In the present work we have considered a non-trivial numerical scenario, namely the evolution of a non-linearly perturbed black hole using Bondi coordinates, in order to show the importance of the tetrad choice for the calculation of wave related quantities. This particular scenario is well suited for a practical demonstration of the problems one would encounter if a careful choice of the tetrad for the Weyl scalar computation is not done. We have in fact shown that the computation of the Weyl scalars in an arbitrary tetrad, chosen by brute-force using mathematical packages like GrTensor, would lead to wrong results for $\Psi_4$, which is the quantity that typically is supposed to contain the (outgoing) gravitational wave degrees of freedom. This fact is evident in our case, where we have compared directly the results for the Weyl scalar in two different tetrads $\mathcal{T}_1$ and $\mathcal{T}_2$, using the Bondi news function in determining that the Weyl scalar corresponding to the quasi-Kinnersley tetrad is the right one. In Section \[sec:results\] we have shown that only the quasi-Kinnersley tetrad $\mathcal{T}_1$ gives results in agreement with the news function. Finally, we emphasize the importance of singling out an appropriate quasi-Kinnersley tetrad from the quasi-Kinnersley frame [@Beetle04; @Nerozzi04]. For instance, the tetrad $\mathcal{T}_2$ after exchange of the $\ell$ and $n$ null basis vectors is related to the tetrad $\mathcal{T}_1$ by a boost. This example indicates that every tetrad in the quasi-Kinnersley frame will give results for $\Psi_0$ and $\Psi_4$ that will show no contribution from the background, so that the wave-like shape of the scalars could lead us to the wrong conclusion of having the right outgoing gravitational signal. As we have shown, this conclusion could well be far from reality. The authors are indebted to Philippos Papadopoulos and Denis Pollney for their help and invaluable discussions. AN is funded by the NASA grant NNG04GL37G to the University of Texas at Austin and by the EU Network Programme (Research Training Network contract HPRN-CT-2000-00137). MB is partly funded by MIUR (Italy). Work on this research started when LMB was at Bates College. [^1]: We want to point out here that the expression “perturbed” could be misleading in this context, as it might suggest we are assuming some kind of approximation. Our numerical simulations are instead fully non-linear evolutions of Einstein’s equations in the Bondi-Sachs formulation.
--- abstract: 'The Coulomb energy of a charge that is uniformly distributed on some set is maximized (among sets of given volume) by balls. It is shown here that near-maximizers are close to balls.' address: - 'A. Burchard, University of Toronto, Department of Mathematics, 40 St. George Street, Room 6290, Toronto, Canada M5S 2E4 ' - 'G.R. Chambers, University of Chicago, Department of Mathematics, 5734 S. University Avenue, Room 208 C, Chicago, IL 60637 ' author: - 'Almut Burchard and Gregory R. Chambers' date: 'July 2, 2015' title: \| Geometric stability\ of the Coulomb energy --- Introduction and main result ============================ The [Coulomb energy]{} of a charge distribution $f$ on $\RR^3$ is — up to a multiplicative physical constant — given by the singular integral $$\E(f)= \int_{\RR^3}\int_{\RR^3} \frac{f(x)f(y)}{\|x-y\|}\, dxdy\,.$$ According to the Riesz-Sobolev inequality, the energy of a positive charge distribution increases under symmetric decreasing rearrangement: If $f^$ is radially decreasing and equimeasurable with $f$, then $$\label{eq:Riesz} \E(f)\le \E(f^)\,.$$ The physical reason is that symmetrization increases the interaction of the charges by reducing the typical distance between them. Equality holds only if the charge distribution is already radially decreasing about some point in $\RR^3$ [@L-Choquard]. Is this characterization of equality cases stable? If the two sides of Eq. (\[eq:Riesz\]) almost agree, how close must $f$ be to a translate of $f^$? We answer this question for charge distributions that are uniform on some set $A\subset \RR^3$ of finite volume. Let $A^$ be the ball of the same volume. With a slight abuse of notation, denote by $$\E(A) = \int_A\int_A \|x-y\|^{-1}\, dxdy$$ the Coulomb energy of the uniform charge distribution on $A$. \[thm:sharp-3\] There exists a constant $c>0$ such that $$\label{eq:sharp-3} \frac{\E(A^) -\E(A)}{\E(A^)} \ge c \left( \inf_\tau \frac{\Vol\bigl((\tau A)\bigtriangleup A^\bigr)} {2\Vol(A)} \right)^2\,.$$ for every $A\subset \RR^3$ of finite positive volume. Here, $\tau$ runs over all translations in $\RR^3$, and $\bigtriangleup$ denotes the symmetric difference. The exponent 2 is best possible; it is achieved for sets constructed from the unit ball by removing an annulus whose outer boundary is the unit sphere, and adding an annulus of the same volume whose inner boundary is the unit sphere. Geometric stability results where a [deficit]{} (the deviation of a functional from its optimal value) controls some measure of [asymmetry]{} (the distance from the manifold of optimizers) have been established for many classical inequalities. The first results in that direction, due to Bonnesen in the 1920s, were quantitative improvements of the isoperimetric inequality for convex sets in the plane. Two papers from the early 1990s have inspired much recent progress. One is Hall’s work on the isoperimetric inequality in $\RR^n$, where he proves stability and raises the question of optimal exponents [@Hall]; the other is the result of Bianchi and Egnell on the stability of the Sobolev inequality for $\|\|\nabla f\|\|^2$ in dimension $n\ge 3$ [@BE]. We refer the interested reader to the surveys [@O; @M]. Less is known for non-local functionals that involve convolutions, even though stability results for those have important applications in Mathematical Physics [@Carlen]. In many variational problems for integral functionals, one can show by compactness arguments that all optimizing sequences must converge — modulo the symmetries of the functional — to extremals [@BG], but bounds for the asymmetry in terms of the deficit are a different matter. Very recently, Christ has introduced tools from additive number theory to prove stability of the Riesz-Sobolev inequality in one dimension [@Christ]. Figalli and Jerison have obtained stability results on the Brunn-Minkowski inequality for non-convex sets in $\RR^n$ [@FJ]. Fusco, Maggi, and Pratelli have proved stability of Talenti’s inequality for the solutions of Poisson’s equation [@FMP-elliptic Theorem 2]. For the Coulomb energy, Guo conjectured that $$\label{eq:YG} \E(f^)-\E(f) \ge c' \inf_{\tau} \E(f\circ\tau^{-1}\!-\!f^)$$ with some constant $c'>0$. (No normalization is required in this inequality, because both sides scale in the same way.) Since the Coulomb kernel is positive definite, the right hand side can be viewed as the square of a distance. The relationship between Eqs. and with $f=\Chi_A$ will be clarified by Lemma \[lem:alpha-YG\]. The proof of Theorem \[thm:sharp-3\] consists of two parts. After some preliminaries, we use the reflection positivity of the functional and a lemma of Fusco, Maggi, and Pratelli [@FMP] to reduce the problem to sets that are symmetric under reflection at the coordinate hyperplanes. The second part of the proof requires an estimate for the Newton potential of symmetric sets. At the end of the paper, we briefly discuss stability for other Riesz kernels and in higher dimensions. Notation, and stability in higher dimensions ============================================ By the [volume]{} of a set $A\subset \RR^n$, denoted $\Vol(A)$, we mean its $n$-dimensional Lebesgue measure. The centered open ball of the same volume is denoted by $A^$; its radius is called the [volume radius]{} of $A$, and denoted by $R_A$. The [Fraenkel asymmetry]{} of $A$ is defined by $$\label{eq:alpha} \alpha(A) = \inf_\tau \frac{\Vol\bigl((\tau A)\bigtriangleup A^\bigr)}{2\Vol(A)} \,.$$ Further, $B_R$ stands for the open ball of radius $R$ centered at the origin, and $\omega_n$ for the volume of the unit ball. The uniform surface measure that is induced on the sphere $\partial B_r\subset\RR^n$ by the ambient Lebesgue measure is denoted by $\sigma$. We consider functionals of the form $$\label{eq:E-lambda} \E(A) = \int_A\int_A \|x-y\|^{-\lambda}\, dxdy$$ with $n\ge 3$ and $\lambda\in [n-2,n)$. (The classical Coulomb energy corresponds to the case $n=3$ and $\lambda=1$.) These functionals share the properties that they are reflection positive as well as positive definite (see [@FL]). Balls uniquely maximize them among sets of given volume [@L-Choquard]; balls are also the unique convex sets for which certain related overdetermined boundary-value problems have solutions [@Reichel]. By scaling, $$\label{eq:scaling} \E(A)\le \E(A^) = \mbox{Constant}\cdot (\Vol(A))^{2-\frac{\lambda}{n}}\,.$$ The [deficit]{} of $A$ is defined by $$\label{def:delta} \delta(A)=\frac{\E(A^) -\E(A)}{\E(A^)}\,.$$ Each of the functionals can be expressed in terms of the corresponding [Riesz potential]{} $$\label{eq:Phi-lambda} \Phi_A(x) = \int_A \|x-y\|^{-\lambda}\, dy\,,\qquad x\in\RR^n$$ as $\E(A)=\int_A \Phi_A$. By the Hardy-Littlewood-Sobolev inequality, $\Phi_A$ lies in $L^p$ for every $p\ge n/\lambda$. It is subharmonic on $\RR^n$ and smooth on the complement of $A$, though discontinuities may occur on $\partial A$. The Riesz potential is the unique solution of the pseudodifferential equation $$(-\Delta)^{\frac{n-\lambda}{2}} \Phi = \mbox{Constant}\cdot \Chi_A$$ that decays at infinity. The constant $c_{n,\lambda}$ can be computed with the help of the Fourier transform (see [@LL Theorem 5.9]). The Riesz-Sobolev inequality implies that $$\label{eq:dom} \int_E \Phi_A(x)\, dx \le \int_{E^} \Phi_{A^}(x)\, dx$$ for every set $E\subset\RR^n$ of finite volume (see [@LL Theorem 3.6]). In particular, $\Phi_{A^}$ is radially decreasing, and $$\label{eq:max} \sup_{x} \Phi_A(x)\le \Phi_{A^}(0) = \int_{A^}\|y\|^{-\lambda}\, dy = \frac{n\omega_n}{n-\lambda}R_A^{n-\lambda}\,.$$ Our proof of Theorem \[thm:sharp-3\] fails in higher dimensions, because the crucial lower bound in Lemma \[lem:key-3\] becomes negative. Nevertheless, we expect that the conclusion should hold — with the sharp exponent 2 and suitable constants $c_{n,\lambda}$ — for the entire family of functionals in Eq. with $n\ge 1$ and positive $\lambda\in [n-2,n)$. When $n\ge 3$ and $\lambda=n-2$, we call $\E(A)$ the [Coulomb energy]{} and $\Phi_A$ the [Newton potential]{} associated with the uniform charge distribution on $A$. The Newton potential has many special properties related to Poisson’s equation $$-\Delta \Phi_A = n(n-2)\omega_n \Chi_A\,.$$ It is harmonic on the complement of $A$, subharmonic on $\RR^n$, and satisfies the Gauss law. For later use, we compute the Newton potential of the centered ball of radius $R$ as $$\label{eq:Phi-sym} \Phi_{B_R}(x) = \omega_n R^2 \cdot \left\{ \begin{array}{ll} \frac{n}{2} - \frac{n-2}{2}\bigl(\frac{\|x\|}{R}\bigr)^2\,, \quad & \|x\|\le R\,, \\[0.2cm] \bigl(\frac{\|x\|}{R}\bigr)^{-(n-2)}\,,& \|x\|\ge R\,, \end{array}\right.$$ and its Coulomb energy as $$\E(B_R) = \frac{2n}{n+2} \omega_n^2 R^{n+2} = \frac{4}{n+2} \Vol(B_R) \cdot \Phi_{B_R}(0)\,.$$ A remarkable fact is [Talenti’s comparison principle]{}, which says that the symmetric decreasing rearrangement of the Newton potential of a charge distribution is [pointwise]{} smaller than the potential resulting from symmetrizing the charge distribution itself [@Talenti], $$\label{eq:Talenti} (\Phi_A)^(x)\le \Phi_{A^}(x)\,,\qquad x\in\RR^n\,.$$ A similar inequality holds between the gradients of these functions. The inequalities are strict, unless $A$ is essentially a ball [@FMP-elliptic Theorem 1]. Eq. is clearly stronger than the integrated version in Eq. . We will use Talenti’s comparison principle to prove the following result. \[thm:main\] Let $\E$ be defined by Eq. on $\RR^n$ with $\lambda=n-2$. For each $n\ge 3$, there exists a constant $c_n$ such that $$\label{eq:main} \frac{\E(A^)-\E(A)}{\E(A^)} \ge c_n \alpha(A) ^{n+2}$$ for every $A\subset\RR^n$ of finite positive volume. Note that the conclusion for $n=3$ is weaker than Theorem \[thm:sharp-3\]. Preliminary estimates ===================== Throughout this section, $A\subset \RR^n$ is a set of finite positive volume, the functional $\E(A)$ is given by Eq. with $\lambda\in [0,n)$, and $\Phi_A$ is the corresponding Riesz potential. We start by sharpening the bound on the maximum of $\Phi_A$ from Eq. . \[lem:max\] If $A\subset\RR^n$ has finite positive volume, then $$\sup_{x\in\RR^n} \Phi_A(x) \le \Phi_{A^}(0) \cdot \left(1- \frac{\lambda(n-\lambda)}{n^2} \alpha(A)^2 \right)\,.$$ By scaling, we may take $A^$ to be the unit ball. For $x\in\RR^n$, $$\Phi_{A^}(0)-\Phi_A(x) = \int_{A^\setminus (x-A)} \|y\|^{-\lambda}\, dy - \int_{(x-A)\setminus A^} \|y\|^{-\lambda}\, dy\,.$$ If $\alpha(A)=\alpha$, then each of the two regions of integration has volume at least $\omega_n \alpha$. The first integral is minimized when $A^\setminus (x-A)$ is an annulus whose outer boundary is the unit sphere, and the second integral is maximized when $(x-A)\setminus A^$ is an annulus whose inner boundary is the unit sphere. Using annuli of the appropriate volume, we calculate in polar coordinates $$\begin{aligned} \Phi_{A^}(0)-\Phi_A(x) &\ge n\omega_n \int_{\left(1-\alpha\right)^{1/n}}^1 \!\!r^{n-1-\lambda}\, dr - n\omega_n \int_1^{\left(1+\alpha\right)^{1/n}} \!\!\!\! r^{n-1-\lambda}\, dr\\ & = \frac{\lambda(n-\lambda)}{n^2} \Phi_{A^}(0) \int_0^\alpha \int_{-s}^s (1+t)^{-1-\frac{\lambda}{n}}\, dtds\,,\end{aligned}$$ where we have used the Fundamental Theorem of Calculus twice. By Jensen’s inequality, the value of the double integral exceeds $\alpha^2$. Lemma \[lem:max\] is needed for the proof of Theorem \[thm:main\]. In the next lemma, we use a similar estimate to relate $\alpha(A)$ to the notion of asymmetry that appears Guo’s conjecture, see Eq. . (It plays no role in the proofs of the main results.) \[lem:alpha-YG\] There exist positive constants $c_{n,\lambda}$ and $C_{n,\lambda}$ such that $$c_{n,\lambda} \alpha(A)^4 \le \inf_\tau \frac{\E(\Chi_A\circ\tau^{-1}\!-\!\Chi_{A})}{\E(A^)} \le C_{n,\lambda}\alpha(A)^{2-\frac{\lambda}{n}}$$ for every $A\subset\RR^n$ of finite positive volume. Assume by scaling that $A^$ is the unit ball, and set $\alpha=\alpha(A)$. For the first inequality, we translate $A$ such that the infimum in the middle term is assumed when $\tau$ is the identity. Since $\E$ extends to a positive definite quadratic form on $L^1\cap L^\infty$, we can use the Cauchy-Schwarz’ inequality to obtain $$\begin{aligned} \E(\Chi_A\!-\!\Chi_{A^})^\frac12 \E(A^)^\frac12 &\ge \int\int \frac{(\Chi_{A^}(x)\!-\!\Chi_A(x))\Chi_{A^}(y)}{ \|x-y\|^{\lambda}} \,dxdy\\ &=\int_{A^\setminus A} \Phi_{A^}(x)\, dx - \int_{A\setminus A^} \Phi_{A^}(x)\, dx\\ &\ge \int_{1-\alpha<\|x\|^n<1} \Phi_{A^}(x)\, dx - \int_{1<\|x\|^n<1+\alpha} \Phi_{A^}(x)\, dx\\ &\ge \mbox{Constant}\cdot\alpha^2\,,\end{aligned}$$ where the constant depends on $n$ and $\lambda$. We have used that $\Phi_{A^}$ is strictly radially decreasing to replace $A^\setminus A$ and $A\setminus A^$ with annuli. The last line follows since the gradient of $\Phi_{A^}$ vanishes only at $x=0$. For the second inequality, we translate $A$ so that the infimum in Eq. is assumed at the identity. The Hardy-Littlewood-Sobolev inequality implies that $$\inf_{\tau} \E(\Chi_{A}\circ\tau^{-1}\!-\!\Chi_{A^}) \le C_{n,\lambda} \|\|\Chi_{A}-\Chi_{A^}\|\|_{\frac{2n}{2n-\lambda}}^2 = C_{n,\lambda} \alpha^{2-\frac{\lambda}{n}}\,.$$ -1.8 We need a few more lemmas for the proof of Theorem \[thm:sharp-3\]. The following integral representation will appear several times. \[lem:outside\] Let $\rho(r)$ denote the volume radius of $A\cap B_r$. For any $R>0$, $$\E(A^) - \E(A) \ge 2\int_R^\infty \int_{A\cap \partial B_r} \Bigl(\Phi_{(A\cap B_r)^}\Big\vert_{\partial B_{\rho(r)}} \!\! - \Phi_{A\cap B_r}(x)\Bigr) \, d\sigma(x)\,dr\,.$$ The functional can be written as $$\begin{aligned} \notag \E(A) &= 2\int_A\int_A \Chi_{\{\|x\|>\|y\|\}}\, \|x-y\|^{-\lambda}\, dydx\\ \label{eq:split} &= 2\int_{A\cap B_R} \Phi_{A\cap B_{\|x\|}}(x)\,dx + 2 \int_{A\setminus B_R} \Phi_{A\cap B_{\|x\|}}(x)\, dx\\ \notag &= \E(A\cap B_R) + 2 \int_R^\infty \int_{A\cap \partial B_r} \Phi_{A\cap B_r}(x) \, d\sigma(x)\,dr\,.\end{aligned}$$ Applying Eq. to $A^$ with $\rho(R)$ in place of $R$, we see that $$\begin{aligned} \E(A^) &= \E(B_{\rho(R)} ) + 2\int_{\rho(R)}^\infty \Phi_{B_\rho}\Big\vert_{\|x\|=\rho}\, n\omega_n \rho^{n-1}\, d\rho\\ &= \E((A\cap B_R)^) + 2\int_R^\infty \Phi_{(A\cap B_r)^} \Big\vert_{\|x\|=\rho(r)} \, \sigma(A\cap \partial B_r)\, dr\,.\end{aligned}$$ In the first line, we have used that $B_{\rho(R)}\subset A^$. The Jacobian for the change of variables in the next step is determined by the relation $n\omega_n\rho^{n-1} d\rho = \sigma(A\cap \partial B_r)\, dr$. Since $\E(A\cap B_R)\le \E((A\cap B_R)^)$ by Eq. , the claim follows upon subtracting Eq. . The next lemma reduces the stability problem to bounded sets. \[lem:bdd\] For every $n\ge 3$ and $\lambda\in [n-2,n)$ there are positive constants $\alpha_{n,\lambda}$ and $c_{n,\lambda}$ with the following property. Given a set $A\subset \RR^n$ of finite positive volume with $\alpha_0:=\Vol(A\bigtriangleup A^)/(2\Vol(A))\le\alpha_{n,\lambda}$, there exists a set $\tilde A$ of the same volume such that $$\tilde A\subset \bigl (1+c_{n,\lambda}\alpha_0^{1-\frac\lambda n}\bigr)A^\,, \qquad \frac{\Vol(\tilde A \bigtriangleup A^)}{2\Vol(\tilde A)} =\alpha_0\,, \qquad \delta(\tilde{A}) \le \delta(A)\,.$$ If $A$ is symmetric about the origin, then so is $\tilde A$. By scaling, we may assume that $A^$ is the unit ball, i.e., $R_A=1$. Given $R>(1+\alpha_0)^{1/n}$, determine $r\in (1,R)$ such that $$\tilde A = (A\cap B_R) \cup (B_r\setminus A^)$$ has the same volume as $A$. By construction, $\Vol(\tilde A\bigtriangleup A^)=\Vol(A\bigtriangleup A^)$, and $r\le (1+\alpha_0)^{1/n}$. We want to choose $R$ so that $\E(\tilde A)\ge \E(A)$. It follows from Eq. that $$\E(A) \le \E(A\cap B_R) + 2 \Vol(A\setminus B_R)\cdot \sup_{\|x\|\ge R}\Phi_A(x)\,.$$ Since $\Phi_A\le \Phi_{A^} + \Phi_{A\setminus A^}$, Eq. implies $$\Phi_A(x) \le \Phi_{A^}(x) + \frac{n\omega_n}{n-\lambda} \alpha_0^{1-\lambda/n}\,.$$ Similarly, since $\tilde A\cap A=A\cap B_R$ by construction, $$\begin{aligned} \E(\tilde A) &= \int_{\tilde A\cap A} \Phi_{\tilde A\cap A}(x)\, dx + \int_{\tilde A\setminus A} 2\Phi_{\tilde A\cap A}(x) + \Phi_{\tilde A\setminus A}(x)\, dx\\ &\ge \E(A\cap B_R) + 2 \Vol(A\setminus B_R)\cdot \inf_{\|x\|\le r}\Phi_{\tilde A}(x) -\E(\tilde A \setminus A)\,,\end{aligned}$$ and $$\Phi_{\tilde A}(x) \ge \Phi_{A^}(x) - \frac{n\omega_n}{n-\lambda} \alpha_0^{1-\lambda/n}\,.$$ We use Eq. and the fact that $\Vol(\tilde A\setminus A)=\Vol(A\setminus B_R)\le \omega_n\alpha_0$ to estimate $$\E(\tilde A\setminus A) \le \mbox{Constant}\cdot\Vol(A\setminus B_R) \alpha_0^{1-\frac{\lambda}{n}}\,.$$ Collecting terms, we obtain that $$\E(\tilde A)-\E(A) \ge 2 \Vol(A\setminus B_R)\cdot \biggl(\!\Phi_{A^}\Big\vert^{\|x\|=(1+\alpha_0)^{1/n}}_{\|x\|=R} \!\!\!\! - \mbox{Constant}\cdot \alpha_0^{1-\frac{\lambda}{n}} \biggr)\,.$$ We have used that $\Phi_{A^}$ is radially decreasing to replace the inner radius $r$ by $(1+\alpha_0)^{1/n}$. Since $\Phi_{A^}$ is a smooth, strictly radially decreasing function whose gradient does not vanish outside $A^$, there exists a constant $c_{n,\lambda}$ such that the right hand side is positive for $$R=1+c_{n,\lambda}\alpha_0^{1-\lambda/n}$$ when $\alpha_0$ is sufficiently small. We now introduce reflection symmetries to the problem. The basic construction is as follows. Given a hyperplane that bisects $A$ into two parts of equal volume, the set $A$ is replaced by the union of one of these parts with its mirror image. We refer to the two sets that can be obtained in this way as [symmetrizations]{} of $A$ at the hyperplane. Clearly, the symmetrizations have the same volume as $A$. [FMP Symmetrization Lemma [@FMP Theorem 2.1].]{} [ For every $n\ge 1$ there is a positive constant $c_n$ with the following property. Given a set $A\subset \RR^n$ of finite positive volume, there exists a set $\tilde A$ obtained by successive symmetrization of $A$ at $n$ orthogonal hyperplanes such that]{} $$\alpha(\tilde A)\ge c_n\alpha(A)\,.$$ \[lem:ref-positive\] If $\lambda\in [n-2,n)$, then the set constructed in the FMP lemma satisfies $$\delta(\tilde A) \le 2^n \delta(A)\,.$$ Consider the two possible symmetrizations $A_+$ and $A_-$ of $A$ at a single hyperplane. Since $\lambda\in [n-2,n)$, the functional is reflection positive, meaning that $$\E(A_+) + \E(A_-) \ge 2 \E(A)\,,$$ see [@FL Section 1.1]. Using that $(A_+)^=(A_-)^=A^$, we subtract both sides of the inequality from $\E(A^)$ to obtain $$\delta(A_+) + \delta(A_-) \le 2 \delta(A)\,,$$ and conclude that the symmetrized sets satisfy $\delta(A_\pm)\le 2 \delta(A)$. The claim follows by repeating the construction $n$ times. We translate and rotate $\tilde A$ to a set that is symmetric at the coordinate hyperplanes, and thus symmetric under $x\mapsto -x$. Such sets have the useful property that their asymmetry is comparable to their symmetric difference from a [centered]{} ball [@FMP Lemma 2.2]. The following estimate for the potential is the key to the proof of Theorem \[thm:sharp-3\]. \[lem:key-3\] If $A\subset B_r$ is symmetric about the origin, then $$\Phi_A(x) \le \Phi_{B_r}(x) - (\sqrt2 r)^{-\lambda} \Vol(B_r\setminus A)$$ for all $x\in\partial B_r$. Let $x\in\partial B_r$ be given. The function $$f(y) = \frac{1}{2}\left( \|y-x\|^{-\lambda} + \|y+x\|^{-\lambda}\right)$$ assumes its minimum at a point on $\partial B_r$ equidistant to $x$ and $-x$, and the minimum value is $(\sqrt2 r)^{-\lambda}$. Since $A$ is symmetric, $$\Phi_A(x) = \int_A \|x-y\|^{-\lambda}\, dy \ge (\sqrt2 r)^{-\lambda} \Vol(A)\,.$$ The claim follows by replacing $A$ with $B_r\setminus A$. For the Newton potential of $A\subset B_r$, Lemma \[lem:key-3\] implies that $$\begin{aligned} \notag \Phi_{A^}\Big\vert_{\partial A^} \!\! - \sup\,\Phi_{A}\Big\vert_{\partial B_r} &\ge \omega_n \left( R_A^2-r^2 + \frac{r^n-R_A^n}{(\sqrt2 r)^{n-2}} \right)\\ \label{eq:key-3} &=\omega_n R_A^2 \bigl(-2+n2^{1-\frac n2}\bigr) \bigl(\tfrac{r}{R_A}-1\bigr) + O\bigl(\tfrac{r}{R_A}-1\bigr)^2\end{aligned}$$ uniformly in $A$ as $\frac{r}{R_A}\to 1$. Note that the leading term is positive in dimension $n=3$. Proof of the main results ========================= We specialize to the case of the Coulomb energy in $\RR^3$, where $\lambda=1$. We want to find a constant $c>0$ such that $\delta(A)\ge c\alpha(A)^2$ for all sets of finite positive volume $A\subset\RR^3$. By scaling, we may assume that $\Vol(A)= \omega_3=4\pi/3$, so that $A^$ is the unit ball. Since $\alpha(A)\le 1$ by definition, it suffices to prove the claim for $\alpha$ sufficiently small. By Lemma \[lem:ref-positive\] we may assume that $A$ is symmetric about the origin. Therefore, by [@FMP Lemma 2.2], $$\alpha_0:= \Vol(A\bigtriangleup A^)/(2\omega_3) \le 3\alpha(A)\,.$$ By Lemma \[lem:bdd\] we may assume furthermore that $A$ lies in the ball of radius $$R_0=1+c_{3,1}\alpha_0^{\frac23}\,,$$ provided that $3\alpha(A)\le \alpha_{3,1}$. We use Lemma \[lem:outside\] with $R=1$ to see that $$\begin{aligned} %\label{eq:integral} \E(A^) - \E(A) \ge 2\int_1^{R_0}\int_{A\cap \partial B_r} \Phi_{(A\cap B_r)^}\Big\vert_{\partial B_{\rho(r)}} \!\!\!\! - \Phi_{A\cap B_r}(x)\, d\sigma(x)\,dr\,,\end{aligned}$$ where $\rho(r)$ is the volume radius of $A\cap B_r$. By Eq. , the integrand is bounded from below by $$\begin{aligned} \Phi_{(A\cap B_r)^}\Big\vert_{\partial B_{\rho(r)}} \!\!\!\! - \Phi_{A\cap B_r}(x) &\ge \omega_3\,\inf_{1\le r\le R_0} \left\{\rho(r)^2-r^2 +\frac{r^3-\rho(r)^3}{\sqrt2 r}\right\}\,.\end{aligned}$$ The function inside the braces can be written as a product $$%\rho^2-r^2 +\frac{r^3-\rho^3}{\sqrt2 r} = (r^3-\rho(r)^3) \left(-\frac{r+\rho(r)}{r^2+r\rho(r)+\rho(r)^2} + \frac{1}{\sqrt{2}r}\right)\,.$$ Since the first factor is non-decreasing in $r$, it is bounded from below by $1-\rho(1)^3=\alpha_0$. This gives for the integral $$\E(A^)-\E(A)\ge \frac{8\pi}{3}\alpha_0^2 \cdot \inf_{(1-\alpha_0)^{1/3}\le \rho\le r\le R_0} \left\{-\frac{r+\rho}{r^2+r\rho+\rho^2} + \frac{1}{\sqrt{2}r} \right\}\,.$$ The infimum is strictly positive for $\alpha_0$ sufficiently small. Since $\alpha_0\ge \alpha(A)$ by definition, the theorem follows. The proof of Theorem \[thm:sharp-3\] used that the Coulomb kernel $\|x\|^{-1}$ is symmetric decreasing and reflection positive, without taking advantage of the special properties of the Newton potential. Since all estimates used in the proof depend continuously on $\lambda$, the conclusion extends to nearby values. [Corollary]{} [ Let $\E_\lambda$ be defined by Eq. for $n=3$ and $\lambda>1$, and let $\delta_\lambda$ be the corresponding deficit given by Eq. . For every $\lambda$ sufficiently close to 1 there exists a constant $c_\lambda$ such that $$\delta_\lambda(A)\ge c_\lambda\alpha(A)^2$$ for all $A\subset\RR^3$.]{} Finally we turn to the Coulomb energy in dimension $n\ge 3$. Let $n\ge 3$ and $\lambda=n-2$. Assume, by scaling, that $A^$ is the unit ball, and let $\alpha=\alpha(A)$ be the asymmetry of $A$. Since $\int_{A} \Phi_A \le \int_{A^}(\Phi_A)^$, $$\E(A^)-\E(A)\ge \int_{A^} \Phi_{A^} -\bigl(\Phi_A\bigr)^\,dx\,.$$ By Talenti’s comparison principle, the integrand is nonnegative. Moreover, by Lemma \[lem:max\] and Eq. , $$\begin{aligned} \Phi_{A^}(x) -\bigl(\Phi_A\bigr)^(x) &\ge \Phi_{A^}(x) - \sup_y \Phi_A(y)\\ &\ge \frac{n-2}{2} \omega_n \left(\frac{2\alpha^2}{n} -\|x\|^2 \right)\,.\end{aligned}$$ Integration yields $$\begin{aligned} \E(A^)-\E(A) &\ge \frac{n-2}{2}\omega_n \int_{A^} \Bigl[\frac{2\alpha^2}{n}-\|x\|^2\Bigr]_+\, dx\\ & =\frac{n-2}{2n}\E(A^)\cdot \biggl(\frac{\sqrt2 \alpha}{\sqrt n}\biggr)^{n+2} \,,\end{aligned}$$ which proves Eq. with $c_n = (n-2)2^{n/2}/n^{2+n/2}$. Acknowledgments {#acknowledgments .unnumbered} =============== The authors thank Nicola Fusco, Francesco Maggi, and Aldo Pratelli for sharing their manuscript [@FMP-elliptic], and for their hospitality on several occasions. 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--- author: - 'D. Péquignot' date: 'Received ? / Accepted ?' title: 'Heating of blue compact dwarf galaxies: gas distribution and photoionization by stars in IZw 18' --- [Photoionization models so far are unable to account for the high electron temperature $T_e$(\[\]) implied by the line intensity ratio \[\]$\lambda$4363Å/\[\]$\lambda$5007Å in low-metallicity blue compact dwarf galaxies, casting doubts on the assumption of photoionization by hot stars as the dominant source of heating of the gas in these objects of large cosmological significance.]{} [Combinations of runs of the 1-D photoionization code NEBU are used to explore alternative models for the prototype giant region shell IZw18NW, with no reference to the filling factor concept and with due consideration for geometrical and stellar evolution constraints.]{} [Acceptable models for IZw18NW are obtained, which represent schematically an incomplete shell comprising radiation-bounded condensations embedded in a low-density matter-bounded diffuse medium. The thermal pressure contrast between gas components is about a factor 7. The diffuse phase can be in pressure balance with the hot superbubble fed by mechanical energy from the inner massive star cluster. The failure of previous modellings is ascribed to (1) the adoption of an inadequate small-scale gas density distribution, which proves critical when the collisional excitation of hydrogen contributes significantly to the cooling of the gas, and possibly (2) a too restrictive implementation of Wolf-Rayet stars in synthetic stellar cluster spectral energy distributions. A neutral gas component heated by soft X-rays, whose power is less than 1% of the star cluster luminosity and consistent with CHANDRA data, can explain the low-ionization fine-structure lines detected by SPITZER. \[O/Fe\] is slightly smaller in IZw18NW than in Galactic Halo stars of similar metallicity and \[C/O\] is correlatively large.]{} [Extra heating by, , dissipation of mechanical energy is not required to explain $T_e$(\[\]) in IZw18. Important astrophysical developments are at stakes in the 5% uncertainty attached to collision strengths.]{} Introduction {#intro} ============ The optical properties of Blue Compact Dwarf (BCD) galaxies are similar to those of Giant Extragalactic Regions (GEHIIR). Their blue continuum arises from one or several young Massive Star Clusters (MSC), which harbour extremely large numbers of massive stars. BCDs are relatively isolated, small-sized, metal-poor galaxies (Kunth & Östlin 2000) and may be the rare ‘living fossils’ of a formerly common population. BCDs can provide invaluable pieces of information about the primordial abundance of helium (, Davidson & Kinman 1985), the chemical composition of the InterStellar Medium (ISM, , Izotov et al. 2006), the formation and evolution of massive stars, and the early evolution of galaxies at large redshift. Among them, stands out as one of the most oxygen-poor BCDs known (, Izotov et al. 1999) and a young galaxy candidate in the Local Universe (, Izotov & Thuan 2004). The line emission of regions is believed to be governed by radiation from massive stars, but spectroscopic diagnostics most often indicate spatial fluctuations of the electron temperature (see the dimensionless parameter $t^2$, Peimbert, 1967), that appear larger than those computed in [usual]{} photoionization models, suggesting an [extra heating]{} of the emitting gas (, Peimbert, 1995; Luridiana et al. 1999). Until the cause(s) of this failure of photoionization models can be identified, a sword of Damocles is hung over a basic tool of astrophysics. Tsamis & Péquignot (2005) showed that, in the GEHIIR 30Dor of the LMC, the various diagnostics could be made compatible with one another if the ionized gas were [chemically inhomogeneous]{} over small spatial scales. A pure photoionization model could then account for the spectrum of a bright filament of this nebula. Although this new model needs confirmation, it is in suggestive agreement with a scenario by Tenorio-Tagle (1996) of a recycling of supernova ejecta through a rain of metal-rich droplets cooling and condensing in the Galaxy halo, then falling back on to the Galactic disc and incorporating into the ISM without significant mixing until a new region eventually forms. If this class of photoionization models is finally accepted, extra heating will not be required for objects like 30Dor, with near Galactic metallicity. Another problem is encountered in low-metallicity (‘low-Z’) BCDs (Appendix A). In BCDs, available spectroscopic data do not provide signatures for $t^2$’s, but a major concern of photoionization models is explaining the high temperature () infered from the observed intensity ratio $r$() = 4363/(5007+4959). Thus, Stasińska & Schaerer (1999, SS99) conclude that photoionization by stars fails to explain $r$() in the GEHIIR NW and that photoionization must be supplemented by other heating mechanisms. A requirement for extra heating is indirectly stated by Luridiana et al. (1999) for NGC2363. A possible heating mechanism is conversion of mechanical energy provided by stellar winds and supernovae, although a conclusion of Luridiana et al. (2001) does not invite to optimism. A limitation of this mechanism is that most of this mechanical energy is likely to dissipate in hot, steadily expanding superbubbles (Martin, 1996; Tenorio-Tagle et al. 2006). It is doubtful that heat conduction from this coronal gas could induce enough localized enhancement of in the photoionized gas (, Maciejewski et al. 1996), even though Slavin et al. (1993) suggest that turbulent mixing may favour an energy transfer. Martin (1997) suggests that shocks could help to explain the trend of ionization throughout the diffuse interstellar gas of BCDs, but concedes that “shocks are only being invoked as a secondary signal in gas with very low surface brightness”. Finally, photoelectric heating from dust is inefficient in metal-poor hot gas conditions (Bakes & Tielens 1994). Nevertheless, the conclusion of SS99 is now accepted in many studies of GEHIIRs. It entails so far-reaching consequences concerning the physics of galaxies at large redshifts as to deserve close scrutiny. If, for exemple, the difference between observed and computed () in the model by SS99 were to be accounted for by artificially raising the heat input proportionally to the photoionization heating, then the total heat input in the emitting gas should be doubled. [This problem therefore deals with the global energetics of the early universe]{}. After reviewing previous models for NW (Sect. \[prev\_IZ\]), observations and new photoionization models are described in Sects. \[obs\] & \[newmod\]. Results presented in Sect. \[res\] are discussed in Sect. \[disc\]. Concluding remarks appear in Sect. \[concl\]. Models for other GEHIIRs are reviewed in Appendix A. Concepts undelying the new photoionization models are stated in Appendices B and C. Photoionization models for NW {#prev_IZ} ============================= Early models are reviewed by SS99. Dufour et al. (1988) envisioned a collection of small regions of different excitations. Campbell (1990) proposed to enhance $r$() by collisional quenching of 5007 in an ultra-compact structure (electron density = 10$^5$). Stevenson et al. (1993) modelled a uniform sphere of radius $\sim$ 0.4. Fairly satisfactory computed emission-line spectra were obtained, but the model regions were inacceptably compact according to subsequent imaging. Firstly, NW is essentially an incomplete region shell of some 5 in diameter surrounding a young MSC, which is [not spatially coincident]{} with the ionized gas. Secondly, both the highly -sensitive line 4363 and the high-ionization line 4686 are detected throughout the whole shell and beyond. Recent modelling attempts {#prev_recent} ------------------------- According to SS99, a ‘model’ is basically a uniform, matter-bounded spherical shell, whose only free parameter is a [filling factor]{} $\epsilon$. The hydrogen density is = 10$^2$, inspired by the electron density () derived from the observed doublet intensity ratio $r$() = 6716/6731. The central ionizing source is a synthetic stellar cluster, which fits the observed continuum flux at 3327Å and maximizes the nebular emission. The inner angular radius is 1.5. The outer radius $r_{\rm out}$ is defined by the condition that the computed 5007/3727 ratio fits the observed one. For increasing $\epsilon$, the material is on average closer to the source and more ionized, which must be compensated for by increasing the optical depth to keep / constant, so that the computed flux increases and / decreases. For $\epsilon$ $>$ 0.1, the shell becomes radiation bounded, / grows larger than it should and $r_{\rm out}$ becomes less than the observed value ($\sim$ 2.5). Because of these trends, SS99 discard large-$\epsilon$ models and select a model with $\epsilon$ $\sim$ 0.01, on the basis that the computed $r_{\rm out}$, and are roughly acceptable[^1]. This ‘best model’ presents two major drawbacks: (1) as stated by SS99, the computed $r$() is too small by a highly significant factor 1.3, and (2) and are grossly underestimated. The line problem {#prev_oiii} ----------------- Concerning $r$(), SS99 note without justification that, for different values of , “no acceptable solution is found”. It will become clear that this cursory statement is central for concluding that extra heating is required. In response, Viegas (2002, hereafter V02) makes the correct point that adopting a density less than 10$^2$ (and $\epsilon$ = 1) can help improving the computed $r$(). However, in the example shown by V02 ( = 30), $r$() is still 10% low[^2] and $r$() is somewhat off. Moreover, not only and , but now [as well]{} is strongly underpredicted, in accordance with the analysis of SS99. V02 then proposes that radiation-bounded filaments with density 10$^4$ are embedded in the low-density gas at different distances from the source. If the emission of and (together with ) can indeed be much increased in this way, this denser component entails serious difficulties. Firstly, since the computed / ratio is not very much less than the observed one in these filaments, a sizeable fraction of must come from them (together with ) and since, due to enhanced cooling (Appendix\[filling\]), $r$() is now [half]{} the observed value, any composite model accounting simultaneously for 5007 and 3727 will underpredict $r$() in the same manner as the uniform model of SS99. Secondly, since at least half the emission should come from the filaments, in which $r$() is again only half the observed value (large ), the composite $r$() will be inacceptably off. Thirdly, no explicit solution is exhibited and it is unclear how a composite model of the kind envisaged by V02 will simultaneouly match all lines. From the evidences she presents, V02 is not founded to claim that “pure photoionization can explain observations”. The inconclusiveness of the alternative she proposes effectively reinforces the standpoint of SS99. The and line problem {#prev_oi} ---------------------- If SS99 regard the discrepancy as a highly significant feature, they are optimistic concerning the photoionization origin of 6300+63, underpredicted by 2 in their model. Two configurations are envisaged by SS99. ### Extremely dense filaments? {#prev_oidens} In a first configuration, radiation-bounded filaments of density 10$^6$ are embedded in the region: the density is so high as to severely quench most lines other than and : only $\sim$ 10% of 6716+31 arises from these filaments, ensuring that $r$() is not much influenced. This attempt to solve in anticipation the problem met by V02 (Sect. \[prev\_oiii\]) raises three difficulties, however: (1) condensations which contrast in density by a factor of 10$^4$ with their surroundings and present a large enough covering factor ($\sim$10% according to SS99) as to intercept a significant fraction of the primary radiation would probably represent most of the mass; (2) since the main body of the model region produces only one quarter of the observed 6716+31 flux, it is not clear where this doublet would be emitted[^3]; and (3) this highly artificial, strictly dual density distribution is not the schematic, first-approximation representation of some more complex reality, rather it is a completely essential feature of the model since [any material at intermediate densities]{} would usefully emit plenty of 6716+31, but lead to a totally wrong $r$() as in the description by V02. ### Extremely distant filaments? {#prev_oidist} The second configuration proposed by SS99 involves radiation bounded ‘ filaments’ of density 10$^2$, located at $\sim$ 20 from the source. If the spectroscopic objections of Sects. \[prev\_oiii\] & \[prev\_oidens\] are now removed since density is moderate and ionization is low in the filaments, new difficulties arise, notably with [geometry]{}: (1) the filaments observed at $>$ 10 or more from the NW MSC of have such a low surface brightness as to contribute negligibly to the brightness of the main shell (if they were projected upon it); (2) the spectrum of this weak emission up to $\sim$ 15 $-$ “Halo” of Vílchez & Iglesias-Páramo (1998), “ Arc” and “Loop” of Izotov et al. (2001) $-$ shows a flux ratio 5007/3727 of order unity, whereas this ratio is 1/300 in the putative filaments, suggesting that the bulk of the emission observed at these distances arises from a gas, whose density is much less than 10$^2$; (3) accepting all the same the existence of distant emitting regions, a very peculiar geometry would be required to project these regions [precisely and uniquely]{} upon the material of the irregular bright NW shell to be modelled; and (4) in projection, this shell appears as a 1.5$-$2.5 ‘ring’, which intercepts 1/200 of a 20-radius sphere (including both the front and rear sides), incommensurable with the covering factor $\sim$ 1/10 assumed by SS99. Previous models: conclusion {#prev_IZ_concl} --------------------------- Attempts to model fail to explain not only $r$() but the and lines as well. It is difficult to follow SS99 when they claim that they are “not too far from a completely satisfactory photoionization model” of . The explanatory value of their description is so loose as to jeopardize any inference drawn from it, including the requirement for extra heating in . In Appendix\[prevG\], a review of models obtained for other GEHIIRs reveals general trends and problems, which can be valuably analysed using the example of . Observations of {#obs} ================ Basic properties {#basic} ---------------- Two bright regions 5 apart, NW and SE, correspond to two young MSCs associated with two distinct GEHIIRs, surrounded by a common irregular, filamentary halo of diffuse ionized gas (, Izotov et al. 2001), immersed in a radio 21cm envelope rotating around the centre of mass located in between the GEHIIRs (, van Zee et al. 1998). Although the column density peaks in the central region, large structures have no stellar counterparts. A fainter cluster, ‘Component C’, deprived of massive stars (no prominent region), appears at 22 to the NW of the main body. The two young MSCs, 1$-$5Myrs old, are the recent manifestations of a larger starburst, which started some 15Myrs ago in Component C and 20Myrs ago in the central region (Izotov & Thuan 2004, IT04). A 20-25Myrs age is consistent with the dynamics of the superbubble studied by Martin (1996). In a radio study, Hirashita & Hunt (2006) suggest 12$-$15Myrs. is classified as a ‘passive BCD’ (, Hirashita & Hunt 2006), that is, the MSCs themselves are relatively diffuse, the stellar formation rate (SFR) is relatively low (Sect. \[disc\_gas\]) and the starburst is not instantaneous. That a background population 300$-$500Myrs old may be the first generation of stars in this galaxy (Papaderos et al. 2002; IT04) is contested by Aloisi et al. (2007). The extended optical halo of is mostly due to ionized gas emission. Unlike for usual BCDs, the bulk of the stars in is highly concentrated, suggesting perhaps a young structure (Papaderos et al. 2002). The distance to , first quoted as 10Mpc, has been revised to $\sim$ 13Mpc (Östlin 2000) after correcting the Hubble flow for the attraction of the Virgo cluster. From AGB star magnitudes, IT04 obtain 14$\pm$1.5Mpc. At the distance $D$ = 4.0$\times$10$^{25}$cm (12.97Mpc) adopted here, the diameter of the bright region NW (5) is over 300pc. From new deep [hst]{} photometry revealing a red giant branch and Cepheid variables, Aloisi et al. (2007) obtain 18$\pm$2Mpc. Except for scaling, present results are just marginally changed if this larger distance is confirmed (Sect. \[disc\_eva\]). Absolute line fluxes and reddening {#abs_red} ----------------------------------- According to Cannon et al. (2002, CSGD02), the absolute fluxes in the 5 polygons paving the NW region and the 7 polygons paving the SE region are 4.9 and 1.7 respectively in units of 10$^{-14}$. Polygons NW D6 and SE D8 do not exactly belong to the main body of the regions and are dismissed. Tenuous emission around the polygons is also neglected. The excess over the Case B recombination value of the observed average / ratios, 2.94 and 2.97 in the NW and SE respectively, is attributed to dust reddening by CSGD02, who rightly doubt the large collisional excitation obtained by SS99 (Appendix\[fiat\_h\]). It remains that the Balmer decrement is influenced by collisions and that the reddening correction to the observed spectrum of has been overestimated. Collisional excitation results from a subtle anticorrelation between and N(H$^0$)/N(H$^+$) within the nebula and can only be determined from a photoionization model (Contrary to a statement by CSGD02, the maximum effect does [not]{} correspond to the hottest gas). The usually adopted recombination ratio is / = 2.75$\pm$0.01 (Izotov et al. 1999). It is anticipated that, according to present models (Sect. \[res\]), a better / is 2.83$\pm$0.02. For use in the present study, published dereddened intensities (also corrected for stellar absorption lines) have been re-reddened by $\Delta$E(B-V) = $-$0.04 (in view of final results, a more nearly accurate correction could be $\Delta$E(B-V) = $-$0.03). Then the typical E(B-V) for NW shifts from 0.08 to 0.04, out of which the foreground Galactic contribution is about 0.02 (Schlegel et al. 1998). The reddening corrected fluxes for the main NW and SE regions are I() = 5.6 and 2.0 respectively in units of 10$^{-14}$. In these units, the flux is 33.0 over the central 13.7$\times$10.5 [hst]{} field (Hunter & Thronson 1995; CSGD02) and 42.0 over a 60$\times$60 field (Dufour & Hester 1990). Adopting overall averages / = 2.8 and E(B-V) = 0.06, the total dereddened flux for is 18.3. Assuming that all of the ionizing photon sources belong to the bright NW and SE MSCs and that is globally radiation bounded, the fraction of photons absorbed in the two main regions is 0.41. Let $Q$ and $Q_{\rm abs}$ be respectively the number of photons (s$^{-1}$) emitted by the MSC and absorbed by the main shell of NW alone. The fraction $Q_{\rm abs}$/$Q$ may be smaller than 0.41 for two reasons. Firstly, as expansion proceeds, the shells around the starbursts become more ‘porous’ due to instabilities and the more evolved NW shell may be more affected. Assuming that no photon escape from the SE shell leads to a minimum $Q_{\rm abs}$/$Q$ = 0.34. A more realistic value is probably $Q_{\rm abs}$/$Q$ = 0.39$\pm$0.02, since a complete absorption in the SE would result in a strong asymmetry of the diffuse halo, which is not observed. Secondly, photons may escape from . This effect is probably weak, given the amount and extension of in . The adopted nominal absorbed fraction for the NW shell will be $Q_{\rm abs}$/$Q$ = 0.37$\pm$0.03, with 0.30 a conservative lower limit, obtained for a 25-30% escape from . Spectroscopic observation summary {#spectr} --------------------------------- The optical spectrum of has been observed for decades (Sargent & Searle 1970; Skillman & Kennicutt, 1993, SK93; Legrand et al. 1997; Izotov et al. 1997, 1997b; Izotov & Thuan, 1998; Vílchez & Iglesias-Páramo, 1998, VI98; Izotov et al. 1999, ICF99; Izotov et al. 2001; Thuan & Izotov 2005, TI05; Izotov et al. 2006) with many instruments ([hale]{}, [kpno]{}, [mmt]{}, [keck]{}, [cfht]{}, etc.), the UV spectrum with [iue]{} (Dufour et al. 1988) and [hst]{} (Garnett et al. 1997; Izotov & Thuan, 1999, IT99), the IR spectrum with [spitzer]{} (Wu et al. 2006, 2007), and the radio continuum with [vla]{} (Hunt et al. 2005; Cannon et al. 2005). The [iue]{} aperture encompasses all of the bright regions. In addition to 1909, there are indications for the presence of 1549 and 1883+92. The [hst]{} spectrum allows a direct comparison of with optical lines, but corresponds to such a limited area (0.86) as to raise the question of the representativeness of the observation for NW as a whole. Nonetheless, / is identical within 10% in available measurements, once the re-evaluation of the flux within the [iue]{} aperture is taken into account (Dufour & Hester 1990). The high-resolution mid-IR spectra of (Wu et al. 2007, Wu07) are secured with a 4.7$\times$11.3 slit. Over the 13.7$\times$10.5 [hst]{} field, the de-redenned flux is 13.4, while the flux from strictly the two central regions, which fill only part of the [spitzer]{} slit, is 7.6. The adopted flux corresponding to the mid-IR spectra is taken as 10$\pm$1, the value also used by Dufour & Hester (1990) for the (partial) [iue]{} aperture. Measuring line fluxes on the published tracings show excellent agreement with tabulated values, except for 18.7$\mu$, whose flux is tentatively shifted from 2.3 to 2.8$\times$10$^{-15}$. The UV and mid-IR spectra are not fully specific to NW. An average de-reddened emission line spectrum for NW, close to the one secured by ICF99 in the optical range, is presented in Col.2 (‘Obs.’) of Table \[tab\_res\] (Line identifications in Col.1; Cols.3$-$6 are presented in Sect. \[res\]). This spectrum differs little from those by Izotov & Thuan (1998) and SK93. A rather deep, high-resolution red spectrum is presented by SK93. A few weak lines are taken from a deep blue MMT spectrum by TI05, who however quote an 3727 flux larger than in earlier studies. Absolute fluxes for and the radio continuum are given on top of Table \[tab\_res\]. The 21cm and 3.6cm fluxes, obtained from Cannon et al. (2005) as a sum of 3 contours for the NW shell, partly originate in non-thermal processes, not considered here. Line intensities are relative to =1000. The intensity ratio 6584/6548 quoted by SK93 is smaller than the theoretical value: this is presumably due to the presence of a broad component (VI98). Correcting for the pseudo-continuum, the theoretical ratio is recovered and a new, smaller value is obtained for the sum of the doublet. 7320+30 (SK93) is uncertain and difficult to link to . Taking into account weak (undetected) lines, such as 4724, 4702, 4734, 4755, a continuum slightly lower than the one adopted by TI05 leads to a moderate increase of the line fluxes. Lines 4906, 5158 and 5176 are seen in the tracing by IT05, with tentative intensities 3, 2 and 2 (=1000) respectively. Only is considered in Table \[tab\_res\] (It is noted that the predicted intensities for these and lines will be $\sim$1). The most critical (de-reddened) line ratio is $r$() = 0.0246, the value also adopted by SS99. This is 3.1% larger than the often quoted value by SK93 (2slit), 0.6% smaller than the value by ICF99 (1.5slit) and 2.3% smaller than in the blue spectrum by TI05 (2slit). New photoionization models for NW {#newmod} ================================= Models are computed using the standard photoionization code [nebu]{} (Péquignot et al. 2001) in spherical symmetry with a central point-like source, suited to the apparent geometry of NW since the bulk of the stars of the NW MSC belongs to a cavity surrounded by the GEHIIR shell. Radiation-bounded filaments embedded in a diffuse medium are modelled. The reader is referred to Appendices \[gas\_distrib\] & \[WRstars\] for a perspective to the present approach. Atomic data are considered in Appendix \[fiat\]. Stellar ionizing radiation {#mod_star} -------------------------- The central source Spectral Energy Distribution (SED) is treated analytically, with no precise reference to existing synthetic stellar cluster SEDs (Appendix\[WRstars\]). No effort is done to describe the optical+UV continuum. The continuum flux at 3327Å (de Mello et al. 1998) is not used to constrain the power of the MSC. Here, this constraint can be replaced to great advantage by the fraction of ionizing photons absorbed in the shell (Sect. \[abs\_red\]). The continuous distribution of stellar masses most often results in an approximately exponential decrease of flux with photon energy from 1 to 4ryd in the SED of current synthetic MSCs (, Luridiana et al. 2003). The sum of two black bodies at different temperatures can mimic this shape, yet providing flexibility to study the influence of the SED. The source of ionizing radiation is described as the sum of a hot black body, BB1 (temperature $T1$$\geq$60kK; luminosity $L1$), and a cooler one, BB2 ($T2$ = 40$-$50kK; $L2$). A constant scaling factor $\delta_4$ ($\leq$1), reminiscent of the discontinuity appearing in the SED of model stars (, Leitherer et al. 1999) and constrained by the observed intensity of 4686, is applied to the BB1 flux at $\geq$ 4ryd. The ionizing continuum depends on [five free parameters]{}. The adopted $T2$ range is reminiscent of massive main sequence stars and lower $T2$’s need not be considered. A sufficiently large range of $T1$ values ought to be considered, as the high-energy tail of the intrinsic SED is influenced by quite a few WR stars, whose properties are either uncertain or unknowable (Appendix \[WRstars\]). Ionized shell {#mod_shell} ------------- The NW shell extends from $R_{\rm i}$ = 2.85$\times10^{20}$cm to $R_{\rm f}$ = 4.75$\times10^{20}$cm (1.5 and 2.5 at $D$ = 4.0$\times$10$^{25}$cm). In ‘genuine’ models a smooth small-scale density distribution is assumed (gas filling factor $\epsilon$ unity). The gas density is defined by means of the following general law for a variable gas pressure $P$, given as a function of the radial optical depth, $\tau$, at 13.6eV: $$P(\tau)=\frac{P_{\rm out}+P_{\rm in}}{2}+\frac{P_{\rm out}-P_{\rm in}}{\pi} \tan^{-1}\Bigl[\kappa\log\Bigl(\frac{\tau}{\tau_{\rm c}}\Bigl)\Bigr].$$ This law is a convenient tool to explore the effects of the density distribution on the model predictions. $P$ is related to the pair of (, ) via the ideal gas law, with derived from solving the statistical equilibrium equations at each step. At the first step of the computation ($\tau = 0$), the initial pressure is $P_{\rm in}$, while at the last step ($\tau = \tau_{\rm m} \sim \infty$) the final pressure is $P_{\rm out}$. A smooth, rapid transition is obtained here by adopting $\kappa = 30$ in all computations. Eq.1 introduces [three free parameters]{}: $P_{\rm in}$, $P_{\rm out}$, and the optical depth $\tau_{\rm c}$ at which the transition from inner to outer pressure occurs. The picture of a filament core embedded in a dilute medium dictates that $P_{\rm in}$ $<$ $P_{\rm out}$. Each filament produces a radial shadow, which emits much less than the material in front of the filament and the filament itself, since it is only subject to the weak, very soft, diffuse field from the rest of the nebula. The shadows are neglected. In order to represent radiation-bounded filaments embedded in a low density medium (Appendix\[filling\]), at least two sectors are needed: a ‘Sector 1’ with $\tau_{\rm m}\gg\tau_{\rm c}$ (radial directions crossing a filament) and a ‘Sector 2’ with $\tau_{\rm m} < \tau_{\rm c}$. To first order, only two sectors are considered. Observation shows that the emission, although definitely extended, is relatively weaker in the filaments surrounding the main shell (VI98; Izotov et al. 2001). This deficit of , unrelated to an outward decrease of the ionization parameter since is a pure ‘photon counting’ line above 4ryd, suggests instead that in no radial direction is the main shell totally deprived of absorbing gas. With the concern of reaching a more significant description, the same small $\tau_{\rm m}$(3) = 0.05 will be attached to the remaining ‘Sector 3’ required to make up the covering factor of the source to unity in all complete models. The emission of Sector 3, a moderate contribution to the intensity, does not impact on conclusions concerning the main shell and the source. For simplicity, in any given run, the values of the three defining parameters of Eq.1 are assumed to be shared by all three sectors. Note that $\tau_{\rm c}$ and $P_{\rm out}$ act only in Sector 1. The [topology]{} (Appendix\[sphere\_slit\]) of the model shell is fully determined by specifying in addition the covering factors $f^{cov}_1$ of Sector 1 (radiation-bounded) and $f^{cov}_2$ of Sector 2 (matter-bounded), with the condition: $$f^{cov}_3 = 1 - (f^{cov}_1 + f^{cov}_2) > 0,$$ and finally the optical depth $\tau_{\rm m}$(2) ($< \tau_{\rm c}$) of Sector 2. The full model shell structure depends on [six free parameters]{}. Adopting the same $R_{\rm i}$ and the same parameters for $P(\tau)$ in the three sectors and assuming that the outer radius of Sector 1 is $R_{\rm f}$ make the computed outer radii of other sectors to be smaller than $R_{\rm f}$. If, however, one would like Sector 2 to extend up to $R_{\rm f}$ and perhaps beyond, models should be re-run for this sector using now a $P(\tau)$ with $P_{\rm out}(2) < P_{\rm in}$ and $\tau_{\rm c}$$<$1. No significant consequences on the computed spectrum result from this change, as the increase of radius and the decrease of density in the outermost layers of Sector 2 (say, $\tau$$\sim$1) have opposite effects on the ‘local’ ionization. Also, ‘improving’ the artificial geometry of Sector 3 (a thin shell at radius $R_{\rm i}$) by assuming a lower $P_{\rm in}$(3) or else a filling $\epsilon\ll 1$ would not change at all the intensity of , while the emission of other lines from this sector is negligible. Although three sectors are considered, Sector 3 is of no practical consequence for the main shell and no parameter is attached to it. A model based on the above description will be termed a ‘two-sector model’ (Sect. \[res\_M2comp\]). ------------------- ---------------------------------------------------------------- Parameter Constraints \[0.1cm\] E(B-V) /; (No freedom: E(B-V) = 0.04) $R_{\rm i}$ No freedom $R_{\rm i}$ = 2.85$\times10^{20}$cm $f^{cov}$ Absolute I() = 5.6$\times$10$^{-14}$ $L1$, $L2$ Cluster SED; $f^{cov} \leq 1.0$; $Q_{\rm abs}$/$Q$$\sim$0.37 $T1$, $T2$ $\delta_4 \leq 1.0$; (log($Q_{\rm He}$/$Q$)$\sim$$-$0.5) $\delta_4$ 4686 \[0.1cm\] He He/H = 0.08; 5876? C 1909 N 6584 O 5007 Ne 3869 Mg Mg/Ar = 10.; 4571? Al Al/Ar = 1.; 1855? Si Si/Ar = 10.; 1883? S S/Ar = 4.37; $<$$>$? Ar 7135 Fe 4658 \[0.1cm\] $r$() = 6716/6731 ($\pm$) $\tau_{\rm m}$ 3727 $\epsilon$ $R_{\rm f}$ = 4.75$\times10^{20}$cm \[0.1cm\] $\epsilon = 1.0$ No freedom $P_{\rm out}$ $r$() $\tau_{\rm c}$ 3727 $P_{\rm in}$ $R_{\rm f}$ = 4.75$\times10^{20}$cm \[0.1cm\] $f^{cov}_1$ $f^{cov}_2$ $>$ 0; $f^{cov}$ = $f^{cov}_1$ + $f^{cov}_2$ $<$ 1 $f^{cov}_2$ $\tau_{\rm m}$(2) $<$ $\tau_{\rm c}$; global 3-D geometry $\tau_{\rm m}$(2) Fine tuning I() for given $f^{cov}_i$ \[0.1cm\] ------------------- ---------------------------------------------------------------- : Constraints on model parameters$^a$[]{data-label="tab_conv"} $^a$Question marks attached to dismissed constraints (see text).\ Model parameters and constraints {#mod_constraints} -------------------------------- Correspondances between model parameters and constraints are outlined in Table \[tab\_conv\]. The parameters are interrelated and iterations are needed to converge to a solution. The weak dependance of E(B-V) on the model Balmer decrement (Sect. \[abs\_red\]) is neglected. The SED is not fully determined by the major constraint $Q_{\rm abs}$/$Q$. Other constraints are in the form of inequalities, some are half-quantitative or deal with ‘plausibility’ arguments. One emission line is selected to constrain each elemental abundance. In Table \[tab\_conv\], a question mark is appended to those lines with unreliable intensities (Table \[tab\_res\]): the intensity of 4571+62 is given as an upper limit as the lines are barely detected (TI05) and suspected to be blended with a WR feature (Guseva et al. 2000); detection of 1855 is a simple guess from a tracing of the [hst]{} spectrum; 1883+92 is barely seen in the [iue]{} spectrum and only the first component of the doublet is detected in the [hst]{} spectrum (IT99). The abundances of Mg, Al and Si are arbitrarily linked to that of argon (Table \[tab\_conv\]), assuming abundance ratios close to solar (Lodders 2003). For simplicity, the solar S/Ar ratio is also adopted and the computed sulfur line intensities can be used to scale S/H according to any preferred criterion (Sect. \[disc\_ab\]). emission lines are blended with strong stellar absorption lines (, ICF99). He/H is set at 0.08 by number. [l\|c\|rrrr]{} Parameters & Run$^a$ &\ of model & N0-N1 & M1 & M2 & M3 & M4\ & 2 $-$ 3 & 4 & 5 & 6 & 7\ \ $T1$/10$^4$K & 10. & 10. & 8. & 8. & 12.\ $L1$/10$^{41}$& 3.5-1.6 & 3.5 & 2.0 & 1.6 & 1.25\ $\delta_4$ & .12-.67 & 0.73 & 0.83 & 0.93 &0.24\ $T2$/10$^4$K & 4. & 4. & 4. & 5. & 4.\ $L2$/10$^{41}$& 3.5-1.6 & 3.5 & 2.0 & 1.6 & 2.5\ log($Q$) $-$ 51. & 1.04-.71 & 1.054 & 0.829 & 0.779 & 0.734\ $-$log($Q_{\rm He}$/$Q$) &.462-.447 &0.445 &0.523 &0.493 &0.539\ \ $\epsilon$ & .0042-.31 & 1.00 & 1.00 & 1.00 & 1.00\ $P_{\rm in}/k$/10$^5$[cgs]{} &38.-8.3 & 5.01 & 3.40 & 2.96 & 2.71\ $P_{\rm out}/k$/10$^5$[cgs]{} & - & 23.2 & 21.7 & 25.4 & 26.8\ $\tau_{\rm c}$ & - & 5.7 & 4.9 & 4.0 & 4.3\ $f^{cov}_1$ & 1.-0.43 & 0.20 & 0.26 & 0.23 & 0.29\ $f^{cov}_2$ & - & - & 0.30 & 0.50 & 0.60\ $\tau_{\rm m}$ or $\tau_{\rm m}$(2) & .96-300.& 270. & 1.21 & 1.46 & 1.00\ \ C & 60-52 & 45.8 & 38.8 & 35.3 & 47.4\ N & 8.1-5.3 & 3.9 & 4.1 & 4.0 & 3.8\ O & 198-192 & 172. & 168. & 162. & 173.\ Ne & 33-30 & 26.4 & 25.7 & 24.8 & 26.9\ S (Table \[tab\_conv\])& 3.4-3.9 & 5.0 & 4.3 & 4.3 & 5.0\ Ar & .77-.90 & 1.14 & 0.99 & 0.98& 1.13\ Fe & 4.8-5.8 & 5.8 & 6.1 & 6.0 & 6.5\ \ $Q_{\rm abs}$/$Q$ & .20-.43 & .200 & .343 & .380 & .426\ $M_{\rm gas}$/10$^6$& .15-.92 & 1.02 & 1.52 & 1.74 & 1.79\ H$^{+}$/H & .998-.98 & .957 & .961 & .963 & .948\ O$^{0}$/O & .00-.017 & .041 & .038 & .037 & .052\ O$^{+}$/O & .076-.10 & .122 & .125 & .129 & .131\ O$^{2+}$/O & .910-.85 & .784 & .793 & .790 & .773\ O$^{3+}$/O & .014-.03 & .049 & .043 & .043 & .042\ (H$^{+}$)/10$^4$K & 1.65-1.73 & 1.873 &1.859 &1.896 &1.839\ (O$^{0}$)/10$^4$K & 1.61-1.14 & 1.040 &1.013 &1.012 &1.003\ (O$^{+}$)/10$^4$K & 1.63-1.43 & 1.320 &1.315 &1.309 &1.272\ (O$^{2+}$)/10$^4$K & 1.66-1.75 & 1.911 &1.915 &1.961 &1.898\ (O$^{3+}$)/10$^4$K & 1.76-2.1 & 2.576 &2.402 &2.449 &2.479\ (H$^{+}$)/& 99-18 & 17.3 & 11.1 & 9.8 & 9.4\ (O$^{0}$)/& 99-8.5 & 46.7 & 34.0 & 37.8 & 41.1\ (O$^{+}$)/& 99-16 & 62.7 & 48.0 & 54.0 & 54.8\ (O$^{2+}$)/& 99-18 & 15.4 & 10.2 & 8.9 & 8.5\ (O$^{3+}$)/& 100-19 & 10.1 & 7.4 & 6.3 & 5.7\ $t^2$(H$^{+}$) & .002-.014 & .032 & .026 & .028 & .030\ $t^2$(O$^{0}$) & .001-.018 & .022 & .024 & .023 & .020\ $t^2$(O$^{+}$) & .0013-.02 & .021 & .024 & .024 & .023\ $t^2$(O$^{2+}$) & .002-.008 & .013 & .010 & .010 & .010\ $t^2$(O$^{3+}$) & .0010-05 &.0008 &.0006 &.0007 &.0016\ \[0.1cm\] $^a$Constant . $N0$: =92; $N1$: =17.0.\ $^b$ from thermal pressure of Eq. (1).\ \[tab\_mod\] In the lower part of Table \[tab\_conv\] are given observational constraints for the structural parameters of the shell, depending on assumptions. In preliminary constant-density ‘runs’ ($N0$, $N1$, not genuine models; Sect. \[res\_N\]), a generalization of the approach of SS99 (Sect. \[prev\_recent\]) is adopted. A one-component model ($M1$; Sect. \[res\_M1comp\]) shows the influence of Eq. (1). Two-component models ($M2$, $M3$ and $M4$; Sect. \[res\_M2comp\]) generalize $M1$ according to Sect. \[mod\_shell\]. Results {#res} ======= [l\|r\|rc\|rc]{} Line id./Models & Obs. & N0 & N0/O & M2 & M2/O\ \ I() & 5.6 & 5.6 & 1.00 & 5.6 & 1.00\ 1.43$\:$GHz$\:$/mJy & 0.433 & 0.240& 0.55 & 0.256 & 0.59\ 8.45$\:$GHz$\:$/mJy & 0.286 & 0.202& 0.71 & 0.216 & 0.76\ \ 4861 & 1000. &1000. & 1.00 & 1000. & 1.00\ 6563 & 2860. &2840. & 0.99 & 2830. & 0.99\ 4340 & 461. & 473. & 1.03 & 473. & 1.03\ 4102 & 266. & 267. & 1.00 & 267. & 1.00\ 1215 (/10) & - &2950. & - & 3010. & -\ 2$h\nu$ (/10)& - &1550. & - & 1620. & -\ 3888 & 90.4 & 89.6 & 0.99 & 90.5 & 1.00\ 4471 & 21.4 & 34.9 & 1.63 & 35.2 & 1.64\ 5876 & 67.7 & 92.0 & 1.36 & 91.3 & 1.35\ 6678 & 25.3 & 25.3 & 1.00 & 25.6 & 1.01\ 7065 & 24.4 & 23.4 & 0.96 & 22.9 & 0.94\ 10830 & - & 251. & - & 190. & -\ 4686 & 36.8 & 36.8 & 1.00 & 36.8 & 1.00\ 1909+07 & 467. & 467. & 1.00 & 467. & 1.00\ 1882+92 & 270.: & 229. & 0.85 & 340. & 1.26\ 1855+63 & 111.:& 42.9 & 0.39 & 82.9 & 0.75\ 1664 &$<$230 & 127. & $>$.5 & 208. & $>$.9\ 1549 & 510.:& 74.2 & 0.14 & 334. & 0.65\ 1397 $\rbrace$$^a$ &$<$300&22.7& $>$.1 &127. & $>$.5\ 1398 $\rbrace$ & \* & 2.0 & \* & 24.9 & \\ 6584+48 & 9.2 & 9.2 & 1.00 & 9.2 & 1.00\ 6300+63 & 8.5 & 0.12 & 0.01 & 8.6 & 1.01\ 3726+29 & 238. & 238. & 1.00 & 238. & 1.00\ 7320+30 & 6.3: & 9.5 & 1.50 & 7.5 & 1.18\ 5007+.. &2683. &2680. & 1.00 &2680. & 1.00\ 4363 & 65.9 & 47.8 & 0.73 & 63.2 & 0.96\ 51.8$\:\mu$m & - & 174. & - & 137. & -\ 88.3$\:\mu$m & - & 216. & - & 213. & -\ 25.9$\:\mu$m &49.1 & 18.2 & 0.37 & 47.8 & 0.97\ 12.8$\:\mu$m &9.0: & 1.3 & 0.14 & 1.9 & 0.21\ 3868+.. & 191. & 191. & 1.00 & 191. & 1.00\ 15.5$\:\mu$m& 45.7& 60.0 & 1.31 & 48.9 & 1.07\ 4571+62 &$<$3.0 & 1.5 &$>$.5 & 1.2 & $>$.4\ 34.8$\:\mu$m & 157. & 4.7 & 0.03 & 22.0 & 0.14\ 6716 & 22.5 & 6.7 & 0.30 & 17.6 & 0.78\ 6731 & 16.9 & 5.1 & 0.30 & 13.1 & 0.78\ 4068 $\:\rbrace$& 3.7& 1.1& 0.41& 2.2 & 0.99\ 4071 $\rbrace$ & \ & 0.4 & \* & 1.5 & \\ 9531+.. & 114. & 130. & 1.15 & 113. & 0.99\ 6312 & 6.7 & 6.0 & 0.89 & 5.7 & 0.85\ 18.7$\:\mu$m& 28.0 & 32.2 & 1.15 & 26.2 & 0.95\ 33.5$\:\mu$m& 120. & 54.5 & 0.45 & 48.0 & 0.40\ 10.5$\:\mu$m & 48.0 & 41.7 & 0.87 & 92.6 & 1.93\ 7136+.. & 23.5 & 23.5 & 1.00 & 23.5 & 1.00\ 8.99$\:\mu$m & - & 8.6 & - & 8.1 & -\ 4711 $\rbrace$& 8.6 & 1.5 & 0.76 & 8.2 & 1.53\ 4713 $\:\:\rbrace$& \&5.0& \* & 5.0 & \\ 4740 & 4.5 & 1.2 & 0.26 & 6.2 & 1.39\ 5.34$\:\mu$m & - & 0.1 & - & 10.8 & -\ 26.0$\:\mu$m & 34. & 0.0 & 0.00 & 3.4 & 0.10\ 4658 & 4.5 & 4.5 & 1.00 & 4.5 & 1.00\ 4986 & 7.4 & 5.8 & 0.78 & 7.0 & 0.94\ 22.9$\:\mu$m & - & 2.2 & - & 3.2 & -\ 4906 & 3.0: & 1.6 & 0.54 & 3.1 & 1.02\ 4227 & 1.8 & 1.5 & 0.84 & 5.5 & 3.10\ $^a$ In Col.1, blends are indicated by braces. The observed intensity of a blend is attributed to the first line and an asterisk to the second line.\ \[tab\_res\] ----------------------- --------- ------------- ------ ------ ------ ------ Line ident. Obs. N0$-$N1 M1 M2 M3 M4 2 3 $-$ 4 5 6 7 8 $Q_{\rm abs}$/$Q$$^a$ 0.37 .20$-$.43 0.20 0.34 0.38 0.43 1549 500: .14$-$.41 1.12 0.65 0.64 1.08 $\:$6300+ 8.5 .01$-$.59 1.19 1.01 0.95 1.38 $\:$7320+ 6.3: 1.5$-$1.24 1.19 1.18 1.19 1.15 $\:$4363 65.9 .73$-$.82 0.96 0.96 1.00 0.94 $\:$25.9$\mu$ 49.1 .37$-$.84 1.08 0.97 0.94 0.98 $\:$12.8$\mu$ 9.0: .14$-$.10 0.14 0.21 0.22 0.23 15.5$\mu$ 45.7 1.31$-$1.22 1.09 1.07 1.02 1.11 $\:$34.8$\mu$ 157. .03$-$.10 0.17 0.14 0.14 0.18 $\:$1882+ 270: .85$-$.99 1.12 1.26 1.34 1.16 $\:$6716 22.5 .30$-$.54 0.96 0.78 0.77 1.10 $\:$6731 16.9 .30$-$.51 0.96 0.78 0.77 1.10 $\:$4068$^b$ 3.7$^b$ .30$-$.43 0.73 0.58 0.58 0.80 $\:$9531+ 114. 1.15$-$1.05 0.95 0.99 1.00 0.92 $\:$6312 6.7 .89$-$.84 0.77 0.85 0.87 0.75 $\:$18.7$\mu$ 28. 1.15$-$1.01 0.94 0.95 0.94 0.91 $\:$33.5$\mu$ 120. .45$-$.44 0.39 0.40 0.40 0.39 $\:$10.5$\mu$ 48. .87$-$1.50 2.15 1.93 1.97 2.15 $\:$4740 4.5 .26$-$.73 1.87 1.39 1.47 1.76 $\:$26.0$\mu$ 34. .00$-$.05 0.10 0.10 0.10 0.14 $\:$4986 7.4 .78$-$1.15 0.90 0.94 0.89 0.89 $\:$4906 3.0: .54$-$.70 0.96 1.02 1.09 1.02 $\:$4227 1.8: .84$-$2.3 2.1 3.1 3.2 3.1 \[0.1cm\] ----------------------- --------- ------------- ------ ------ ------ ------ : Model ‘predictions’ for NW: model intensities divided by observed intensities $^a$$Q_{\rm abs}$/$Q$: absolute values.\ $^b$4068 intensity not corrected for 4071 (Table \[tab\_res\]). \[tab\_compa\] Input and output model properties are listed in the first column of Table \[tab\_mod\] as: (1) five primary ionizing source parameters (Sect. \[mod\_star\]); (2) resulting numbers of photons (s$^{-1}$) $Q$ and $Q_{\rm He}$ emitted by the source above 13.6 and 24.6eV respectively; (3) four ($N0$, $N1$, $M1$) to six ($M2$, $M3$, $M4$) shell parameters (Sect. \[mod\_shell\]); (4) elemental abundances; (5) photon fraction $Q_{\rm abs}$/$Q$ absorbed in the shell; (6) mass $M_{\rm gas}$ of ionized gas in units of 10$^6$; (7) mean ionic fractions of H$^+$ and oxygen ions weighted by ; (8) average and $t^2$ weighted by $\times$$N_{\rm ion}$ and average weighted by $N_{\rm ion}$ for H$^+$ and oxygen ions. The model SEDs (Sect. \[mod\_star\]) are shown in Fig. \[fig\_SED\]: panel (a) is common to $N0$, $N1$ and $M1$; panels (b), (c) and (d) correspond to $M2$, $M3$ and $M4$ respectively. Radiation is harder and stronger in panel (a) (see ‘hardness coefficient’ $\alpha$ in caption to Fig. \[fig\_SED\]). The gas pressure laws $P(\tau)$, drawn in Fig. \[fig\_P\] (parameters in Table \[tab\_mod\]), illustrate the contrast between preliminary runs and adopted models. Line identifications and observed de-reddened intensities are provided in Cols.1 and 2 of Table \[tab\_res\]. Computed intensities appear in Col.3 and Col.5 for Run $N0$ and Model $M2$ respectively. Predictions are given for some unobserved lines (intensities are 10 times the quoted values for 1215Å and 2$h\nu$). The ratios of computed to observed intensities, noted ‘$N0$/O’ and ‘$M2$/O’ appear in Col.4 and Col.6 for $N0$ and $M2$ respectively. Ideally, these ratios should be 1.00 for all observed lines. Inasmuch as the convergence is completed, at least [all]{} lines which were used as model constraints (Table \[tab\_conv\]) must be exactly matched by construction (Table \[tab\_res\]). For the sake of evaluating the models, these lines are therefore useless. Similarly, ‘redundent’ lines ( and series, etc.), which carry no astrophysically significant information in the context, as well as unobserved lines, can be discarded. Remaining ‘useful’ lines are listed in Cols.1$-$2 of Table \[tab\_compa\] and model intensities divided by observed intensities are displayed in Cols.3$-$8 for $N0$–$N1$, $M1$–$M4$ respectively. These intensities are ‘predictions’ in that they are not considered at any step of the convergence. In Table \[tab\_compa\][^4], 4363Å stands out as the strongest, accurately measured optical line. $Q_{\rm abs}$/$Q$ is repeated in Table \[tab\_compa\]. Constant density runs with filling factor: $N0$, $N1$ {#res_N} ----------------------------------------------------- $N0$ is a preliminary run (Col.2 of Table \[tab\_mod\]) in which is constant and $Q$, $Q_{\rm He}$, $R_{\rm i}$ and $R_{\rm f}$ are about as in the description by SS99 (corrected for the larger $D$). The convergence process, involving O/H, Ne/H, etc. (Table \[tab\_conv\]), is more complete than the one performed by SS99, but the differences of procedures do not change the conclusions. If is in principle derived from $r$(), the sensitivity of $r$() to is relatively weak at the low density prevailing in the shell, while the exact value adopted for may, [in this particular structure]{}, strongly influence the computed spectrum. By changing coherently , $\epsilon$ and $f^{cov}$, the three constraints I(), / and $R_{\rm f}$ can be fulfilled along a sequence. $N0$ is extracted from this sequence by assuming, as in the SS99 run, a covering factor $f^{cov}$ = 1. The solution is close to the one chosen by SS99, with =92, $\epsilon$=0.0042, (radial) $\tau$$\sim$1 ($\equiv$ $\tau_{\rm m}$(2) in Table \[tab\_mod\]) and $r$() only $-$2.1% off the observed value. $N0$ (Cols.3$-$4 of Table \[tab\_res\]; Col.3 of Table \[tab\_compa\]) [fully confirms the very large problems met by SS99]{} with 4363 and 6300 (Sect. \[prev\_recent\]). $N0$ also fails in that $Q_{\rm abs}$/$Q$ is half the expected value. Decreasing $Q$ (SED luminosity) implies to decrease (for /) and increase $\epsilon$ (for I()). Decreasing should help increasing , thus $r$(), and the high ionization lines, largely underestimated in $N0$. Correlatively, / is restored for a larger $\tau_{\rm m}$, which helps increasing and other low-ionization lines. By further decreasing and increasing $\epsilon$, whilst fine-tuning $f^{cov}$ to keep the outer shell radius $R_{\rm f}$ (and I()) and $\delta_4$ (for ), it is possible to further increase $\tau_{\rm m}$ until the shell eventually gets [radiation bounded]{}. The resulting (unique) solution is $N1$ (Col.3 of Table \[tab\_mod\]; Col.4 of Table \[tab\_compa\]), which much improves upon $N0$ concerning and high-ionization lines, while $Q_{\rm abs}$/$Q$ $\equiv$ $f^{cov}$ = 0.43 is just a bit large. Because of the much lower density, =17, the ratio $r$() is now $+$4.7% off, worse than in $N0$, yet not decisively inacceptable. Nevertheless, the normalized $r$(), enhanced from 0.73 to 0.82, is still very significantly [too small]{}. This failure of $N1$ is illustrated in the upper panels of Figs. \[fig\_Ne\]$-$\[fig\_Oxy\]. The runs of and with nebular radius $R$ are shown in Figs. \[fig\_Ne\]a–\[fig\_Te\]a for $N0$ and Figs. \[fig\_Ne\]b–\[fig\_Te\]b for $N1$. Ionic fractions of oxygen [versus]{} $\tau$ are shown in Figs. \[fig\_Oxy\]a and \[fig\_Oxy\]b for $N0$ and $N1$ respectively. In $N0$, is everywhere above 1.55$\times 10^{4}$K. In $N1$, the inner is 3800K higher than in $N0$, but O$^{2+}$ is abundant up to $\tau$ = 15, where is below 1.40$\times 10^{4}$K and the [average]{} () is not much increased. This generalization shows that no solution with constant exists even for the rather hard SED adopted by SS99. In an extreme variant of $N1$, the SED is just one 10$^5$K black body (converged $L$ = 2.7$\times$10$^{41}$, $\delta_4$ = 0.41, log($Q_{\rm He}$/$Q$) = $-$0.27), but the normalized $r$() = 0.87 is still too small, despite the unrealistically hard SED. A one-sector photoionization model: $M1$ {#res_M1comp} ---------------------------------------- Model $M1$ includes the same primary source as Run $N0$ and again only one sector, but with controlled by Eq. 1 (Fig. \[fig\_P\]). Parameters appear in Col.4 of Table \[tab\_mod\] and predictions in Col.5 of Table \[tab\_compa\]. The litigious lines 4363 and 6300 are [very much improved]{} compared to Run $N0$ (and even $N1$), as are and . The decisive merit of Model $M1$ is to demonstrate that, with no extra free parameter, no change of shell size and no significant change of source SED, [the ‘$r$() problem’ met in $N0$ can be essentially solved]{} just by considering radiation-bounded filaments embedded in a lower density (higher ionization) medium instead of a clumped shell at constant density. While the normalized $r$() is 0.96 ( in Table \[tab\_compa\]), no dense or distant clumps of the kind postulated by SS99 (Sect. \[prev\_IZ\]) are needed to account for low-ionization lines. The pressure contrast is $<$ 5. $Q_{\rm abs}$/$Q$ is again too small, but $Q$ cannot decrease because $\delta_4$ is close to unity (Table \[tab\_mod\]). In $N0$, $\delta_4$ was (perhaps anomalously) small, enabling a shift from $N0$ to $N1$. Hardening the already hard primary ($\alpha$ = 0.7 due to large $T1$; Fig. \[fig\_SED\]) would enhance , predicted too strong. Also, $f^{cov}$ is only 0.20, resulting in an artificial cigar-like radial distribution, in which the low-density gas exactly shields the denser filaments from direct primary radiation. Obviously, the limits of the one-sector model are being reached. A matter-bounded sector is to be added for the sake of a larger absorbed fraction of photons in the shell, but not principally to improve the already quite satisfactory intensities of 4363 and 6300. Two-sector photoionization models: $M2$, $M3$, $M4$ {#res_M2comp} --------------------------------------------------- The enhancement of and related to the matter-bounded sector must be balanced by a weaker/softer SED. Models illustrate the influence of the SED (Fig. \[fig\_SED\]). Given a SED and both covering factors, then $\tau_{\rm m}$(2), $\tau_{\rm c}$ and $\delta_4$ can be fine-tuned to account for I(), / and . Iterations along the same lines as for Model $M1$ eventually lead to a model, provided that the limits on parameters are respected (Table \[tab\_conv\]). A two-parameter model sequence can be attached to any SED by considering several pairs ($f^{cov}_1$, $f^{cov}_2$), but little freedom is attached to $f^{cov}_1$, as Sector1 is where $\sim$75% of and most of come from. Then $f^{cov}_1$ must be on the order of, or moderately larger than the $f^{cov}$ of the one-sector model, say, in the range 0.2$-$0.3. Also $f^{cov}_2$ cannot be small since one-sector models are rejected (Sect. \[res\_M1comp\]) and $f^{cov}_1$+$f^{cov}_2$ must be kept significantly less than unity to enable excitation beyond the shell (Sect. \[mod\_shell\]), implying $f^{cov}_2$ $\sim$ 0.3$-$0.6. ### Model $M2$ and variants {#res_M2} Model $M2$ (Col.5 of Table \[tab\_mod\]) is the first ‘complete’ model. $Q_{\rm abs}$/$Q$ is at the low end of the nominal interval. Ouput of line intensities (Cols.5–6 of Table \[tab\_res\]; Col.6 of Table \[tab\_compa\]) is to be contrasted to the $N0$ output. Runs of and with $R$ are shown in Figs. \[fig\_Ne\]c–\[fig\_Te\]c. The sharp ‘spike’ of the curve shows how thin a radiation-bounded filament is compared to the shell. Runs of and are best seen in plots [versus]{} $\tau$ (Figs. \[fig\_Ne\]d–\[fig\_Te\]d). In plots for $M2$, vertical arrows mark the outer boundary of Sector 2. Ion fractions O$^{n+}$/O [versus]{} $R$ and $\tau$ are shown in Figs. \[fig\_Oxy\]c$-$\[fig\_Oxy\]d. Sufficient ionization is maintained in $M2$ owing to the lower average density, which also helps increasing in the high-ionization layers, despite the significantly softer radiation field (smaller $Q_{\rm He}$/$Q$ and larger $\alpha$, Fig. \[fig\_SED\]b): the inner is now 2.5$\times$10$^4$K. In Fig. \[fig\_Te\]d, the jumps of at $\tau$$\sim$0.1 and $\sim$4.7 correspond to the boundaries of the He$^{2+}$ shell (fairly well traced by O$^{3+}$ in Fig. \[fig\_Oxy\]d) and the filament respectively. Comparing Fig. \[fig\_Oxy\]d to Fig. \[fig\_Oxy\]b, the ionic fractions are qualitatively similar in Model $M2$ and Run $N1$, but the transition from O$^{2+}$ to O$^{+}$ is sharper and occurs at a smaller optical depth in $M2$. Average properties and abundances of Model $M2$ are quite similar to those of $M1$ and the predicted 4363 intensity is again slighly weak, although the score of $M2$ is significantly better for and (Col.6 of Table \[tab\_compa\]). Variants to Model $M2$ can be obtained by changing $f^{cov}_1$ and $f^{cov}_2$ within limits, while retaining source parameters (except for minute fine-tuning of $\delta_4$). In Col.1 of Table \[tab\_vari\] are listed 7 shell parameters and 6 lines extracted from Table \[tab\_compa\]. $M2$ (Col.2 of Table \[tab\_vari\]) is compared to re-converged models $M2b$ (Col.3) and $M2c$ (Col.4). Increasing $f^{cov}_2$ from a small to a large value, with $f^{cov}_1$ left unchanged, structure parameters are not much changed except for a decrease of $\tau_{\rm m}$(2) and a small decrease of O/H due to the larger weight of the hot high-ionization zone. Accordingly, is increased, but 4363 is increased by only 1%. Decreasing $f^{cov}_1$ from 0.26 to 0.22, is recovered by increasing $\tau_{\rm m}$(2) and 3727 by decreasing $\tau_{\rm c}$, with the consequence that $P_{\rm in}$ must decrease, thus increases and O/H decreases. Finally the 4363 intensity increases up to the observed value, 4740 and 10.5$\mu$ increase and 6300 decreases. ------------------------ ------ ------ ------ ------ ------ ------ Param. / line M2 M2b M2c M4b M4 M4c 2 3 4 5 6 7 $f^{cov}_1$ 0.26 0.26 0.22 0.29 0.29 0.22 $f^{cov}_2$ 0.30 0.60 0.60 0.30 0.60 0.60 $\tau_{\rm m}$(2) 1.21 0.62 0.92 2.17 1.01 1.57 $\tau_{\rm c}$ 4.9 4.7 3.8 4.6 4.3 3.0 $P_{\rm in}/k$/10$^5$ 3.4 3.4 3.1 2.7 2.7 2.4 $P_{\rm out}/k$/10$^5$ 21.7 22.5 24.6 26.8 26.8 26.8 O/H$\times$10$^7$ 168. 165. 162. 178. 173. 171. $\:$6300+ 1.01 1.00 0.92 1.42 1.37 1.06 $\:$4363 0.96 0.97 1.00 0.92 0.94 0.95 $\:$25.9$\mu$ 0.97 0.91 0.92 1.05 0.98 0.99 $\:$6716 0.78 0.81 0.79 1.05 1.10 0.84 $\:$10.5$\mu$ 1.93 1.97 2.05 2.07 2.15 1.92 $\:$4740 1.39 1.48 1.62 1.55 1.77 1.81 \[0.1cm\] ------------------------ ------ ------ ------ ------ ------ ------ : Influence of shell parameters on selected line intensity predictions for models $M2$ and $M4$ \[tab\_vari\] ### Models $M3$ and $M4$ {#res_M34} In Model $M3$, $T2$/10$^4$K is enhanced from 4 to 5. $Q_{\rm abs}$/$Q$ is larger than in $M2$ due to lower luminosity. The larger $T2$ increases the average energy of photons absorbed in the O$^{2+}$ region and the intensity of 4363 is slightly larger. In the selected example, is again exactly matched as in $M2c$, but for more ‘standard’ $f^{cov}_1$ and $f^{cov}_2$ (Col.6 of Table \[tab\_mod\]). Line intensity predictions are slightly improved (Col.7 of Table \[tab\_compa\] [versus]{} Col.4 of Table \[tab\_vari\]). In Model $M4$, $T2$ is again as in $M2$, while $T1$/10$^4$K is enhanced from 8 to 12 and $L1$/$L2$ is halved. $Q_{\rm abs}$/$Q$ is close to its allowed maximum due to radiation hardening, which also leads to $\delta_4$$\ll$1 (Col.7 of Table \[tab\_mod\]). The large flux just below 4ryd enhances simultaneously the high and low ionization lines, but the heating of the O$^{2+}$ region is lesser and the large $T1$ (positive curvature of the SED) does not favour a large $r$() (Col.8 of Table \[tab\_compa\]). From Table \[tab\_vari\], variants $M4b$ (Col.5) and $M4c$ (Col.7) of $M4$ (Col.6) fail to enhance 4363 up to the observed value. Increasing the source luminosity by 20%, thus decreasing $Q_{\rm abs}$/$Q$ from 0.43 to 0.36, has strictly no effect on the predicted after convergence. Discussion {#disc} ========== Irrespective of the ‘technical’ demand raised by $Q_{\rm abs}$/$Q$ in Sect. \[res\_M1comp\], a two-sector model is the minimum complexity of any shell topology (Sect. \[mod\_shell\]). The two degrees of freedom attached to the matter-bounded sector are inescapable. Model $M4$ (and variants) appears slightly less successful than $M2$ and $M3$ concerning 4363 and the high-ionization lines (4740, 10.5$\mu$). $M4$ has a possibly less likely SED and presents the largest $P_{\rm out}$/$P_{\rm in}$. The discussion focuses on Models $M2$ and $M3$, with $M2$ the ‘standard’ from which variants are built. Spectral energy distribution {#disc_SED} ---------------------------- Accounting for a $Q_{\rm abs}$/$Q$ larger than, say, 1/3 turns out to be demanding. Selected models correspond to nearly maximum possible values for each SED. Acceptable $Q_{\rm abs}$/$Q$ can indeed be obtained, but the latitude on the SED and power of the ionizing source is narrow. The uncertainties in evolutionary synthetic cluster models (Appendix\[WRstars\]) and in the evolutionnary status of NW itself are sufficient to provisionally accept the ‘empirical’ SED corresponding to preferred models $M2$ or $M3$ (Fig. \[fig\_SED\]) as plausible. The ‘predicted’ typical trend is $L_{\nu}$($h\nu$=1$\rightarrow$4ryd) $\propto$ exp($-h\nu$/ryd). Ionized gas distribution {#disc_gas} ------------------------ The model is most specific in that emission lines partly arise from a low density gas, while the largest is (). A density $\leq$ 10 (Table \[tab\_mod\]) appears very low by current standards of photoionization models for BCDs (Sect. \[prev\_indiv\]). Nevertheless, the superbubble model of Martin (1996) is consistent with a current SFR = 0.02yr$^{-1}$ for the whole NW+SE complex. The two best estimates in the compilation by Wu07 are 0.03 and 0.02yr$^{-1}$. Adopting half the Martin (1996)’s rate and a wind injection radius of 0.1kpc (1.5 at 13Mpc) for the NW cluster alone, expression (10) in Veilleux et al. (2005) suggests an inner pressure of the coronal gas $P/k$ $\sim$ 3$\times$10$^{5}$K, hence an ambient ISM number density $\sim$ 12 in the inner region ($\sim$ 2.5$\times$10$^{4}$K; Sect. \[res\]), or else $\sim$ 12/2.3 = 5 for a photoionized gas in pressure balance with the coronal phase which presumably permeats the shell. Although $\epsilon$ is unity in models, this phase can fill in the volume corresponding to Sector3. Optical and UV lines {#disc_opt} -------------------- lines are discussed in Sects. \[disc\_ab\] & \[disc\_eva\]. and are accounted for within uncertainties. Other UV lines are elusive (Col.6 of Table \[tab\_res\]). Computed fluxes for 8 optical lines are within 20% of observation (Table \[tab\_compa\]), which is satisfactory considering the weakness of some of the lines. The 10$-$15% discrepancy on the ratio 6312/9531 does not challenge the model itself, given the various uncertainties. The $\sim$20% underestimation of should be considered with respect to . The line 9531 is matched, but the far-red flux may be less reliable, and 6312 departs from observation about in the same way as , but 6312 is a weak line. The exact status of is undecided. The -sensitive intensity ratio 4986/4658 is somewhat small in $N0$, large in $N1$ and more nearly correct in $Mi$ models. Would be emitted in a high- gas component as suggested by SS99 and V02, then 4986, roughly co-extensive with , would be undetectable. The weak line 4906, co-extensive with , confirms the ion distribution of the $Mi$ models and the iron abundance, although the agreement with observation is partly fortuitous. 4227 is overpredicted by a factor $\sim$3, but the observed intensity is very uncertain and could be 2–3 times stronger than the quoted value, as judged from published tracing (TI05). Also, only one computation of collision strengths has ever been done for the optical lines of the difficult ion (Appendix\[fiat\_misc\]). Finally, the ionic fraction Fe$^{4+}$/Fe, less than 5%, is subject to ionization balance inaccuracy. The predicted intensity $\sim$ I(4227Å)/4 of 4071 enhances the computed flux of 4068 up to the observed value (Tables \[tab\_res\] & \[tab\_compa\]). The observed line intensities are inconsistent (Table \[tab\_res\]), due to stellar lines (Sect. \[mod\_constraints\]). 4711, blended with 4713, is therefore useless. The weak 4740 tends to be overestimated by $\sim$ 50% in the preferred models. Trial calculations show that, adopting a recombination coefficient 12 times the radiative one (instead of 8 times, Appendix\[fiat\_misc\]) and dividing Ar/H by 1.13, and would be matched in $M2$. Infrared fine-structure lines {#disc_ir} ----------------------------- ### and {#disc_ir_nesiii} The reliably observed IR lines with optical counterparts, 15.5$\mu$ and 18.7$\mu$, are very well matched, confirming the scaling adopted for the [spitzer]{} fluxes and the model temperatures. The $Mi$ models, globally hotter, are more successful than the $Ni$ runs. No $t^2$ in excess of the one of the adopted configuration (Fig. \[fig\_t2\]d) is required. The predicted intensity of 33.5$\mu$ is only 40% of the observed value. Since the theoretical ratio of the IR lines is insensitive to conditions in , looking for alternative models is hopeless. The collision strengths $\Omega$ for the lines may not be of ultimate accuracy, as the results of Tayal & Gupta (1999) and Galavis et al. (1995) differ, but the more recent $\Omega$’s are likely more accurate. Also, the predicted 33.5$\mu$ is even worse using older data. Since Wu07 cast doubts on the accuracy of the flux calibration at the end of the [spitzer]{} spectrum, it is assumed that the atomic data are accurate and that the observed fluxes around 34$\mu$ should be divided by 2.3. ### {#disc_ir_oiv} If the drift of flux calibration at 34$\mu$ (Sect. \[disc\_ir\_nesiii\]) smoothly vanishes towards shorter wavelengths, the 26$\mu$ flux may still be overestimated. Conversely, the [spitzer]{} field of view encompasses SE, which emits little , leading to underestimate / in NW. Since these effects act in opposite directions, the original / is adopted for NW. As shown in Fig. \[fig\_Oxy\], O$^{3+}$ and O$^{2+}$ coexist in the He$^{2+}$ zone. O$^{3+}$/O$^{2+}$ and therefore 25.88$\mu$/4686 as well are sensitive to . In $N0$, is matched and is strongly underpredicted (Table \[tab\_compa\]). The predicted flux improves in the conditions of $N1$ and even by-pass observation in $M1$, whose ionizing flux is however too large (Sect. \[res\_M1comp\]). In the standard Model $M2$, the predicted exactly matches observation after adding the blended line 25.91$\mu$, whose computed flux is $\sim$ 2% of . Since relevant atomic data are reliable, [25.88$\mu$ indicates that must be on the order of 10 in the emitting region of the NW shell]{}. The model density results from general assumptions (photoionization by stars, shell geometry, $\epsilon$ = 1, Eq. 1, etc.) and a requirement to match a few basic line intensities with no reference to high-ionization lines, but . The computed intensity is a true prediction, especially as the models were essentially worked out prior to IR observations: the spectrum presented by Wu et al. (2006) showed the predicted line, finally noted by Wu07. ### {#disc_ir_siv} The predicted 10.5$\mu$ flux is twice the observed one. The collision strengths obtained by Tayal (2000) and Saraph & Storey (1999) for this line are in good agreement. The average fractional concentration of S$^{3+}$, $\sim$ 1/3, is stable in different models because sulfur is mostly distributed among the three ions S$^{2+}$$-$S$^{4+}$. Displacing the ionization balance by changing, , the gas density tends to make either S$^{2+}$ or S$^{4+}$ to migrate to S$^{3+}$. Only in the unsatisfactory run $N0$ is accounted for. Two-sector, constant-pressure ‘models’ allowing $\epsilon$$<$1 and using the SED of Model $M2$ were run with the conditions $Q_{\rm abs}$/$Q$ $>$ 0.3 and O/H $<$ 1.7$\times$10$^{-5}$. In these trials, the 5007 and $r$() constraints (Table \[tab\_conv\]) are relaxed and the observed / ratio is exactly matched by playing with (gas pressure) and $\epsilon$. Despite ample freedom and because of the higher $\sim$ 25, the computed flux is at most 60% of the observed one. Thus, forgetting other difficulties, the suggestion is that the excess flux can only be cured at the expense of . A broader exploration of the SED (discontinuities?) and the gas distribution could be undertaken. The 10.5$\mu$ flux published by Wu et al. (2006) was 25% larger than according to Wu07. The new value should be preferred, but this difference is at least indicative of possible uncertainties. The ratio / may also be intrinsically larger in NW than in SE. The theoretical ionization balance of some ions of sulfur (and argon) is subject to uncertainties (Appendix\[fiat\_misc\]). The observed / ratio can be recovered in $M2$ if the S$^{2+}$ recombination coefficient is multiplied by a factor 2.3, which is perhaps too large a correction: then, both and are matched if S/H is divided by 1.33, with the caveat that the predicted intensities are divided by 1.3. A combination of observational and theoretical effects just listed could alleviate the ‘ problem’. Low ionization fine-structure lines {#disc_ir_low} ----------------------------------- Although the error bars of order 10% quoted by Wu07 may not include all sources of uncertainties, both 12.78$\mu$ and 25.98$\mu$ are detected in high-resolution mode. 34.80$\mu$ is strong, even though the flux quoted by Wu07 (Table \[tab\_compa\]) may be too large (Sect. \[disc\_ir\_nesiii\]). Usually, the bulks of 35$\mu$ and 26$\mu$ arise from a Photon-Dominated Region (PDR), at the warm interface between an ionization front and a molecular cloud (, Kaufman et al. 2006). Schematically in a PDR, the photo-electric heating by UV radiation on dust grains (and other molecular processes) is balanced by fine-structure (and molecular) line emission. The small reddening intrinsic to (Sect. \[abs\_red\]) and the ‘large’ gaseous iron content (Sect. \[disc\_ab\]) imply that little dust is available. Molecules and PAHs are not detected in (Vidal-Madjar et al. 2000; Leroy et al. 2007; Wu07). The classical PDR concept may therefore not apply to , raising the question of the origin of 35$\mu$ and 26$\mu$, both underpredicted by factors 5–10 in the models (Table \[tab\_compa\]). A way to produce a ‘pseudo-PDR’ is X-ray heating. Two new variants of $M2$ are considered, in which a hot black body representing a soft X-ray emission from NW is added to the original SED. The adopted temperature is = 2$\times$10$^{6}$K and the luminosity = 4 and 8$\times$10$^{39}$ for variants $M2_{\rm X}$ and $M2_{\rm X2}$ respectively (Fig. \[fig\_SEDX\]). The actual X-ray luminosity of , $\sim$ 1.6$\times$10$^{39}$ in the 0.5–10 kev range of [chandra]{}, mainly arises from the centre of NW and is consistent with a power law of slope $-1$ (Thuan et al. 2004), drawn in Fig. \[fig\_SEDX\]. The SED for $M2_{\rm X}$ is a relatively high, yet plausible extrapolation of the [chandra]{} data. $M2_{\rm X2}$ is considered for comparison purpose. , still governed by Eq. (1), is leveled out at 300. The run of physical conditions with $\tau$ is shown in Fig. \[fig\_M2X\] for $M2_{\rm X}$. While $<$$>$ is increased by only $\sim$ 100K in the region, a warm low-ionization layer develops beyond the ionization front. Computation is stopped at = 100K ($\tau$ $\sim$ 7$\times$10$^{4}$, compared to 250 in $M2$). In $M2_{\rm X2}$ (dashed lines in Figs. \[fig\_M2X\]a–\[fig\_M2X\]c), the new layer is hotter and more ionized (final $\tau$ $\sim$ 1.2$\times$10$^{5}$). The geometrical thickness of the layer is 21% and 35% of the shell in $M2_{\rm X}$ and $M2_{\rm X2}$ respectively. In the zone, O$^+$/O does not exceed $\sim$ 10$^{-3}$, in marked contrast with Ne$^+$/Ne, overplotted as a thin solid line in Fig. \[fig\_M2X\]d. Lines 12.8$\mu$ and 7.0$\mu$ are usually discarded in PDR models on the basis that the ionization limits of Ne$^0$ and Ar$^0$ exceed 1ryd. Here, owing to the scarcity of free electrons and the lack of charge exchange with H$^0$, photoionization by soft X-rays can keep 1–10% of these elements ionized. In regions, cooling is due to inelastic collisions with H$^0$. Reliable collisional rates exist for the main coolents 157$\mu$ (Barinovs et al. 2005) and 63$\mu$ (Abrahamsson et al. 2007; several processes need be considered for : see Chambaud et al. 1980; Péquignot 1990) and for 35$\mu$ (Barinovs et al. 2005), but not for, , 26$\mu$. Following Kaufman et al. (2006), it is assumed that the cross-section for H$^0$ + Si$^+$ also applies to fine-structure transitions of other singly ionized species. Concerning (ground state $^6$D$_{9/2}$), collisions to $^6$D$_{7/2}$ follow the above rule, but cross-sections for transitions to the next $^6$D$_{J}$ are taken as 2/3, 2/4, etc. of the first one. ------------- --------- ----- ----- ----- ----- ----- ----- Line$^b$ Obs. colls.$^c$ - no yes no yes no yes $\:$158. - 2.1 2.1 124 119 224 204 $\:$63.2 - 4.7 4.6 274 210 624 450 12.8 9 1.9 3.6 1.9 5.5 2.0 16 $\:$34.8 157$^d$ 21 22 37 80 55 190 $\:$7.0 - 1.2 1.3 1.4 2.2 2.2 7.9 $\:$26.0 34 3.4 4.8 5.8 38 9.3 108 $\:$35.3 - 0.8 1.4 1.1 9.5 1.7 33 \[0.1cm\] ------------- --------- ----- ----- ----- ----- ----- ----- : Fine-structure line intensities$^a$ and X-rays $^a$ In units = 1000.\ $^b$ Wavelengths in $\mu$m.\ $^c$ Inelastic collisions of , , and with omitted (‘no’) or included (‘yes’).\ $^d$ The re-calibrated flux is 68 after Sect. \[disc\_ir\_oiv\].\ \[tab\_fs\] Low-ionization IR lines, including dominant coolents and other unobserved lines, are considered in Table \[tab\_fs\]. Line identifications and observed intensities appear in Cols.1 & 2. The results of two computations are provided for $M2$ (Cols.3-4), $M2_{\rm X}$ (Cols.5-6) and $M2_{\rm X2}$ (Cols.7-8). In the first one, the excitations of , , , and (but not and ) by collisions with are inhibited. In the second one, they are included according to the above prescription. Comparing different odd columns (‘no’) of Table \[tab\_fs\], the rise of line intensities as the zone develops shows that the excitation of, , by free electrons is still active. is stable, due to the scarcity of Ne$^+$ (Fig. \[fig\_M2X\]d) relative to Si$^+$. Comparing now odd columns to even columns, it is seen that collisions, ineffective in the ‘normal’ region model $M2$ (except, quite interestingly, for ), strongly enhance the excitation rates. After subtracting emission from the region ($M2$), collisions contribute 75–80% of the excitation (virtually 100% in the case of ). Concerning 35$\mu$, the atomic data are not controversial and the re-calibrated flux of 68 is reasonably well defined (Sect. \[disc\_ir\_oiv\]). Then $M2_{\rm X2}$ (Col.8 of Table \[tab\_fs\]) is excluded, while $M2_{\rm X}$ (Col.6) or an even weaker soft X-ray source (closer to the [chandra]{} extrapolation) can account for . Although the remarkably coherent predictions for 35$\mu$ and 26$\mu$ are partly fortuitous, they are consistent with (1) the collision strength is correctly guessed, (2) Fe/Si is the same in the ionized and neutral gas, and (3) the excitation by soft X-ray heating is viable. Concerning 12.8$\mu$, the discrepancy with observation (factor 0.6) may not be significant, as the line is weak and its detection in the [low]{}-resolution mode is not taken as certain by Wu07. The collision strength may be too small. Summarizing, a plausible extrapolation to soft X-rays of the [chandra]{} flux can provide an explanation to the relatively large intensity of 35$\mu$ and other fine-structure lines in . This is considerable support to the general picture of photoionization as the overwhelmingly dominant cause of heating of the region, since [heating by conversion of mechanical energy appears unnecessary even in regions protected from ionization and heating by star radiation]{}. Full confirmation should await reliable collisional excitation rates by for fine-structure lines of [all]{} singly ionized species. The soft X-rays from NW have little effect on the region: both 6300 and 25.9$\mu$ are increased by 3% and by 30%. The 1.7% enhancement of 4363 is of interest (Sect. \[disc\_eva\]). Stability of results {#disc_stab} -------------------- The relative stability of the predicted line intensities is a consequence of the set of constraints (Table \[tab\_conv\]). Allowing for a range of values, a broader variety of results could be obtained. Are conclusions dependent on input data? The only basic line showing substantial variability in different spectroscopic studies is 3727. This line is sometimes found to be stronger than the adopted value (VI98; TI05). In a new variant $M2v$ of Model $M2$, the observed intensity is assumed to be 20% larger than in Tables \[tab\_res\] & \[tab\_compa\] and the covering factors are left unchanged. The inevitable 18% increase of the already too strong line 7320+30 is not too significant, considering that the 7325 flux is very uncertain and may not correspond to a slit position with stronger 3727. Owing to the larger fractional abundance of O$^+$, O/H is increased by 5%. The larger weight of low-ionization layers induces a 3% increase of Ne/H and a 13% decrease of N/H and Fe/H, since the and intensities were left unchanged. Both and decrease by a few %, whilst increases by 6% and both and increase by $\sim$ 11%. Ar/H and the lines increase by only 1% and 4363 [is unchanged]{}. In a more extreme example, $M2cv_{\rm X}$, with an assumed intensity of 322 instead of 238 (factor 1.35), the $f$’s as in $M2c$ and the SED as in $M2_{\rm X}$ (reconverged $\tau_{\rm c}$ = 2.5 and $P_{\rm in}$/$P_{\rm out}$ = 9), is exactly matched again, is +2% off, +6%, +87% and only $-$8%. Thus, changes are moderate and turn out to alleviate difficulties noted in Sects. \[disc\_opt\] & \[disc\_ir\], , the weakness of the doublet. The computed $r$ is robust. Elemental abundances {#disc_ab} -------------------- To first order, O/H reflects $<$(O$^{2+}$)$>$, related to $r$(), , the predicted 4363 intensity. Models $M3$ and $M2c$ both almost exactly fit 4363 and share the same O/H = 1.62$\times$10$^{-5}$, which is the best estimate, provided that (1) oxygen lines have been given optimal observed intensities, (2) these models faithfully represent the region, and (3) collision strengths are accurate. Concerning line intensities, 5007 is quite stable in different spectra of NW and the reasonably large, yet representative, ratio $r$() (Sect. \[spectr\]) is taken for granted, since our objective is deciding whether this specific ratio can be consistently explained assuming photoionization by stars. In model $M2cv_{\rm X}$ (Sect. \[disc\_stab\]), which also fits exactly the lines, 3727 was assumed to be enhanced by 35%, leading to O/H = 1.74$\times$10$^{-5}$. Concerning models, the difference between the directly derived from $r$(), () = 19850K, and the $\times$$N$(O$^{2+}$) weighted average, $<$(O$^{2+}$)$>$ = 19650K, corresponds to a formal $t^2$() = 0.012, similar to the computed $t^2$(O$^{2+}$) = 0.010. This difference makes only 1% difference for O$^{2+}$/H$^{+}$ and an empirical estimate neglecting $t^2$ should nearly coincide with model results for this ion. A major feature of the profile is that the difference $<$(O$^{2+}$)$>$ – $<$(O$^{+}$)$>$, which was only 300K in $N0$ and 3200K in $N1$, is 6600K in best models. Models are essential in providing a to derive O$^{+}$/H$^{+}$ from 3727, as () is poorly determined from the uncertain 7325 and $t^2$() is inaccessible. A ‘canonical’ O/H for NW is 1.46$\times$10$^{-5}$ (SK93, ICF99), 11% less than the present 1.62$\times$10$^{-5}$, out of which 4% are due to collisional excitation of and the remaining 7% could be a non-trivial consequence of the relatively large $t^2$’s obtained for some ions in the present models (Table \[tab\_mod\]; Fig. \[fig\_t2\]), although differences in collision strengths may also intervene at the 2% level (Appendix\[fiat\_oiii\]). The silicium and sulfur abundances were not fine-tuned in models. The computed flux suggests dividing the assumed Si/H by 1.25. Concerning S/H, is underestimated, globally underestimated and neatly overestimated. Correcting the ionization balance /, S/H should be divided by 1.3 in best models (Sect. \[disc\_ir\_siv\]), but is then underestimated. Since a combination of effects may explain the overestimation of , the model S/H is tentatively divided by 1.15. Similarly, / is best accounted for if Ar/H is divided by 1.13 (Sect. \[disc\_opt\]), but the line is weak and the adopted correction factor is 1.06. Thus, S/Ar in is within 10% of the solar value, in agreement with a conclusion of Stevenson et al. (1993). The iron abundance relies on the lines, since 26$\mu$ does not arise from the region, while the and intensities are uncertain. The intensities are from a spectrum in which 3727 is stronger than on average (TI05). In variant $M2cv_{\rm X}$ (Sect. \[disc\_stab\]), where the intensity of is multiplied by 1.35, both the predicted 4906 and Fe/H are divided by 1.4. This lower Fe/H is adopted. Solar abundances are tabulated by Asplund et al. (2005, AGS05). The compilation by Lodders (2003) is in substantial agreement with AGS05 ($+0.03$dex for all O–S elements of interest here and $+0.02$dex for Fe relative to H), except for Ar/H ($+0.37$dex). The larger argon abundance is convincingly advocated by Lodders (2003). Ar/O is adopted from this reference. Then Ar/H coincides with the value listed by Anders & Grevesse (1989). The shift of O/H from Anders & Grevesse (1989) to AGS05 is $-0.27$dex, out of which $-$0.07dex corresponds to the change from proto-solar to solar abundances. Shifts for X/Fe are $\sim$ $-0.20$dex for N, O, Ne, $-0.11$dex for C, $-0.07$dex for S and $\sim$ 0.0 for other elements of interest. In Table \[tab\_ab\], the present model abundances by number 12+log(X/H) for NW (‘$M$’) are provided in Col.2. The abundances X/O relative to oxygen from models (Col.3, ‘$M$’) are compared to empirical values obtained by IT99 (Col.4, ‘IT’). The brackets \[X/Y\] = log(X/Y) – log(X/Y)$_{\odot}$ from models are given in Cols.5 & 6 (Y $\equiv$ H and Fe). Finally \[X/Fe\] is provided for Galactic Halo stars with \[Fe/H\] $\sim$ $-1.8$ (Col.7, ‘H’). Despite considerable efforts to include 3D and non-LTE effects in the study of line formation in cool stars, astrophysical descriptions and atomic data may still entail uncertainties in stellar abundances (, Fabbian et al. 2006), particularly for nitrogen. Also, a subpopulation of N-rich stars is well identified (, Carbon et al. 1987). Comparing Cols.3 & 4, the model and empirical X/O agree to about 0.1dex. The present C/O is close to the one obtained by Garnett et al. (1997), who claim that C/O is anomalously large in . IT99 argue that the subregion of NW observed with the [hst]{} is especially hot according to spatially resolved MMT data and that C/O is therefore small. Nonetheless, IT99 also derive an exceedingly low O/H at the same position “because of the higher ”, which poses a problem of logics since there is [a priori]{} no link of causality between and O/H within . The \[C/O\] = $-0.39$ resulting from the present model is indeed marginally incompatible with the up-to-date \[C/O\] = $-0.57\pm0.15$ corresponding to Galactic Halo stars with \[O/H\] = $-1.45$ (Fabbian et al. 2006). This ‘large’ \[C/O\] is analysed by Garnett et al. (1997) in terms of carbon excess, suggesting that an old stellar population managed to produce this element, then challenging the view that is genuinely young (, IT04), a view also challenged by Aloisi et al. (2007). From models, \[C/Fe\] appears to be identical in and halo stars of similar metallicity (Cols.6 & 7). [The relatively large \[C/O\] in is due to a relatively small \[O/Fe\]]{}. This is indirectly confirmed by the agreement between and halo stars for all elements beyond neon (argon should follow lighter $\alpha$-elements). The \[X/O\]’s ($\equiv$ \[X/H\] + 1.45) are the usual basis to discuss elemental abundances in BCD’s and nebulae. Exceptionally, in , the abundances of iron and heavy $\alpha$-elements are in harmony, allowing to consider the oxygen abundance with respect to metallicity, instead of defining metallicity by means of oxygen itself. Any iron locked into dust grains would further decrease \[O/Fe\] in . Apparently, for sufficiently low metallicity of the ISM and/or sufficient youth of the host galaxy, iron does not find paths to efficiently condense into dust. Alternatively, dust grains may be destroyed by shocks. Overall evaluation of models {#disc_eva} ---------------------------- Line intensities are generally well accounted for, although 10.5$\mu$ is overpredicted by 90–100% (Sect. \[disc\_ir\_siv\]). ----------- --------- --------- --------- --------- ---------- ------------------ X/O X/O \[X/H\] \[X/Fe\] \[X/Fe\] El. $M$$^a$ $M$ IT$^b$ $M$ $M$ H$^c$ C 6.55 $-$0.66 $-$0.77 $-$1.84 $-$0.02 $-$0.05$\pm$0.15 N 5.60 $-$1.61 $-$1.56 $-$2.18 $-$0.36 $-$0.40$\pm$0.30 O 7.21 $-$ $-$ $-$1.45 0.37 0.52$\pm$0.15 Ne 6.39 $-$0.82 $-$0.80 $-$1.45 0.37 $-$ Si 5.89 $-$1.32 $-$1.46 $-$1.62 0.20 0.21$\pm$0.08 S 5.57 $-$1.64 $-$1.55 $-$1.57 0.25 0.21$\pm$0.07 Ar 4.97 $-$2.24 $-$2.16 $-$1.55 0.27 $-$ Fe 5.63 $-$1.58 $-$1.45 $-$1.82 $-$ $-$ \[0.1cm\] ----------- --------- --------- --------- --------- ---------- ------------------ : Abundances in NW compared to solar and Galactic halo stars $^a$ 12 + log(X/H) by number in present model $M$.\ $^b$ Empirical abundance: Izotov & Thuan (1998, 1999) with 12 + log(O/H) = 7.16; Fe: Thuan & Izotov (2005).\ $^c$ Halo stars \[Fe/H\] $\sim$ $-1.8$: Fabbian et al. (2006), García Pérez et al. (2006), Nissen et al. (2007), Carbon et al. (1987).\ \[tab\_ab\] The freedom left in the parameters describing the SED and the shell acts at the few % level upon the 4363 predicted intensity. Thus, the calculated 4363 shifts from 96.0% of the observed intensity in $M2$ ($T2$ = 4$\times$10$^{4}$K, $f^{cov}_i$ = 0.26, 0.30) to 99.6% in $M2c$ ($f^{cov}_i$ = 0.22, 0.60) and 99.8% in M3 ($T2$ = 5$\times$10$^{4}$K, $f^{cov}_i$ = 0.23, 0.50). Adding less than 1% luminosity as soft X-rays (, $M2_{\rm X}$, compared to $M2$, Sect. \[disc\_ir\_low\]), results in +1.7% for 4363. Also, increasing He/H from 0.080 to the possibly more realistic value 0.084 (Peimbert et al. 2007), 4363 is enhanced by a further +0.6%. Since both the X-ray and He/H corrections are more than plausible, it is relatively easy to reach 100–102% of the observed 4363 intensity in the assumed configurations. However, models tell us that $r$() can hardly be larger than the observed value. It may prove necessary to consider alternative gas distributions, , in case the misfit is confirmed by more accurate observational and theoretical data. Assuming larger densities in the diffuse medium (Sect. \[disc\_ir\_siv\]) and/or considering thick filaments closer to the source tend to penalize 4363. At this point, uncertainties on collision strengths $\Omega$ need be considered (Appendix\[fiat\_oiii\]). Using $\Omega$’s by Lennon & Burke (1994, LB94) instead of Aggarwal (1993, Ag93), the computed 4363 would be 2.1% smaller and more difficult to explain. On the other hand, using $\Omega$($^3$P$-$$^1$D) from Ag93 and $\Omega$($^3$P$-$$^1$S) from LB94 would enhance all computed 4363 intensities by 2.1%. Concerning transition $^3$P$-$$^1$S which controls 4363, both Ag93 and LB94 find a 5–6% increase of $\Omega$ from 2$\times$10$^{4}$K to 3$\times$10$^{4}$K, suggesting the influence of resonances. If for some reason the energy of these resonances could be shifted down, there would be room for a few % increase of $\Omega$ at 2$\times$10$^{4}$K compared to the current value, then introducing more flexibility in the present model of . Summarizing, the hypothesis of pure photoionization by stars in the form explored here is perfectly tractable, but the models approach a limit. This is in a sense satisfactory, considering that is an extreme object among BCD’s, but sufficient flexibility in choosing solutions is worthwile. An analysis of how the computed $r$() can be influenced shows that, in the case of , possible variations of ‘astrophysical’ origin are of the same order as the uncertainties affecting the $\Omega$’s. Since the set of computed $r$() tends to be down by 2–3% relative to observation, it is legitimate to question the $\Omega$’s. Now, the recent re-evaluation of the distance to by Aloisi et al. (2007) may offer an ‘astrophysical alternative’: multiplying $D$, $R_{\rm i}$ and $R_{\rm f}$ by 2$^{1/2}$ and the luminosty by 2, the relative volume increase leads to smaller $P_{\rm in}$ and $\tau_{\rm c}$, and, after reconvergence, 4363 is enhanced by +2.4%. Noneteless, 10.5$\mu$ is enhanced too. Concluding remarks {#concl} ================== Owing to its small heavy element content, stands at the high- boundary of photoionized nebulae. Where ionization and temperature are sufficiently high, the cooling is little dependent on conditions, except through the concentration of H$^0$, controlled by density. Therefore, in the photoionization model logics, [ is then a density indicator]{}, in the same way it is an O/H indicator in usual regions. It is for not having recognized implications of this new logics, that low-metallicity BCD models failed. In a photoionization model study of NW, SS99 employed a filling-factor description and concluded that () was fundamentally unaccountable. The [vogue]{} for this simple description of the ionized gas distribution resides in its apparent success for usual regions, a success falling in fact to the strong dependence of cooling on abundances. Universally adopted along past decades, this concept led all authors to conclude that photoionization by hot stars did not provide enough energy to low-Z GEHIIRs. This conclusion is in line with a movement of calling into question photoionization by stars as the overwhelmingly dominant source of heat and ionization in gaseous nebulae, a movement cristalizing on the ‘$t^2$ problem’ (Esteban et al. 2002; Peimbert et al. 2004), since the presence of fluctuations [supposedly larger]{} than those reachable assuming photoionization by stars implies additional heating. A conclusion of the present study is that the gas distribution is no less critical than the radiation source in determining the line spectrum of regions. Assuming pure photoionization by stars, the remarkable piece of information carried by the large () of NW is that the mean density of the emitting region is much less than (), a low confirmed by line ratios 25.9$\mu$/4686 and 4986/4658. NW models comprising a plausible SED and respecting geometrical constraints can closely match almost all observed lines from UV to IR, including the crucial 4363 (10.5$\mu$ is a factor 2 off, however). Thus, extra heating by, , dissipation of mechanical energy in the photoionized gas of low-metallicity BCD galaxies like is [not]{} required to solve the ‘() problem’. Moreover, since low-ionization fine-structure lines can be explained by soft X-rays, (hydrodynamical) heating is [not even]{} required in warm regions protected from ionization and heating by star radiation. As a final note, on close scrutiny, the solutions found here [tend to be just marginally consistent]{} with observed $r$(). Given the claimed accuracy in the different fields of physics and astrophysics involved, postulating a mechanical source of heating is premature, whereas a [2–3% upward correction]{} to the collision strength for transition O$^{2+}$($^3$P–$^1$S) at $\sim$ 2$\times$10$^4$K is an alternative worth exploring by atomic physics. Yet, another possibility is a substantial increase of the distance to . Owing to accurate spectroscopy and peculiar conditions in , important astrophysical developments are at stakes in the 5% uncertainty attached to collision strengths. If photoionized nebulae are shaped by shocks and other hydrodynamic effects, this does not imply that the emission-line intensities are detectably influenced by the thermal energy deposited by these processes. Unravelling this extra thermal energy by means of spectroscopic diagnostics and models is an exciting prospect, whose success depends on a recognition of all resources of the photoionization paradigm. 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Individual GEHIIRs {#prev_indiv} ------------------ ### O/H $\geq$ 1.5$\times$10$^{-4}$ {#prev_indiv_go} González Delgado & Pérez (2000) successfully model NGC604 (O/H = 3$\times$10$^{-4}$) in M33 as a radiation-bounded sphere (radius 20$-$110pc) of density 30 and filling $\epsilon$ = 0.1, both 6300 and 4363 being explained. García-Vargas et al. (1997) successfully model circumnuclear GEHIIRs in NGC 7714 as thin, constant-density ( $\sim$ 200), radiation-bounded shells with O/H = (2$-$3)$\times$10$^{-4}$: $r$() is accounted for within errors and is just moderately underestimated. The nuclear GEHIIR of NGC 7714 is modelled by González Delgado et al. (1999) as a full sphere with very small $\epsilon$. The adopted O/H = 3$\times$10$^{-4}$ is too large since 4363 is underpredicted. Obviously, a better fit to the available optical line spectrum could again be achieved for the nucleus. González Delgado et al. (1999) are probably not well founded to invoke extra heating by shocks. Luridiana & Peimbert (2001, LP01) propose a photoionization model for NGC 5461 (O/H = 2.5$\times$10$^{-4}$), a GEHIIR in M101. As for NGC 2363 (Appendix\[prev\_indiv\_po\]), LP01 apply an aperture correction to their spherical model. The and $r$() spatial profiles are reproduced with a Gaussian density distribution of very small $\epsilon$ and high inner density $-$ 500 compared to ()$\sim$150 $-$ [meant to increase the inner O$^+$ fraction]{}. In this way, the 5007, 4363 and 3727 fluxes restricted to the theoretical slit can be accounted for[^5], but not , “a not unusual fact”, nor , which, unlike a belief of LP01, is not enhanced by increasing the primary flux below 1.0ryd. As noted by LP01, the outputs of their model are strongly dependent on the density structure. From their sophisticated study of NGC 588 (O/H = 2$\times$10$^{-4}$), Jamet et al. (2005, JS05) conclude that “the energy balance remains unexplained”. This negative conclusion is essentially based on the fact that (), from the ratio 4363Å/5007Å, is observed to be larger than (, IR), from 5007Å/88$\mu$m, by $\Delta$$T_{obs}$ = 2700$\pm$700K, while the corresponding $\Delta$$T_{mod}$ is only 1400$\pm$200K in models, which are otherwise satisfactory, accounting reasonably well for () and the distribution of ionization (models DD1 and DDH exhibited by JS05, who carefully consider uncertainties related to the SED and the small-scale gas distribution). Considering the difficulty of calibrating the ISO-LWS fluxes relative to the optical and the sensitivity of the 88$\mu$m emissivity to , the 400K gap between $\Delta$$T_{obs}$ and $\Delta$$T_{mod}$ is [not a sound basis to claim the existence of an energy problem]{}. If diagnostics based on IR lines are desirable, the energy problem raised so far in GEHIIR studies is not related to these lines. Instead of a heating problem as in , the model presented by JS05 could be facing a [cooling]{} problem, since the computed (, IR) is too high. An additional energy source is not clearly needed in the cases of NGC 588, NGC 5461, NGC 7714 and NGC 604. ### O/H $<$ 1.5$\times$10$^{-4}$ {#prev_indiv_po} Relaño et al. (2002) provide an inventory of NGC 346, a GEHIIR of the SMC (O/H = 1.3$\times$10$^{-4}$). Their spherical, constant-density, matter-bounded photoionization model, whose only free parameter is a filling factor $\epsilon$ ([alla]{} SS99), accounts for the escape of ionizing photons, but underpredicts collisional lines, especially 4363. After unsuccessful variations on geometry, the authors preconize, following SS99, an additional source of energy. After an extensive exploration of photoionization models with filling factor for the bright GEHIIR NGC 2363 (O/H = 8$\times$10$^{-5}$), Luridiana et al. (1999, LPL99) conclude that they cannot find a solution unless they introduce fluctuations by hand, , they assume a larger $t^2$ than the one intrinsic to their model. This $t^2$, intended to enhance the computed 4363, is justified by the fact that the observed Paschen jump temperature is less than () and supported by a self-consistency argument: a larger $t^2$ leads to a larger O/H, hence a larger number of WR stars, hence (1) a larger injection of mechanical energy, supposed to feed the temperature fluctuations themselves, and (2) a larger photon flux above the He$^+$ ionization limit, useful to increase 4686. However, as acknowledged by Luridiana et al. (2001), the WR star winds make a poor job to generate a significant $t^2$ in NGC 2363. Also, present views suggest that arguments based on WR stars in low-Z galaxies were illusory just a few years ago (, Leitherer 2006; Appendix\[WRstars\]). Finally, $r$(), underestimated by only $\sim$ 12% in the ‘standard’ low-Z model by LPL99, is divided by 2 on using the larger O/H, so that a relatively minor difficulty is first made much worse and then solved by means of a $t^2$. The slit correction advocated by LPL99 is considered in Appendix\[sphere\_slit\]. Luridiana et al. (2003, LPPC03) consider a spherical model for a GEHIIR of SBS 0335-052 (O/H = 2$\times$10$^{-5}$). A Gaussian distribution with large maximum density and small $\epsilon$ proves unsatisfactory. LPPC03 then consider a 10-shell model (over 50 free parameters, most of which pre-defined), in which [each shell is radiation bounded]{} and is characterized by a covering factor. Although each shell is still given an $\epsilon$, the new model is equivalent to a collection of geometrically thin radiation-bounded sectors at different distances from the source (see also Giammanco et al. 2004) and “gracefully reproduces the constancy of the ionization degree along the diameter of the nebula”. Hence, the authors are forced by observational evidences to [implicitly abandon]{} the classical filling-factor approach. Nonetheless, whatever the complexity of these models, all of them fail to account for the high (). LPPC03 consider a Gaussian model for SE (O/H = 1.7$\times$10$^{-5}$) with again a relatively large maximum density and, unlike for SBS 0335-052, a relatively large $\epsilon$, resulting in a rather compact model nebula, in which the computed () compares quite well with the observed one. Unlike for the NW, the [hst]{} image (Cannon et al. 2002) of the younger SE region does not show a shell surrounding an MSC. Nonetheless, considering the strong output of mechanical energy from massive stars, it is likely that inner cavities already developped. The strong indirect evidence for too compact a gas distribution in the model by LPPC03 is the notable weakness of the computed intensity of and other low-ionization lines. Adopting a more expanded structure in order to increase , yet keeping the general trend of the gas distribution, the computed () would be forseeably lower than in the model by LPPC03. Individual GEHIIRs: discussion {#prev_indiv_disc} ------------------------------ LPPC03 describe the ‘() problem’ they face in their study of SBS0335-052 (Appendix\[prev\_indiv\_po\]) as “a systematic feature” of region models and, following SS99, they state that this problem “can be ascribed to an additional energy source acting in photoionization regions, other than photoionization itself”. Nevertheless, () seems to be accountable in existing photoionization models for GEHIIRs with, say, O/H $\geq$ 1.5$\times$10$^{-4}$ (Appendix\[prev\_indiv\_go\]). Similarly, the computed 4363 is correct, possibly even too large, for regions of the LMC (Oey et al. 2000). If, despite apparent complementarities, the () and $t^2$ problems have different origins (Sect. \[intro\]), no ‘systematic feature’ can be invoked. In modelling objects with near solar abundances, 4363 is controlled by O/H, 5007 by the ‘color temperature’ of the ionizing radiation, 3727 by the ionization parameter, while is maximized in radiation-bounded conditions. For these objects, assuming a ‘large’ (constant) density, , $\sim$ (), associated to an [ad hoc]{} $\epsilon\ll$ 1, is often successful, [although this does not prejudge of the relevance of the model found]{}. Indeed, this assumption proves to be at the heart of the () problem met in low-Z BCDs (Appendix\[filling\]). Extensive analyses of BCDs {#prev_extens} -------------------------- The conclusion of an early extensive analysis based on radiation-bounded, low-density full sphere models for low-Z BCDs (Stasińska & Leitherer 1996) is optimistic concerning (), whereas is then qualitatively explained in terms of shock heating. Nevertheless, in an extension of this study to large-Z objects with no measured (), Stasińska et al. (2001) reinforce the energy problem raised by SS99 (Sect. \[prev\_oiii\]) when they conclude that “a purely ‘stellar’ solution seems now clearly excluded for the problem of / [versus]{} / as well as /”, while, conversely, they still endorse the unproved statement of SS99 (Sect. \[prev\_oi\]) that “strong emission is easily produced by photoionization models in dense filaments”. The sequence of photoionization models proposed by Stasińska & Izotov (2003, SI03) for a large sample of low-Z BCDs (divided in three abundance bins) illustrates views expressed after the failure of models acknowledged by SS99 for (Sect. \[prev\_IZ\]). In the description by SI03, an evolving synthetic stellar cluster (10$^5$, instantaneous burst) photoionizes a spherical shell of constant density = 10$^2$ at the boundary of an adiabatically expanding hot bubble. With suitable bubble properties, underlying old stellar population, aperture correction and time evolution of the covering factor, the range of equivalent width (EW()) and the trends of 5007, 3727, 6300 [versus]{} EW() can be reproduced for the high-Z bin (O/H $\sim$ 1.5$\times$10$^{-4}$) within the scatter of the data. Applying similar prescriptions to the intermediate-Z bin, the oxygen lines and 4686 ( was just fair in the first bin) are underpredicted. SI03 diagnose an insufficient average energy per absorbed photon and assume that the stellar cluster is supplemented by a [strong 10$^6$K bremsstrahlung-like radiation source]{}, which solves the problem (He$^+$ is further ionized by extra 4–5ryd photons; see, however, Appendix\[WRstars\]) and alleviates the problem (the soft X-rays further heat and widen the ionization front), but barely improves and . Agreement of the model sequence with observation is finally restored by supposing [in addition]{} that the shell includes a [time-variable oxygen-rich gas component]{} attributed to self-enrichment: in the example shown by SI03, this component is 4-fold enriched in CNO, etc. relative to the original abundance and encompasses half of the shell mass after a few , so that one generation of stars produced a 2.5-fold enhancement of the average abundance in the photoionized gas. This description essentially applies to the low metallicity bin (O/H $\sim$ 2$\times$10$^{-5}$) of particular concern for , but with [even more extreme properties]{} for the O-rich component, since it should be overabundant by 1, resulting in a [5-fold enhancement of the final average abundance]{}. Extensive analyses of BCDs: discussion {#prev_extens_disc} -------------------------------------- The time scale of 0.5 for the growth of the O-rich component in the description by SI03 cannot directly fit in the self-pollution scenario since it is shorter than the stellar evolution time scale. Also, a sudden [oxygen]{} self-pollution of the gas is not observed in supernova remnants. The assumed X-ray power is $\sim$10% of the cluster luminosity or $\sim$2 times the estimated X-ray [rosat]{} power (0.07$-$2.4 keV) of the hot bubble fed by stellar winds and supernovae around a usual MSC (Strickland & Stevens 1999; Cerviño et al. 2002). Moreover, the hot gas is generally raised at [several]{} 10$^6$K (Stevens & Strickland 1998). Adopting a larger temperature, the X-ray power should be even larger, as only the softer radiation interacts usefully with the ionized gas. itself is a rather strong X-ray emitter in the 0.5–10 keV range, yet 20 times weaker than the source assumed by SI03 (Thuan et al. 2004; Sect. \[disc\_ir\_low\]). Apart from these problems, SI03 do not address the question of the intensity of 4363. The narrow radiation-bounded shell adopted by SI03 usefully favours 6300, but makes the computed intensity of 4363 even worse than the one obtained by, , SS99 (Sect. \[prev\_IZ\]). Moreover, adding the prominent O-rich component advocated by SI03 will (1) decrease 4363 by a further 30$-$40% on average and (2) conflict with the existence of very low-Z BCDs, since any of them will be condamned to shift to the intermediate class defined by SI03 after just 1 or 2. If what SI03 qualify as “appealing explanations” presents any character of necessity for BCD models, then the hypothesis of photoionization by stars, which was already given a rough handling by SS99 in their analysis of , should be considered as definitively burried for the whole class of low-Z BCDs. The fact that SI03 discard 4363 in their analysis confirms that they endorse and reinforce views expressed by SS99 or Stasińska et al. (2001) and give up explaining () in low-metallicity GEHIIRs by means of stellar radiation. However, [the same restrictive assumption]{} as for individual GEHIIR studies (Appendices\[prev\_indiv\]–\[prev\_indiv\_disc\]) bears on the gas distribution adopted by SI03, since their (geometrically thin) ‘high’ constant- model sphere is nothing but a zero-order approximation to a model shell with classical filling factor (Appendix\[filling\]). On the gas distribution in GEHIIRs {#gas_distrib} ================================== Misadventures of the filling factor concept {#filling} ------------------------------------------- For the sake of reproducing the surface brightness of regions, asssuming a gas density much larger than $<$$^2$$>$$^{1/2}$, the ‘filling factor paradigm’ posits that the emitting gas belongs to [optically thin]{}, ‘infinitesimal’ clumps, filling altogether a fraction $\epsilon\ll 1$ of the volume. Given that the stellar evolution timescale exceeds the sound crossing time of regions, small optically thin ionized clumps will have time to expand and merge into finite-size structures. [If]{} these structures have the original density, they are likely to have [finite or large optical depths, in contradiction with the filling factor concept]{}. Filamentary structures, ubiquitous in images of nearby GEHIIRs, are often taken as justifications for introducing $\epsilon$ in photoionization models. However, (1) the geometrical thickness of observed filaments is consistent with radiation-bounded structures and (2) an individual filament most often emits both high and low ionization lines (, Tsamis & Péquignot 2005). [The filling-factor description is flawed]{}. A GEHIIR may well be a collection of [radiation-bounded]{} filaments embedded in coronal [and]{} photoionized diffuse media. The idea behind assuming this configuration is that only the ionized ‘atmospheres’ of long-lived, radiation-bounded, evaporating structures will possibly maintain a substantial overpressure relative to their surroundings (a similar idea applies to the “proplyds” found in Orion; , Henney & O’Dell 1999). The filling factor concept fails on both theoretical and observational grounds. Nevertheless, introduced as a technical tool to manage diagnostics like $r$(), $\epsilon$ came to be improperly used to adjust the [local]{} ionization equilibrium of the gas through , in an effort to overcome problems of ion stratification generated by the filling factor description itself (Appendix\[sphere\_slit\]). The () problem met in oxygen-poor GEHIIRs may relate to the [loss of plasticity]{} affecting photoionization models, as the dependence of gas cooling on abundances vanishes. Then, [cooling]{} depends on the relative concentration of H$^0$ (collisional excitation of ), [controlled by the local* ]{}. Hence, the (improper) freedom on $\epsilon$ is eroded. Moreover, if density is not uniform, () is a biased estimate for in the bulk of the emitting gas, since S$^+$ ions will belong to dense, optically thicker clumps. Emission from an interclump medium with $<$() will selectively enhance the computed 4363 intensity. LPL99 state promisingly that their model includes “denser condensations uniformly distributed in a more tenuous gas”, but in practice only the condensations emit. This restriction is shared by virtually all published models for low-Z GEHIIRs. While the assumed density of the emitting gas can be orders of magnitude larger than $<$$^2$$>$$^{1/2}$, emission from a lower density gas is neglected [by construction]{} (The study by JS05 is an exception, but NGC 588 is not low-Z; Appendix\[prev\_indiv\_go\]). The () problem suggests lifting this restriction. Spheres, slits, filling factor and stratification {#sphere_slit} ------------------------------------------------- Spherical models raise the question of how to compare computed spectra with nebular spectra observed through, , a narrow slit. LPL99 advocate extracting emission from that part of the sphere which would project on the slit. Despite obvious problems with non-sphericity, LPL99 and others argue that this procedure would at least allow weighting the contributions from low- and high-ionization zones in a more realistic manner. Using the classical filling factor concept (Appendix\[filling\]) in GEHIIR models, ion stratification spreads over the whole nebula and the emission is effectively confined to outer layers, in which the primary radiation eventually vanishes. If, on the contrary, the emitting gas belongs to [radiation-bounded]{} filaments distributed within the nebula, then [ion stratification disappears to first order]{}. Radial ionization gradients, if any, are no more related to a progressive destruction of primary photons along the full radial extension of the nebula, but to changes in (local) average ionization parameter. LPL99 conclude that is due to shock excitation in NGC 2363 (Sect. \[prev\_indiv\_po\]) because the computed intensity is weak in their theoretical slit extraction. Nonetheless, the intensity is fairly correct in their global spectrum. This apparent failure of their photoionization model may well be due to the unfortunate combination of (1) a very small $\epsilon$ and (2) the extraction of a slit shorter than the diameter of the model sphere. Along the same line, LPPC03 are confronted to undesirable consequences of the filling factor assumption on the variation of ionization along a slit crossing SBS 0335-052 (Appendix\[prev\_indiv\_po\]). If a geometrically defined model can hardly provide an approximation to a complex region, thus casting doubts on theoretical slit extractions, [global spectra]{} are less sensitive to geometry, owing to conservation laws. Moreover, in computing 1-D photoionization models, the (spherical) symmetry enters [only]{} in the treatment of the diffuse ionizing radiation field, which is generally not dominant in the total field. The diffuse field, most effective just above the ionization limits of H, He and He$^+$, is relatively [local]{} at these photon energies (in accordance with the ‘Case B’ approximation) and little dependent on global geometry. Let us define an “elementary spherical model” (for given SED) by a radial density distribution of whatever complexity. Since the local state of the gas is chiefly related to the primary (radial) radiation, a composite model made of a judicious combination of elementary spherical models, each of them restricted to a sector characterized by a [covering factor]{}, can provide [topologically significant and numerically accurate]{} descriptions of global spectra for nebulae with complex structures. Defining a ‘topology’ as a particular set of spherical models with their attached covering factors, any given topology is in one-to-one correspondence with a global spectrum [and a full class of geometries]{}, since any sector can be replaced by an arbitrary set of subsectors, provided that the sum of the covering factors of these subsectors is conserved. Thus, a good modelling strategy for a GEHIIR is one in which a global (probably composite) model spectrum is compared to the observed global spectrum. If only one slit observation is available, given that the ion stratification tends to be relatively loose and erratic in GEHIIRs, it is wise to directly use this spectrum as the average spectrum (together with scaling by the absolute flux), with the understanding that the resulting photoionization model will represent a ‘weighted average’ of the real object. For many practical purposes, this weighting may not significantly impact on the inferences made from the model, unless the slit position is exceedingly unrepresentative. 4686, WR stars and SEDs {#WRstars} ======================= harbours Wolf-Rayet (WR) stars (Legrand et al. 1997; Izotov et al. 1997; de Mello et al. 1998; Brown et al. 2002). WR stars have been challenged as the sole/main cause of nebular 4686 in BCDs on the basis of a lack of correlation between the occurence of this line and the broad ‘WR bumps’ (, Guseva et al. 2000). The study of WR stars is experiencing a revolution (Maeder et al. 2005; Meynet & Maeder 2005; Gräfener & Hamann 2005; Vink & de Koter 2005; Crowther 2007) after the realization that (1) [rotation]{} of massive stars favours enhanced equatorial mass loss, element mixing by shears, and angular momentum transport by meridian circulation, (2) low-Z massive stars tend to be fast rotators and accelerate as they evolve off the main sequence, so that the lower mass limit for a star to become a WNE star is much reduced, and (3) for a given type of WR star, the mass loss is lower for lower metallicity (Fe/H, not O/H), with three consequences: the broad WR features are less evident for low metallicity (weaker optical continuum [and]{} smaller EW of WR bumps), the duration of the WR stage can be longer, and the EUV luminosity is larger due to reduced blanketing effect. Thus, the above lack of correlation can now be partly ascribed to a [bias]{}, related to the tendency of WR star atmospheres to display less prominent optical signature when they emit more EUV radiation. [The WR star population of and the ability of these stars to emit radiation beyond 4ryd have almost certainly been grossly underestimated]{} (Crowther & Hadfield 2006). Other observations, , for SBS 0335 052E (Izotov et al. 2001b; Izotov et al. 2006b) are still taken as evidence for excitation by radiation from very fast shocks: (1) the line is broader than other nebular lines, (2) the emission is spread out far away from the main MSCs, and (3) is larger in emitting area, hence at large distances from the main ionizing sources. These findings are definitely [no]{} compelling arguments against photoionization by WR stars. The larger line width indicates larger turbulence and/or velocity gradients, not necessarily shocks. That is observed to be larger in emitting gas is in agreement with photoionization models. The spatial extent of may reflect the distribution of a few WR stars, which may not belong to the main cluster and may not be easily detected (Crowther & Hadfield 2006). Alternatively, can be produced far from the ionizing stars if the medium is porous and permeated by low density, optically thin gas, , along a galactic wind outflow (Izotov et al. 2006b). The picture of a galactic wind also suggests an explanation for the width. Photoionization models are test beds for ionizing radiation sources, but inferences on the physics of GEHIIRs should not depend on uncertain SEDs. Existing synthetic star clusters are inadequate to model . Apart from known problems with star sampling (Cerviño et al. 2003; Cerviño & Luridiana 2006), limited knowledge of the history of actual MSCs and current uncertainties about WR stars, new free parameters (initial angular momentum and magnetic field of individual stars; rate of binarity) will broaden the range of possible SED evolutions, while collective effects in a compact cluster of massive stars may influence the output of ionizing radiation far from it, due to high-density stellar winds (Thompson et al. 2006). These comments justify (1) the assumption of an excitation of solely by WR stars and (2) the use of a flexible analytical SED for NW (Sect. \[mod\_star\]). Atomic data {#fiat} =========== Collisional excitation of {#fiat_h} -------------------------- Collision strengths $\Omega$($1s$–$nl$) ($n<6; l<n$) for are taken from Anderson et al. (2000, ABBS00). The $\Omega$’s for $1s$–$2s$ and $1s$–$2p$ are much larger than for the next transitions $1s$–$nl$ and are not controversial. The main coolent agent in low-Z BCDs should be correctly implemented in all codes. Nonetheless, in the conditions of , the results for transitions 1–2 by ABBS00 are about 10% larger than those carefully fitted by Callaway (1994), giving an estimate of possible uncertainties. The adopted data tend to enhance the cooling with respect to earlier data and to (conservatively) worsen the ‘() problem’. Total $\Omega$(1–$n$)’s listed by Przybilla & Butler (2004) virtually coincide with ABBS00 values for 1–2, confirming the cooling rate, but diverge from ABBS00 for $n>2$ and increasing similarly to early, probably wrong, data (see Péquignot & Tsamis, 2005). /10$^4$K: 0.5 1.0 2.0 3.0 -------------------------- -------- -------- -------- -------- Reference:$^a$ \[0.1cm\] Sea58 - 1.59 - - SSS69 - 2.39 - - ENS69 1.85 2.50 2.91 2.96 ES74 2.17 2.36 2.55 - Men83 2.02 2.17 2.39 - Ag83 2.035 2.184 2.404 2.511 BLS89 2.10 2.29 2.51 2.60 Ag93 2.039 2.191 2.414 2.519 LB94 2.1268 2.2892 2.5174 2.6190 AgK99$^b$ 2.0385 2.1906 2.4147 2.5191 \[0.1cm\] LB94/AgK99$^c$ 1.0435 1.0450 1.0425 1.0397 \[0.1cm\] \[0.1cm\] Sea58 - 0.220 - - SSS69 - 0.335 - - ENS69 0.255 0.298 0.331 0.339 ES74 0.276 0.325 0.356 - Men83 0.248 0.276 0.314 - Ag83 0.2521 0.2793 0.3162 0.3315 BLS89 0.260 0.287 0.318 0.331 Ag93 0.2732 0.2885 0.3221 0.3404 LB94 0.2720 0.2925 0.3290 0.3466 AgK99$^b$ 0.2732 0.2885 0.3221 0.3404 \[0.1cm\] LB94/AgK99$^c$ 0.9956 1.0139 1.0214 1.0182 \[0.1cm\] \[0.1cm\] Sea58 - 0.640 - - SSS69 - 0.310 - - ENS69 0.483 0.578 0.555 0.510 ES74 0.807 0.856 0.752 - Men83 0.516 0.617 0.634 - Ag83 0.5463 0.6468 0.6670 0.6524 BLS89 0.59 0.677 0.664 0.634 Ag93 0.4312 0.5227 0.5769 0.5812 LB94 0.4942 0.5815 0.6105 0.6044 AgK99$^b$ 0.4312 0.5227 0.5769 0.5812 \[0.1cm\] LB94/AgK99$^c$ 1.1461 1.1125 1.0582 1.0399 \[0.1cm\] : Effective collision strengths for $^a$ Refs: Sea58: Seaton(1958); SSS69: Saraph et al. (1969); ENS69: Eissner et al. (1969); ES74: Eissner & Seaton (1974); Men83: Mendoza (1983); Ag83: Aggarwal (1983); BLS89: Burke et al. (1989); Ag93: Aggarwal (1993); LB94: Lennon & Burke (1994); AgK99: Aggarwal & Keenan (1999).\ $^b$ Results from Aggarwal (1993).\ $^c$ Collision strength ratio.\ \[tab\_oiii\] Collisional excitation of {#fiat_oiii} -------------------------- Effective collision strengths $\Omega$ obtained over past 50 years are listed in Table \[tab\_oiii\] at four ’s for transitions $^3$P$-$$^1$D, $^3$P$-$$^1$S and $^1$D$-$$^1$S. Aggarwal & Keenan (1999) did not feel it necessary to update earlier values by Aggarwal (1993; Ag93), almost contemporary with Lennon & Burke (1994). The ratios of the recent values are given in Table \[tab\_oiii\]. The differences are over 4% for $^3$P$-$$^1$D and 10% for $^1$D$-$$^1$S (6% in conditions), but the latter has no influence at low . [nebu]{} includes a fit better than 0.5% to Ag93 data. The transition probabilities used in [nebu]{} are from Galavis et al. (1997, GMZ97). The accuracy of the Opacity Project (OP) data for these transitions is 8$-$10% (Wiese et al. 1996). Coherently, the much more elaborate results by GMZ97 differ from the OP results by 9.6% and 5.5% for A($^1$D$-$$^1$S) and A($^1$P$-$$^1$S) respectively. Would A($^1$D$-$$^1$S) change by as much as 5%, the branching ratio of 4363 would change by 0.6%. Thus, discrepancies not exceeding 5% exist among different calculations (3% for $\Omega$ [ratio]{}s), suggesting that uncertainties on the computed $r$() are probably $<$ 5%. The 25–30% underestimation found by SS99 is not due to erroneous atomic data. Miscellaneous data {#fiat_misc} ------------------ The adopted table for radiative and dielectronic recombinations is limited to the 11 sequences H-like$-$Na-like (Badnell 2006). Dielectronic rates for $-$ are obtained by Badnell (1991), but total recombination coefficients for (recombined ions) , , , , $-$, are taken from Nahar and co-workers (Nahar, 2000 and references cited). The recombination rate for used in this and previous [nebu]{} computations is 1.15 times the Nahar’s value. Empirical total rate coefficients based on PN models (Péquignot, unpublished), implemented in [nebu]{} for a decade, are 5 and 8 times the radiative ones for and respectively. A larger factor is suspected for at high . Collision strengths of special mention include those for (Pradhan et al. 2006; also Tayal 2006b), (Tayal 2006), (Tayal & Gupta 1999), (Tayal 2000) and (Wöste et al. 2002). Collisions with H$^0$ are considered in Sect. \[disc\_ir\_low\]. Charge exchange rates with H$^0$ for O$^{2+}$ and N$^{2+}$ are now from Barragán et al. (2006). [^1]: In the context, the and fluxes are poor selection criteria for $\epsilon$, since the former is proportional to the (unknown) covering factor of the shell and the latter depends much on questionable synthetic stellar cluster spectra (Appendix \[WRstars\]). [^2]: The tolerance of 9% allowed by SS99 (adopted by V02) is probably too large (Sect. \[spectr\]). This error bar is justified in the logics of SS99, who aim to demonstrate an [absence]{} of solution. [^3]: SS99 prefer casting doubts on the ionization balance of S$^+$. [^4]: Since 5007+4959 is exactly matched, the entry 4363 in Tables \[tab\_compa\] and \[tab\_vari\] is the ‘normalized $r$()’, , the ratio of the computed $r$() to the observed $r$(). [^5]: LP01 state [a priori]{} that 4363 “almost surely has a contribution from processes other than photoionization” and, consistently, conclude that their model “fails to reproduce the observed 4363 intensity”, but both statements seem to be refuted by pieces of evidence they present.
--- abstract: 'The Noether symmetry issue for Horndeski Lagrangian has been studied. We have been proven a series of theorems about the form of Noether conserved charge (current) for irregular (not quadratic) dynamical systems. Special attentions have been made on Horndeski Lagrangian. We have been proven that for Horndeski Lagrangian always is possible to find a way to make symmetrization.' author: - 'D. Momeni,[^1]' - 'R. Myrzakulov' title: Noether symmetry in Horndeski Lagrangian --- arXiv:1410.1520 \[gr-qc\] \[b1\]Introduction ================== Canonical scalar fields are so popluar in theoretical physics because of their simplicity and easy way to interpret. Naturally if we use the Kaluza-Klein reduction for Einstein-Hilbert action in higher dimensions, the reduced lower dimensional action is equal to a scalar theory which is coupled to an abelian gauge field. As an example, we can obtain Bergmann-Wagoner bi-scalar general action of scalar-tensor gravity [@Bamba:2014jua]. If we apply this reduction scheme on a more generalized model of gravity in the form of Lovelock gravity, we obtain more terms of scalar fields,which are now coupled to the gravity or to its second order invariants. Thechnically, as we know, when the brane model of Dvali-Gabadadze-Porrati (DGP) [@dgp] is decoupled, the resulted model is the scalar theory but with nonlinear terms of interaction [@declim]. The idea of nonlinear scalar models are older than this new motivated idea. Indeed,Horndeski was who proposed the most general scalar field theory which its equation of motion (Euler-Lagrange (EL)) remains second order [@G.; @W.; @Horndeski] . As a fully covariant extension of the original Horndeski models, recently the idea of Galileon was introduced as the scalar theory with Galileon symmetry [@Galileon]. This idea has been extended and developed through recent years to explain different aspects of gravitational theory from black hole physics to cosmology [@gcos]-[@Naruko]. The models are written in a such way that they remain invariant under a local transformation of fields $\phi\to \phi+ \partial_{\mu}b$, here $b$ is gauge field. A remarkable note about Galileon models is that they are represented the most generalized form of any other modified theory in the literature. The idea of Galileon is proposed in [@Galileon] and later it was extended to covariant form [@CovariantGalileon]. Other extensions have been followed [@GeneralizedGalileon]. In particular, the first two terms of Horndeski Lagrangian are very important to study. These terms are constructed from the second order forms like $(\nabla_{\mu}\phi)^2$ and $(\nabla_{\mu}\phi)^2\nabla_{\mu}\nabla^{\mu}\phi$. We would like to write these types of the Lagrangian densities in following forms [@KGB; @G-inf]: $$\begin{aligned} {\cal L}_2&=&k(\phi, X), \\ {\cal L}_3&=&-G(\phi, X)\nabla_{\mu}\nabla^{\mu}\phi,\end{aligned}$$ Here $k$ and $ G$ are arbitary functions of field $\phi$ and its kinetic part $X\equiv -\partial_\mu\phi \partial^\mu\phi/2$. Other higher order terms can be constructed using different geometric quantities like $R$ (the Ricci tensor), $G_{\mu\nu}$( the Einstein tensor),and higher derivatives of field. Furthermore, we know that $G=X$, we obtain covariant Galileons [@CovariantGalileon]. In this paper, we consider a class of Horndeski Lagrangian, which is presented by the following action $$\begin{aligned} \label{Horndeski Lagrangian} S_{tot}=\int{\frac{R}{2}\sqrt{-g}d^4x}+\sum_{i=2}^{5}\int{d^4x\sqrt{-g}{\cal L}_i}, \label{S}\end{aligned}$$ Where diffrent Lagrangian densities have been defined by the following: $$\begin{aligned} &&{\cal L}_2=G_2(\phi,X),\\&& {\cal L}_3=G_3(\phi,X)\nabla_{\mu}\nabla^{\mu} \phi\\&& {\cal L}_4=G_{4,X}(\phi,X)\Big[\{\nabla_{\mu}\nabla^{\mu} \phi\}^2\\&&\nonumber-\nabla_{\alpha} \nabla_{\beta}\phi\nabla^{\alpha}\nabla^{\beta} \phi\Big]+RG_{4}(\phi,X),\\&& {\cal L}_5=G_{5,X}(\phi,X)\Big[ \{\nabla_{\mu}\nabla^{\mu} \phi\}^3\\&&\nonumber-3\nabla_{\mu}\nabla^{\mu}\phi\nabla_{\alpha}\nabla_{\beta}\phi\nabla^{\alpha}\nabla^{\beta} \phi\\&&\nonumber+2\nabla_{\alpha}\nabla_{\beta}\phi\nabla^{\alpha} \nabla^{\rho}\phi\nabla _{\rho}\nabla^{\beta}\phi\Big]\\&&\nonumber-6G_{\mu\nu}\nabla^{\mu}\nabla^{\nu}\phi G_{5}(\phi,X)\end{aligned}$$ Here $G_{i,X}(\phi,X)\equiv\frac{\partial G_{i}(\phi,X)}{\partial X}$ ,$R$ is the Ricci tensor, $G_{\mu\nu}$ is the Einstein tensor, also we set $\kappa^2=8\pi G=1,c=1$. Our aim here is to addressee symmetry issue for Horndeski Lagrangian given in (\[Horndeski Lagrangian\]). In literature a paper [@Alberto; @Nicolis] existed that specifically discusses constraints on a general scalar field if you enforce only the literal galilean symmetry. In our work we’ll consider Horndeski models and not Galilleons. In particular, the model doesn’t respect Galileon symmetry . The functions $G_i$ given in (\[Horndeski Lagrangian\]) are arbitrary functions of $\phi,X$. Furthermore, because Horndeski Lagrangian is constructed in a covariant form, so it is manifestly Lorentz invarint. We have been investigated all possible Noether symmetries of such models in the cosmological FLRW model. Our plan in this work is as the following:\ In Sec. \[NS\] we review the fundamental theory of Noether symmetry for regular dynamical systems. In Sec. \[second-action\] we are considering Horndeski Lagrangian with Noether symmetries. In Sec. \[new\] we have been proven a sequence of theorems about Noether symmetries for higher order derivatives models, including Horndeski Lagrangian. We conclude in Sec. \[conclusion\]. \[NS\] Review of Noether symmetry ================================= Let us to consider a dynamical system with $N$ configurational cordinates $q_i$ is defined by the Lagrangian $L\equiv L(q_i, \dot{q}_i;t),\ \ 1\leq i\leq N$. The set of EL equations for this dynamical system is written as $\dot{p}_i-\frac{\partial L}{\partial q_{i}}=0,\ \ p_{i}\equiv\frac{\partial L}{\partial \dot{q}_i}$. We mention here that up or down the index has the same meaning since we are working in the flat space. What we called it as [Noether Symmetry Approach]{} is the existence of a vector, Noether vector $\vec{X}$ [@noether2],[@cap3],[@noether3],[@noether4],[@cap4]: $$\label{17} X=\Sigma_{i=1}^{N}\alpha^i(q)\frac{\partial}{\partial q^i}+ \dot{\alpha}^i(q)\frac{\partial}{\partial\dot{q}^i}\,{,}$$ and a set of non-singular functions $\alpha_{i}(q_j)$, in a such a way that the Lie derivative of Lagrangian vanishes on all points of the manifold (the tangent space of configurations $T{\cal Q}\equiv\{q_i, \dot{q}_i\}$): $$\label{19} L_X{\cal L}=0\,$$ The mentioned condition can be written in the following expanded form: $$\label{18} L_X{\cal L}=X{\cal L}=\Sigma_{i=1}^{N}\alpha^i(q)\frac{\partial {\cal L}}{\partial q^i}+ \dot{\alpha}^i(q)\frac{\partial {\cal L}}{\partial\dot{q}^i}\,{.}$$ From the phase-space point of view, existence of $\vec{X}$ implies that the total phase flux enclosed in a region of space, is conserved along $X$. In fact, it is an easy task to show that (by taking into account the EL equations): $$\label{20} \frac{d}{dt}\frac{\partial {\cal L}}{\partial\dot{q}^i}- \frac{\partial {\cal L}}{\partial q^i}=0\,{,}\ \ 1\leq i \leq N.$$ Consequently, we have: $$\label{21} \Sigma_{i=1}^{N}\frac{d}{dt}\left(\alpha^i\frac{\partial {\cal L}}{\partial\dot{q}^i}\right)=L_X{\cal L}\,{.}$$ If we can find $\alpha_{i}$ by vanishing the coefficents of all powers of $\dot{q}^i$, then we will show that there exist a `global` conserved charge as the following: $$\label{22} \Sigma_0=\Sigma_{i=1}^{N}\alpha^ip_i$$ In other words, the existence of Noether symmetry implies that the [Lie derivative of the Lagrangian]{} on a given vector field ${\bf X} $ vanishes, i.e. $$\pounds_{\bf X} L = 0. \label{noether}$$ It has been proven that Noether symmetry is a powerful tool to study cosmological models in different models [@cap3]-[@hann]. In our article we explore Noether symmetries (\[noether\]) for the Horndeski Lagrangian, given by (\[S\]). Noether symmetry for third order Horndeski Lagrangian {#second-action} ====================================================== To have a more comprehensive result, let us to consider the following Lagrangian which was proposed as minimal G-inflation [@Kobayashi:2011nu]: $$\begin{aligned} \mathcal{L}_3=k(\phi,X)-G(\phi,X)Y\label{model2}.\end{aligned}$$ Where we denote by $Y=\nabla_{\mu}\nabla^{\mu}\phi$. There is no simple way to reduce this Lagrangian to a simpler quadratic form, because of the appearence of the highly nonlinear term $Y$. To resolve this problem, we propose a couple of Lagrange multipliers $\{\lambda,\mu\}$ in the following forms: $$\begin{aligned} &&L=3a\dot{a}^2+a^3\Big[k(\phi,X)-G(\phi,X)Y\Big]\\&&\nonumber-a^3\Big(\lambda(X-\frac{1}{2}\dot{\phi}^2)+\mu(Y-\nabla_{\mu}\nabla^{\mu}\phi)\Big).\end{aligned}$$ By varying the Lagrangian $L$ w.r.t to the $\{X,Y\}$ we obtain $\lambda=k_{,X}-YG_{,X},\mu=-G$, so the reduced Lagrangian is written as the following: $$\begin{aligned} &&L(a,\phi,X,Y;\dot{a},\dot{\phi})=3a\dot{a}^2+ a^3\Big[k(\phi,X)\\&&\nonumber-G(\phi,X)Y\Big]-a^3(X-\frac{1}{2}\dot{\phi}^2)(k_{,X}-YG_{,X}).\end{aligned}$$ The appropriate set of the coordinates for configuration space is $q^i=\{a,\phi,X,Y\}$. We define a vector field $\vec{X}=\alpha\frac{\partial}{\partial a}+\beta\frac{\partial}{\partial \phi}+\gamma\frac{\partial}{\partial X}+\theta\frac{\partial}{\partial Y}+\dot{\alpha}\frac{\partial}{\partial \dot{a}}+\dot{\beta}\frac{\partial}{\partial \dot{\phi}}$, here the functions $\alpha^i=\{\alpha,\beta,\gamma,\theta\}$ are defined on configuration space, so we have the following system of PDEs as a result of (\[noether\]) for the above point-like Lagrangian: $$\begin{aligned} &&\frac{\partial G}{\partial a}=\frac{\partial G}{\partial Y}=0,\\&& \frac{\partial k}{\partial a}=\frac{\partial k}{\partial Y}=0,\\&& 3\alpha a^{-1}(k-X(k_{,X}-YG_{,X})-YG)\\&&\nonumber+\beta(k_{,\phi}-X(k_{,X\phi}-YG_{,X\phi})-YG_{,\phi})\\&&\nonumber+\gamma(-X(k_{,XX}-YG_{,XX})+YG_{,X})\\&&\nonumber -\theta(XG_{,X}+G)=0\\&& \alpha+2a\alpha_{,a}=0\\&& 3\alpha a^{-1}(k_{,X}-YG_{,X})+\beta (k_{,X\phi}-YG_{,X\phi})\\&&\nonumber+\gamma(k_{,XX}-YG_{,XX})\\&&\nonumber-\theta G_{,X}+2\beta_{,\phi}(k_{,X}-YG_{,X})=0\\&& 6\alpha_{,\phi}+a^2\beta_{,a}(k_{,X}-YG_{,X})=0\\&& \alpha_{,X}=\alpha_{,Y}=0\\&& \beta_{,X}(k_{,X}-YG_{,X})=0\\&& \beta_{,Y}(k_{,X}-YG_{,X})=0.\end{aligned}$$ We know that $p_a=6a\dot{a},\ \ p_{\phi}=a^3\dot{\phi}(k_{,X}-YG_{,X})$ and the corressponding Noether charge is written as the following: $$\begin{aligned} \Sigma=6\alpha a \dot{a}+\beta a^3\dot{\phi}(k_{,X}-YG_{,X})=\Sigma_{0}.\end{aligned}$$ The system of PDEs has three major class of exact solutions. Class A: The system has the following exact solutions if we impose $k_{,X}-YG_{,X}=0,\ \ G_{,X}\neq0$: $$\begin{aligned} &&\alpha=\frac{\alpha_0}{\sqrt{a}},\ \ \beta=\frac{\alpha_0 Y}{a\sqrt{a}}\frac{\beta_0(ch(\phi)-3f(\phi))}{f'(\phi)},\\&& \gamma=-\frac{\alpha_0ch(\phi)}{a\sqrt{a}G_{,X}},\\&& \theta=0.\end{aligned}$$ Where $\{h(\phi),f(\phi)\}$ are arbitrary functions of $ \phi$. The associated conserved Noether charge is $\Sigma_{A}=6\alpha_0 \sqrt{a} \dot{a}$ from here we find $a(t)=\Big[a_{0}^{3/2}+\frac{\Sigma_{A}}{4\alpha_0}(t-t_0)\Big]^{2/3}$. So we conclude here that: The third order action of G-inflation presented by (\[model2\]) has Noether symmetry vector field: $$\begin{aligned} &&\vec{X}=\frac{\alpha_0}{\sqrt{a}}\frac{\partial}{\partial a}+Y\frac{\alpha_0}{a\sqrt{a}}\frac{\beta_0(ch(\phi)-3f(\phi))}{f'(\phi)}\frac{\partial}{\partial \phi}\\&&\nonumber-\frac{\alpha_0ch(\phi)}{a\sqrt{a}G_{,X}}\frac{\partial}{\partial X}+\frac{d}{dt}\Big[\frac{\alpha_0}{\sqrt{a}}\Big]\frac{\partial}{\partial \dot{a}}\\&&\nonumber+\frac{d}{dt}\Big[Y\frac{\alpha_0}{a\sqrt{a}}\frac{\beta_0(ch(\phi)-3f(\phi))}{f'(\phi)}\Big]\frac{\partial}{\partial \dot{\phi}}.\end{aligned}$$ So, the action is in the following form: $$\begin{aligned} &&S=\int{\sqrt{-g}d^4x \Big(\frac{R}{2}+f(\phi)\nabla_{\mu}\nabla^{\mu}\phi\Big)}\\&&\nonumber=\int{\sqrt{-g}d^4x\Big(\frac{R}{2}+2Xf'(\phi)\Big)}\end{aligned}$$ [It is important to mention here that the above Noether symmetrized model is written in the following equivalent form:]{} $$\begin{aligned} &&S=\int{\sqrt{-g}d^4x \Big(\frac{R}{2}-\frac{1}{2}\nabla_{\mu}\psi\nabla^{\mu}\psi\Big)}\end{aligned}$$ Where $\psi=\pm\int{\sqrt{f'(\phi)}d\phi}$. Equation of motion of a scalar field is obtained: $$\begin{aligned} &&\ddot{\Psi}+3H\dot{\Psi}=0\end{aligned}$$ Which can be solved by $\psi=\frac{\psi_0}{\Big[a_{0}^{3/2}+\frac{\Sigma_{A}}{4\alpha_0}(t-t_0)\Big]^{2}}$. So,if we apply Noether symmetry method to the $\mathcal{L}_3$ , the set of the EOMs is completely integrable. Class B: We suppose that $k_{,X}=0,\ \ G_{,X}=0$. Consequently we have $$\begin{aligned} \alpha=\alpha(a,\phi),\ \ \beta=\beta(a,\phi),G=G(\phi),k=k(\phi).\end{aligned}$$ Exact solution for PDEs are given by: $$\begin{aligned} &&\alpha=\alpha(a)=\frac{\alpha_0}{\sqrt{a}}\\&& \beta=\beta(a)=\beta_0\frac{\alpha(a)}{a}\\&& \theta=\theta_0g(\phi)Y\frac{\alpha(a)}{a}.\end{aligned}$$ Where $$\begin{aligned} &&k(\phi)=k_0e^{-3\phi/\beta_0},\\&& G(\phi)=Ce^{-3\phi/\beta_0}+\frac{\theta_0}{\beta_0}e^{-3\phi/\beta_0}\int{d\phi g(\phi)e^{3\phi/\beta_0}}.\end{aligned}$$ Noeher vector is: $$\begin{aligned} &&\vec{X}=\frac{\alpha_0}{\sqrt{a}}\frac{\partial}{\partial a}+\beta_0\frac{\alpha(a)}{a}\frac{\partial}{\partial \phi}+\theta_0g(\phi)Y\frac{\alpha(a)}{a}\frac{\partial}{\partial Y}\\&&\nonumber+\frac{d}{dt}\Big[\frac{\alpha_0}{\sqrt{a}}\Big]\frac{\partial}{\partial \dot{a}}+\frac{d}{dt}\Big[\beta_0\frac{\alpha(a)}{a}\Big]\frac{\partial}{\partial \dot{\phi}}\end{aligned}$$ So, the following third order Galileon Lagrangian has Noether symmetry: $$\begin{aligned} \mathcal{L}=k(\phi)-G(\phi)\nabla_{\mu}\nabla^{\mu}\phi.\end{aligned}$$ or equivalently: $$\begin{aligned} S=-\frac{1}{2}\int{d^4x \sqrt{-g}\Big(XG_{,\phi}-\frac{1}{2}k(\phi)\Big)}\end{aligned}$$ [This form is the k-inflation model in the standard canonical form .]{}[@kinflation] Case C: There is another interesting analytic class of solutions when we put $\gamma=\theta=0,k_X-YG_X=\Psi_{,X}$. In this case, we have the following solutions: - $\alpha=\beta=0,\Psi=F_1(a,\phi)$ In this family , the scalar field has no dynamics, because the kinetic term is absent. Consequently the solutions have no physical application as cosmological models and we can ignore them. - $\alpha=0,\beta=F_2(a,\phi),\Psi=F_{3}(a)$ This class is potentially very interesting case, because the physical action is reduced to the Einstein-Hilbert action in the presence of a type of fluid in which the pressure is given by $p\equiv \Psi$. It is easy to show that all the EOMs are integrable with this type of Noether symmetry. - $\alpha=\beta=0,\Psi=\Psi(a,\phi,X)$ This solution corresponds to a generalized k-inflationary model in which the scalar field is non-minimally coupled to the cosmological background via the arbitrary function $=\Psi(a,\phi,X)$. - $\alpha=0,\beta=c_1,\Psi=F_4(a,X)$ The model is a purely kinetic k-inflation which is non-minimally coupled to the background. The model is well studied in literature [@Scherrer:2004au]. Such models are deserved to be considered as a unified model for dark matter and dark energy. The model is considered as a valid alternative to the generalized Chaplygin gas models. Such models are considered as dark energy component with a very slow varying sound speed and are compatible with the cosmic microwave background fluctuations on large angular scales. Furthermore, recently the authors showed that [@Barausse:2015wia], such models areable to explain the sensitivity parameters for emission by compact-star binaries. - $\alpha=0,\beta=F_5(\phi),\Psi=X\frac{F_{6}(a)}{F_{7}(a)^2}+F_{8}(a)$ We have a type of fluid in the cosmological background. It ihas been shown that the model is completely integrable for a set of the EOMs. It has been shown that [@Gomes:2015dhl], the model has scaling solutions when Horndeski’s field is coupled to the background. Scaling solutions are those that $\frac{\rho_{\phi}}{\rho_m}$ is constant. So, we can find a solution of the following equivalent equation $\frac{d\rho_{\phi}}{d\ln a}=\frac{d\rho_m}{d\ln a}$. Noether symmetry for higher derivative Horndeski Lagrangian {#new} =========================================================== We saw in the previous sectionsthat all types of the modified gravity theories always have nonlinearity, due to the higher order derivatives of the fields. Historically Ostrogradski [@ostrogradeski] was the first who studied canonical formalism (Hamiltonian formalism) for a class of models with higher order derivatives (see [@prd1990] for a comprehensive review). In the case of Galileon, even if we work at the level of minimal models, with $\mathcal{L}_3$, there is no simple way to reduce point-like Lagrangian to the standard quadratic form $\mathcal{L}=\mathcal{L}(\phi,X^m)$. A way is to introduce a set of appropriate Lagrange multipliers. But in this section we introduce an alternative to work with higher derivative Lagrangians. The simplest case which we are particularly interesting , is a class of Lagrangian functions $L(q_i,\dot{q}_{i},(\partial_{t})^{n}[q_{i}]),\ \ n\leq2$. It is possible to extend it to $n\leq3$, but such cases doesn’t have simple physical interpretations. We would like to see how we can find generalized Noether symmetry for a Lagrangian with second order time derivatives , namely $L=L(q_i,\dot{q}_i,\ddot{q}_i),\ \ 1\leq i\leq N$. We are interesting in the cases in which by integration part-by-part one can not reduce the Lagrangian to the standard quadratic form $L'=L'(q_i,\dot{q_i})$. The first simple example is point-like Lagrangian of standard GR, that it contains $\ddot{a}$ due to the curvature term $R=\pm6(\frac{\ddot{a}}{a}+\frac{\dot{a}^2}{a^2})$ but we can omit second order derivative term $\ddot{a}$ using an integration part-by part. For G-inflation models if we pass to the higher terms, we need a generalized Noether symmetry. This is one of the most important motivation for us in this work. Let us start to define a set of appropriate conjugate momenta: $$\begin{aligned} &&p_i=\frac{\partial L}{\partial \dot{q}^i},\ \ r_i=\frac{\partial L}{\partial \ddot{q}^i}.\end{aligned}$$ The generalized EL equation is given by: $$\begin{aligned} &&\frac{\partial L}{\partial q^i}-\frac{d}{dt}\Big(\frac{\partial L}{\partial \dot{q}^i}\Big)+\frac{d^2}{dt^2}\Big(\frac{\partial L}{\partial \ddot{q}^i}\Big)=0.\end{aligned}$$ Or equivalently we write down it in the following form: $$\begin{aligned} &&\frac{\partial L}{\partial q^i}-\dot{p}_{i}+\ddot{r}_i=0\label{EL2}.\end{aligned}$$ We define a vector field: $$\begin{aligned} &&\vec{X}=\Sigma_{i=1}^{N}\Big(\alpha_i\frac{\partial}{\partial q^i}+\dot{\alpha}_i\Big[\frac{\partial}{\partial \dot{q}^i}-2\frac{d}{dt}\Big(\frac{\partial}{\partial \ddot{q}^i}\Big)\Big]\\&&\nonumber-\ddot{\alpha}_i\frac{\partial L}{\partial \ddot{q}^i}\Big)\label{v2}.\end{aligned}$$ We call it as generalized Noether symmetry if and only if it satisfies the following algebrically vector equation: $$L_X{\cal L}=0\,,$$ In this case we find that the following polynomial should be vanish: $$\begin{aligned} &&\Sigma_{i=1}^{N}\alpha_i\frac{\partial L}{\partial q_i}+\Sigma_{i=1}^{N}\Sigma_{j=1}^{N}(\alpha_i)_{,j}\dot{q}_j\Big[\frac{\partial L}{\partial \dot{q}_i}\\&&\nonumber-2\frac{d}{dt}\Big(\frac{\partial L}{\partial \ddot{q}_i}\Big)\Big]-\Sigma_{i=1}^{N}\Big(\Sigma_{j=1}^{N}(\alpha_i)_{,j}\ddot{q}_j\\&&\nonumber+\Sigma_{j,k=1}^{N}(\alpha_i)_{,j,k}\dot{q}_j\dot{q}_k\Big)\frac{\partial L}{\partial \ddot{q}_i}=0.\end{aligned}$$ If we collect all terms of different powers of $\{\dot{q}_i,\ddot{q}_j\}$ we obtain a system of second order PDEs (not first order like the standard Lagrangians) for $\{a_{i}(q^k)\}.$ Consequently, the associated Noether charge is obtained by the following theorem: `Theorem`: [For Lagrangian $L=L(q_i,\dot{q}_i,\ddot{q}_i)$, there exists a Noether vector symmetry given by (\[v2\]) and a Noether conserved charge: ]{} $$\begin{aligned} &&K=\Sigma_{i=1}^{N}(\alpha_i p_i-\partial_t(\alpha_i r_i)).\end{aligned}$$ `Proof`: Using (\[EL2\]) it is easy to show that $\dot{K}=0$, because we’ve: $$\begin{aligned} &&\dot{K}=\Sigma_{i=1}^{N}(\partial_t(\alpha_i p_i)-\partial_{tt}(\alpha_i r_i)) \\&&\nonumber=\Sigma_{i=1}^{N}(\alpha_i\dot{p}_i+\dot{\alpha}_i p_i\-(\alpha_i \ddot{r}_{i}+2\dot{\alpha}_i\dot{r}_i+\ddot{\alpha}_ir_i))\\&&\nonumber \Longrightarrow \Sigma_{i=1}^{N}( \alpha_i(\dot{p}_i-\ddot{r}_i)+\dot{\alpha}_i(p_i-2\dot{r}_i) -\ddot{\alpha}_ir_i)\\&&\nonumber=\Sigma_{i=1}^{N}\Big(\alpha_i\frac{\partial L}{\partial q_i}+\dot{\alpha}_i\Big[\frac{\partial L}{\partial \dot{q}_i}-2\frac{d}{dt}\Big(\frac{\partial L}{\partial \ddot{q}_i}\Big)\Big]-\ddot{\alpha}_i\frac{\partial L}{\partial \ddot{q}_i}\Big)\\&&\nonumber \Longrightarrow L_X{\cal L}=0.\end{aligned}$$ This is our Q.E.D. It is adequate to present the following generalized theorem for the dynaminal system in the form $L=L (q_i, q_ {I} ^ {(a)}; t), \ \ q_{i}^{(a)}=(\partial t) ^a [q_ {I}], \ \ 1\leq i\leq N, \ \ 1\leq a\leq s, \ \ s\leq N$. We are remembering to the mind that in this class of models, the generalized EL equation is written as the following: $$\begin{aligned} \frac{\partial L}{\partial q_{i}}+\Sigma_{a=1}^{s}(-1)^a(\partial t)^a(p^{a}_i)=0,\end{aligned}$$ [ Here $p^{a}_i\equiv \frac{\partial L}{\partial q_{i}^{(a)}}$ is the new set of conjugate momenta]{}.\ \ `Theorem`: For Lagrangian $L=L (q_i, q_ {I} ^ {(a)}; t) $, there exists a generalized conserved Noether current given by: $$\begin{aligned} \mathcal{K}=\Sigma_{i=1}^{N}\Sigma_{a=1}^{s}(-1)^{a+1}(\partial t)^{a-1}\Big[\alpha_i p^{a}_i\Big].\end{aligned}$$ `Proof`: Using the Leibniz rule for derivatives we obtain: $$\begin{aligned} &&\partial_{t}(\mathcal{K})=\Sigma_{i=1}^{N}\Sigma_{a=1}^{s}(-1)^{a+1}(\partial t)^{a}\Big[\alpha_i p^{a}_i\Big] \\&&\nonumber=\Sigma_{i=1}^{N}\Sigma_{a=1}^{s}\Sigma_{k=1}^{a}(-1)^{a+1}\frac{a!}{k!(a-k)!}\\&&\nonumber(\partial t)^k[\alpha_i](\partial t)^{a-k}[p^{a}_i]\\&&\nonumber =\Sigma_{i=1}^{N}\Sigma_{a=1}^{s}\Sigma_{k=1}^{a}\Big((-1)^{a+1}\frac{a!}{k!(a-k)!}(\partial t)^k\\&&\nonumber\times[\alpha_i](\partial t)^{a-k}[\frac{\partial }{\partial q_{i}^{(a)}}]\Big) L \Longrightarrow L_{\vec{X}}L=0\end{aligned}$$ Where the following vector field, $$\begin{aligned} &&\vec{X}=\Sigma_{i=1}^{N}\Sigma_{a=1}^{s}\Sigma_{k=1}^{a}\Big((-1)^{a+1}\frac{a!}{k!(a-k)!}(\partial t)^k\\&&\nonumber[\alpha_i](\partial t)^{a-k}\times[\frac{\partial }{\partial q_{i}^{(a)}}]\Big)\end{aligned}$$ is the Noether vector symmetry. `Coroally`: [For a general Horndeski model with the following Lagrangian]{} $$\begin{aligned} L=L(\phi,(\nabla)^a\phi), 1\leq a<s.\end{aligned}$$ The following vector is conserved: $$\begin{aligned} \mathcal{K}^{\mu}=\Sigma_{a=1}^{s}(-1)^{a+1}(\nabla)^{a-1}\Big[\alpha(\phi) \frac{\partial L}{\partial [(\nabla_{\mu})^a\phi]}\Big].\end{aligned}$$ i.e. $\nabla_{\mu}\mathcal{K}^{\mu}=0$. In the above theorem if we use the conventional Horndeski’s notations we should identify , $\nabla\equiv \nabla_{\mu},\ \ X\sim (\nabla_{\mu}\phi)^2,\ \ Y\sim (\nabla_{\mu})^2\phi=\nabla_{\mu}\nabla^{\mu}\phi$ and etc.\ \ `Illustrative example`: If we consider the minimal model of Horndeski theory, $\mathcal{L}_2+\mathcal{L}_3=k(\phi,X)-G(\phi,X)\nabla_{\mu}\nabla^{\mu}\phi$, we obtain: $$\begin{aligned} &&\mathcal{K}^{\mu}=\nabla^{\mu}\Big[\alpha(\phi)G(\phi,X)\Big] -\alpha(\phi)(k_{,X}\\&&\nonumber-G_{,X}\nabla_{\mu}\nabla^{\mu}\phi)\nabla^{\mu}\phi.\end{aligned}$$ Which is trivially conserved if we fix $\alpha$ by Noether conservation vector condition (\[noether\]). Conclusions {#conclusion} =========== The most general form of scalar-tensor theory for gravity in the covariant form was proposed by Horndeski. The significant feature is that the set of equations of motion remains second order. Because Horndeski Lagrangian was constructed in a covariant form, consequently it is manifestly Lorentz invariant. In this work we addressed Noether symmetry of point like Lagrangian in the framework of Horndeski theory. We proposed a theorem about the Noether symmetry for a general highr order Lagrangian, specially in the form of Horndeski models. We extended the idea of the Lie generator of the normal tangent space. We have been proven that a vector field : $$\begin{aligned} &&\vec{X}=\Sigma_{i=1}^{N}\Sigma_{a=1}^{s} \\&&\nonumber\Sigma_{k=1}^{a}\Big((-1)^{a+1}\frac{a!}{k!(a-k)!}(\partial t)^k[\alpha_i](\partial t)^{a-k}[\frac{\partial }{\partial q_{i}^{(a)}}]\Big) \end{aligned}$$ is the Noether vector symmetry for $L=L (q_i, q_ {I} ^ {(a)}; t) $. For a general Galileon model $\mathcal{L}_2+\mathcal{L}_3$,we have been proven that there exists conserved current. 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--- abstract: \| In image morphing, a sequence of plausible frames are synthesized and composited together to form a smooth transformation between given instances. Intermediates must remain faithful to the input, stand on their own as members of the set, and maintain a well-paced visual transition from one to the next. In this paper, we propose a conditional GAN morphing framework operating on a pair of input images. The network is trained to synthesize frames corresponding to temporal samples along the transformation, and learns a proper shape prior that enhances the plausibility of intermediate frames. While individual frame plausibility is boosted by the adversarial setup, a special training protocol producing sequences of frames, combined with a perceptual similarity loss, promote smooth transformation over time. Explicit stating of correspondences is replaced with a grid-based freeform deformation spatial transformer that predicts the geometric warp between the inputs, instituting the smooth geometric effect by bringing the shapes into an initial alignment. We provide comparisons to classic as well as latent space morphing techniques, and demonstrate that, given a set of images for self-supervision, our network learns to generate visually pleasing morphing effects featuring believable in-betweens, with robustness to changes in shape and texture, requiring no correspondence annotation. <ccs2012> <concept> <concept\_id>10010147.10010371.10010382.10010383</concept\_id> <concept\_desc>Computing methodologies Image processing</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012> author: - \| \ [ ]{} bibliography: - 'egbib.bib' title: Image Morphing with Perceptual Constraints and STN Alignment --- Introduction ============ Morphing is the process of transformation between states of appearance, and may involve operations ranging from basic translation and rotation, to changes in color and texture, and, perhaps most iconically, shape shifting. In the era of big data and deep learning, the ability to morph between objects could have an impact beyond the generation of the visual effect itself. For instance, synthesized intermediate frames depicting novel variations of input objects, may be added to existing datasets for densification and enrichment. Traditional morphing techniques rely on correspondences between relevant features across the participating instances, to drive an operation of warp and cross-dissolve [@beier1992feature]. These methods are mostly invariant to the semantics of the underlying objects and are therefore prone to artifacts such as ghosting and implausible intermediates. Correspondence points are normally user-provided, or are automatically computed assuming sufficient similarity. Recently, an abundance of available data has given rise to its utilization as guidance proxies for extraction of short or smooth paths between the two endpoints [@averbuch2016smooth], thus providing more plausible in-betweens. In this paper, we aim to further tap into the data-driven morphing paradigm, and leverage the power of deep neural networks to learn a shape prior befitting a given source dataset, catering to the task of image morphing. Specifically, we employ a generative adversarial network (GAN) [@goodfellow2014generative] combined with a spatial transformer [@jaderberg2015spatial] for shape alignment, for mitigation of the challenges associated with morphing. GANs are known for displaying impressive generative capabilities by their capacity to learn and model a given distribution, a particularly lucrative attribute for a task for which realism and plausibility is crucial. Accordingly, we opt to design a GAN framework to learn the space of natural images of a given class so that intermediate frames appear to be realistic, and to enforce sufficient similarity between sufficiently close frames, to maintain smoothness of transformation. We present a conditional GAN framework trained to generate sequences of transformations between two or more inputs, and further integrate it with a grid-based freeform spatial transformer network to alleviate large discrepancies in shape. The generated output sequences are constrained by a perceptual loss, culminating in an end-to-end solution that encourages transitions that are both plausible and smooth, with a gradual and realistic change in shape and texture. The result is a trained generator specializing in a certain family of shapes, that, given a pair of inputs and a desired point in time, outputs the appropriate in-between frame. A full morphing effect can then be synthesized by requisitioning a reasonably dense sequence of frames, which yield a smooth transformation (see Figure \[fig:teaser\]). During training, each sampled set of inputs is first processed by a spatial transformer network, which computes an alignment allowing a feature-based warp operation to map each input to the other. Next, our conditional generator processes the warped inputs, and outputs a sequence of frames, each corresponding to a given point in time. A reconstruction loss encourages the two endpoint frames to match the inputs. Meanwhile, a GAN loss pushes the generated frames towards the natural image manifold of the training set. Finally, a perceptual transition loss [@zhang2018unreasonable] constrains the transformation over time to be smooth and gradual. We demonstrate the competence of our generator and its ability to produce visually pleasing morphing effects with smooth transitions and plausible in-betweens, on different sets of objects, both real and computer rendered. We conduct a thorough ablation study to examine the individual contributions of our design components, and perform comparisons to traditional morphing, as well as GAN-based latent space interpolation. We show that our framework, uniting the GAN paradigm with shape alignment and perceptually constrained transitions, provides a solution that is robust to significant changes in shape, a challenging setup that commonly induces ghosting artifacts in morphs. Related Work ============ Classic morphing. Pioneering morphing techniques combine correspondence-driven bidirectional warping with blending operations to generate a sequence of images depicting a transformation between the entities in play. The approach by Beier and Neely [@beier1992feature] leverages user-defined line segments to establish corresponding feature points that are used to distort each endpoint towards the other, and proceeds to apply a cross-dissolve operation on respective pairs of warped images to obtain a transformation sequence. More recently, Liao et al. [@liao2014automating] automatically extract correspondences for morphs by performing an optimization of a term similar to structural image similarity [@wang2004image] on a halfway domain. Averbuch-Elor et al. [@averbuch2016smooth] adopt a data-driven approach where a collection of images from a specific class of objects is used to locate smooth sequences of images. A morphing effect is then generated from source to target via in-betweens that are smoothed with a global similarity transform. In deep image analogies [@liao2017visual], deep features are leveraged for bidirectional correspondences supporting bidirectional attribute transfer for synthesis of style and content hybrids. Similarly, Aberman et al. [@aberman2018neural] focus on cross-domain correspondences extracted using a coarse-to-fine search of mutual nearest neighbor features, and show that this can produce cross-domain morphs. Shechtman et al. [@shechtman2010regenerative] introduced an alternative way to morph between different images using patch-based synthesis that did not rely on correspondences and cross-blending, and Darabi et al. [@darabi2012image] extended it by allowing patches to rotate and scale. While these methods produced nice transitions that look different than the traditional warp+blend effect, the method is limited to patches drawn from the two sources and does not produce new content. Deep interpolations. Neural networks have been previously trained to synthesize novel views of objects using interpolation. Given two images of the same object from two different viewpoints as input, Ji et al. [@ji2017deep] generate a new image of the object from an in-between viewpoint. The images are first brought into horizontal alignment, and are then processed by an encoder-decoder network that predicts dense correspondences used to compute an interpolated view. General image interpolations are commonly demonstrated within the VAE and GAN realms. A notable by-product of a trained GAN is its rich latent embedding space that facilitates linear interpolation between data points. Such interpolations drive a generation of morphing sequences, by producing a series of interpolated latent vectors that are decoded to images that appear to smoothly morph from source to target [@brock2018large; @berthelot2017began; @karras2017progressive; @donahue2019large]. To perform interpolation between existing instances, one must obtain their corresponding latent codes in order to compute interpolated vectors and their decoded images. This is commonly accomplished with an optimization process that starts from a random code, which is updated to minimize a loss such as $L2$ on the desired image [@webster2019detecting]. However, in practice, the learned manifold may not be able to reconstruct any given test set image, and obtaining the corresponding code to a given image may also be challenging. Solutions that combine an encoder mapping existing instances to the learned space, such as VAE-GAN [@larsen2015autoencoding] and CVAE-GAN [@bao2017cvae], which is trained simultaneously, and iGAN [@zhu2016generative], which is trained successively, alleviate this difficulty, but the crux of the problem remains, particularly when one seeks to map more unique entities. Method ====== Our system combines several key components that together provide a robust solution for morphing effect generation. We henceforth present these components and address the manner in which they cater to the three requirements, namely, frame realism with respect to the training set, smooth transitions, and input fidelity at the endpoints. Basic setup ----------- We use a convolutional GAN approach [@goodfellow2014generative; @radford2015unsupervised] for our morphing. GANs have been demonstrated to perform highly sophisticated modeling of image training data [@brock2018large]. This characteristic is appealing for our endeavor, as we seek to create sequences of transformation between entities belonging to a specific family of objects, i.e., our target distribution. Therefore, a GAN loss can help fulfill our first requirement of realism. In our implementation, we combine the Least-Squares GAN loss [@mao2017least] with two discriminators: a local PatchGAN [@li2016precomputed] discriminator and a global discriminator. We denote by $\mathcal{L}_D$ and $\mathcal{L}_G$ the GAN losses used to train $D$ and $G$ respectively, each by minimization of the corresponding sum: $$\mathcal{L}_D = {\mathcal{L}_{D}}_{\text{local}}^{\text{real}} + {\mathcal{L}_{D}}_{\text{global}}^{\text{real}} + {\mathcal{L}_{D}}_{\text{local}}^{\text{fake}} + {\mathcal{L}_{D}}_{\text{global}}^{\text{fake}}$$ $$\label{eq:lossdg} \mathcal{L}_G = {\mathcal{L}^_{G}}_{\text{local}}^{\text{fake}} + {\mathcal{L}^_{G}}_{\text{global}}^{\text{fake}}$$ The asterisk in Equation \[eq:lossdg\] indicates the inversion of labels when $D$ is used to evaluate $\mathcal{L}_G$. Common image morphing operates on existing instances given as input, thus, accordingly, we opt for a special type of GAN known as the conditional GAN [@mirza2014conditional; @isola2017image; @zhu2017unpaired; @kim2017learning], whose output is directly influenced by one or more signals given as input. In our case, those signals include the two input images that are to be morphed, $I_A, I_B$, as well as a scalar $t$ specifying the desired time sample of the output in-between frame. We note that this could also be generalized to an arbitrary number $k \ge 2$ of input images to be morphed, along with a vector of interpolation parameters with $L_1$ norm of unity. Our conditional GAN consists of an encoder followed by a generator. Our second requirement is smoothness of transitions. This is dealt with by a combination of a special training protocol and a suitable loss component. To better control and guide the generation to comply with our aim, at training time, for each input pair $I_A, I_B$, we generate a sequence of frames of length $k$. Each of these frames correspond to a predetermined time sample, and are uniformly sampled on the unit interval $[0, 1]$. This approach allows us to apply a loss component, $\mathcal{L}_T$, designed to constrain the similarity between frames, and encourage smooth transitions. More specifically, we make use of a pretrained neural network (VGG-16 [@simonyan2014very]) to obtain deep features of generated frames upon which perceptual similarity (PS) is computed [@zhang2018unreasonable]. As a frame-of-reference, we compute input pair PS: $\text{PS}_{4,5}(I_A, I_B) = \text{MSE}(\text{VGG}_{4,5}(I_A), \text{VGG}_{4,5}(I_B))$, where $\text{VGG}_{4,5}(I_A)$ are all VGG features of $I_A$ extracted from layer groups 4 and 5 (out of 5). Using that, we define $\mathcal{L}_T$ as: $$\mathcal{L}_T = \max_{i=2..k}\{ \\|\text{PS}_{4,5}(I_{i-1}, I_i) - (t_i - t_{i-1}) \cdot \text{PS}_{4,5}(I_A, I_B)\\|^2\} \label{eq:local_ps}$$ That is, we constrain each frame to be a certain distance, in semantic feature space, from its preceding frame. This distance should ideally match the feature distance $PS_{4,5}(I_A, I_B)$ between the input images, after rescaling by the time between adjacent frames $t_i - t_{i-1}$. The final component in our basic setup is a reconstruction loss, which encourages the endpoint frames in the sequence to match the inputs: $$\mathcal{L}_R = \text{MSE}(I_1,I_A) + \text{MSE}(I_k,I_B)$$ Alignment {#sec:alignment} --------- The characteristic locality of convolutional networks is a known hindrance in situations where changes in shape are required. To support a wide range of inputs of varying shapes, we recognize the need for higher-level, semantic information to establish the relationship between the inputs, much like classic morphing techniques that rely on correspondences between points and features to drive a warping operation. Manually collecting correspondence points between instances in large datasets such as ours is intractable. Although it is possible to incorporate an automatic correspondence computation [@aberman2018neural], we opt for an integrated end-to-end solution which is both computationally faster, and as we show later, can be more robust in cases where there are significant differences in shape. A spatial transformer network (STN) [@jaderberg2015spatial] is a component that can be added to a neural network as a means to learn and apply transformations to the data to assist the main learning task. In our setting, we seek to compute an alignment between the inputs, and apply it onto them to be given to the generator for further processing. For greater flexibility and range of deformation, we add a spatial transformer component that computes a grid-based freeform deformation warp field [@hanocka2018alignet]. This component is placed before the encoder-generator component of our main network, and is composed of two convolutional blocks and a fully-connected block predicting the warp grid, whose size is a parameter set to 5x5 in our experiments. The inputs $I_A, I_B$ are concatenated channel-wise before passing through this component, which outputs two grids (for $x$ and $y$) indicating the warp from $I_A$ to $I_B$ - $\mathcal{W}_{AB}$. Likewise, $\mathcal{W}_{BA}$ is obtained by switching the order between $I_A$ and $I_B$. See Figure \[fig:stn\_arch\] for an illustration, and our supplementary material for specific design details. ![Grid-based freeform spatial transformer. The two inputs are concatenated and processed by the network which outputs a 5x5 grid aligning the first to the second. The deformed first image is compared against the second image using perceptual similarity. The grid is compared to the identity grid for regularization.[]{data-label="fig:stn_arch"}](stn3.png){width="8.5cm"} We combine the STN with our sequence generation scheme, by applying a series of partial deformations to the inputs, each corresponding to a certain time stamp. The partial deformation for $\mathcal{W}_{AB}$ at time $t$ is $\mathcal{W}_{AB}^t = \mathcal{I} + t \cdot (\mathcal{W}_{AB} - \mathcal{I})$, and $\mathcal{W}_{BA}^t = \mathcal{I} + (1-t) \cdot (\mathcal{W}_{BA} - \mathcal{I})$ for $\mathcal{W}_{BA}$, where $\mathcal{I}$ is the identity warp grid. The grids are upsampled to the input image size using bilinear interpolation, and are applied onto $I_A$ and $I_B$ to obtain a sequence of warped inputs $\{I_A^t\}_{t=t_1}^{t_k}, \{I_B^t\}_{t=t_1}^{t_k}$, that are passed on to the encoder. See figure \[fig:stn\] for three examples of partial to full deformations computed by our STN. We add two losses tailored to our spatial transformer network. The first is a shape warp loss, $\mathcal{L}_W$, comparing the warped $I_A$, denoted by $I_A^{t_k}$, to $I_B$, and the second, $\mathcal{L}_I$, compares the predicted grid to the identity grid, for regularization. $\mathcal{L}_W$ makes another use of perceptual similarity by using the deep VGG features of layer group 5. These provide a higher level of abstraction that encourages the overall shape of the warped image to match the other endpoint, as opposed to stylistic details. The two losses are given by: $$\begin{split} \mathcal{L}_W = \text{PS}_5(I_A^{t_k}, I_B) \\ \mathcal{L}_I = \text{MSE}(\mathcal{W_{AB}},\mathcal{I}) \end{split}$$ We note that the losses we have described thus far, do not directly bind the inner frames to the inputs $I_A, I_B$. With the addition of the alignment computation, we are able to add a final perceptual similarity loss, $\mathcal{L}_E$, that draws a connection between each frame and its corresponding warped inputs, without restricting the ability of the frame to shift the shape of its underlying object. We choose layer group 4 for this purpose, to benefit from a combination of abstraction and a notion of finer detail, and compute a blend of similarities dependent upon the time stamp: $$\mathcal{L}_E = \sum_{i=1}^k (1-t_i) \cdot \text{PS}_4(I_{t_i}, I_A^{t_i}) + t_i \cdot \text{PS}_4(I_{t_i}, I_B^{t_i}) \\ \label{eq:global_ps}$$ The total loss function of our generator is thus: $$\mathcal{L}_G = \lambda_G\mathcal{L}_G + \lambda_T\mathcal{L}_T + \lambda_R\mathcal{L}_R + \lambda_W\mathcal{L}_W + \lambda_I\mathcal{L}_I + \lambda_E\mathcal{L}_E$$ Network structure ----------------- The architectures of $G$ and $D$ are similar to those of DiscoGAN [@kim2017learning]. $G$ is composed of an encoder containing blocks of conv and ReLU followed by a decoder, containing blocks of tconv (transposed convolution) and ReLU. Both local and global $D$ contain blocks of conv and ReLU with a final Sigmoid. In both $G$ and $D$, the number of blocks depends on the input image resolution. For more details please refer to our supplementary material. ![image](arch4.png){width="15cm"} We employ a late fusion protocol, where the inputs $I_A^{t_i}, I_B^{t_i}$ are first processed separately by the encoder of $G$, which outputs feature maps $F_A^{t_i}, F_B^{t_i}$ respectively. An adaptive instance normalization component [@huang2017arbitrary] blends the mean and standard deviation of the feature maps according to the input time stamp $t_i$. That is, for given statistics $\mu_A^{t_i}, \mu_B^{t_i}$ and $\sigma_A^{t_i}, \sigma_B^{t_i}$, we compute the blended statistics for time $t_i$: $$\begin{split} \mu_{t_i} = (1-t_i) \cdot \mu_A^{t_i} + t_i \cdot \mu_B^{t_i} \\ \sigma_{t_i} = \sqrt{(1-t_i) \cdot (\sigma_A^{t_i})^2 + t_i \cdot (\sigma_B^{t_i})^2} \end{split} \label{eq:adain}$$ $F_A^{t_i}$ is then updated as: ${F_A^{t_i}}^* = \sigma_{t_i} \cdot \frac{(F_A^{t_i} - \mu_A^{t_i})}{\sigma_A^{t_i}} + \mu_{t_i}$, and $F_B^{t_i}$ similarly. Next, ${F_A^{t_i}}^, {F_B^{t_i}}^$ are concatenated channel-wise, along with an additional channel containing the time stamp $t_i$ expanded to the appropriate spatial resolution – $F_{t_i}$. The resulting block of data, ${F_A^{t_i}}^* {F_B^{t_i}}^* F_{t_i}$, is processed by the decoder which outputs the corresponding generated frame. During training we generate $k$ frames, thus we prepare $k$ such blocks $\{{F_A^{t_i}}^* {F_B^{t_i}}^* F_{t_i}\}_{i=1}^k$, all of which are passed through the decoder. At train time, we randomly draw another instance from within the set for each input in the batch, and together these make up the input pairs. At each iteration, we also draw at random a pool of images to be shown to $D$ as real data. Since each pair of inputs spawns $k$ frames, the real pool for each pair is of size $k$ as well. See Figure \[fig:pipeline\] for a high level illustration of our pipeline. Content and style {#sec:costl} ----------------- We extend our solution to address the problem of content and style separation [@gatys2015neural; @johnson2016perceptual; @dumoulin2017learned; @huang2017arbitrary] within the morphing scope, to allow greater control over the desired outcome and provide increased freedom of creativity. Instead of a single axis of transformation between our two inputs $I_A, I_B$, we seek to engage two axes corresponding to disentangled transitions of content and style. This can be viewed as a 2D morphing effect taking place within the unit square, such that at coordinate $(t_{c_i},t_{s_j})$, the content of the generated frame reflects an interpolation of $(1-t_{c_i}) \cdot I_A^c + t_{c_i} \cdot I_B^c$ and its style a similar interpolation of $(1-t_{s_j}) \cdot I_A^s + t_{s_j} \cdot I_B^s$, where $t_{c_1},t_{s_1}=0$ and $t_{c_k},t_{s_k}=1$ ($k$ samples along both axes), and $I_A^c, I_B^c$ and $I_A^s, I_B^s$ are the content and style characteristics of the inputs respectively. We recognize the inherent capacity of the various components in our pipeline towards the distinction between the manifestation of content in our setup, i.e., overall shape and geometric detail, and stylistic attributes such as color and texture. Specifically, we observe that our local and global perceptual similarity losses can be employed in such a way as to encourage one aspect or the other by demand. Combining these with the initial warping mechanism catering to content (shape) rather than style, and the adaptive instance normalization component favoring style over content, we are able to formulate a disentangled solution dependent upon the two axes of transformation. Alignment. Initial alignment is carried out as before, but is only governed by the content axis, disregarding the style axis completely. Training. The new training protocol resembles our original one in that for each input pair, we generate $k$ frames. We randomly sample $k-2$ points along each axis, and keep $t_{c_1},t_{s_1}=0$ and $t_{c_k},t_{s_k}=1$. As the feature maps corresponding to frame $i$, $F_A^{t_{c_i}}, F_B^{t_{c_i}}$, exit the encoder, we perform adaptive instance normalization according to the style axis alone, such that $t_i$ in Equation \[eq:adain\] is replaced with $t_{s_i}$. We then concatenate the two samples associated with frame $i$ – $t_{c_i}, t_{s_i}$, each expanded to the appropriate spatial resolution as before, to the normalized feature stack. The stack given to the decoder is thus: ${F_A^{t_{c_i}}}^* {F_B^{t_{c_i}}}^* F_{t_{c_i}} F_{t_{s_i}}$. Perceptual similarity losses. We create a hard separation between the authorities of the two PS losses with respect to content and style. The local PS loss $\mathcal{L}_T$ is assigned to the content whereas the global loss $\mathcal{L}_E$ is assigned to style. For $\mathcal{L}_T$, $t_i$ in Equation \[eq:local\_ps\] is replaced with $t_{c_i}$. Similarly, for $\mathcal{L}_E$, $t_i$ in Equation \[eq:global\_ps\] is replaced with $t_{s_i}$. Additionally, to increase the emphasis upon stylistic elements, we compute $\mathcal{L}_E$ with VGG layer group 3 instead of 4. Evaluation ========== In this section we perform various experiments to evaluate our method, both within its own scope (\[sec:ablation\]), and externally (\[sec:res\]). We experiment on four datasets - boots [@finegrained; @semjitter] ($\sim10\text{k}$) and handbags [@zhu2016generative] ($\sim12\text{k}$), depicting real-world objects, and cars ($\sim7\text{k}$) and airplanes ($\sim4\text{k}$), featuring renders of objects from ShapeNet [@chang2015shapenet]. For each dataset, we randomly draw 100 pairs of inputs from a separate test set upon which we conduct all our experiments. For each pair, we generate a sequence of 11 frames. Our model is trained on a 128x128 image resolution for 200 epochs, except for the variations trained for the ablation study, which were trained on a 64x64 resolution for computational efficiency. Ablation study {#sec:ablation} -------------- We explore the individual contributions of our various design components by conducting an ablation study. For this purpose, we train six variations of our network, aside from the proposed solution. Each variation excludes one component: GAN loss (adversary), local perceptual similarity, global perceptual similarity, reconstruction loss, adaptive instance normalization and STN (which also excludes global perceptual similarity, see Subsection \[sec:alignment\]). We compute the Fréchet Inception Distance (FID) [@heusel2017gans] between the generated frames of each version in each dataset, and its respective training set, resized to a resolution of 96x96. The overall trend of the scores, summarized in Table \[tbl:fid\_ablation\], indicates that our main solution generates images that are generally in-line with the training set distribution. Additionally, in Figure \[fig:ablation\], which contains visual examples for generated sequences obtained with the six variations, we note the various shortcomings characterizing the five ablation variations. The “w/o GAN” version does not preserve object detail, the “w/o PS” versions do not appropriately combine characteristics from both inputs, the “w/o recon” version does not adhere to the two endpoints and neither does the “w/o adaIn” version, and the “w/o STN” version is characterized by a serious degeneration, exhibiting little to no deformation in shape, resulting in a preference of one endpoint over the other. Note that as part of our earlier experiments, we did not experience a similar degeneration with a baseline system that did not incorporate an STN. However, these earlier versions naturally produced substantially lower quality results (due to their lack of advanced image alignment), and their far-removed architecture places them are outside the scope of this ablation study. ![image](abl_fig2.pdf){width="16cm"} ---------------- ------- ------- ------- -------- ------- Ablation Bags Boots Cars Planes Mean \[0.5ex\] Main 31.96 27.75 34.90 44.18 34.70 w/o GAN 30.71 27.98 37.10 44.52 35.08 w/o local PS 31.67 27.32 29.72 43.79 33.13 w/o global PS 36.17 31.85 38.61 49.19 38.96 w/o recon 33.18 29.03 36.13 41.17 34.88 w/o adaIn 34.40 32.57 40.29 44.51 37.94 w/o STN 53.68 57.72 64.18 57.26 58.21 ---------------- ------- ------- ------- -------- ------- : Ablation FID scores on four datasets. We compute FID scores for different versions of our method, on a test set of 100 input pairs per dataset with 9 frames each, totaling at 900 frames per dataset. These generated frames are compared against the corresponding training set.[]{data-label="tbl:fid_ablation"} ![image](comp_fig1.pdf){width="16cm"} ![image](comp_fig2.pdf){width="16cm"} Results and comparisons {#sec:res} ----------------------- We compare our results to three other methods. The first is simple linear blending. We take the two sequences of warped inputs that our STN outputs, and blend each pair of corresponding frames according to their respective time stamp. The second is the morphing method by Liao et al. [@liao2014automating] (termed “Halfway” in Table \[tbl:fid\_comparison\] and Figures \[fig:comparison2\] and \[fig:comparison3\]). The final method is GAN-based latent space interpolation. Although recent high resolution GAN solutions such as BigGAN [@brock2018large] have been shown to produce impressively high quality generation and interpolation results, they are not as readily available to train, thus we opt for the well-known WGAN-GP [@gulrajani2017improved] solution for which we make use of the official implementation. We also experimented with VAE-GAN [@larsen2015autoencoding] and IntroVAE [@huang2018introvae], but found WGAN-GP to provide superior results on our data. After training WGAN-GP on each of our four datasets, we train an encoder per trained model, to assist in our efforts to recover latent codes of existing instances. To obtain the latent codes of our test input images, we first pass them through the trained encoder, and then proceed to optimize the code further with an $L2$ loss on the input image. Table \[tbl:fid\_comparison\] summarizes the FID scores obtained by comparing the generated frames of each method on each of the test sets, with the corresponding training set. Note that all methods except WGAN-GP, which is compared at a 64x64 resolution, are compared at a resolution of 192x192. The presented scores show that the classic techniques we compared to produce images that are closer to the “real" data distribution than those generated by our method and WGAN-GP. These results are not surprising, since the classic techniques operate on the original images and perform operations of warp and cross dissolve, while generator-based methods procure the entire image every time, and are therefore bound to stray farther from the original distribution. Thus, even when an intermediate frame features ghosting artifacts, it may not incur a high FID score when it is essentially a blend of the two original inputs, as is the case in both of the classic methods. ---------------- ----------- ----------- ----------- ----------- ----------- Comparison Bags Boots Cars Planes Mean \[0.5ex\] Ours 29.12 25.78 28.94 50.35 33.55 Linear blend 29.75 23.97 28.04 45.14 31.72 Halfway 22.72 21.47 23.06 39.61 26.71 WGAN-GP 68.91 83.71 54.38 55.53 65.63 ---------------- ----------- ----------- ----------- ----------- ----------- : Comparing FID scores on four datasets. All methods were given the same set of 100 input pairs yielding morphing sequences of length 9, totaling at 900 frames per method. []{data-label="tbl:fid_comparison"} Figures \[fig:comparison2\] and \[fig:comparison3\] present qualitative examples of our generated sequences compared to those of the other methods. We observe that classic techniques exhibit excellent adherence to the original inputs as well as smooth transitions, however, at times they suffer from ghosting artifacts and exaggerated deformations due to incorrect correspondences. In contrast, our method is able to overcome differences in the overall shape, supporting a plausible transformation between the inputs. Specifically, we note that Liao et al. [@liao2014automating] (Halfway) produce high quality effects composed of visually pleasing frames when the correspondence is accurate (many examples are available in our supplementary material). The difficulty arises when the two input images depict objects of larger shape offsets (see the Boots example in Figure \[fig:comparison2\] and the Planes example in Figure \[fig:comparison3\]), or are somewhat lacking in texture and color (see the wheels in the Cars example in Figure \[fig:comparison3\]). The baseline, linear blending, uses the alignment computed by our STN, and therefore benefits from its robustness to large differences in shape. However, the alignment provides general cues for warping, and further processing is often needed in order to promote smoother transitions. See Figure \[fig:comparison2\], where ghosting artifacts are visible just above the opening of the bag, and at the tip and back of the boots. Lastly, our experiments with WGAN-GP [@gulrajani2017improved] show that generation quality as well as latent space encoding of existing instances, is still insufficient for high quality morphing effect creation. However, despite the artifacts that often appear in the generated frames, a strong advantage of latent space interpolation is its manner of frame creation. Frames are generated independently of one another, unlike approaches that are based on warp and cross-dissolve operations, and thus, ghosting artifacts are naturally avoided. ### User study To obtain user perspective, we designed a survey that presents the user with 36 pairs of morphing effects (9 of each dataset), such that each pair is composed of our result vs. that of one of the compared methods (in arbitrary order). For each pair, the users were asked to select the one they preferred of the two (subjectively), as well as the one that exhibits a more plausible transformation of shape (objectively). Users were able to select ’no preference’ whenever they wished. A total of 50 participants took part in our study. The results are shown in Table \[tbl:user\_study\], where the column ’Ours’ contains the portion of morphing effects where our method was selected over the other method (appearing in the ’Compared to’ column). Likewise, the column ’Theirs’ contains the portion where the other method triumphed, and the ’Tie’ column specifies the remaining portion, where ’no preference’ was selected. The statistics of the two questions appear in the same cell in the format $q_1/q_2$, such that $q_1$ corresponds to the statistics of the first question. These results show that in all sets except for Planes, users prefer Liao et al. [@liao2014automating] (Halfway) over ours, with larger margins in the real image datasets (Bags and Boots), where faithfulness to the original image statistics is more crucial. The Planes dataset contains instances with highly distinct silhouettes that prove challenging for all methods, but are slightly better handled by our method, which is able to reliably compute the alignment between the inputs. Our method had the upper hand over Linear blend and WGAN-GP in all datasets, with a smaller margin against Linear blend, whose performance is satisfactory when the two input shapes are sufficiently similar in shape, but otherwise produces ghosting artifacts. Note that all morphing effects were taken from the pool of 100 effects per dataset that we generated from the test set, all of which are available for viewing in our supplementary material. While the classic method of Liao et al. [@liao2014automating] has the overall upper hand in terms of user preference, the advantage of our method is its consistency and robustness to different shape silhouettes and textures, and its speedy inference time (see our supplementary material for run time comparisons). Our main limitation is individual frame quality which relies on network generation, thus, latest and future advances in neural generation may help alleviate this, although at a probable training time penalty. Set Compared to Ours Theirs Tie -------- -------------- ---------------- ---------------- ------------- Bags Halfway 0.25/0.27 0.625/0.59 0.125/0.13 Bags Linear blend 0.51/0.49 0.22/0.25 0.27/0.26 Bags WGAN-GP 0.88/0.875 0.08/0.08 0.04/0.046 Boots Halfway 0.29/0.27 0.53/0.53 0.18/0.2 Boots Linear blend 0.39/0.41 0.32/0.28 0.29/0.3 Boots WGAN-GP 0.93/0.86 0.007/0.05 0.066/0.083 Cars Halfway 0.34/0.33 0.45/0.45 0.21/0.23 Cars Linear blend 0.38/0.375 0.3/0.3 0.32/0.32 Cars WGAN-GP 0.89/0.88 0.01/0.04 0.09/0.08 Planes Halfway 0.43/0.43 0.41/0.41 0.16/0.16 Planes Linear blend 0.47/0.48 0.3/0.3 0.23/0.21 Planes WGAN-GP 0.86/0.81 0.04/0.086 0.11/0.11 : User study results. Refer to main text for details. []{data-label="tbl:user_study"} ![image](costl_fig.pdf){width="16cm"} ### Content and style {#content-and-style} Figure \[fig:costl\] contains two examples for content and style disentangled morphing as described in Subsection \[sec:costl\]. For a given input pair, we generate each frame in a 6x6 grid, such that for cell $(i,j)$, the coordinate $i$ represents the desired location on the content axis, and similarly for coordinate $j$ with the style axis. For more results, please see our supplementary material. For our full implementation please see our GitHub page. Conclusion ========== We presented a new approach for morphing effect generation, combining the conditional GAN paradigm with a grid-based freeform deformation STN and a set of perceptual similarity losses. The components that make up our pipeline have been carefully curated to promote the generation of realistic in-betweens with smooth and gradual transitions, resulting in a solution that is robust to inputs exhibiting differences in shape and texture. Particularly, shape misalignments are overcome automatically by the integrated STN that learns a strong shape prior based on semantic features, rather than on potentially misleading low-level features. In a world that is constantly hungry for more visual data, the ability to generate high-fidelity image instances is particularly beneficial. These can be used not only for artistic purposes, but also to enrich and augment existing datasets in support of various endeavors requiring substantial amounts of information. Moreover, as a frame generation framework, a natural and potentially advantageous connection ties us to the field of video processing and synthesizing, one that may establish a bidirectional exchange of ideas with the prospect of mutual gain. We note that our current setup is composed of simple building blocks – a no-frills generator and discriminator that maintain a balance of good performance with low computational cost. Despite that, potential improvements and extensions to these components may further increase the quality of the generated frames, which are not always free of common morphing maladies such as ghosting and blurring. The addition of supervision to the pipeline may broaden the scope of our approach, and allow various types of transitions such as rotations. Similarly, morphing between images with arbitrary backgrounds may call for an integration of a dedicated segmentation component, one that is either pretrained, or trained within the entire framework in an end-to-end manner. Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported by Adobe and the Israel Science Foundation (grant no. 2366/16 and 2472/17).
--- author: - \| Dennis Weyland\ Università della Svizzera italiana, Lugano, Switzerland\ Department of Economics and Management, University of Brescia, Italy\ [[email protected]]([email protected]) bibliography: - 'paper.bib' title: Some Comments on the Stochastic Eulerian Tour Problem --- Introduction ============ The <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> was introduced in 2008 [@mohan2008stochastic] as a stochastic variant of the well-known <span style="font-variant:small-caps;">Eulerian Tour Problem</span>. In a follow-up paper the same authors investigated some heuristics for solving the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> [@mohan2010heuristics]. After a thorough study of these two publications a few issues emerged. In this short research commentary we would like to discuss these issues. For this purpose, we will first introduce the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> using the original formulation of [@mohan2008stochastic]. Afterwards we will discuss the following issues more in detail: - The formal definition of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> seems quite cumbersome. In fact, there are many components which are totally meaningless and which distract from the actual problem. - The <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> has been introduced as a stochastic variant of the well-known <span style="font-variant:small-caps;">Eulerian Tour Problem</span>. This is rather misleading, since the non-stochastic variant of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is completely different from the <span style="font-variant:small-caps;">Eulerian Tour Problem</span>. - The non-stochastic variant of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> can be easily seen to be a generalization of the famous <span style="font-variant:small-caps;">Traveling Salesman Problem</span>. Hardness and inapproximability results are therefore passed over from the <span style="font-variant:small-caps;">Traveling Salesman Problem</span> to the non-stochastic variant of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> and finally to the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span>. - In [@mohan2008stochastic] it has been stated that the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is an NP-hard stochastic variant of the polynomially solvable <span style="font-variant:small-caps;">Eulerian Tour Problem</span>. This is quite an interesting claim, but based on the previous two issues it has to be rejected. The <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is NP-hard but it is neither the stochastic variant of the <span style="font-variant:small-caps;">Eulerian Tour Problem</span> nor the stochastic variant of a polynomially solvable problem. The Stochastic Eulerian Tour Problem ==================================== According to [@mohan2008stochastic] the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is defined in the following way. We are given an undirected Eulerian graph $G = (V,E)$, a subset $R \subseteq E$ of the edges with cardinality $n$ and a function $d: E \rightarrow \mathbb{R}$ representing distances on the edges. Additionally, one of the vertices is the depot, where the Eulerian tour is supposed to start and end. In [@mohan2008stochastic] this node is duplicated as $v_0$, which then serves as the depot and is connected to the original depot by two edges of length $0$. Furthermore, each edge of the set $R$ has associated a probability which indicates the likelihood that this edge actually requires to be served. Finally, it is assumed that if the number of edges that require to be served in a particular realization of the stochastic data is $k$, then every set of $k$ edges from $R$ is equally likely. The optimization goal is to compute an a priori Eulerian tour for the graph $G$, such that the expected costs of the a posteriori tour with respect to the given probabilities is minimized. From a given a priori Eulerian tour, the a posteriori tour is derived in the following way. The tour starts at the depot, visits the edges that require to be served in the order and orientation given by the a priori Eulerian tour, and finishes the tour at the depot. For movements between the edges that require to be served, the shortest paths with respect to the given distances are chosen. The cost of such an a posteriori tour is the sum of the traveled distances. Comments on the Problem Definition ================================== There are a few issues with the definition of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> which we will discuss in this section. First of all, let us have a look at the a priori solutions. These are Eulerian tours on the given graph $G$. But the only properties of such an a priori solution that are of interested, are the order and orientation of the edges from the set $R$. In fact, the rest of the graph is just used to compute the distances between any of the edges in $R$. After updating the distances between any pair of vertices that belong to edges in $R$ to their shortest path distances within the given graph $G$, we could safely remove the edges which do not belong to the set $R$, that means we could remove all the edges in the set $E \setminus R$. Additionally, we could safely remove all the vertices which do not belong to any of the edges in $R$, except the depot of course. The resulting graph still captures the basic essentials of the original problem. It no longer has to be Eulerian and the a priori solutions are no longer Eulerian tours, but they simply specify an order and an orientation of the remaining edges. It is not clear at all, why the problem was originally defined in such a cumbersome way which even distracts from the important features. Apart from that, it was originally assumed that if the number of edges that require service in a particular realization of the stochastic data is $k$, then every set of $k$ edges from $R$ is equally likely. This assumption seems really weird, as it immediately implies that the probabilities that edges require service are the same for all edges, which is a highly objectionable restriction of this problem. It might be interesting to investigate this special case, but the problem definition should be more general. As it is done for the presence of vertices for the <span style="font-variant:small-caps;">Probabilistic Traveling Salesman Problem</span>, the presence of edges for the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> could be modeled as independent stochastic events. There is no need for the additional assumption given in [@mohan2008stochastic]. On top of these two issues, it is possible to perform another simplification. It definitely makes sense for practical applications to have a depot where the tour starts and ends. But in our simplified definition of this problem such an explicit definition of a depot is no longer necessary, since it can be simulated by an edge of negligible length which always requires service, that means which requires service with a probability of $1$. All in all, the definition of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> can be simplified to the following definition, which still captures all the basic essentials. We are given an undirected and complete (no longer necessarily Eulerian) graph $G = (V, E)$ and a function $d: E \rightarrow \mathbb{R}$ which represents the distances on the edges. Additionally, we are given a set of edges $R$, such that each vertex from $V$ occurs in exactly one of the edges of $R$, and a function $p: R \rightarrow [0,1]$ which represents the probabilities that the edges require service. Now the objective is simply to find an order and orientation of the edges of $R$, which is the a priori solution, such that the expected costs of the a posteriori solution is minimized. For a particular realization of the stochastic events, the a posteriori solution is derived from the a priori solution by visiting the edges that require service in the order and orientation given by the a priori solution while skipping all the other edges. The cost for such an a posteriori tour is the sum of the traveled distances. We believe that this definition is much more simple and straightforward than the original definition. We no longer need a depot in this definition. We also do not need the weird assumption about the probabilities. And most important, we do not require the given graph to be Eulerian and the a priori solutions to be Eulerian tours. At the same time it still captures all the basic essentials of the original definition. The Relation to the Eulerian Tour Problem ========================================= The problem has been named <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> since it was claimed to be a stochastic variant of the <span style="font-variant:small-caps;">Eulerian Tour Problem</span>. But the main issue here is that the non-stochastic variant of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is not the <span style="font-variant:small-caps;">Eulerian Tour Problem</span>. This is apparent even for the original definition of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span>. Fixing all the probabilities to $1$, we arrive at a problem, where we have to compute a tour of minimum cost, starting and finishing at a depot and visiting certain edges in a specific order and orientation. This is definitely not the <span style="font-variant:small-caps;">Eulerian Tour Problem</span> and as we will see in the next section, this problem differs significantly from the <span style="font-variant:small-caps;">Eulerian Tour Problem</span>. Therefore, it seems very unfortunate that the problem was named the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span>. Hardness and Inapproximability of the Stochastic\ Eulerian Tour Problem ================================================= In [@mohan2008stochastic] it was shown that the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is NP-hard by a reduction from the <span style="font-variant:small-caps;">Probabilistic Traveling Salesman Problem</span>. It was not too difficult to prove this result, but with our simplified definition of the problem we can easily show that the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is a generalization of the famous <span style="font-variant:small-caps;">Traveling Salesman Problem</span>. This does not only imply NP-hardness, but it also allows us to transfer inapproximability results from the <span style="font-variant:small-caps;">Traveling Salesman Problem</span> to the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span>. For a given instance of the <span style="font-variant:small-caps;">Traveling Salesman Problem</span>, we just have to replace each vertex by two vertices which are sufficiently close to each other and an edge connecting them. These new edges are those which actually require service and we additionally assign a probability of $1$ to them. It is easy to see that there is a straightforward bijection between solutions for this new instance of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> and solutions for the original instance of the <span style="font-variant:small-caps;">Traveling Salesman Problem</span>. Moreover, the costs between such two solutions differ only by a negligible amount. We think that this relation is quite obvious and we do not think that in addition a formal proof is required at this point. Based on these findings, we have to reject a claim that has been made in [@mohan2008stochastic], namely that the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is an NP-hard stochastic variant of the polynomially solvable <span style="font-variant:small-caps;">Eulerian Tour Problem</span>. Well, the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is NP-hard, but it is neither the stochastic variant of the <span style="font-variant:small-caps;">Eulerian Tour Problem</span> nor the stochastic variant of a polynomially solvable problem. The latter statement follows from the fact that our reduction from the <span style="font-variant:small-caps;">Traveling Salesman Problem</span> was in fact to the non-stochastic variant of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span>, since all the probabilities were fixed at $1$. Discussion and Conclusions ========================== As we have seen, the definition of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> was in fact quite cumbersome and could be reduced to a much simpler formulation which still captures the basic essentials of this problem. Additionally, we could show that the non-stochastic variant of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is not the <span style="font-variant:small-caps;">Eulerian Tour Problem</span> but some sort of generalization of the <span style="font-variant:small-caps;">Traveling Salesman Problem</span>. Therefore, hardness and inapproximability results are passed over from the <span style="font-variant:small-caps;">Traveling Salesman Problem</span> to the non-stochastic variant of the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> and finally to the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span>. Based on this result we also have to reject the claim that the <span style="font-variant:small-caps;">Stochastic Eulerian Tour Problem</span> is an NP-hard stochastic variant of a polynomially solvable problem. Acknowledgments {#acknowledgments .unnumbered} --------------- This research has been supported by the Swiss National Science Foundation as part of the Early Postdoc.Mobility grant 152293.